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\begin{isabellebody}%
\def\isabellecontext{Refinement}%
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\isatagtheory
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\ Refinement\isanewline
\isakeyword{imports}\ Setup\isanewline
\isakeyword{begin}%
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\isamarkupsection{Program and datatype refinement \label{sec:refinement}%
}
\isamarkuptrue%
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\begin{isamarkuptext}%
Code generation by shallow embedding (cf.~\secref{sec:principle})
allows to choose code equations and datatype constructors freely,
given that some very basic syntactic properties are met; this
flexibility opens up mechanisms for refinement which allow to extend
the scope and quality of generated code dramatically.%
\end{isamarkuptext}%
\isamarkuptrue%
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\isamarkupsubsection{Program refinement%
}
\isamarkuptrue%
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\begin{isamarkuptext}%
Program refinement works by choosing appropriate code equations
explicitly (cf.~\label{sec:equations}); as example, we use Fibonacci
numbers:%
\end{isamarkuptext}%
\isamarkuptrue%
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\isatagquote
\isacommand{fun}\isamarkupfalse%
\ fib\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
\ \ \ \ {\isachardoublequoteopen}fib\ {\isadigit{0}}\ {\isacharequal}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
\ \ {\isacharbar}\ {\isachardoublequoteopen}fib\ {\isacharparenleft}Suc\ {\isadigit{0}}{\isacharparenright}\ {\isacharequal}\ Suc\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
\ \ {\isacharbar}\ {\isachardoublequoteopen}fib\ {\isacharparenleft}Suc\ {\isacharparenleft}Suc\ n{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ fib\ n\ {\isacharplus}\ fib\ {\isacharparenleft}Suc\ n{\isacharparenright}{\isachardoublequoteclose}%
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\begin{isamarkuptext}%
\noindent The runtime of the corresponding code grows exponential due
to two recursive calls:%
\end{isamarkuptext}%
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\begin{isamarkuptext}%
\isatypewriter%
\noindent%
\hspace*{0pt}fib ::~Nat -> Nat;\\
\hspace*{0pt}fib Zero{\char95}nat = Zero{\char95}nat;\\
\hspace*{0pt}fib (Suc Zero{\char95}nat) = Suc Zero{\char95}nat;\\
\hspace*{0pt}fib (Suc (Suc n)) = plus{\char95}nat (fib n) (fib (Suc n));%
\end{isamarkuptext}%
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\begin{isamarkuptext}%
\noindent A more efficient implementation would use dynamic
programming, e.g.~sharing of common intermediate results between
recursive calls. This idea is expressed by an auxiliary operation
which computes a Fibonacci number and its successor simultaneously:%
\end{isamarkuptext}%
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\isatagquote
\isacommand{definition}\isamarkupfalse%
\ fib{\isacharunderscore}step\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat\ {\isasymtimes}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
\ \ {\isachardoublequoteopen}fib{\isacharunderscore}step\ n\ {\isacharequal}\ {\isacharparenleft}fib\ {\isacharparenleft}Suc\ n{\isacharparenright}{\isacharcomma}\ fib\ n{\isacharparenright}{\isachardoublequoteclose}%
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\begin{isamarkuptext}%
\noindent This operation can be implemented by recursion using
dynamic programming:%
\end{isamarkuptext}%
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\isacommand{lemma}\isamarkupfalse%
\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
\ \ {\isachardoublequoteopen}fib{\isacharunderscore}step\ {\isadigit{0}}\ {\isacharequal}\ {\isacharparenleft}Suc\ {\isadigit{0}}{\isacharcomma}\ {\isadigit{0}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
\ \ {\isachardoublequoteopen}fib{\isacharunderscore}step\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}let\ {\isacharparenleft}m{\isacharcomma}\ q{\isacharparenright}\ {\isacharequal}\ fib{\isacharunderscore}step\ n\ in\ {\isacharparenleft}m\ {\isacharplus}\ q{\isacharcomma}\ m{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
\ \ \isacommand{by}\isamarkupfalse%
\ {\isacharparenleft}simp{\isacharunderscore}all\ add{\isacharcolon}\ fib{\isacharunderscore}step{\isacharunderscore}def{\isacharparenright}%
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\begin{isamarkuptext}%
\noindent What remains is to implement \isa{fib} by \isa{fib{\isacharunderscore}step} as follows:%
\end{isamarkuptext}%
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\isacommand{lemma}\isamarkupfalse%
\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
\ \ {\isachardoublequoteopen}fib\ {\isadigit{0}}\ {\isacharequal}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
\ \ {\isachardoublequoteopen}fib\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isacharequal}\ fst\ {\isacharparenleft}fib{\isacharunderscore}step\ n{\isacharparenright}{\isachardoublequoteclose}\isanewline
\ \ \isacommand{by}\isamarkupfalse%
\ {\isacharparenleft}simp{\isacharunderscore}all\ add{\isacharcolon}\ fib{\isacharunderscore}step{\isacharunderscore}def{\isacharparenright}%
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\begin{isamarkuptext}%
\noindent The resulting code shows only linear growth of runtime:%
\end{isamarkuptext}%
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\begin{isamarkuptext}%
\isatypewriter%
\noindent%
\hspace*{0pt}fib{\char95}step ::~Nat -> (Nat,~Nat);\\
\hspace*{0pt}fib{\char95}step (Suc n) = let {\char123}\\
\hspace*{0pt} ~~~~~~~~~~~~~~~~~~~~(m,~q) = fib{\char95}step n;\\
\hspace*{0pt} ~~~~~~~~~~~~~~~~~~{\char125}~in (plus{\char95}nat m q,~m);\\
\hspace*{0pt}fib{\char95}step Zero{\char95}nat = (Suc Zero{\char95}nat,~Zero{\char95}nat);\\
\hspace*{0pt}\\
\hspace*{0pt}fib ::~Nat -> Nat;\\
\hspace*{0pt}fib (Suc n) = fst (fib{\char95}step n);\\
\hspace*{0pt}fib Zero{\char95}nat = Zero{\char95}nat;%
\end{isamarkuptext}%
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\isamarkupsubsection{Datatypes \label{sec:datatypes}%
}
\isamarkuptrue%
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\begin{isamarkuptext}%
Conceptually, any datatype is spanned by a set of
\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
\isa{{\isasymtau}}. The HOL datatype package by default registers any new
datatype in the table of datatypes, which may be inspected using the
\hyperlink{command.print-codesetup}{\mbox{\isa{\isacommand{print{\isacharunderscore}codesetup}}}} command.
In some cases, it is appropriate to alter or extend this table. As
an example, we will develop an alternative representation of the
queue example given in \secref{sec:queue_example}. The amortised
representation is convenient for generating code but exposes its
\qt{implementation} details, which may be cumbersome when proving
theorems about it. Therefore, here is a simple, straightforward
representation of queues:%
\end{isamarkuptext}%
\isamarkuptrue%
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\isatagquote
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\ {\isacharprime}a\ queue\ {\isacharequal}\ Queue\ {\isachardoublequoteopen}{\isacharprime}a\ list{\isachardoublequoteclose}\isanewline
\isanewline
\isacommand{definition}\isamarkupfalse%
\ empty\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ queue{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
\ \ {\isachardoublequoteopen}empty\ {\isacharequal}\ Queue\ {\isacharbrackleft}{\isacharbrackright}{\isachardoublequoteclose}\isanewline
\isanewline
\isacommand{primrec}\isamarkupfalse%
\ enqueue\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ queue\ {\isasymRightarrow}\ {\isacharprime}a\ queue{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
\ \ {\isachardoublequoteopen}enqueue\ x\ {\isacharparenleft}Queue\ xs{\isacharparenright}\ {\isacharequal}\ Queue\ {\isacharparenleft}xs\ {\isacharat}\ {\isacharbrackleft}x{\isacharbrackright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
\isanewline
\isacommand{fun}\isamarkupfalse%
\ dequeue\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ queue\ {\isasymRightarrow}\ {\isacharprime}a\ option\ {\isasymtimes}\ {\isacharprime}a\ queue{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
\ \ \ \ {\isachardoublequoteopen}dequeue\ {\isacharparenleft}Queue\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}None{\isacharcomma}\ Queue\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
\ \ {\isacharbar}\ {\isachardoublequoteopen}dequeue\ {\isacharparenleft}Queue\ {\isacharparenleft}x\ {\isacharhash}\ xs{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}Some\ x{\isacharcomma}\ Queue\ xs{\isacharparenright}{\isachardoublequoteclose}%
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\begin{isamarkuptext}%
\noindent This we can use directly for proving; for executing,
we provide an alternative characterisation:%
\end{isamarkuptext}%
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\isacommand{definition}\isamarkupfalse%
\ AQueue\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ queue{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
\ \ {\isachardoublequoteopen}AQueue\ xs\ ys\ {\isacharequal}\ Queue\ {\isacharparenleft}ys\ {\isacharat}\ rev\ xs{\isacharparenright}{\isachardoublequoteclose}\isanewline
\isanewline
\isacommand{code{\isacharunderscore}datatype}\isamarkupfalse%
\ AQueue%
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\begin{isamarkuptext}%
\noindent Here we define a \qt{constructor} \isa{AQueue} which
is defined in terms of \isa{Queue} and interprets its arguments
according to what the \emph{content} of an amortised queue is supposed
to be. Equipped with this, we are able to prove the following equations
for our primitive queue operations which \qt{implement} the simple
queues in an amortised fashion:%
\end{isamarkuptext}%
\isamarkuptrue%
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\isacommand{lemma}\isamarkupfalse%
\ empty{\isacharunderscore}AQueue\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
\ \ {\isachardoublequoteopen}empty\ {\isacharequal}\ AQueue\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharbrackleft}{\isacharbrackright}{\isachardoublequoteclose}\isanewline
\ \ \isacommand{unfolding}\isamarkupfalse%
\ AQueue{\isacharunderscore}def\ empty{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
\ simp\isanewline
\isanewline
\isacommand{lemma}\isamarkupfalse%
\ enqueue{\isacharunderscore}AQueue\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
\ \ {\isachardoublequoteopen}enqueue\ x\ {\isacharparenleft}AQueue\ xs\ ys{\isacharparenright}\ {\isacharequal}\ AQueue\ {\isacharparenleft}x\ {\isacharhash}\ xs{\isacharparenright}\ ys{\isachardoublequoteclose}\isanewline
\ \ \isacommand{unfolding}\isamarkupfalse%
\ AQueue{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
\ simp\isanewline
\isanewline
\isacommand{lemma}\isamarkupfalse%
\ dequeue{\isacharunderscore}AQueue\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
\ \ {\isachardoublequoteopen}dequeue\ {\isacharparenleft}AQueue\ xs\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\ {\isacharequal}\isanewline
\ \ \ \ {\isacharparenleft}if\ xs\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}\ then\ {\isacharparenleft}None{\isacharcomma}\ AQueue\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\isanewline
\ \ \ \ else\ dequeue\ {\isacharparenleft}AQueue\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharparenleft}rev\ xs{\isacharparenright}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
\ \ {\isachardoublequoteopen}dequeue\ {\isacharparenleft}AQueue\ xs\ {\isacharparenleft}y\ {\isacharhash}\ ys{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}Some\ y{\isacharcomma}\ AQueue\ xs\ ys{\isacharparenright}{\isachardoublequoteclose}\isanewline
\ \ \isacommand{unfolding}\isamarkupfalse%
\ AQueue{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
\ simp{\isacharunderscore}all%
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\begin{isamarkuptext}%
\noindent For completeness, we provide a substitute for the
\isa{case} combinator on queues:%
\end{isamarkuptext}%
\isamarkuptrue%
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\isacommand{lemma}\isamarkupfalse%
\ queue{\isacharunderscore}case{\isacharunderscore}AQueue\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
\ \ {\isachardoublequoteopen}queue{\isacharunderscore}case\ f\ {\isacharparenleft}AQueue\ xs\ ys{\isacharparenright}\ {\isacharequal}\ f\ {\isacharparenleft}ys\ {\isacharat}\ rev\ xs{\isacharparenright}{\isachardoublequoteclose}\isanewline
\ \ \isacommand{unfolding}\isamarkupfalse%
\ AQueue{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
\ simp%
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\begin{isamarkuptext}%
\noindent The resulting code looks as expected:%
\end{isamarkuptext}%
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\begin{isamarkuptext}%
\isatypewriter%
\noindent%
\hspace*{0pt}structure Example :~sig\\
\hspace*{0pt} ~val foldl :~('a -> 'b -> 'a) -> 'a -> 'b list -> 'a\\
\hspace*{0pt} ~val rev :~'a list -> 'a list\\
\hspace*{0pt} ~val null :~'a list -> bool\\
\hspace*{0pt} ~datatype 'a queue = AQueue of 'a list * 'a list\\
\hspace*{0pt} ~val empty :~'a queue\\
\hspace*{0pt} ~val dequeue :~'a queue -> 'a option * 'a queue\\
\hspace*{0pt} ~val enqueue :~'a -> 'a queue -> 'a queue\\
\hspace*{0pt}end = struct\\
\hspace*{0pt}\\
\hspace*{0pt}fun foldl f a [] = a\\
\hspace*{0pt} ~| foldl f a (x ::~xs) = foldl f (f a x) xs;\\
\hspace*{0pt}\\
\hspace*{0pt}fun rev xs = foldl (fn xsa => fn x => x ::~xsa) [] xs;\\
\hspace*{0pt}\\
\hspace*{0pt}fun null [] = true\\
\hspace*{0pt} ~| null (x ::~xs) = false;\\
\hspace*{0pt}\\
\hspace*{0pt}datatype 'a queue = AQueue of 'a list * 'a list;\\
\hspace*{0pt}\\
\hspace*{0pt}val empty :~'a queue = AQueue ([],~[]);\\
\hspace*{0pt}\\
\hspace*{0pt}fun dequeue (AQueue (xs,~y ::~ys)) = (SOME y,~AQueue (xs,~ys))\\
\hspace*{0pt} ~| dequeue (AQueue (xs,~[])) =\\
\hspace*{0pt} ~~~(if null xs then (NONE,~AQueue ([],~[]))\\
\hspace*{0pt} ~~~~~else dequeue (AQueue ([],~rev xs)));\\
\hspace*{0pt}\\
\hspace*{0pt}fun enqueue x (AQueue (xs,~ys)) = AQueue (x ::~xs,~ys);\\
\hspace*{0pt}\\
\hspace*{0pt}end;~(*struct Example*)%
\end{isamarkuptext}%
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\noindent From this example, it can be glimpsed that using own
constructor sets is a little delicate since it changes the set of
valid patterns for values of that type. Without going into much
detail, here some practical hints:
\begin{itemize}
\item When changing the constructor set for datatypes, take care
to provide alternative equations for the \isa{case} combinator.
\item Values in the target language need not to be normalised --
different values in the target language may represent the same
value in the logic.
\item Usually, a good methodology to deal with the subtleties of
pattern matching is to see the type as an abstract type: provide
a set of operations which operate on the concrete representation
of the type, and derive further operations by combinations of
these primitive ones, without relying on a particular
representation.
\end{itemize}%
\end{isamarkuptext}%
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