[matthijs/master-project/report.git] / Chapters / Prototype.tex

\chapter[chap:prototype]{Prototype}
  An important step in this research is the creation of a prototype compiler.
  Having this prototype allows us to apply the ideas from the previous chapter
  to actual hardware descriptions and evaluate their usefulness. Having a
  prototype also helps to find new techniques and test possible
  interpretations.

  Obviously the prototype was not created after all research
  ideas were formed, but its implementation has been interleaved with the
  research itself. Also, the prototype described here is the final version, it
  has gone through a number of design iterations which we will not completely
  describe here.

  \section{Choice of language}
    When implementing this prototype, the first question to ask is: What
    (functional) language will we use to describe our hardware? (Note that
    this does not concern the \emph{implementation language} of the compiler,
    just the language \emph{translated by} the compiler).

    On the highest level, we have two choices:

    \startitemize
      \item Create a new functional language from scratch. This has the
      advantage of having a language that contains exactly those elements that
      are convenient for describing hardware and can contain special
      constructs that might.
      \item Use an existing language and create a new backend for it. This has
      the advantage that existing tools can be reused, which will speed up
      development.
    \stopitemize

    Considering that we required a prototype which should be working quickly,
    and that implementing parsers, semantic checkers and especially
    typcheckers isn't exactly the core of this research (but it is lots and
    lots of work!), using an existing language is the obvious choice. This
    also has the advantage that a large set of language features is available
    to experiment with and it is easy to find which features apply well and
    which don't. A possible second prototype could use a custom language with
    just the useful features (and possibly extra features that are specific to
    the domain of hardware description as well).

    The second choice is to pick one of the many existing languages. As
    mentioned before, this language is Haskell.  This choice has not been the
    result of a thorough comparison of languages, for the simple reason that
    the requirements on the language were completely unclear at the start of
    this language. The fact that Haskell is a language with a broad spectrum
    of features, that it is commonly used in research projects and that the
    primary compiler, GHC, provides a high level API to its internals, made
    Haskell an obvious choice.

    TODO: Was Haskell really a good choice? Perhaps say this somewhere else?

  \section{Prototype design}
    As stated above, we will use the Glasgow Haskell Compiler (\small{GHC}) to
    implement our prototype compiler. To understand the design of the
    compiler, we will first dive into the \small{GHC} compiler a bit. It's
    compilation consists of the following steps (slightly simplified):

    \startuseMPgraphic{ghc-pipeline}
      % Create objects
      save inp, front, desugar, simpl, back, out;
      newEmptyBox.inp(0,0);
      newBox.front(btex Parser etex);
      newBox.desugar(btex Desugarer etex);
      newBox.simpl(btex Simplifier etex);
      newBox.back(btex Backend etex);
      newEmptyBox.out(0,0);

      % Space the boxes evenly
      inp.c - front.c = front.c - desugar.c = desugar.c - simpl.c 
        = simpl.c - back.c = back.c - out.c = (0, 1.5cm);
      out.c = origin;

      % Draw lines between the boxes. We make these lines "deferred" and give
      % them a name, so we can use ObjLabel to draw a label beside them.
      ncline.inp(inp)(front) "name(haskell)";
      ncline.front(front)(desugar) "name(ast)";
      ncline.desugar(desugar)(simpl) "name(core)";
      ncline.simpl(simpl)(back) "name(simplcore)";
      ncline.back(back)(out) "name(native)";
      ObjLabel.inp(btex Haskell source etex) "labpathname(haskell)", "labdir(rt)";
      ObjLabel.front(btex Haskell AST etex) "labpathname(ast)", "labdir(rt)";
      ObjLabel.desugar(btex Core etex) "labpathname(core)", "labdir(rt)";
      ObjLabel.simpl(btex Simplified core etex) "labpathname(simplcore)", "labdir(rt)";
      ObjLabel.back(btex Native code etex) "labpathname(native)", "labdir(rt)";

      % Draw the objects (and deferred labels)
      drawObj (inp, front, desugar, simpl, back, out);
    \stopuseMPgraphic
    \placefigure[right]{GHC compiler pipeline}{\useMPgraphic{ghc-pipeline}}

    \startdesc{Frontend}
      This step takes the Haskell source files and parses them into an
      abstract syntax tree (\small{AST}). This \small{AST} can express the
      complete Haskell language and is thus a very complex one (in contrast
      with the Core \small{AST}, later on). All identifiers in this
      \small{AST} are resolved by the renamer and all types are checked by the
      typechecker.
    \stopdesc
    \startdesc{Desugaring}
      This steps takes the full \small{AST} and translates it to the
      \emph{Core} language. Core is a very small functional language with lazy
      semantics, that can still express everything Haskell can express. Its
      simpleness makes Core very suitable for further simplification and
      translation. Core is the language we will be working on as well.
    \stopdesc
    \startdesc{Simplification}
      Through a number of simplification steps (such as inlining, common
      subexpression elimination, etc.) the Core program is simplified to make
      it faster or easier to process further.
    \stopdesc
    \startdesc{Backend}
      This step takes the simplified Core program and generates an actual
      runnable program for it. This is a big and complicated step we will not
      discuss it any further, since it is not required for our prototype.
    \stopdesc

    In this process, there a number of places where we can start our work.
    Assuming that we don't want to deal with (or modify) parsing, typechecking
    and other frontend business and that native code isn't really a useful
    format anymore, we are left with the choice between the full Haskell
    \small{AST}, or the smaller (simplified) core representation.

    The advantage of taking the full \small{AST} is that the exact structure
    of the source program is preserved. We can see exactly what the hardware
    descriiption looks like and which syntax constructs were used. However,
    the full \small{AST} is a very complicated datastructure. If we are to
    handle everything it offers, we will quickly get a big compiler.

    Using the core representation gives us a much more compact datastructure
    (a core expression only uses 9 constructors). Note that this does not mean
    that the core representation itself is smaller, on the contrary. Since the
    core language has less constructs, a lot of things will take a larger
    expression to express.

    However, the fact that the core language is so much smaller, means it is a
    lot easier to analyze and translate it into something else. For the same
    reason, \small{GHC} runs its simplifications and optimizations on the core
    representation as well.

    However, we will use the normal core representation, not the simplified
    core. Reasons for this are detailed below.
    
    The final prototype roughly consists of three steps:
    
    \startuseMPgraphic{ghc-pipeline}
      % Create objects
      save inp, front, norm, vhdl, out;
      newEmptyBox.inp(0,0);
      newBox.front(btex \small{GHC} frontend + desugarer etex);
      newBox.norm(btex Normalization etex);
      newBox.vhdl(btex \small{VHDL} generation etex);
      newEmptyBox.out(0,0);

      % Space the boxes evenly
      inp.c - front.c = front.c - norm.c = norm.c - vhdl.c 
        = vhdl.c - out.c = (0, 1.5cm);
      out.c = origin;

      % Draw lines between the boxes. We make these lines "deferred" and give
      % them a name, so we can use ObjLabel to draw a label beside them.
      ncline.inp(inp)(front) "name(haskell)";
      ncline.front(front)(norm) "name(core)";
      ncline.norm(norm)(vhdl) "name(normal)";
      ncline.vhdl(vhdl)(out) "name(vhdl)";
      ObjLabel.inp(btex Haskell source etex) "labpathname(haskell)", "labdir(rt)";
      ObjLabel.front(btex Core etex) "labpathname(core)", "labdir(rt)";
      ObjLabel.norm(btex Normalized core etex) "labpathname(normal)", "labdir(rt)";
      ObjLabel.vhdl(btex \small{VHDL} description etex) "labpathname(vhdl)", "labdir(rt)";

      % Draw the objects (and deferred labels)
      drawObj (inp, front, norm, vhdl, out);
    \stopuseMPgraphic
    \placefigure[right]{GHC compiler pipeline}{\useMPgraphic{ghc-pipeline}}

    \startdesc{Frontend}
      This is exactly the frontend and desugarer from the \small{GHC}
      pipeline, that translates Haskell sources to a core representation.
    \stopdesc
    \startdesc{Normalization}
      This is a step that transforms the core representation into a normal
      form. This normal form is still expressed in the core language, but has
      to adhere to an extra set of constraints. This normal form is less
      expressive than the full core language (e.g., it can have limited higher
      order expressions, has a specific structure, etc.), but is also very
      close to directly describing hardware.
    \stopdesc
    \startdesc{\small{VHDL} generation}
      The last step takes the normal formed core representation and generates
      \small{VHDL} for it. Since the normal form has a specific, hardware-like
      structure, this final step is very straightforward.
    \stopdesc
    
    The most interesting step in this process is the normalization step. That
    is where more complicated functional constructs, which have no direct
    hardware interpretation, are removed and translated into hardware
    constructs. This step is described in a lot of detail at
    \in{chapter}[chap:normalization].
    
  \section{The Core language}
    Most of the prototype deals with handling the program in the Core
    language. In this section we will show what this language looks like and
    how it works.

    The Core language is a functional language that describes
    \emph{expressions}. Every identifier used in Core is called a
    \emph{binder}, since it is bound to a value somewhere. On the highest
    level, a Core program is a collection of functions, each of which bind a
    binder (the function name) to an expression (the function value, which has
    a function type).

    The Core language itself does not prescribe any program structure, only
    expression structure. In the \small{GHC} compiler, the Haskell module
    structure is used for the resulting Core code as well. Since this is not
    so relevant for understanding the Core language or the Normalization
    process, we'll only look at the Core expression language here.

    Each Core expression consists of one of these possible expressions.

    \startdesc{Variable reference}
\startlambda
a
\stoplambda
      This is a simple reference to a binder. It's written down as the
      name of the binder that is being referred to, which should of course be
      bound in a containing scope (including top level scope, so a reference
      to a top level function is also a variable reference). Additionally,
      constructors from algebraic datatypes also become variable references.

      The value of this expression is the value bound to the given binder.

      Each binder also carries around its type, but this is usually not shown
      in the Core expressions. Occasionally, the type of an entire expression
      or function is shown for clarity, but this is only informational. In
      practice, the type of an expression is easily determined from the
      structure of the expression and the types of the binders and occasional
      cast expressions. This minimize the amount of bookkeeping needed to keep
      the typing consistent.
    \stopdesc
    \startdesc{Literal}
\startlambda
10
\stoplambda
      This is a simple literal. Only primitive types are supported, like
      chars, strings, ints and doubles. The types of these literals are the
      \quote{primitive} versions, like \lam{Char\#} and \lam{Word\#}, not the
      normal Haskell versions (but there are builtin conversion functions).
    \stopdesc
    \startdesc{Application}
\startlambda
func arg
\stoplambda
      This is simple function application. Each application consists of two
      parts: The function part and the argument part. Applications are used
      for normal function \quote{calls}, but also for applying type
      abstractions and data constructors.
      
      The value of an application is the value of the function part, with the
      first argument binder bound to the argument part.
    \stopdesc
    \startdesc{Lambda abstraction}
\startlambda
λbndr.body
\stoplambda
      This is the basic lambda abstraction, as it occurs in labmda calculus.
      It consists of a binder part and a body part.  A lambda abstraction
      creates a function, that can be applied to an argument. 
     
      Note that the body of a lambda abstraction extends all the way to the
      end of the expression, or the closing bracket surrounding the lambda. In
      other words, the lambda abstraction \quote{operator} has the lowest
      priority of all.

      The value of an application is the value of the body part, with the
      binder bound to the value the entire lambda abstraction is applied to.
    \stopdesc
    \startdesc{Non-recursive let expression}
\startlambda
let bndr = value in body
\stoplambda
      A let expression allows you to bind a binder to some value, while
      evaluating to some other value (where that binder is in scope). This
      allows for sharing of subexpressions (you can use a binder twice) and
      explicit \quote{naming} of arbitrary expressions. Note that the binder
      is not in scope in the value bound to it, so it's not possible to make
      recursive definitions with the normal form of the let expression (see
      the recursive form below).

      Even though this let expression is an extension on the basic lambda
      calculus, it is easily translated to a lambda abstraction. The let
      expression above would then become:

\startlambda
(λbndr.body) value
\stoplambda

      This notion might be useful for verifying certain properties on
      transformations, since a lot of verification work has been done on
      lambda calculus already.

      The value of a let expression is the value of the body part, with the
      binder bound to the value. 
    \stopdesc
    \startdesc{Recursive let expression}
\startlambda
let 
  bndr1 = value1
  \vdots
  bndrn = valuen
in 
  body
\stoplambda

      This is the recursive version of the let expression. In \small{GHC}'s
      Core implementation, non-recursive and recursive lets are not so
      distinct as we present them here, but this provides a clearer overview.
      
      The main difference with the normal let expression is that each of the
      binders is in scope in each of the values, in addition to the body. This
      allows for self-recursive definitions or mutually recursive definitions.

      It should also be possible to express a recursive let using normal
      lambda calculus, if we use the \emph{least fixed-point operator},
      \lam{Y}.
    \stopdesc
    \startdesc{Case expression}
\startlambda
  case scrut of bndr
    DEFAULT -> defaultbody
    C0 bndr0,0 ... bndr0,m -> body0
    \vdots
    Cn bndrn,0 ... bndrn,m -> bodyn
\stoplambda

TODO: Define WHNF

    A case expression is the only way in Core to choose between values. A case
    expression evaluates its scrutinee, which should have an algebraic
    datatype, into weak head normal form (\small{WHNF}) and (optionally) binds
    it to \lam{bndr}. It then chooses a body depending on the constructor of
    its scrutinee. If none of the constructors match, the \lam{DEFAULT}
    alternative is chosen. 
    
    Since we can only match the top level constructor, there can be no overlap
    in the alternatives and thus order of alternatives is not relevant (though
    the \lam{DEFAULT} alternative must appear first for implementation
    efficiency).
    
    Any arguments to the constructor in the scrutinee are bound to each of the
    binders after the constructor and are in scope only in the corresponding
    body.

    To support strictness, the scrutinee is always evaluated into WHNF, even
    when there is only a \lam{DEFAULT} alternative. This allows a strict
    function argument to be written like:

\startlambda
function (case argument of arg
  DEFAULT -> arg)
\stoplambda

    This seems to be the only use for the extra binder to which the scrutinee
    is bound. When not using strictness annotations (which is rather pointless
    in hardware descriptions), \small{GHC} seems to never generate any code
    making use of this binder. The current prototype does not handle it
    either, which probably means that code using it would break.

    Note that these case statements are less powerful than the full Haskell
    case statements. In particular, they do not support complex patterns like
    in Haskell. Only the constructor of an expression can be matched, complex
    patterns are implemented using multiple nested case expressions.

    Case statements are also used for unpacking of algebraic datatypes, even
    when there is only a single constructor. For examples, to add the elements
    of a tuple, the following Core is generated:

\startlambda
sum = λtuple.case tuple of
  (,) a b -> a + b
\stoplambda
  
    Here, there is only a single alternative (but no \lam{DEFAULT}
    alternative, since the single alternative is already exhaustive). When
    it's body is evaluated, the arguments to the tuple constructor \lam{(,)}
    (\eg, the elements of the tuple) are bound to \lam{a} and \lam{b}.
  \stopdesc
  \startdesc{Cast expression}
\startlambda
body :: targettype
\stoplambda
    A cast expression allows you to change the type of an expression to an
    equivalent type. Note that this is not meant to do any actual work, like
    conversion of data from one format to another, or force a complete type
    change. Instead, it is meant to change between different representations
    of the same type, \eg switch between types that are provably equal (but
    look different).
    
    In our hardware descriptions, we typically see casts to change between a
    Haskell newtype and its contained type, since those are effectively
    different representations of the same type.

    More complex are types that are proven to be equal by the typechecker,
    but look different at first glance. To ensure that, once the typechecker
    has proven equality, this information sticks around, explicit casts are
    added. In our notation we only write the target type, but in reality a
    cast expressions carries around a \emph{coercion}, which can be seen as a
    proof of equality. TODO: Example

    The value of a cast is the value of its body, unchanged. The type of this
    value is equal to the target type, not the type of its body.

    Note that this syntax is also used sometimes to indicate that a particular
    expression has a particular type, even when no cast expression is
    involved. This is then purely informational, since the only elements that
    are explicitely typed in the Core language are the binder references and
    cast expressions, the types of all other elements are determined at
    runtime.
  \stopdesc
  \startdesc{Note}

    The Core language in \small{GHC} allows adding \emph{notes}, which serve
    as hints to the inliner or add custom (string) annotations to a core
    expression. These shouldn't be generated normally, so these are not
    handled in any way in the prototype.
  \stopdesc
  \startdesc{Type}
\startlambda
@type
\stoplambda
    It is possibly to use a Core type as a Core expression. This is done to
    allow for type abstractions and applications to be handled as normal
    lambda abstractions and applications above. This means that a type
    expression in Core can only ever occur in the argument position of an
    application, and only if the type of the function that is applied to
    expects a type as the first argument. This happens for all polymorphic
    functions, for example, the \lam{fst} function:

\startlambda
fst :: \forall a. \forall b. (a, b) -> a
fst = λtup.case tup of (,) a b -> a

fstint :: (Int, Int) -> Int
fstint = λa.λb.fst @Int @Int a b
\stoplambda
    
    The type of \lam{fst} has two universally quantified type variables. When
    \lam{fst} is applied in \lam{fstint}, it is first applied to two types.
    (which are substitued for \lam{a} and \lam{b} in the type of \lam{fst}, so
    the type of \lam{fst} actual type of arguments and result can be found:
    \lam{fst @Int @Int :: (Int, Int) -> Int}).
  \stopdesc

  TODO: Core type system

  \section[sec:prototype:statetype]{State annotations in Haskell}
      Ideal: Type synonyms, since there is no additional code overhead for
      packing and unpacking. Downside: there is no explicit conversion in Core
      either, so type synonyms tend to get lost in expressions (they can be
      preserved in binders, but this makes implementation harder, since that
      statefulness of a value must be manually tracked).

      Less ideal: Newtype. Requires explicit packing and unpacking of function
      arguments. If you don't unpack substates, there is no overhead for
      (un)packing substates. This will result in many nested State constructors
      in a nested state type. \eg: 

  \starttyping
  State (State Bit, State (State Word, Bit), Word)
  \stoptyping

      Alternative: Provide different newtypes for input and output state. This
      makes the code even more explicit, and typechecking can find even more
      errors. However, this requires defining two type synomyms for each
      stateful function instead of just one. \eg:
  \starttyping
  type AccumStateIn = StateIn Bit
  type AccumStateOut = StateOut Bit
  \stoptyping
      This also increases the possibility of having different input and output
      states. Checking for identical input and output state types is also
      harder, since each element in the state must be unpacked and compared
      separately.

      Alternative: Provide a type for the entire result type of a stateful
      function, not just the state part. \eg:

  \starttyping
  newtype Result state result = Result (state, result)
  \stoptyping
      
      This makes it easy to say "Any stateful function must return a
      \type{Result} type, without having to sort out result from state. However,
      this either requires a second type for input state (similar to
      \type{StateIn} / \type{StateOut} above), or requires the compiler to
      select the right argument for input state by looking at types (which works
      for complex states, but when that state has the same type as an argument,
      things get ambiguous) or by selecting a fixed (\eg, the last) argument,
      which might be limiting.

      \subsubsection{Example}
      As an example of the used approach, a simple averaging circuit, that lets
      the accumulation of the inputs be done by a subcomponent.

      \starttyping
        newtype State s = State s

        type AccumState = State Bit
        accum :: Word -> AccumState -> (AccumState, Word)
        accum i (State s) = (State (s + i), s + i)

        type AvgState = (AccumState, Word)
        avg :: Word -> AvgState -> (AvgState, Word)
        avg i (State s) = (State s', o)
          where
            (accums, count) = s
            -- Pass our input through the accumulator, which outputs a sum
            (accums', sum) = accum i accums
            -- Increment the count (which will be our new state)
            count' = count + 1
            -- Compute the average
            o = sum / count'
            s' = (accums', count')
      \stoptyping

      And the normalized, core-like versions:

      \starttyping
        accum i spacked = res
          where
            s = case spacked of (State s) -> s
            s' = s + i
            spacked' = State s'
            o = s + i
            res = (spacked', o)

        avg i spacked = res
          where
            s = case spacked of (State s) -> s
            accums = case s of (accums, \_) -> accums
            count = case s of (\_, count) -> count
            accumres = accum i accums
            accums' = case accumres of (accums', \_) -> accums'
            sum = case accumres of (\_, sum) -> sum
            count' = count + 1
            o = sum / count'
            s' = (accums', count')
            spacked' = State s'
            res = (spacked', o)
      \stoptyping



      As noted above, any component of a function's state that is a substate,
      \eg passed on as the state of another function, should have no influence
      on the hardware generated for the calling function. Any state-specific
      \small{VHDL} for this component can be generated entirely within the called
      function. So,we can completely leave out substates from any function.
      
      From this observation, we might think to remove the substates from a
      function's states alltogether, and leave only the state components which
      are actual states of the current function. While doing this would not
      remove any information needed to generate \small{VHDL} from the function, it would
      cause the function definition to become invalid (since we won't have any
      substate to pass to the functions anymore). We could solve the syntactic
      problems by passing \type{undefined} for state variables, but that would
      still break the code on the semantic level (\ie, the function would no
      longer be semantically equivalent to the original input).

      To keep the function definition correct until the very end of the process,
      we will not deal with (sub)states until we get to the \small{VHDL} generation.
      Here, we are translating from Core to \small{VHDL}, and we can simply not generate
      \small{VHDL} for substates, effectively removing the substate components
      alltogether.

      There are a few important points when ignore substates.

      First, we have to have some definition of "substate". Since any state
      argument or return value that represents state must be of the \type{State}
      type, we can simply look at its type. However, we must be careful to
      ignore only {\em substates}, and not a function's own state.

      In the example above, this means we should remove \type{accums'} from
      \type{s'}, but not throw away \type{s'} entirely. We should, however,
      remove \type{s'} from the output port of the function, since the state
      will be handled by a \small{VHDL} procedure within the function.

      When looking at substates, these can appear in two places: As part of an
      argument and as part of a return value. As noted above, these substates
      can only be used in very specific ways.

      \desc{State variables can appear as an argument.} When generating \small{VHDL}, we
      completely ignore the argument and generate no input port for it.

      \desc{State variables can be extracted from other state variables.} When
      extracting a state variable from another state variable, this always means
      we're extracting a substate, which we can ignore. So, we simply generate no
      \small{VHDL} for any extraction operation that has a state variable as a result.

      \desc{State variables can be passed to functions.} When passing a
      state variable to a function, this always means we're passing a substate
      to a subcomponent. The entire argument can simply be ingored in the
      resulting port map.

      \desc{State variables can be returned from functions.} When returning a
      state variable from a function (probably as a part of an algebraic
      datatype), this always mean we're returning a substate from a
      subcomponent. The entire state variable should be ignored in the resulting
      port map. The type binder of the binder that the function call is bound
      to should not include the state type either.

      \startdesc{State variables can be inserted into other variables.} When inserting
      a state variable into another variable (usually by constructing that new
      variable using its constructor), we can identify two cases: 

      \startitemize
        \item The state is inserted into another state variable. In this case,
        the inserted state is a substate, and can be safely left out of the
        constructed variable.
        \item The state is inserted into a non-state variable. This happens when
        building up the return value of a function, where you put state and
        retsult variables together in an algebraic type (usually a tuple). In
        this case, we should leave the state variable out as well, since we
        don't want it to be included as an output port.
      \stopitemize

      So, in both cases, we can simply leave out the state variable from the
      resulting value. In the latter case, however, we should generate a state
      proc instead, which assigns the state variable to the input state variable
      at each clock tick.
      \stopdesc
      
      \desc{State variables can appear as (part of) a function result.} When
      generating \small{VHDL}, we can completely ignore any part of a function result
      that has a state type. If the entire result is a state type, this will
      mean the entity will not have an output port. Otherwise, the state
      elements will be removed from the type of the output port.


      Now, we know how to handle each use of a state variable separately. If we
      look at the whole, we can conclude the following:

      \startitemize
      \item A state unpack operation should not generate any \small{VHDL}. The binder
      to which the unpacked state is bound should still be declared, this signal
      will become the register and will hold the current state.
      \item A state pack operation should not generate any \small{VHDL}. The binder th
      which the packed state is bound should not be declared. The binder that is
      packed is the signal that will hold the new state.
      \item Any values of a State type should not be translated to \small{VHDL}. In
      particular, State elements should be removed from tuples (and other
      datatypes) and arguments with a state type should not generate ports.
      \item To make the state actually work, a simple \small{VHDL} proc should be
      generated. This proc updates the state at every clockcycle, by assigning
      the new state to the current state. This will be recognized by synthesis
      tools as a register specification.
      \stopitemize


      When applying these rules to the example program (in normal form), we will
      get the following result. All the parts that don't generate any value are
      crossed out, leaving some very boring assignments here and there.
      
    
  \starthaskell
    avg i --spacked-- = res
      where
        s = --case spacked of (State s) -> s--
        --accums = case s of (accums, \_) -> accums--
        count = case s of (--\_,-- count) -> count
        accumres = accum i --accums--
        --accums' = case accumres of (accums', \_) -> accums'--
        sum = case accumres of (--\_,-- sum) -> sum
        count' = count + 1
        o = sum / count'
        s' = (--accums',-- count')
        --spacked' = State s'--
        res = (--spacked',-- o)
  \stophaskell
          
      When we would really leave out the crossed out parts, we get a slightly
      weird program: There is a variable \type{s} which has no value, and there
      is a variable \type{s'} that is never used. Together, these two will form
      the state proc of the function. \type{s} contains the "current" state,
      \type{s'} is assigned the "next" state. So, at the end of each clock
      cycle, \type{s'} should be assigned to \type{s}.

      Note that the definition of \type{s'} is not removed, even though one
      might think it as having a state type. Since the state type has a single
      argument constructor \type{State}, some type that should be the resulting
      state should always be explicitly packed with the State constructor,
      allowing us to remove the packed version, but still generate \small{VHDL} for the
      unpacked version (of course with any substates removed).
      
      As you can see, the definition of \type{s'} is still present, since it
      does not have a state type (The State constructor. The \type{accums'} substate has been removed,
      leaving us just with the state of \type{avg} itself.
    \subsection{Initial state}
      How to specify the initial state? Cannot be done inside a hardware
      function, since the initial state is its own state argument for the first
      call (unless you add an explicit, synchronous reset port).

      External init state is natural for simulation.

      External init state works for hardware generation as well.

      Implementation issues: state splitting, linking input to output state,
      checking usage constraints on state variables.

        Implementation issues
          \subsection[sec:prototype:separate]{Separate compilation}
          - Simplified core?

        Haskell language coverage / constraints
                Recursion
                Builtin types
                Custom types (Sum types, product types)
                Function types / higher order expressions