theory Program
imports Introduction
begin
section {* Turning Theories into Programs \label{sec:program} *}
subsection {* The @{text "Isabelle/HOL"} default setup *}
text {*
We have already seen how by default equations stemming from
@{command definition}/@{command primrec}/@{command fun}
statements are used for code generation. This default behaviour
can be changed, e.g. by providing different code equations.
All kinds of customisation shown in this section is \emph{safe}
in the sense that the user does not have to worry about
correctness -- all programs generatable that way are partially
correct.
*}
subsection {* Selecting code equations *}
text {*
Coming back to our introductory example, we
could provide an alternative code equations for @{const dequeue}
explicitly:
*}
lemma %quote [code]:
"dequeue (AQueue xs []) =
(if xs = [] then (None, AQueue [] [])
else dequeue (AQueue [] (rev xs)))"
"dequeue (AQueue xs (y # ys)) =
(Some y, AQueue xs ys)"
by (cases xs, simp_all) (cases "rev xs", simp_all)
text {*
\noindent The annotation @{text "[code]"} is an @{text Isar}
@{text attribute} which states that the given theorems should be
considered as code equations for a @{text fun} statement --
the corresponding constant is determined syntactically. The resulting code:
*}
text %quote {*@{code_stmts dequeue (consts) dequeue (Haskell)}*}
text {*
\noindent You may note that the equality test @{term "xs = []"} has been
replaced by the predicate @{term "null xs"}. This is due to the default
setup in the \qn{preprocessor} to be discussed further below (\secref{sec:preproc}).
Changing the default constructor set of datatypes is also
possible. See \secref{sec:datatypes} for an example.
As told in \secref{sec:concept}, code generation is based
on a structured collection of code theorems.
For explorative purpose, this collection
may be inspected using the @{command code_thms} command:
*}
code_thms %quote dequeue
text {*
\noindent prints a table with \emph{all} code equations
for @{const dequeue}, including
\emph{all} code equations those equations depend
on recursively.
Similarly, the @{command code_deps} command shows a graph
visualising dependencies between code equations.
*}
subsection {* @{text class} and @{text instantiation} *}
text {*
Concerning type classes and code generation, let us examine an example
from abstract algebra:
*}
class %quote semigroup =
fixes mult :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<otimes>" 70)
assumes assoc: "(x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
class %quote monoid = semigroup +
fixes neutral :: 'a ("\<one>")
assumes neutl: "\<one> \<otimes> x = x"
and neutr: "x \<otimes> \<one> = x"
instantiation %quote nat :: monoid
begin
primrec %quote mult_nat where
"0 \<otimes> n = (0\<Colon>nat)"
| "Suc m \<otimes> n = n + m \<otimes> n"
definition %quote neutral_nat where
"\<one> = Suc 0"
lemma %quote add_mult_distrib:
fixes n m q :: nat
shows "(n + m) \<otimes> q = n \<otimes> q + m \<otimes> q"
by (induct n) simp_all
instance %quote proof
fix m n q :: nat
show "m \<otimes> n \<otimes> q = m \<otimes> (n \<otimes> q)"
by (induct m) (simp_all add: add_mult_distrib)
show "\<one> \<otimes> n = n"
by (simp add: neutral_nat_def)
show "m \<otimes> \<one> = m"
by (induct m) (simp_all add: neutral_nat_def)
qed
end %quote
text {*
\noindent We define the natural operation of the natural numbers
on monoids:
*}
primrec %quote (in monoid) pow :: "nat \<Rightarrow> 'a \<Rightarrow> 'a" where
"pow 0 a = \<one>"
| "pow (Suc n) a = a \<otimes> pow n a"
text {*
\noindent This we use to define the discrete exponentiation function:
*}
definition %quote bexp :: "nat \<Rightarrow> nat" where
"bexp n = pow n (Suc (Suc 0))"
text {*
\noindent The corresponding code:
*}
text %quote {*@{code_stmts bexp (Haskell)}*}
text {*
\noindent This is a convenient place to show how explicit dictionary construction
manifests in generated code (here, the same example in @{text SML}):
*}
text %quote {*@{code_stmts bexp (SML)}*}
text {*
\noindent Note the parameters with trailing underscore (@{verbatim "A_"})
which are the dictionary parameters.
*}
subsection {* The preprocessor \label{sec:preproc} *}
text {*
Before selected function theorems are turned into abstract
code, a chain of definitional transformation steps is carried
out: \emph{preprocessing}. In essence, the preprocessor
consists of two components: a \emph{simpset} and \emph{function transformers}.
The \emph{simpset} allows to employ the full generality of the Isabelle
simplifier. Due to the interpretation of theorems
as code equations, rewrites are applied to the right
hand side and the arguments of the left hand side of an
equation, but never to the constant heading the left hand side.
An important special case are \emph{inline theorems} which may be
declared and undeclared using the
\emph{code inline} or \emph{code inline del} attribute respectively.
Some common applications:
*}
text_raw {*
\begin{itemize}
*}
text {*
\item replacing non-executable constructs by executable ones:
*}
lemma %quote [code inline]:
"x \<in> set xs \<longleftrightarrow> x mem xs" by (induct xs) simp_all
text {*
\item eliminating superfluous constants:
*}
lemma %quote [code inline]:
"1 = Suc 0" by simp
text {*
\item replacing executable but inconvenient constructs:
*}
lemma %quote [code inline]:
"xs = [] \<longleftrightarrow> List.null xs" by (induct xs) simp_all
text_raw {*
\end{itemize}
*}
text {*
\noindent \emph{Function transformers} provide a very general interface,
transforming a list of function theorems to another
list of function theorems, provided that neither the heading
constant nor its type change. The @{term "0\<Colon>nat"} / @{const Suc}
pattern elimination implemented in
theory @{text Efficient_Nat} (see \secref{eff_nat}) uses this
interface.
\noindent The current setup of the preprocessor may be inspected using
the @{command print_codesetup} command.
@{command code_thms} provides a convenient
mechanism to inspect the impact of a preprocessor setup
on code equations.
\begin{warn}
The attribute \emph{code unfold}
associated with the @{text "SML code generator"} also applies to
the @{text "generic code generator"}:
\emph{code unfold} implies \emph{code inline}.
\end{warn}
*}
subsection {* Datatypes \label{sec:datatypes} *}
text {*
Conceptually, any datatype is spanned by a set of
\emph{constructors} of type @{text "\<tau> = \<dots> \<Rightarrow> \<kappa> \<alpha>\<^isub>1 \<dots> \<alpha>\<^isub>n"} where @{text
"{\<alpha>\<^isub>1, \<dots>, \<alpha>\<^isub>n}"} is exactly the set of \emph{all} type variables in
@{text "\<tau>"}. The HOL datatype package by default registers any new
datatype in the table of datatypes, which may be inspected using the
@{command print_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:intro}. 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 a simple, straightforward
representation of queues:
*}
datatype %quote 'a queue = Queue "'a list"
definition %quote empty :: "'a queue" where
"empty = Queue []"
primrec %quote enqueue :: "'a \<Rightarrow> 'a queue \<Rightarrow> 'a queue" where
"enqueue x (Queue xs) = Queue (xs @ [x])"
fun %quote dequeue :: "'a queue \<Rightarrow> 'a option \<times> 'a queue" where
"dequeue (Queue []) = (None, Queue [])"
| "dequeue (Queue (x # xs)) = (Some x, Queue xs)"
text {*
\noindent This we can use directly for proving; for executing,
we provide an alternative characterisation:
*}
definition %quote AQueue :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a queue" where
"AQueue xs ys = Queue (ys @ rev xs)"
code_datatype %quote AQueue
text {*
\noindent Here we define a \qt{constructor} @{const "AQueue"} which
is defined in terms of @{text "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:
*}
lemma %quote empty_AQueue [code]:
"empty = AQueue [] []"
unfolding AQueue_def empty_def by simp
lemma %quote enqueue_AQueue [code]:
"enqueue x (AQueue xs ys) = AQueue (x # xs) ys"
unfolding AQueue_def by simp
lemma %quote dequeue_AQueue [code]:
"dequeue (AQueue xs []) =
(if xs = [] then (None, AQueue [] [])
else dequeue (AQueue [] (rev xs)))"
"dequeue (AQueue xs (y # ys)) = (Some y, AQueue xs ys)"
unfolding AQueue_def by simp_all
text {*
\noindent For completeness, we provide a substitute for the
@{text case} combinator on queues:
*}
lemma %quote queue_case_AQueue [code]:
"queue_case f (AQueue xs ys) = f (ys @ rev xs)"
unfolding AQueue_def by simp
text {*
\noindent The resulting code looks as expected:
*}
text %quote {*@{code_stmts empty enqueue dequeue (SML)}*}
text {*
\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 @{text 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}
*}
subsection {* Equality and wellsortedness *}
text {*
Surely you have already noticed how equality is treated
by the code generator:
*}
primrec %quote collect_duplicates :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
"collect_duplicates xs ys [] = xs"
| "collect_duplicates xs ys (z#zs) = (if z \<in> set xs
then if z \<in> set ys
then collect_duplicates xs ys zs
else collect_duplicates xs (z#ys) zs
else collect_duplicates (z#xs) (z#ys) zs)"
text {*
\noindent The membership test during preprocessing is rewritten,
resulting in @{const List.member}, which itself
performs an explicit equality check.
*}
text %quote {*@{code_stmts collect_duplicates (SML)}*}
text {*
\noindent Obviously, polymorphic equality is implemented the Haskell
way using a type class. How is this achieved? HOL introduces
an explicit class @{class eq} with a corresponding operation
@{const eq_class.eq} such that @{thm eq [no_vars]}.
The preprocessing framework does the rest by propagating the
@{class eq} constraints through all dependent code equations.
For datatypes, instances of @{class eq} are implicitly derived
when possible. For other types, you may instantiate @{text eq}
manually like any other type class.
Though this @{text eq} class is designed to get rarely in
the way, a subtlety
enters the stage when definitions of overloaded constants
are dependent on operational equality. For example, let
us define a lexicographic ordering on tuples
(also see theory @{theory Product_ord}):
*}
instantiation %quote "*" :: (order, order) order
begin
definition %quote [code del]:
"x \<le> y \<longleftrightarrow> fst x < fst y \<or> fst x = fst y \<and> snd x \<le> snd y"
definition %quote [code del]:
"x < y \<longleftrightarrow> fst x < fst y \<or> fst x = fst y \<and> snd x < snd y"
instance %quote proof
qed (auto simp: less_eq_prod_def less_prod_def intro: order_less_trans)
end %quote
lemma %quote order_prod [code]:
"(x1 \<Colon> 'a\<Colon>order, y1 \<Colon> 'b\<Colon>order) < (x2, y2) \<longleftrightarrow>
x1 < x2 \<or> x1 = x2 \<and> y1 < y2"
"(x1 \<Colon> 'a\<Colon>order, y1 \<Colon> 'b\<Colon>order) \<le> (x2, y2) \<longleftrightarrow>
x1 < x2 \<or> x1 = x2 \<and> y1 \<le> y2"
by (simp_all add: less_prod_def less_eq_prod_def)
text {*
\noindent Then code generation will fail. Why? The definition
of @{term "op \<le>"} depends on equality on both arguments,
which are polymorphic and impose an additional @{class eq}
class constraint, which the preprocessor does not propagate
(for technical reasons).
The solution is to add @{class eq} explicitly to the first sort arguments in the
code theorems:
*}
lemma %quote order_prod_code [code]:
"(x1 \<Colon> 'a\<Colon>{order, eq}, y1 \<Colon> 'b\<Colon>order) < (x2, y2) \<longleftrightarrow>
x1 < x2 \<or> x1 = x2 \<and> y1 < y2"
"(x1 \<Colon> 'a\<Colon>{order, eq}, y1 \<Colon> 'b\<Colon>order) \<le> (x2, y2) \<longleftrightarrow>
x1 < x2 \<or> x1 = x2 \<and> y1 \<le> y2"
by (simp_all add: less_prod_def less_eq_prod_def)
text {*
\noindent Then code generation succeeds:
*}
text %quote {*@{code_stmts "op \<le> \<Colon> _ \<times> _ \<Rightarrow> _ \<times> _ \<Rightarrow> bool" (SML)}*}
text {*
In some cases, the automatically derived code equations
for equality on a particular type may not be appropriate.
As example, watch the following datatype representing
monomorphic parametric types (where type constructors
are referred to by natural numbers):
*}
datatype %quote monotype = Mono nat "monotype list"
(*<*)
lemma monotype_eq:
"eq_class.eq (Mono tyco1 typargs1) (Mono tyco2 typargs2) \<equiv>
eq_class.eq tyco1 tyco2 \<and> eq_class.eq typargs1 typargs2" by (simp add: eq)
(*>*)
text {*
\noindent Then code generation for SML would fail with a message
that the generated code contains illegal mutual dependencies:
the theorem @{thm monotype_eq [no_vars]} already requires the
instance @{text "monotype \<Colon> eq"}, which itself requires
@{thm monotype_eq [no_vars]}; Haskell has no problem with mutually
recursive @{text instance} and @{text function} definitions,
but the SML serialiser does not support this.
In such cases, you have to provide your own equality equations
involving auxiliary constants. In our case,
@{const [show_types] list_all2} can do the job:
*}
lemma %quote monotype_eq_list_all2 [code]:
"eq_class.eq (Mono tyco1 typargs1) (Mono tyco2 typargs2) \<longleftrightarrow>
eq_class.eq tyco1 tyco2 \<and> list_all2 eq_class.eq typargs1 typargs2"
by (simp add: eq list_all2_eq [symmetric])
text {*
\noindent does not depend on instance @{text "monotype \<Colon> eq"}:
*}
text %quote {*@{code_stmts "eq_class.eq :: monotype \<Rightarrow> monotype \<Rightarrow> bool" (SML)}*}
subsection {* Explicit partiality *}
text {*
Partiality usually enters the game by partial patterns, as
in the following example, again for amortised queues:
*}
definition %quote strict_dequeue :: "'a queue \<Rightarrow> 'a \<times> 'a queue" where
"strict_dequeue q = (case dequeue q
of (Some x, q') \<Rightarrow> (x, q'))"
lemma %quote strict_dequeue_AQueue [code]:
"strict_dequeue (AQueue xs (y # ys)) = (y, AQueue xs ys)"
"strict_dequeue (AQueue xs []) =
(case rev xs of y # ys \<Rightarrow> (y, AQueue [] ys))"
by (simp_all add: strict_dequeue_def dequeue_AQueue split: list.splits)
text {*
\noindent In the corresponding code, there is no equation
for the pattern @{term "AQueue [] []"}:
*}
text %quote {*@{code_stmts strict_dequeue (consts) strict_dequeue (Haskell)}*}
text {*
\noindent In some cases it is desirable to have this
pseudo-\qt{partiality} more explicitly, e.g.~as follows:
*}
axiomatization %quote empty_queue :: 'a
definition %quote strict_dequeue' :: "'a queue \<Rightarrow> 'a \<times> 'a queue" where
"strict_dequeue' q = (case dequeue q of (Some x, q') \<Rightarrow> (x, q') | _ \<Rightarrow> empty_queue)"
lemma %quote strict_dequeue'_AQueue [code]:
"strict_dequeue' (AQueue xs []) = (if xs = [] then empty_queue
else strict_dequeue' (AQueue [] (rev xs)))"
"strict_dequeue' (AQueue xs (y # ys)) =
(y, AQueue xs ys)"
by (simp_all add: strict_dequeue'_def dequeue_AQueue split: list.splits)
text {*
Observe that on the right hand side of the definition of @{const
"strict_dequeue'"} the constant @{const empty_queue} occurs
which is unspecified.
Normally, if constants without any code equations occur in a
program, the code generator complains (since in most cases this is
not what the user expects). But such constants can also be thought
of as function definitions with no equations which always fail,
since there is never a successful pattern match on the left hand
side. In order to categorise a constant into that category
explicitly, use @{command "code_abort"}:
*}
code_abort %quote empty_queue
text {*
\noindent Then the code generator will just insert an error or
exception at the appropriate position:
*}
text %quote {*@{code_stmts strict_dequeue' (consts) empty_queue strict_dequeue' (Haskell)}*}
text {*
\noindent This feature however is rarely needed in practice.
Note also that the @{text HOL} default setup already declares
@{const undefined} as @{command "code_abort"}, which is most
likely to be used in such situations.
*}
end