\chapter[chap:description]{Hardware description}
  This chapter will provide an overview of the hardware description language
  that was created and the issues that have arisen in the process. It will
  focus on the issues of the language, not the implementation.

  When translating Haskell to hardware, we need to make choices in what kind
  of hardware to generate for what Haskell constructs. When faced with
  choices, we've tried to stick with the most obvious choice wherever
  possible. In a lot of cases, when you look at a hardware description it is
  comletely clear what hardware is described. We want our translator to
  generate exactly that hardware whenever possible, to minimize the amount of
  surprise for people working with it.
   
  In this chapter we try to describe how we interpret a Haskell program from a
  hardware perspective. We provide a description of each Haskell language
  element that needs translation, to provide a clear picture of what is
  supported and how.

  \section{Function application}
  The basic syntactic element of a functional program are functions and
  function application. These have a single obvious \small{VHDL} translation: Each
  function becomes a hardware component, where each argument is an input port
  and the result value is the output port.

  Each function application in turn becomes component instantiation. Here, the
  result of each argument expression is assigned to a signal, which is mapped
  to the corresponding input port. The output port of the function is also
  mapped to a signal, which is used as the result of the application.

  \in{Example}[ex:And3] shows a simple program using only function
  application and the corresponding architecture.

\startbuffer[And3]
-- | A simple function that returns 
--   the and of three bits
and3 :: Bit -> Bit -> Bit -> Bit
and3 a b c = and (and a b) c
\stopbuffer

  \startuseMPgraphic{And3}
    save a, b, c, anda, andb, out;

    % I/O ports
    newCircle.a(btex $a$ etex) "framed(false)";
    newCircle.b(btex $b$ etex) "framed(false)";
    newCircle.c(btex $c$ etex) "framed(false)";
    newCircle.out(btex $out$ etex) "framed(false)";

    % Components
    newCircle.anda(btex $and$ etex);
    newCircle.andb(btex $and$ etex);

    a.c    = origin;
    b.c    = a.c + (0cm, 1cm);
    c.c    = b.c + (0cm, 1cm);
    anda.c = midpoint(a.c, b.c) + (2cm, 0cm);
    andb.c = midpoint(b.c, c.c) + (4cm, 0cm);

    out.c   = andb.c + (2cm, 0cm);

    % Draw objects and lines
    drawObj(a, b, c, anda, andb, out);

    ncarc(a)(anda) "arcangle(-10)";
    ncarc(b)(anda);
    ncarc(anda)(andb);
    ncarc(c)(andb);
    ncline(andb)(out);
  \stopuseMPgraphic

  \placeexample[here][ex:And3]{Simple three port and.}
    \startcombination[2*1]
      {\typebufferhs{And3}}{Haskell description using function applications.}
      {\boxedgraphic{And3}}{The architecture described by the Haskell description.}
    \stopcombination

  TODO: Define top level function and subfunctions/circuits.

  \subsection{Partial application}
  It should be obvious that we cannot generate hardware signals for all
  expressions we can express in Haskell. The most obvious criterium for this
  is the type of an expression. We will see more of this below, but for now it
  should be obvious that any expression of a function type cannot be
  represented as a signal or i/o port to a component.

  From this, we can see that the above translation rules do not apply to a
  partial application. \in{Example}[ex:Quadruple] shows an example use of
  partial application and the corresponding architecture.

\startbuffer[Quadruple]
-- | Multiply the input word by four.
quadruple :: Word -> Word
quadruple n = mul (mul n)
  where
    mul = (*) 2
\stopbuffer

  \startuseMPgraphic{Quadruple}
    save in, two, mula, mulb, out;

    % I/O ports
    newCircle.in(btex $n$ etex) "framed(false)";
    newCircle.two(btex $2$ etex) "framed(false)";
    newCircle.out(btex $out$ etex) "framed(false)";

    % Components
    newCircle.mula(btex $\times$ etex);
    newCircle.mulb(btex $\times$ etex);

    two.c    = origin;
    in.c     = two.c + (0cm, 1cm);
    mula.c  = in.c + (2cm, 0cm);
    mulb.c  = mula.c + (2cm, 0cm);
    out.c   = mulb.c + (2cm, 0cm);

    % Draw objects and lines
    drawObj(in, two, mula, mulb, out);

    nccurve(two)(mula) "angleA(0)", "angleB(45)";
    nccurve(two)(mulb) "angleA(0)", "angleB(45)";
    ncline(in)(mula);
    ncline(mula)(mulb);
    ncline(mulb)(out);
  \stopuseMPgraphic

  \placeexample[here][ex:Quadruple]{Simple three port and.}
    \startcombination[2*1]
      {\typebufferhs{Quadruple}}{Haskell description using function applications.}
      {\boxedgraphic{Quadruple}}{The architecture described by the Haskell description.}
    \stopcombination

  Here, the definition of mul is a partial function application: It applies
  \hs{2 :: Word} to the function \hs{(*) :: Word -> Word -> Word} resulting in
  the expression \hs{(*) 2 :: Word -> Word}. Since this resulting expression
  is again a function, we can't generate hardware for it directly. This is
  because the hardware to generate for \hs{mul} depends completely on where
  and how it is used. In this example, it is even used twice!

  However, it is clear that the above hardware description actually describes
  valid hardware. In general, we can see that any partial applied function
  must eventually become completely applied, at which point we can generate
  hardware for it using the rules for function application above. It might
  mean that a partial application is passed around quite a bit (even beyond
  function boundaries), but eventually, the partial application will become
  completely applied.

  \section{State}
    A very important concept in hardware designs is \emph{state}. In a
    stateless (or, \emph{combinatoric}) design, every output is a directly and solely dependent on the
    inputs. In a stateful design, the outputs can depend on the history of
    inputs, or the \emph{state}. State is usually stored in \emph{registers},
    which retain their value during a clockcycle, and are typically updated at
    the start of every clockcycle. Since the updating of the state is tightly
    coupled (synchronized) to the clock signal, these state updates are often
    called \emph{synchronous}.
  
    To make our hardware description language useful to describe more that
    simple combinatoric designs, we'll need to be able to describe state in
    some way.

    \subsection{Approaches to state}
      In Haskell, functions are always pure (except when using unsafe
      functions like \hs{unsafePerformIO}, which should be prevented whenever
      possible). This means that the output of a function solely depends on
      its inputs. If you evaluate a given function with given inputs, it will
      always provide the same output.

      TODO: Define pure

      This is a perfect match for a combinatoric circuit, where the output
      also soley depend on the inputs. However, when state is involved, this
      no longer holds. Since we're in charge of our own language, we could
      remove this purity constraint and allow a function to return different
      values depending on the cycle in which it is evaluated (or rather, the
      current state). However, this means that all kinds of interesting
      properties of our functional language get lost, and all kinds of
      transformations and optimizations might no longer be meaning preserving.

      Provided that we want to keep the function pure, the current state has
      to be present in the function's arguments in some way. There seem to be
      two obvious ways to do this: Adding the current state as an argument, or
      including the full history of each argument.

      \subsubsection{Stream arguments and results}
        Including the entire history of each input (\eg, the value of that
        input for each previous clockcycle) is an obvious way to make outputs
        depend on all previous input. This is easily done by making every
        input a list instead of a single value, containing all previous values
        as well as the current value.

        An obvious downside of this solution is that on each cycle, all the
        previous cycles must be resimulated to obtain the current state. To do
        this, it might be needed to have a recursive helper function as well,
        wich might be hard to properly analyze by the compiler.

        A slight variation on this approach is one taken by some of the other
        functional \small{HDL}s in the field (TODO: References to Lava,
        ForSyDe, ...): Make functions operate on complete streams. This means
        that a function is no longer called on every cycle, but just once. It
        takes stream as inputs instead of values, where each stream contains
        all the values for every clockcycle since system start. This is easily
        modeled using an (infinite) list, with one element for each clock
        cycle. Since the funciton is only evaluated once, its output is also a
        stream. Note that, since we are working with infinite lists and still
        want to be able to simulate the system cycle-by-cycle, this relies
        heavily on the lazy semantics of Haskell.

        Since our inputs and outputs are streams, all other (intermediate)
        values must be streams. All of our primitive operators (\eg, addition,
        substraction, bitwise operations, etc.) must operate on streams as
        well (note that changing a single-element operation to a stream
        operation can done with \hs{map}, \hs{zipwith}, etc.).

        Note that the concept of \emph{state} is no more than having some way
        to communicate a value from one cycle to the next. By introducing a
        \hs{delay} function, we can do exactly that: Delay (each value in) a
        stream so that we can "look into" the past. This \hs{delay} function
        simply outputs a stream where each value is the same as the input
        value, but shifted one cycle. This causes a \quote{gap} at the
        beginning of the stream: What is the value of the delay output in the
        first cycle? For this, the \hs{delay} function has a second input
        (which is a value, not a stream!).

        \in{Example}[ex:DelayAcc] shows a simple accumulator expressed in this
        style.

\startbuffer[DelayAcc]
acc :: Stream Word -> Stream Word
acc in = out
  where
    out = (delay out 0) + in
\stopbuffer

\startuseMPgraphic{DelayAcc}
  save in, out, add, reg;

  % I/O ports
  newCircle.in(btex $in$ etex) "framed(false)";
  newCircle.out(btex $out$ etex) "framed(false)";

  % Components
  newReg.reg("") "dx(4mm)", "dy(6mm)", "reflect(true)";
  newCircle.add(btex + etex);
  
  in.c    = origin;
  add.c   = in.c + (2cm, 0cm);
  out.c   = add.c + (2cm, 0cm);
  reg.c   = add.c + (0cm, 2cm);

  % Draw objects and lines
  drawObj(in, out, add, reg);

  nccurve(add)(reg) "angleA(0)", "angleB(180)", "posB(d)";
  nccurve(reg)(add) "angleA(180)", "angleB(-45)", "posA(out)";
  ncline(in)(add);
  ncline(add)(out);
\stopuseMPgraphic


        \placeexample[here][ex:DelayAcc]{Simple accumulator architecture.}
          \startcombination[2*1]
            {\typebufferhs{DelayAcc}}{Haskell description using streams.}
            {\boxedgraphic{DelayAcc}}{The architecture described by the Haskell description.}
          \stopcombination


        This notation can be confusing (especially due to the loop in the
        definition of out), but is essentially easy to interpret. There is a
        single call to delay, resulting in a circuit with a single register,
        whose input is connected to \hs{outl (which is the output of the
        adder)}, and it's output is the \hs{delay out 0} (which is connected
        to one of the adder inputs).

        This notation has a number of downsides, amongst which are limited
        readability and ambiguity in the interpretation. TODO: Reference
        Christiaan.
        
      \subsubsection{Explicit state arguments and results}
        A more explicit way to model state, is to simply add an extra argument
        containing the current state value. This allows an output to depend on
        both the inputs as well as the current state while keeping the
        function pure (letting the result depend only on the arguments), since
        the current state is now an argument.

        In Haskell, this would look like \in{example}[ex:ExplicitAcc].

\startbuffer[ExplicitAcc]
-- input -> current state -> (new state, output)
acc :: Word -> Word -> (Word, Word)
acc in (State s) = (State s', out)
  where
    out = s + in
    s'  = out
\stopbuffer

        \placeexample[here][ex:ExplicitAcc]{Simple accumulator architecture.}
          \startcombination[2*1]
            {\typebufferhs{ExplicitAcc}}{Haskell description using explicit state arguments.}
            % Picture is identical to the one we had just now.
            {\boxedgraphic{DelayAcc}}{The architecture described by the Haskell description.}
          \stopcombination

        This approach makes a function's state very explicit, which state
        variables are used by a function can be completely determined from its
        type signature (as opposed to the stream approach, where a function
        looks the same from the outside, regardless of what state variables it
        uses (or wether it's stateful at all).

        This approach is the one chosen for Cλash and will be examined more
        closely below.

    \subsection{Explicit state specification}
      We've seen the concept of explicit state in a simple example below, but
      what are the implications of this approach?

      \subsubsection{Substates}
        Since a function's state is reflected directly in its type signature,
        if a function calls other stateful functions (\eg, has subcircuits) it
        has to somehow know the current state for these called functions. The
        only way to do this, is to put these \emph{substates} inside the
        caller's state. This means that a function's state is the sum of the
        states of all functions it calls, and its own state.

        This also means that the type of a function (at least the "state"
        part) is dependent on its implementation and the functions it calls.
        This is the major downside of this approach: The separation between
        interface and implementation is limited. However, since Cλash is not
        very suitable for separate compilation (see
        \in{section}[sec:prototype:separate]) this is not a big problem in
        practice. Additionally, when using a type synonym for the state type
        of each function, we can still provide explicit type signatures
        while keeping the state specification for a function near its
        definition only.
    
      \subsubsection{...}
        We need some way to know which arguments should become input ports and
        which argument(s?) should become the current state (\eg, be bound to
        the register outputs). This does not hold holds not just for the top
        level function, but also for any subfunctions. Or could we perhaps
        deduce the statefulness of subfunctions by analyzing the flow of data
        in the calling functions?

        To explore this matter, we make an interesting observation: We get
        completely correct behaviour when we put all state registers in the
        top level entity (or even outside of it). All of the state arguments
        and results on subfunctions are treated as normal input and output
        ports. Effectively, a stateful function results in a stateless
        hardware component that has one of its input ports connected to the
        output of a register and one of its output ports connected to the
        input of the same register.

        TODO: Example?

        Of course, even though the hardware described like this has the
        correct behaviour, unless the layout tool does smart optimizations,
        there will be a lot of extra wire in the design (since registers will
        not be close to the component that uses them). Also, when working with
        the generated \small{VHDL} code, there will be a lot of extra ports
        just to pass one state values, which can get quite confusing.

        To fix this, we can simply \quote{push} the registers down into the
        subcircuits. When we see a register that is connected directly to a
        subcircuit, we remove the corresponding input and output port and put
        the register inside the subcircuit instead. This is slightly less
        trivial when looking at the Haskell code instead of the resulting
        circuit, but the idea is still the same.

        TODO: Example?

        However, when applying this technique, we might push registers down
        too far. When you intend to store a result of a stateless subfunction
        in the caller's state and pass the current value of that state
        variable to that same function, the register might get pushed down too
        far. It is impossible to distinguish this case from similar code where
        the called function is in fact stateful. From this we can conclude
        that we have to either:

        \startitemize
        \item accept that the generated hardware might not be exactly what we
        intended, in some specific cases. In most cases, the hardware will be
        what we intended.
        \item explicitely annotate state arguments and results in the input
        description.
        \stopitemize

        The first option causes (non-obvious) exceptions in the language
        intepretation. Also, automatically determining where registers should
        end up is easier to implement correctly with explicit annotations, so
        for these reasons we will look at how this annotations could work.


      TODO: Note about conditions on state variables and checking them.

    \subsection{Explicit state implementation}
      Recording state variables at the type level.

      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.

  \section[sec:recursion]{Recursion}
  An import concept in functional languages is recursion. In it's most basic
  form, recursion is a function that is defined in terms of itself. This
  usually requires multiple evaluations of this function, with changing
  arguments, until eventually an evaluation of the function no longer requires
  itself.

  Recursion in a hardware description is a bit of a funny thing. Usually,
  recursion is associated with a lot of nondeterminism, stack overflows, but
  also flexibility and expressive power.

  Given the notion that each function application will translate to a
  component instantiation, we are presented with a problem. A recursive
  function would translate to a component that contains itself. Or, more
  precisely, that contains an instance of itself. This instance would again
  contain an instance of itself, and again, into infinity. This is obviously a
  problem for generating hardware.

  This is expected for functions that describe infinite recursion. In that
  case, we can't generate hardware that shows correct behaviour in a single
  cycle (at best, we could generate hardware that needs an infinite number of
  cycles to complete).
  
  However, most recursive hardware descriptions will describe finite
  recursion. This is because the recursive call is done conditionally. There
  is usually a case statement where at least one alternative does not contain
  the recursive call, which we call the "base case". If, for each call to the
  recursive function, we would be able to detect which alternative applies,
  we would be able to remove the case expression and leave only the base case
  when it applies. This will ensure that expanding the recursive functions
  will terminate after a bounded number of expansions.

  This does imply the extra requirement that the base case is detectable at
  compile time. In particular, this means that the decision between the base
  case and the recursive case must not depend on runtime data.

  \subsection{List recursion}
  The most common deciding factor in recursion is the length of a list that is
  passed in as an argument. Since we represent lists as vectors that encode
  the length in the vector type, it seems easy to determine the base case. We
  can simply look at the argument type for this. However, it turns out that
  this is rather non-trivial to write down in Haskell in the first place. As
  an example, we would like to write down something like this:
 
  \starthaskell
    sum :: Vector n Word -> Word
    sum xs = case null xs of
      True -> 0
      False -> head xs + sum (tail xs)
  \stophaskell

  However, the typechecker will now use the following reasoning (element type
  of the vector is left out):

  \startitemize
  \item tail has the type \hs{(n > 0) => Vector n -> Vector (n - 1)}
  \item This means that xs must have the type \hs{(n > 0) => Vector n}
  \item This means that sum must have the type \hs{(n > 0) => Vector n -> a}
  \item sum is called with the result of tail as an argument, which has the
  type \hs{Vector n} (since \hs{(n > 0) => n - 1 == m}).
  \item This means that sum must have the type \hs{Vector n -> a}
  \item This is a contradiction between the type deduced from the body of sum
  (the input vector must be non-empty) and the use of sum (the input vector
  could have any length).
  \stopitemize

  As you can see, using a simple case at value level causes the type checker
  to always typecheck both alternatives, which can't be done! This is a
  fundamental problem, that would seem perfectly suited for a type class.
  Considering that we need to switch between to implementations of the sum
  function, based on the type of the argument, this sounds like the perfect
  problem to solve with a type class. However, this approach has its own
  problems (not the least of them that you need to define a new typeclass for
  every recursive function you want to define).

  Another approach tried involved using GADTs to be able to do pattern
  matching on empty / non empty lists. While this worked partially, it also
  created problems with more complex expressions.

  TODO: How much detail should there be here? I can probably refer to
  Christiaan instead.

  Evaluating all possible (and non-possible) ways to add recursion to our
  descriptions, it seems better to leave out list recursion alltogether. This
  allows us to focus on other interesting areas instead. By including
  (builtin) support for a number of higher order functions like map and fold,
  we can still express most of the things we would use list recursion for.
 
  \subsection{General recursion}
  Of course there are other forms of recursion, that do not depend on the
  length (and thus type) of a list. For example, simple recursion using a
  counter could be expressed, but only translated to hardware for a fixed
  number of iterations. Also, this would require extensive support for compile
  time simplification (constant propagation) and compile time evaluation
  (evaluation constant comparisons), to ensure non-termination. Even then, it
  is hard to really guarantee termination, since the user (or GHC desugarer)
  might use some obscure notation that results in a corner case of the
  simplifier that is not caught and thus non-termination.

  Due to these complications, we leave other forms of recursion as
  future work as well.
