doc-src/Codegen/Refinement.thy

+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 \<Rightarrow> 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 %quotetypewriter {*
+  @{code_stmts fib (consts) 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 \<Rightarrow> nat \<times> 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 %quotetypewriter {*
+  @{code_stmts fib (consts) 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 \<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.
+
+  The prerequisite for datatype constructors is only syntactical: a
+  constructor must be 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>"}; then @{text "\<kappa>"} 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 queue_case}:
+*}
+
+lemma %quote Queue_AQueue [code]:
+  "Queue = AQueue []"
+  by (simp add: AQueue_def fun_eq_iff)
+
+lemma %quote queue_case_AQueue [code]:
+  "queue_case f (AQueue xs ys) = f (ys @ rev xs)"
+  by (simp add: AQueue_def)
+
+text {*
+  \noindent The resulting code looks as expected:
+*}
+
+text %quotetypewriter {*
+  @{code_stmts empty enqueue dequeue Queue queue_case (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 @{text "'a dlist"} of list consisting
+  of distinct elements.
+
+  The first step is to decide on which representation the abstract
+  type (in our example @{text "'a dlist"}) should be implemented.
+  Here we choose @{text "'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 @{text "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 abstract]:
+  "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 abstract]:
+  "list_of_dlist (Dlist.insert x dxs) = List.insert x (list_of_dlist dxs)"
+  by (fact list_of_dlist_insert)
+
+lemma %quote [code abstract]:
+  "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 %quotetypewriter {*
+  @{code_stmts Dlist.empty Dlist.insert Dlist.remove list_of_dlist (Haskell)}
+*} (*(types) dlist (consts) dempty dinsert dremove list_of List.member insert remove *)
+
+text {*
+  Typical data structures implemented by representations involving
+  invariants are available in the library, theory @{theory Mapping}
+  specifies key-value-mappings (type @{typ "('a, 'b) mapping"});
+  these can be implemented by red-black-trees (theory @{theory RBT}).
+*}
+
+end
+