doc-src/Codegen/Thy/Refinement.thy

more efficient Markup_Tree, based on branches sorted by quasi-order;
renamed markup_node.scala to markup_tree.scala and classes/objects accordingly;
Position.Range: produce actual Text.Range;
Symbol.Index.decode: convert 1-based Isabelle offsets here;
added static Command.range;
simplified Command.markup;
Document_Model.token_marker: flatten markup at most once;
tuned;

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.~\label{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 %quote {*@{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 %quote {*@{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
  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 [] []"
  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 {*
  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 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 *}

text {*
  FIXME
*}

end