diff --git a/src/Doc/Codegen/Refinement.thy b/src/Doc/Codegen/Refinement.thy --- a/src/Doc/Codegen/Refinement.thy +++ b/src/Doc/Codegen/Refinement.thy @@ -1,277 +1,277 @@ theory Refinement imports Setup begin section \Program and datatype refinement \label{sec:refinement}\ text \ 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. \ subsection \Program refinement\ text \ Program refinement works by choosing appropriate code equations explicitly (cf.~\secref{sec:equations}); as example, we use Fibonacci numbers: \ fun %quote fib :: "nat \ nat" where "fib 0 = 0" | "fib (Suc 0) = Suc 0" | "fib (Suc (Suc n)) = fib n + fib (Suc n)" text \ \noindent The runtime of the corresponding code grows exponential due to two recursive calls: \ text %quote \ @{code_stmts fib constant: fib (Haskell)} \ text \ \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: \ definition %quote fib_step :: "nat \ nat \ nat" where "fib_step n = (fib (Suc n), fib n)" text \ \noindent This operation can be implemented by recursion using dynamic programming: \ lemma %quote [code]: "fib_step 0 = (Suc 0, 0)" "fib_step (Suc n) = (let (m, q) = fib_step n in (m + q, m))" by (simp_all add: fib_step_def) text \ \noindent What remains is to implement \<^const>\fib\ by \<^const>\fib_step\ as follows: \ lemma %quote [code]: "fib 0 = 0" "fib (Suc n) = fst (fib_step n)" by (simp_all add: fib_step_def) text \ \noindent The resulting code shows only linear growth of runtime: \ text %quote \ @{code_stmts fib constant: fib fib_step (Haskell)} \ subsection \Datatype refinement\ text \ Selecting specific code equations \emph{and} datatype constructors leads to datatype refinement. 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: \ datatype %quote 'a queue = Queue "'a list" definition %quote empty :: "'a queue" where "empty = Queue []" primrec %quote enqueue :: "'a \ 'a queue \ 'a queue" where "enqueue x (Queue xs) = Queue (xs @ [x])" fun %quote dequeue :: "'a queue \ 'a option \ '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 \ 'a list \ '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 \Queue\ and interprets its arguments according to what the \emph{content} of an amortised queue is supposed to be. The prerequisite for datatype constructors is only syntactical: a constructor must be of type \\ = \ \ \ \\<^sub>1 \ \\<^sub>n\ where \{\\<^sub>1, \, \\<^sub>n}\ is exactly the set of \emph{all} type variables in \\\; then \\\ is its corresponding datatype. The HOL datatype package by default registers any new datatype with its constructors, but this may be changed using @{command_def code_datatype}; the currently chosen constructors can be inspected using the @{command print_codesetup} command. 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 [] []" by (simp add: AQueue_def empty_def) lemma %quote enqueue_AQueue [code]: "enqueue x (AQueue xs ys) = AQueue (x # xs) ys" by (simp add: AQueue_def) 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)" by (simp_all add: AQueue_def) text \ \noindent It is good style, although no absolute requirement, to provide code equations for the original artefacts of the implemented type, if possible; in our case, these are the datatype constructor \<^const>\Queue\ and the case combinator \<^const>\case_queue\: \ lemma %quote Queue_AQueue [code]: "Queue = AQueue []" by (simp add: AQueue_def fun_eq_iff) lemma %quote case_queue_AQueue [code]: "case_queue f (AQueue xs ys) = f (ys @ rev xs)" by (simp add: AQueue_def) text \ \noindent The resulting code looks as expected: \ text %quote \ @{code_stmts empty enqueue dequeue Queue case_queue (SML)} \ text \ The same techniques can also be applied to types which are not specified as datatypes, e.g.~type \<^typ>\int\ is originally specified as quotient type by means of @{command_def typedef}, but for code generation constants allowing construction of binary numeral values are used as constructors for \<^typ>\int\. This approach however fails if the representation of a type demands invariants; this issue is discussed in the next section. \ subsection \Datatype refinement involving invariants \label{sec:invariant}\ text \ Datatype representation involving invariants require a dedicated setup for the type and its primitive operations. As a running example, we implement a type \<^typ>\'a dlist\ of lists consisting of distinct elements. The specification of \<^typ>\'a dlist\ itself can be found in theory \<^theory>\HOL-Library.Dlist\. The first step is to decide on which representation the abstract type (in our example \<^typ>\'a dlist\) should be implemented. Here we choose \<^typ>\'a list\. Then a conversion from the concrete type to the abstract type must be specified, here: \ text %quote \ \<^term_type>\Dlist\ \ text \ \noindent Next follows the specification of a suitable \emph{projection}, i.e.~a conversion from abstract to concrete type: \ text %quote \ \<^term_type>\list_of_dlist\ \ text \ \noindent This projection must be specified such that the following \emph{abstract datatype certificate} can be proven: \ lemma %quote [code abstype]: "Dlist (list_of_dlist dxs) = dxs" by (fact Dlist_list_of_dlist) text \ \noindent Note that so far the invariant on representations (\<^term_type>\distinct\) has never been mentioned explicitly: the invariant is only referred to implicitly: all values in set \<^term>\{xs. list_of_dlist (Dlist xs) = xs}\ are invariant, and in our example this is exactly \<^term>\{xs. distinct xs}\. The primitive operations on \<^typ>\'a dlist\ are specified indirectly using the projection \<^const>\list_of_dlist\. For the empty \dlist\, \<^const>\Dlist.empty\, we finally want the code equation \ text %quote \ \<^term>\Dlist.empty = Dlist []\ \ text \ \noindent This we have to prove indirectly as follows: \ lemma %quote [code]: "list_of_dlist Dlist.empty = []" by (fact list_of_dlist_empty) text \ \noindent This equation logically encodes both the desired code equation and that the expression \<^const>\Dlist\ is applied to obeys the implicit invariant. Equations for insertion and removal are similar: \ lemma %quote [code]: "list_of_dlist (Dlist.insert x dxs) = List.insert x (list_of_dlist dxs)" by (fact list_of_dlist_insert) lemma %quote [code]: "list_of_dlist (Dlist.remove x dxs) = remove1 x (list_of_dlist dxs)" by (fact list_of_dlist_remove) text \ \noindent Then the corresponding code is as follows: \ text %quote \ - @{code_stmts Dlist.empty Dlist.insert Dlist.remove list_of_dlist (Haskell)} + @{code_stmts Dlist.empty Dlist.insert Dlist.remove list_of_dlist (SML)} \ text \ See further @{cite "Haftmann-Kraus-Kuncar-Nipkow:2013:data_refinement"} for the meta theory of datatype refinement involving invariants. Typical data structures implemented by representations involving invariants are available in the library, theory \<^theory>\HOL-Library.Mapping\ specifies key-value-mappings (type \<^typ>\('a, 'b) mapping\); these can be implemented by red-black-trees (theory \<^theory>\HOL-Library.RBT\). \ end