Fix reference.

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

\chapter[chap:normalization]{Normalization}
  % A helper to print a single example in the half the page width. The example
  % text should be in a buffer whose name is given in an argument.
  %
  % The align=right option really does left-alignment, but without the program
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  \define[1]\example{
    \framed[offset=1mm,align=right,strut=no,background=box,frame=off]{
      \setuptyping[option=LAM,style=sans,before=,after=,strip=auto]
      \typebuffer[#1]
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    }
  }

  \define[4]\transexample{
    \placeexample[here][ex:trans:#1]{#2}
    \startcombination[2*1]
      {\example{#3}}{Original program}
      {\example{#4}}{Transformed program}
    \stopcombination
  }

  The first step in the core to \small{VHDL} translation process, is normalization. We
  aim to bring the core description into a simpler form, which we can
  subsequently translate into \small{VHDL} easily. This normal form is needed because
  the full core language is more expressive than \small{VHDL} in some areas and because
  core can describe expressions that do not have a direct hardware
  interpretation.

  \todo{Describe core properties not supported in \VHDL, and describe how the
  \VHDL we want to generate should look like.}

  \section{Normal form}
    \todo{Refresh or refer to distinct hardware per application principle}
    The transformations described here have a well-defined goal: To bring the
    program in a well-defined form that is directly translatable to hardware,
    while fully preserving the semantics of the program. We refer to this form as
    the \emph{normal form} of the program. The formal definition of this normal
    form is quite simple:

    \placedefinition{}{A program is in \emph{normal form} if none of the
    transformations from this chapter apply.}

    Of course, this is an \quote{easy} definition of the normal form, since our
    program will end up in normal form automatically. The more interesting part is
    to see if this normal form actually has the properties we would like it to
    have.

    But, before getting into more definitions and details about this normal form,
    let's try to get a feeling for it first. The easiest way to do this is by
    describing the things we want to not have in a normal form.

    \startitemize
      \item Any \emph{polymorphism} must be removed. When laying down hardware, we
      can't generate any signals that can have multiple types. All types must be
      completely known to generate hardware.
      
      \item Any \emph{higher order} constructions must be removed. We can't
      generate a hardware signal that contains a function, so all values,
      arguments and returns values used must be first order.

      \item Any complex \emph{nested scopes} must be removed. In the \small{VHDL}
      description, every signal is in a single scope. Also, full expressions are
      not supported everywhere (in particular port maps can only map signal
      names and constants, not complete expressions). To make the \small{VHDL}
      generation easy, a separate binder must be bound to ever application or
      other expression.
    \stopitemize

    \todo{Intermezzo: functions vs plain values}

    A very simple example of a program in normal form is given in
    \in{example}[ex:MulSum]. As you can see, all arguments to the function (which
    will become input ports in the final hardware) are at the outer level.
    This means that the body of the inner lambda abstraction is never a
    function, but always a plain value.

    As the body of the inner lambda abstraction, we see a single (recursive)
    let expression, that binds two variables (\lam{mul} and \lam{sum}). These
    variables will be signals in the final hardware, bound to the output port
    of the \lam{*} and \lam{+} components.

    The final line (the \quote{return value} of the function) selects the
    \lam{sum} signal to be the output port of the function. This \quote{return
    value} can always only be a variable reference, never a more complex
    expression.

    \todo{Add generated VHDL}

    \startbuffer[MulSum]
    alu :: Bit -> Word -> Word -> Word
    alu = λa.λb.λc.
        let
          mul = (*) a b
          sum = (+) mul c
        in
          sum
    \stopbuffer

    \startuseMPgraphic{MulSum}
      save a, b, c, mul, add, sum;

      % 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.sum(btex $res$ etex) "framed(false)";

      % Components
      newCircle.mul(btex * etex);
      newCircle.add(btex + etex);

      a.c      - b.c   = (0cm, 2cm);
      b.c      - c.c   = (0cm, 2cm);
      add.c            = c.c + (2cm, 0cm);
      mul.c            = midpoint(a.c, b.c) + (2cm, 0cm);
      sum.c            = add.c + (2cm, 0cm);
      c.c              = origin;

      % Draw objects and lines
      drawObj(a, b, c, mul, add, sum);

      ncarc(a)(mul) "arcangle(15)";
      ncarc(b)(mul) "arcangle(-15)";
      ncline(c)(add);
      ncline(mul)(add);
      ncline(add)(sum);
    \stopuseMPgraphic

    \placeexample[here][ex:MulSum]{Simple architecture consisting of a
    multiplier and a subtractor.}
      \startcombination[2*1]
        {\typebufferlam{MulSum}}{Core description in normal form.}
        {\boxedgraphic{MulSum}}{The architecture described by the normal form.}
      \stopcombination

    The previous example described composing an architecture by calling other
    functions (operators), resulting in a simple architecture with components and
    connections. There is of course also some mechanism for choice in the normal
    form. In a normal Core program, the \emph{case} expression can be used in a
    few different ways to describe choice. In normal form, this is limited to a
    very specific form.

    \in{Example}[ex:AddSubAlu] shows an example describing a
    simple \small{ALU}, which chooses between two operations based on an opcode
    bit. The main structure is similar to \in{example}[ex:MulSum], but this
    time the \lam{res} variable is bound to a case expression. This case
    expression scrutinizes the variable \lam{opcode} (and scrutinizing more
    complex expressions is not supported). The case expression can select a
    different variable based on the constructor of \lam{opcode}.

    \startbuffer[AddSubAlu]
    alu :: Bit -> Word -> Word -> Word
    alu = λopcode.λa.λb.
        let
          res1 = (+) a b
          res2 = (-) a b
          res = case opcode of
            Low -> res1
            High -> res2
        in
          res
    \stopbuffer

    \startuseMPgraphic{AddSubAlu}
      save opcode, a, b, add, sub, mux, res;

      % I/O ports
      newCircle.opcode(btex $opcode$ etex) "framed(false)";
      newCircle.a(btex $a$ etex) "framed(false)";
      newCircle.b(btex $b$ etex) "framed(false)";
      newCircle.res(btex $res$ etex) "framed(false)";
      % Components
      newCircle.add(btex + etex);
      newCircle.sub(btex - etex);
      newMux.mux;

      opcode.c - a.c   = (0cm, 2cm);
      add.c    - a.c   = (4cm, 0cm);
      sub.c    - b.c   = (4cm, 0cm);
      a.c      - b.c   = (0cm, 3cm);
      mux.c            = midpoint(add.c, sub.c) + (1.5cm, 0cm);
      res.c    - mux.c = (1.5cm, 0cm);
      b.c              = origin;

      % Draw objects and lines
      drawObj(opcode, a, b, res, add, sub, mux);

      ncline(a)(add) "posA(e)";
      ncline(b)(sub) "posA(e)";
      nccurve(a)(sub) "posA(e)", "angleA(0)";
      nccurve(b)(add) "posA(e)", "angleA(0)";
      nccurve(add)(mux) "posB(inpa)", "angleB(0)";
      nccurve(sub)(mux) "posB(inpb)", "angleB(0)";
      nccurve(opcode)(mux) "posB(n)", "angleA(0)", "angleB(-90)";
      ncline(mux)(res) "posA(out)";
    \stopuseMPgraphic

    \placeexample[here][ex:AddSubAlu]{Simple \small{ALU} supporting two operations.}
      \startcombination[2*1]
        {\typebufferlam{AddSubAlu}}{Core description in normal form.}
        {\boxedgraphic{AddSubAlu}}{The architecture described by the normal form.}
      \stopcombination

    As a more complete example, consider \in{example}[ex:NormalComplete]. This
    example contains everything that is supported in normal form, with the
    exception of builtin higher order functions. The graphical version of the
    architecture contains a slightly simplified version, since the state tuple
    packing and unpacking have been left out. Instead, two seperate registers are
    drawn. Also note that most synthesis tools will further optimize this
    architecture by removing the multiplexers at the register input and
    instead put some gates in front of the register's clock input, but we want
    to show the architecture as close to the description as possible.

    As you can see from the previous examples, the generation of the final
    architecture from the normal form is straightforward. In each of the
    examples, there is a direct match between the normal form structure,
    the generated VHDL and the architecture shown in the images.

    \startbuffer[NormalComplete]
      regbank :: Bit 
                 -> Word 
                 -> State (Word, Word) 
                 -> (State (Word, Word), Word)

      -- All arguments are an inital lambda (address, data, packed state)
      regbank = λa.λd.λsp.
      -- There are nested let expressions at top level
      let
        -- Unpack the state by coercion (\eg, cast from
        -- State (Word, Word) to (Word, Word))
        s = sp ▶ (Word, Word)
        -- Extract both registers from the state
        r1 = case s of (a, b) -> a
        r2 = case s of (a, b) -> b
        -- Calling some other user-defined function.
        d' = foo d
        -- Conditional connections
        out = case a of
          High -> r1
          Low -> r2
        r1' = case a of
          High -> d'
          Low -> r1
        r2' = case a of
          High -> r2
          Low -> d'
        -- Packing a tuple
        s' = (,) r1' r2'
        -- pack the state by coercion (\eg, cast from
        -- (Word, Word) to State (Word, Word))
        sp' = s' ▶ State (Word, Word)
        -- Pack our return value
        res = (,) sp' out
      in
        -- The actual result
        res
    \stopbuffer

    \startuseMPgraphic{NormalComplete}
      save a, d, r, foo, muxr, muxout, out;

      % I/O ports
      newCircle.a(btex \lam{a} etex) "framed(false)";
      newCircle.d(btex \lam{d} etex) "framed(false)";
      newCircle.out(btex \lam{out} etex) "framed(false)";
      % Components
      %newCircle.add(btex + etex);
      newBox.foo(btex \lam{foo} etex);
      newReg.r1(btex $\lam{r1}$ etex) "dx(4mm)", "dy(6mm)";
      newReg.r2(btex $\lam{r2}$ etex) "dx(4mm)", "dy(6mm)", "reflect(true)";
      newMux.muxr1;
      % Reflect over the vertical axis
      reflectObj(muxr1)((0,0), (0,1));
      newMux.muxr2;
      newMux.muxout;
      rotateObj(muxout)(-90);

      d.c               = foo.c + (0cm, 1.5cm); 
      a.c               = (xpart r2.c + 2cm, ypart d.c - 0.5cm);
      foo.c             = midpoint(muxr1.c, muxr2.c) + (0cm, 2cm);
      muxr1.c           = r1.c + (0cm, 2cm);
      muxr2.c           = r2.c + (0cm, 2cm);
      r2.c              = r1.c + (4cm, 0cm);
      r1.c              = origin;
      muxout.c          = midpoint(r1.c, r2.c) - (0cm, 2cm);
      out.c             = muxout.c - (0cm, 1.5cm);

    %  % Draw objects and lines
      drawObj(a, d, foo, r1, r2, muxr1, muxr2, muxout, out);
      
      ncline(d)(foo);
      nccurve(foo)(muxr1) "angleA(-90)", "posB(inpa)", "angleB(180)";
      nccurve(foo)(muxr2) "angleA(-90)", "posB(inpb)", "angleB(0)";
      nccurve(muxr1)(r1) "posA(out)", "angleA(180)", "posB(d)", "angleB(0)";
      nccurve(r1)(muxr1) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(180)";
      nccurve(muxr2)(r2) "posA(out)", "angleA(0)", "posB(d)", "angleB(180)";
      nccurve(r2)(muxr2) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(0)";
      nccurve(r1)(muxout) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(-90)";
      nccurve(r2)(muxout) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(-90)";
      % Connect port a
      nccurve(a)(muxout) "angleA(-90)", "angleB(180)", "posB(sel)";
      nccurve(a)(muxr1) "angleA(180)", "angleB(-90)", "posB(sel)";
      nccurve(a)(muxr2) "angleA(180)", "angleB(-90)", "posB(sel)";
      ncline(muxout)(out) "posA(out)";
    \stopuseMPgraphic

    \todo{Don't split registers in this image?}
    \placeexample[here][ex:NormalComplete]{Simple architecture consisting of an adder and a
    subtractor.}
      \startcombination[2*1]
        {\typebufferlam{NormalComplete}}{Core description in normal form.}
        {\boxedgraphic{NormalComplete}}{The architecture described by the normal form.}
      \stopcombination
    


    \subsection[sec:normalization:intendednormalform]{Intended normal form definition}
      Now we have some intuition for the normal form, we can describe how we want
      the normal form to look like in a slightly more formal manner. The following
      EBNF-like description completely captures the intended structure (and
      generates a subset of GHC's core format).

      Some clauses have an expression listed in parentheses. These are conditions
      that need to apply to the clause.

      \defref{intended normal form definition}
      \todo{Fix indentation}
      \startlambda
      \italic{normal} := \italic{lambda}
      \italic{lambda} := λvar.\italic{lambda} (representable(var))
                      | \italic{toplet} 
      \italic{toplet} := letrec [\italic{binding}...] in var (representable(var))
      \italic{binding} := var = \italic{rhs} (representable(rhs))
                       -- State packing and unpacking by coercion
                       | var0 = var1 ▶ State ty (lvar(var1))
                       | var0 = var1 ▶ ty (var1 :: State ty ∧ lvar(var1))
      \italic{rhs} := userapp
                   | builtinapp
                   -- Extractor case
                   | case var of C a0 ... an -> ai (lvar(var))
                   -- Selector case
                   | case var of (lvar(var))
                      [ DEFAULT -> var ]  (lvar(var))
                      C0 w0,0 ... w0,n -> var0
                      \vdots
                      Cm wm,0 ... wm,n -> varm       (\forall{}i \forall{}j, wi,j \neq vari, lvar(vari))
      \italic{userapp} := \italic{userfunc}
                       | \italic{userapp} {userarg}
      \italic{userfunc} := var (gvar(var))
      \italic{userarg} := var (lvar(var))
      \italic{builtinapp} := \italic{builtinfunc}
                          | \italic{builtinapp} \italic{builtinarg}
      \italic{builtinfunc} := var (bvar(var))
      \italic{builtinarg} := var (representable(var) ∧ lvar(var))
                          | \italic{partapp} (partapp :: a -> b)
                          | \italic{coreexpr} (¬representable(coreexpr) ∧ ¬(coreexpr :: a -> b))
      \italic{partapp} := \italic{userapp} | \italic{builtinapp}
      \stoplambda

      \todo{There can still be other casts around (which the code can handle,
      e.g., ignore), which still need to be documented here}

      When looking at such a program from a hardware perspective, the top level
      lambda's define the input ports. The variable reference in the body of
      the recursive let expression is the output port. Most function
      applications bound by the let expression define a component
      instantiation, where the input and output ports are mapped to local
      signals or arguments. Some of the others use a builtin construction (\eg
      the \lam{case} expression) or call a builtin function (\eg \lam{+} or
      \lam{map}). For these, a hardcoded \small{VHDL} translation is
      available.

  \section[sec:normalization:transformation]{Transformation notation}
    To be able to concisely present transformations, we use a specific format
    for them. It is a simple format, similar to one used in logic reasoning.

    Such a transformation description looks like the following.

    \starttrans
    <context conditions>
    ~
    <original expression>
    --------------------------          <expression conditions>
    <transformed expresssion>
    ~
    <context additions>
    \stoptrans

    This format desribes a transformation that applies to \lam{<original
    expresssion>} and transforms it into \lam{<transformed expression>}, assuming
    that all conditions apply. In this format, there are a number of placeholders
    in pointy brackets, most of which should be rather obvious in their meaning.
    Nevertheless, we will more precisely specify their meaning below:

      \startdesc{<original expression>} The expression pattern that will be matched
      against (subexpressions of) the expression to be transformed. We call this a
      pattern, because it can contain \emph{placeholders} (variables), which match
      any expression or binder. Any such placeholder is said to be \emph{bound} to
      the expression it matches. It is convention to use an uppercase letter (\eg
      \lam{M} or \lam{E}) to refer to any expression (including a simple variable
      reference) and lowercase letters (\eg \lam{v} or \lam{b}) to refer to
      (references to) binders.

      For example, the pattern \lam{a + B} will match the expression 
      \lam{v + (2 * w)} (binding \lam{a} to \lam{v} and \lam{B} to 
      \lam{(2 * w)}), but not \lam{(2 * w) + v}.
      \stopdesc

      \startdesc{<expression conditions>}
      These are extra conditions on the expression that is matched. These
      conditions can be used to further limit the cases in which the
      transformation applies, commonly to prevent a transformation from
      causing a loop with itself or another transformation.

      Only if these conditions are \emph{all} true, the transformation
      applies.
      \stopdesc

      \startdesc{<context conditions>}
      These are a number of extra conditions on the context of the function. In
      particular, these conditions can require some (other) top level function to be
      present, whose value matches the pattern given here. The format of each of
      these conditions is: \lam{binder = <pattern>}.

      Typically, the binder is some placeholder bound in the \lam{<original
      expression>}, while the pattern contains some placeholders that are used in
      the \lam{transformed expression}.
      
      Only if a top level binder exists that matches each binder and pattern,
      the transformation applies.
      \stopdesc

      \startdesc{<transformed expression>}
      This is the expression template that is the result of the transformation. If, looking
      at the above three items, the transformation applies, the \lam{<original
      expression>} is completely replaced with the \lam{<transformed expression>}.
      We call this a template, because it can contain placeholders, referring to
      any placeholder bound by the \lam{<original expression>} or the
      \lam{<context conditions>}. The resulting expression will have those
      placeholders replaced by the values bound to them.

      Any binder (lowercase) placeholder that has no value bound to it yet will be
      bound to (and replaced with) a fresh binder.
      \stopdesc

      \startdesc{<context additions>}
      These are templates for new functions to add to the context. This is a way
      to have a transformation create new top level functions.

      Each addition has the form \lam{binder = template}. As above, any
      placeholder in the addition is replaced with the value bound to it, and any
      binder placeholder that has no value bound to it yet will be bound to (and
      replaced with) a fresh binder.
      \stopdesc

    As an example, we'll look at η-abstraction:

    \starttrans
    E                 \lam{E :: a -> b}
    --------------    \lam{E} does not occur on a function position in an application
    λx.E x            \lam{E} is not a lambda abstraction.
    \stoptrans

    η-abstraction is a well known transformation from lambda calculus. What
    this transformation does, is take any expression that has a function type
    and turn it into a lambda expression (giving an explicit name to the
    argument). There are some extra conditions that ensure that this
    transformation does not apply infinitely (which are not necessarily part
    of the conventional definition of η-abstraction).

    Consider the following function, which is a fairly obvious way to specify a
    simple ALU (Note that \in{example}[ex:AddSubAlu] shows the normal form of this
    function). The parentheses around the \lam{+} and \lam{-} operators are
    commonly used in Haskell to show that the operators are used as normal
    functions, instead of \emph{infix} operators (\eg, the operators appear
    before their arguments, instead of in between).

    \startlambda 
    alu :: Bit -> Word -> Word -> Word
    alu = λopcode. case opcode of
      Low -> (+)
      High -> (-)
    \stoplambda

    There are a few subexpressions in this function to which we could possibly
    apply the transformation. Since the pattern of the transformation is only
    the placeholder \lam{E}, any expression will match that. Whether the
    transformation applies to an expression is thus solely decided by the
    conditions to the right of the transformation.

    We will look at each expression in the function in a top down manner. The
    first expression is the entire expression the function is bound to.

    \startlambda
    λopcode. case opcode of
      Low -> (+)
      High -> (-)
    \stoplambda

    As said, the expression pattern matches this. The type of this expression is
    \lam{Bit -> Word -> Word -> Word}, which matches \lam{a -> b} (Note that in
    this case \lam{a = Bit} and \lam{b = Word -> Word -> Word}).

    Since this expression is at top level, it does not occur at a function
    position of an application. However, The expression is a lambda abstraction,
    so this transformation does not apply.

    The next expression we could apply this transformation to, is the body of
    the lambda abstraction:

    \startlambda
    case opcode of
      Low -> (+)
      High -> (-)
    \stoplambda

    The type of this expression is \lam{Word -> Word -> Word}, which again
    matches \lam{a -> b}. The expression is the body of a lambda expression, so
    it does not occur at a function position of an application. Finally, the
    expression is not a lambda abstraction but a case expression, so all the
    conditions match. There are no context conditions to match, so the
    transformation applies.

    By now, the placeholder \lam{E} is bound to the entire expression. The
    placeholder \lam{x}, which occurs in the replacement template, is not bound
    yet, so we need to generate a fresh binder for that. Let's use the binder
    \lam{a}. This results in the following replacement expression:

    \startlambda
    λa.(case opcode of
      Low -> (+)
      High -> (-)) a
    \stoplambda

    Continuing with this expression, we see that the transformation does not
    apply again (it is a lambda expression). Next we look at the body of this
    lambda abstraction:

    \startlambda
    (case opcode of
      Low -> (+)
      High -> (-)) a
    \stoplambda
    
    Here, the transformation does apply, binding \lam{E} to the entire
    expression and \lam{x} to the fresh binder \lam{b}, resulting in the
    replacement:

    \startlambda
    λb.(case opcode of
      Low -> (+)
      High -> (-)) a b
    \stoplambda

    Again, the transformation does not apply to this lambda abstraction, so we
    look at its body. For brevity, we'll put the case statement on one line from
    now on.

    \startlambda
    (case opcode of Low -> (+); High -> (-)) a b
    \stoplambda

    The type of this expression is \lam{Word}, so it does not match \lam{a -> b}
    and the transformation does not apply. Next, we have two options for the
    next expression to look at: The function position and argument position of
    the application. The expression in the argument position is \lam{b}, which
    has type \lam{Word}, so the transformation does not apply. The expression in
    the function position is:

    \startlambda
    (case opcode of Low -> (+); High -> (-)) a
    \stoplambda

    Obviously, the transformation does not apply here, since it occurs in
    function position (which makes the second condition false). In the same
    way the transformation does not apply to both components of this
    expression (\lam{case opcode of Low -> (+); High -> (-)} and \lam{a}), so
    we'll skip to the components of the case expression: The scrutinee and
    both alternatives. Since the opcode is not a function, it does not apply
    here.

    The first alternative is \lam{(+)}. This expression has a function type
    (the operator still needs two arguments). It does not occur in function
    position of an application and it is not a lambda expression, so the
    transformation applies.

    We look at the \lam{<original expression>} pattern, which is \lam{E}.
    This means we bind \lam{E} to \lam{(+)}. We then replace the expression
    with the \lam{<transformed expression>}, replacing all occurences of
    \lam{E} with \lam{(+)}. In the \lam{<transformed expression>}, the This gives us the replacement expression:
    \lam{λx.(+) x} (A lambda expression binding \lam{x}, with a body that
    applies the addition operator to \lam{x}).

    The complete function then becomes:
    \startlambda
    (case opcode of Low -> λa1.(+) a1; High -> (-)) a
    \stoplambda

    Now the transformation no longer applies to the complete first alternative
    (since it is a lambda expression). It does not apply to the addition
    operator again, since it is now in function position in an application. It
    does, however, apply to the application of the addition operator, since
    that is neither a lambda expression nor does it occur in function
    position. This means after one more application of the transformation, the
    function becomes:

    \startlambda
    (case opcode of Low -> λa1.λb1.(+) a1 b1; High -> (-)) a
    \stoplambda

    The other alternative is left as an exercise to the reader. The final
    function, after applying η-abstraction until it does no longer apply is:

    \startlambda 
    alu :: Bit -> Word -> Word -> Word
    alu = λopcode.λa.b. (case opcode of
      Low -> λa1.λb1 (+) a1 b1
      High -> λa2.λb2 (-) a2 b2) a b
    \stoplambda

    \subsection{Transformation application}
      In this chapter we define a number of transformations, but how will we apply
      these? As stated before, our normal form is reached as soon as no
      transformation applies anymore. This means our application strategy is to
      simply apply any transformation that applies, and continuing to do that with
      the result of each transformation.

      In particular, we define no particular order of transformations. Since
      transformation order should not influence the resulting normal form,
      \todo{This is not really true, but would like it to be...} this leaves
      the implementation free to choose any application order that results in
      an efficient implementation.

      When applying a single transformation, we try to apply it to every (sub)expression
      in a function, not just the top level function body. This allows us to
      keep the transformation descriptions concise and powerful.

    \subsection{Definitions}
      In the following sections, we will be using a number of functions and
      notations, which we will define here.

      \todo{Define substitution (notation)}

      \subsubsection{Concepts}
        A \emph{global variable} is any variable (binder) that is bound at the
        top level of a program, or an external module. A \emph{local variable} is any
        other variable (\eg, variables local to a function, which can be bound by
        lambda abstractions, let expressions and pattern matches of case
        alternatives).  Note that this is a slightly different notion of global versus
        local than what \small{GHC} uses internally.
        \defref{global variable} \defref{local variable}

        A \emph{hardware representable} (or just \emph{representable}) type or value
        is (a value of) a type that we can generate a signal for in hardware. For
        example, a bit, a vector of bits, a 32 bit unsigned word, etc. Values that are
        not runtime representable notably include (but are not limited to): Types,
        dictionaries, functions.
        \defref{representable}

        A \emph{builtin function} is a function supplied by the Cλash framework, whose
        implementation is not valid Cλash. The implementation is of course valid
        Haskell, for simulation, but it is not expressable in Cλash.
        \defref{builtin function} \defref{user-defined function}

      For these functions, Cλash has a \emph{builtin hardware translation}, so calls
      to these functions can still be translated. These are functions like
      \lam{map}, \lam{hwor} and \lam{length}.

      A \emph{user-defined} function is a function for which we do have a Cλash
      implementation available.

      \subsubsection{Predicates}
        Here, we define a number of predicates that can be used below to concisely
        specify conditions.\refdef{global variable}

        \emph{gvar(expr)} is true when \emph{expr} is a variable that references a
        global variable. It is false when it references a local variable.

        \refdef{local variable}\emph{lvar(expr)} is the complement of \emph{gvar}; it is true when \emph{expr}
        references a local variable, false when it references a global variable.

        \refdef{representable}\emph{representable(expr)} or \emph{representable(var)} is true when
        \emph{expr} or \emph{var} is \emph{representable}.

    \subsection[sec:normalization:uniq]{Binder uniqueness}
      A common problem in transformation systems, is binder uniqueness. When not
      considering this problem, it is easy to create transformations that mix up
      bindings and cause name collisions. Take for example, the following core
      expression:

      \startlambda
      (λa.λb.λc. a * b * c) x c
      \stoplambda

      By applying β-reduction (see \in{section}[sec:normalization:beta]) once,
      we can simplify this expression to:

      \startlambda
      (λb.λc. x * b * c) c
      \stoplambda

      Now, we have replaced the \lam{a} binder with a reference to the \lam{x}
      binder. No harm done here. But note that we see multiple occurences of the
      \lam{c} binder. The first is a binding occurence, to which the second refers.
      The last, however refers to \emph{another} instance of \lam{c}, which is
      bound somewhere outside of this expression. Now, if we would apply beta
      reduction without taking heed of binder uniqueness, we would get:

      \startlambda
      λc. x * c * c
      \stoplambda

      This is obviously not what was supposed to happen! The root of this problem is
      the reuse of binders: Identical binders can be bound in different scopes, such
      that only the inner one is \quote{visible} in the inner expression. In the example
      above, the \lam{c} binder was bound outside of the expression and in the inner
      lambda expression. Inside that lambda expression, only the inner \lam{c} is
      visible.

      There are a number of ways to solve this. \small{GHC} has isolated this
      problem to their binder substitution code, which performs \emph{deshadowing}
      during its expression traversal. This means that any binding that shadows
      another binding on a higher level is replaced by a new binder that does not
      shadow any other binding. This non-shadowing invariant is enough to prevent
      binder uniqueness problems in \small{GHC}.

      In our transformation system, maintaining this non-shadowing invariant is
      a bit harder to do (mostly due to implementation issues, the prototype doesn't
      use \small{GHC}'s subsitution code). Also, the following points can be
      observed.

      \startitemize
      \item Deshadowing does not guarantee overall uniqueness. For example, the
      following (slightly contrived) expression shows the identifier \lam{x} bound in
      two seperate places (and to different values), even though no shadowing
      occurs.

      \startlambda
      (let x = 1 in x) + (let x = 2 in x)
      \stoplambda

      \item In our normal form (and the resulting \small{VHDL}), all binders
      (signals) within the same function (entity) will end up in the same
      scope. To allow this, all binders within the same function should be
      unique.

      \item When we know that all binders in an expression are unique, moving around
      or removing a subexpression will never cause any binder conflicts. If we have
      some way to generate fresh binders, introducing new subexpressions will not
      cause any problems either. The only way to cause conflicts is thus to
      duplicate an existing subexpression.
      \stopitemize

      Given the above, our prototype maintains a unique binder invariant. This
      means that in any given moment during normalization, all binders \emph{within
      a single function} must be unique. To achieve this, we apply the following
      technique.

      \todo{Define fresh binders and unique supplies}

      \startitemize
      \item Before starting normalization, all binders in the function are made
      unique. This is done by generating a fresh binder for every binder used. This
      also replaces binders that did not cause any conflict, but it does ensure that
      all binders within the function are generated by the same unique supply.
      \refdef{fresh binder}
      \item Whenever a new binder must be generated, we generate a fresh binder that
      is guaranteed to be different from \emph{all binders generated so far}. This
      can thus never introduce duplication and will maintain the invariant.
      \item Whenever (a part of) an expression is duplicated (for example when
      inlining), all binders in the expression are replaced with fresh binders
      (using the same method as at the start of normalization). These fresh binders
      can never introduce duplication, so this will maintain the invariant.
      \item Whenever we move part of an expression around within the function, there
      is no need to do anything special. There is obviously no way to introduce
      duplication by moving expressions around. Since we know that each of the
      binders is already unique, there is no way to introduce (incorrect) shadowing
      either.
      \stopitemize

  \section{Transform passes}
    In this section we describe the actual transforms.

    Each transformation will be described informally first, explaining
    the need for and goal of the transformation. Then, we will formally define
    the transformation using the syntax introduced in
    \in{section}[sec:normalization:transformation].

    \subsection{General cleanup}
      These transformations are general cleanup transformations, that aim to
      make expressions simpler. These transformations usually clean up the
       mess left behind by other transformations or clean up expressions to
       expose new transformation opportunities for other transformations.

       Most of these transformations are standard optimizations in other
       compilers as well. However, in our compiler, most of these are not just
       optimizations, but they are required to get our program into intended
       normal form.

      \subsubsection[sec:normalization:beta]{β-reduction}
        \defref{beta-reduction}
        β-reduction is a well known transformation from lambda calculus, where it is
        the main reduction step. It reduces applications of lambda abstractions,
        removing both the lambda abstraction and the application.

        In our transformation system, this step helps to remove unwanted lambda
        abstractions (basically all but the ones at the top level). Other
        transformations (application propagation, non-representable inlining) make
        sure that most lambda abstractions will eventually be reducable by
        β-reduction.

        Note that β-reduction also works on type lambda abstractions and type
        applications as well. This means the substitution below also works on
        type variables, in the case that the binder is a type variable and teh
        expression applied to is a type.

        \starttrans
        (λx.E) M
        -----------------
        E[x=>M]
        \stoptrans

        % And an example
        \startbuffer[from]
        (λa. 2 * a) (2 * b)
        \stopbuffer

        \startbuffer[to]
        2 * (2 * b)
        \stopbuffer

        \transexample{beta}{β-reduction}{from}{to}

        \startbuffer[from]
        (λt.λa::t. a) @Int
        \stopbuffer

        \startbuffer[to]
        (λa::Int. a)
        \stopbuffer

        \transexample{beta-type}{β-reduction for type abstractions}{from}{to}
       
      \subsubsection{Empty let removal}
        This transformation is simple: It removes recursive lets that have no bindings
        (which usually occurs when unused let binding removal removes the last
        binding from it).

        Note that there is no need to define this transformation for
        non-recursive lets, since they always contain exactly one binding.

        \starttrans
        letrec in M
        --------------
        M
        \stoptrans

        \todo{Example}

      \subsubsection[sec:normalization:simplelet]{Simple let binding removal}
        This transformation inlines simple let bindings, that bind some
        binder to some other binder instead of a more complex expression (\ie
        a = b).

        This transformation is not needed to get an expression into intended
        normal form (since these bindings are part of the intended normal
        form), but makes the resulting \small{VHDL} a lot shorter.
        
        \starttrans
        letrec
          a0 = E0
          \vdots
          ai = b
          \vdots
          an = En
        in
          M
        -----------------------------  \lam{b} is a variable reference
        letrec                         \lam{ai} ≠ \lam{b}
          a0 = E0 [ai=>b]
          \vdots
          ai-1 = Ei-1 [ai=>b]
          ai+1 = Ei+1 [ai=>b]
          \vdots
          an = En [ai=>b]
        in
          M[ai=>b]
        \stoptrans

        \todo{example}

      \subsubsection{Unused let binding removal}
        This transformation removes let bindings that are never used.
        Occasionally, \GHC's desugarer introduces some unused let bindings.

        This normalization pass should really be unneeded to get into intended normal form
        (since unused bindings are not forbidden by the normal form), but in practice
        the desugarer or simplifier emits some unused bindings that cannot be
        normalized (e.g., calls to a \type{PatError}\todo{Check this name}). Also,
        this transformation makes the resulting \small{VHDL} a lot shorter.

        \todo{Don't use old-style numerals in transformations}
        \starttrans
        letrec
          a0 = E0
          \vdots
          ai = Ei
          \vdots
          an = En
        in
          M                             \lam{ai} does not occur free in \lam{M}
        ----------------------------    \forall j, 0 ≤ j ≤ n, j ≠ i (\lam{ai} does not occur free in \lam{Ej})
        letrec
          a0 = E0
          \vdots
          ai-1 = Ei-1
          ai+1 = Ei+1
          \vdots
          an = En
        in
          M
        \stoptrans

        \todo{Example}

      \subsubsection{Cast propagation / simplification}
        This transform pushes casts down into the expression as far as possible.
        Since its exact role and need is not clear yet, this transformation is
        not yet specified.

        \todo{Cast propagation}

      \subsubsection{Top level binding inlining}
        This transform takes simple top level bindings generated by the
        \small{GHC} compiler. \small{GHC} sometimes generates very simple
        \quote{wrapper} bindings, which are bound to just a variable
        reference, or a partial application to constants or other variable
        references.

        Note that this transformation is completely optional. It is not
        required to get any function into intended normal form, but it does help making
        the resulting VHDL output easier to read (since it removes a bunch of
        components that are really boring).

        This transform takes any top level binding generated by the compiler,
        whose normalized form contains only a single let binding.

        \starttrans
        x = λa0 ... λan.let y = E in y
        ~
        x
        --------------------------------------         \lam{x} is generated by the compiler
        λa0 ... λan.let y = E in y
        \stoptrans

        \startbuffer[from]
        (+) :: Word -> Word -> Word
        (+) = GHC.Num.(+) @Word \$dNum
        ~
        (+) a b
        \stopbuffer
        \startbuffer[to]
        GHC.Num.(+) @ Alu.Word \$dNum a b
        \stopbuffer

        \transexample{toplevelinline}{Top level binding inlining}{from}{to}
       
        \in{Example}[ex:trans:toplevelinline] shows a typical application of
        the addition operator generated by \GHC. The type and dictionary
        arguments used here are described in
        \in{Section}[section:prototype:polymorphism].

        Without this transformation, there would be a \lam{(+)} entity
        in the \VHDL which would just add its inputs. This generates a
        lot of overhead in the \VHDL, which is particularly annoying
        when browsing the generated RTL schematic (especially since most
        non-alphanumerics, like all characters in \lam{(+)}, are not
        allowed in \VHDL architecture names\footnote{Technically, it is
        allowed to use non-alphanumerics when using extended
        identifiers, but it seems that none of the tooling likes
        extended identifiers in filenames, so it effectively doesn't
        work.}, so the entity would be called \quote{w7aA7f} or
        something similarly unreadable and autogenerated).

    \subsection{Program structure}
      These transformations are aimed at normalizing the overall structure
      into the intended form. This means ensuring there is a lambda abstraction
      at the top for every argument (input port or current state), putting all
      of the other value definitions in let bindings and making the final
      return value a simple variable reference.

      \subsubsection[sec:normalization:eta]{η-abstraction}
        This transformation makes sure that all arguments of a function-typed
        expression are named, by introducing lambda expressions. When combined with
        β-reduction and non-representable binding inlining, all function-typed
        expressions should be lambda abstractions or global identifiers.

        \starttrans
        E                 \lam{E :: a -> b}
        --------------    \lam{E} is not the first argument of an application.
        λx.E x            \lam{E} is not a lambda abstraction.
                          \lam{x} is a variable that does not occur free in \lam{E}.
        \stoptrans

        \startbuffer[from]
        foo = λa.case a of 
          True -> λb.mul b b
          False -> id
        \stopbuffer

        \startbuffer[to]
        foo = λa.λx.(case a of 
            True -> λb.mul b b
            False -> λy.id y) x
        \stopbuffer

        \transexample{eta}{η-abstraction}{from}{to}

      \subsubsection[sec:normalization:appprop]{Application propagation}
        This transformation is meant to propagate application expressions downwards
        into expressions as far as possible. This allows partial applications inside
        expressions to become fully applied and exposes new transformation
        opportunities for other transformations (like β-reduction and
        specialization).

        Since all binders in our expression are unique (see
        \in{section}[sec:normalization:uniq]), there is no risk that we will
        introduce unintended shadowing by moving an expression into a lower
        scope. Also, since only move expression into smaller scopes (down into
        our expression), there is no risk of moving a variable reference out
        of the scope in which it is defined.

        \starttrans
        (letrec binds in E) M
        ------------------------
        letrec binds in E M
        \stoptrans

        % And an example
        \startbuffer[from]
        ( letrec
            val = 1
          in 
            add val
        ) 3
        \stopbuffer

        \startbuffer[to]
        letrec
          val = 1
        in 
          add val 3
        \stopbuffer

        \transexample{appproplet}{Application propagation for a let expression}{from}{to}

        \starttrans
        (case x of
          p1 -> E1
          \vdots
          pn -> En) M
        -----------------
        case x of
          p1 -> E1 M
          \vdots
          pn -> En M
        \stoptrans

        % And an example
        \startbuffer[from]
        ( case x of 
            True -> id
            False -> neg
        ) 1
        \stopbuffer

        \startbuffer[to]
        case x of 
          True -> id 1
          False -> neg 1
        \stopbuffer

        \transexample{apppropcase}{Application propagation for a case expression}{from}{to}

      \subsubsection[sec:normalization:letrecurse]{Let recursification}
        This transformation makes all non-recursive lets recursive. In the
        end, we want a single recursive let in our normalized program, so all
        non-recursive lets can be converted. This also makes other
        transformations simpler: They can simply assume all lets are
        recursive.

        \starttrans
        let
          a = E
        in
          M
        ------------------------------------------
        letrec
          a = E
        in
          M
        \stoptrans

      \subsubsection{Let flattening}
        This transformation puts nested lets in the same scope, by lifting the
        binding(s) of the inner let into the outer let. Eventually, this will
        cause all let bindings to appear in the same scope.

        This transformation only applies to recursive lets, since all
        non-recursive lets will be made recursive (see
        \in{section}[sec:normalization:letrecurse]).

        Since we are joining two scopes together, there is no risk of moving a
        variable reference out of the scope where it is defined.

        \starttrans
        letrec 
          a0 = E0
          \vdots
          ai = (letrec bindings in M)
          \vdots
          an = En
        in
          N
        ------------------------------------------
        letrec
          a0 = E0
          \vdots
          ai = M
          \vdots
          an = En
          bindings
        in
          N
        \stoptrans

        \startbuffer[from]
        letrec
          a = 1
          b = letrec
            x = a
            y = c
          in
            x + y
          c = 2
        in
          b
        \stopbuffer
        \startbuffer[to]
        letrec
          a = 1
          b = x + y
          c = 2
          x = a
          y = c
        in
          b
        \stopbuffer

        \transexample{letflat}{Let flattening}{from}{to}

      \subsubsection{Return value simplification}
        This transformation ensures that the return value of a function is always a
        simple local variable reference.

        Currently implemented using lambda simplification, let simplification, and
        top simplification. Should change into something like the following, which
        works only on the result of a function instead of any subexpression. This is
        achieved by the contexts, like \lam{x = E}, though this is strictly not
        correct (you could read this as "if there is any function \lam{x} that binds
        \lam{E}, any \lam{E} can be transformed, while we only mean the \lam{E} that
        is bound by \lam{x}. This might need some extra notes or something).

        Note that the return value is not simplified if its not representable.
        Otherwise, this would cause a direct loop with the inlining of
        unrepresentable bindings. If the return value is not
        representable because it has a function type, η-abstraction should
        make sure that this transformation will eventually apply. If the value
        is not representable for other reasons, the function result itself is
        not representable, meaning this function is not translatable anyway.

        \starttrans
        x = E                            \lam{E} is representable
        ~                                \lam{E} is not a lambda abstraction
        E                                \lam{E} is not a let expression
        ---------------------------      \lam{E} is not a local variable reference
        letrec x = E in x
        \stoptrans

        \starttrans
        x = λv0 ... λvn.E
        ~                                \lam{E} is representable
        E                                \lam{E} is not a let expression
        ---------------------------      \lam{E} is not a local variable reference
        letrec x = E in x
        \stoptrans

        \starttrans
        x = λv0 ... λvn.let ... in E
        ~                                \lam{E} is representable
        E                                \lam{E} is not a local variable reference
        -----------------------------
        letrec x = E in x
        \stoptrans

        \startbuffer[from]
        x = add 1 2
        \stopbuffer

        \startbuffer[to]
        x = letrec x = add 1 2 in x
        \stopbuffer

        \transexample{retvalsimpl}{Return value simplification}{from}{to}
        
        \todo{More examples}

    \subsection[sec:normalization:argsimpl]{Representable arguments simplification}
      This section contains just a single transformation that deals with
      representable arguments in applications. Non-representable arguments are
      handled by the transformations in
      \in{section}[sec:normalization:nonrep]. 
      
      This transformation ensures that all representable arguments will become
      references to local variables. This ensures they will become references
      to local signals in the resulting \small{VHDL}, which is required due to
      limitations in the component instantiation code in \VHDL (one can only
      assign a signal or constant to an input port). By ensuring that all
      arguments are always simple variable references, we always have a signal
      available to map to the input ports.

      To reduce a complex expression to a simple variable reference, we create
      a new let expression around the application, which binds the complex
      expression to a new variable. The original function is then applied to
      this variable.

      \refdef{global variable}
      Note that references to \emph{global variables} (like a top level
      function without arguments, but also an argumentless dataconstructors
      like \lam{True}) are also simplified. Only local variables generate
      signals in the resulting architecture. Even though argumentless
      dataconstructors generate constants in generated \VHDL code and could be
      mapped to an input port directly, they are still simplified to make the
      normal form more regular.

      \refdef{representable}
      \starttrans
      M N
      --------------------    \lam{N} is representable
      letrec x = N in M x     \lam{N} is not a local variable reference
      \stoptrans
      \refdef{local variable}

      \startbuffer[from]
      add (add a 1) 1
      \stopbuffer

      \startbuffer[to]
      letrec x = add a 1 in add x 1
      \stopbuffer

      \transexample{argsimpl}{Argument simplification}{from}{to}

    \subsection[sec:normalization:builtins]{Builtin functions}
      This section deals with (arguments to) builtin functions.  In the
      intended normal form definition\refdef{intended normal form definition}
      we can see that there are three sorts of arguments a builtin function
      can receive.
      
      \startitemize[KR]
        \item A representable local variable reference. This is the most
        common argument to any function. The argument simplification
        transformation described in \in{section}[sec:normalization:argsimpl]
        makes sure that \emph{any} representable argument to \emph{any}
        function (including builtin functions) is turned into a local variable
        reference.
        \item (A partial application of) a top level function (either builtin on
        user-defined). The function extraction transformation described in
        this section takes care of turning every functiontyped argument into
        (a partial application of) a top level function.
        \item Any expression that is not representable and does not have a
        function type. Since these can be any expression, there is no
        transformation needed. Note that this category is exactly all
        expressions that are not transformed by the transformations for the
        previous two categories. This means that \emph{any} core expression
        that is used as an argument to a builtin function will be either
        transformed into one of the above categories, or end up in this
        categorie. In any case, the result is in normal form.
      \stopitemize

      As noted, the argument simplification will handle any representable
      arguments to a builtin function. The following transformation is needed
      to handle non-representable arguments with a function type, all other
      non-representable arguments don't need any special handling.

      \subsubsection[sec:normalization:funextract]{Function extraction}
        This transform deals with function-typed arguments to builtin
        functions. 
        Since builtin functions cannot be specialized (see
        \in{section}[sec:normalization:specialize]) to remove the arguments,
        these arguments are extracted into a new global function instead. In
        other words, we create a new top level function that has exactly the
        extracted argument as its body. This greatly simplifies the
        translation rules needed for builtin functions, since they only need
        to handle (partial applications of) top level functions.

        Any free variables occuring in the extracted arguments will become
        parameters to the new global function. The original argument is replaced
        with a reference to the new function, applied to any free variables from
        the original argument.

        This transformation is useful when applying higher order builtin functions
        like \hs{map} to a lambda abstraction, for example. In this case, the code
        that generates \small{VHDL} for \hs{map} only needs to handle top level functions and
        partial applications, not any other expression (such as lambda abstractions or
        even more complicated expressions).

        \starttrans
        M N                     \lam{M} is (a partial aplication of) a builtin function.
        ---------------------   \lam{f0 ... fn} are all free local variables of \lam{N}
        M (x f0 ... fn)         \lam{N :: a -> b}
        ~                       \lam{N} is not a (partial application of) a top level function
        x = λf0 ... λfn.N
        \stoptrans

        \startbuffer[from]
        addList = λb.λxs.map (λa . add a b) xs
        \stopbuffer

        \startbuffer[to]
        addList = λb.λxs.map (f b) xs
        ~
        f = λb.λa.add a b
        \stopbuffer

        \transexample{funextract}{Function extraction}{from}{to}

        Note that the function \lam{f} will still need normalization after
        this.

    \subsection{Case normalisation}
      \subsubsection{Scrutinee simplification}
        This transform ensures that the scrutinee of a case expression is always
        a simple variable reference.

        \starttrans
        case E of
          alts
        -----------------        \lam{E} is not a local variable reference
        letrec x = E in 
          case E of
            alts
        \stoptrans

        \startbuffer[from]
        case (foo a) of
          True -> a
          False -> b
        \stopbuffer

        \startbuffer[to]
        letrec x = foo a in
          case x of
            True -> a
            False -> b
        \stopbuffer

        \transexample{letflat}{Case normalisation}{from}{to}


      \subsubsection{Case simplification}
        This transformation ensures that all case expressions become normal form. This
        means they will become one of:
        \startitemize
        \item An extractor case with a single alternative that picks a single field
        from a datatype, \eg \lam{case x of (a, b) -> a}.
        \item A selector case with multiple alternatives and only wild binders, that
        makes a choice between expressions based on the constructor of another
        expression, \eg \lam{case x of Low -> a; High -> b}.
        \stopitemize
        
        \defref{wild binder}
        \starttrans
        case E of
          C0 v0,0 ... v0,m -> E0
          \vdots
          Cn vn,0 ... vn,m -> En
        --------------------------------------------------- \forall i \forall j, 0 ≤ i ≤ n, 0 ≤ i < m (\lam{wi,j} is a wild (unused) binder)
        letrec
          v0,0 = case E of C0 v0,0 .. v0,m -> v0,0
          \vdots
          v0,m = case E of C0 v0,0 .. v0,m -> v0,m
          \vdots
          vn,m = case E of Cn vn,0 .. vn,m -> vn,m
          x0 = E0
          \vdots
          xn = En
        in
          case E of
            C0 w0,0 ... w0,m -> x0
            \vdots
            Cn wn,0 ... wn,m -> xn
        \stoptrans
        \todo{Check the subscripts of this transformation}

        Note that this transformation applies to case statements with any
        scrutinee. If the scrutinee is a complex expression, this might result
        in duplicate hardware. An extra condition to only apply this
        transformation when the scrutinee is already simple (effectively
        causing this transformation to be only applied after the scrutinee
        simplification transformation) might be in order. 

        \fxnote{This transformation specified like this is complicated and misses
        conditions to prevent looping with itself. Perhaps it should be split here for
        discussion?}

        \startbuffer[from]
        case a of
          True -> add b 1
          False -> add b 2
        \stopbuffer

        \startbuffer[to]
        letnonrec
          x0 = add b 1
          x1 = add b 2
        in
          case a of
            True -> x0
            False -> x1
        \stopbuffer

        \transexample{selcasesimpl}{Selector case simplification}{from}{to}

        \startbuffer[from]
        case a of
          (,) b c -> add b c
        \stopbuffer
        \startbuffer[to]
        letrec
          b = case a of (,) b c -> b
          c = case a of (,) b c -> c
          x0 = add b c
        in
          case a of
            (,) w0 w1 -> x0
        \stopbuffer

        \transexample{excasesimpl}{Extractor case simplification}{from}{to}

        \refdef{selector case}
        In \in{example}[ex:trans:excasesimpl] the case expression is expanded
        into multiple case expressions, including a pretty useless expression
        (that is neither a selector or extractor case). This case can be
        removed by the Case removal transformation in
        \in{section}[sec:transformation:caseremoval].

      \subsubsection[sec:transformation:caseremoval]{Case removal}
        This transform removes any case statements with a single alternative and
        only wild binders.

        These "useless" case statements are usually leftovers from case simplification
        on extractor case (see the previous example).

        \starttrans
        case x of
          C v0 ... vm -> E
        ----------------------     \lam{\forall i, 0 ≤ i ≤ m} (\lam{vi} does not occur free in E)
        E
        \stoptrans

        \startbuffer[from]
        case a of
          (,) w0 w1 -> x0
        \stopbuffer

        \startbuffer[to]
        x0
        \stopbuffer

        \transexample{caserem}{Case removal}{from}{to}

    \subsection[sec:normalization:nonrep]{Removing unrepresentable values}
      The transformations in this section are aimed at making all the
      values used in our expression representable. There are two main
      transformations that are applied to \emph{all} unrepresentable let
      bindings and function arguments. These are meant to address three
      different kinds of unrepresentable values: Polymorphic values, higher
      order values and literals. The transformation are described generically:
      They apply to all non-representable values. However, non-representable
      values that don't fall into one of these three categories will be moved
      around by these transformations but are unlikely to completely
      disappear. They usually mean the program was not valid in the first
      place, because unsupported types were used (for example, a program using
      strings).
     
      Each of these three categories will be detailed below, followed by the
      actual transformations.

      \subsubsection{Removing Polymorphism}
        As noted in \in{section}[sec:prototype:polymporphism],
        polymorphism is made explicit in Core through type and
        dictionary arguments. To remove the polymorphism from a
        function, we can simply specialize the polymorphic function for
        the particular type applied to it. The same goes for dictionary
        arguments. To remove polymorphism from let bound values, we
        simply inline the let bindings that have a polymorphic type,
        which should (eventually) make sure that the polymorphic
        expression is applied to a type and/or dictionary, which can
        then be removed by β-reduction (\in{section}[sec:normalization:beta]).

        Since both type and dictionary arguments are not representable,
        \refdef{representable}
        the non-representable argument specialization and
        non-representable let binding inlining transformations below
        take care of exactly this.

        There is one case where polymorphism cannot be completely
        removed: Builtin functions are still allowed to be polymorphic
        (Since we have no function body that we could properly
        specialize). However, the code that generates \VHDL for builtin
        functions knows how to handle this, so this is not a problem.

      \subsubsection{Defunctionalization}
        These transformations remove higher order expressions from our
        program, making all values first-order.
      
        Higher order values are always introduced by lambda abstractions, none
        of the other Core expression elements can introduce a function type.
        However, other expressions can \emph{have} a function type, when they
        have a lambda expression in their body. 
        
        For example, the following expression is a higher order expression
        that is not a lambda expression itself:
        
        \refdef{id function}
        \startlambda
          case x of
            High -> id
            Low -> λx.x
        \stoplambda

        The reference to the \lam{id} function shows that we can introduce a
        higher order expression in our program without using a lambda
        expression directly. However, inside the definition of the \lam{id}
        function, we can be sure that a lambda expression is present.
        
        Looking closely at the definition of our normal form in
        \in{section}[sec:normalization:intendednormalform], we can see that
        there are three possibilities for higher order values to appear in our
        intended normal form:

        \startitemize[KR]
          \item[item:toplambda] Lambda abstractions can appear at the highest level of a
          top level function. These lambda abstractions introduce the
          arguments (input ports / current state) of the function.
          \item[item:builtinarg] (Partial applications of) top level functions can appear as an
          argument to a builtin function.
          \item[item:completeapp] (Partial applications of) top level functions can appear in
          function position of an application. Since a partial application
          cannot appear anywhere else (except as builtin function arguments),
          all partial applications are applied, meaning that all applications
          will become complete applications. However, since application of
          arguments happens one by one, in the expression:
          \startlambda
            f 1 2
          \stoplambda
          the subexpression \lam{f 1} has a function type. But this is
          allowed, since it is inside a complete application.
        \stopitemize

        We will take a typical function with some higher order values as an
        example. The following function takes two arguments: a \lam{Bit} and a
        list of numbers. Depending on the first argument, each number in the
        list is doubled, or the list is returned unmodified. For the sake of
        the example, no polymorphism is shown. In reality, at least map would
        be polymorphic.

        \startlambda
        λy.let double = λx. x + x in
             case y of
                Low -> map double
                High -> λz. z
        \stoplambda

        This example shows a number of higher order values that we cannot
        translate to \VHDL directly. The \lam{double} binder bound in the let
        expression has a function type, as well as both of the alternatives of
        the case expression. The first alternative is a partial application of
        the \lam{map} builtin function, whereas the second alternative is a
        lambda abstraction.

        To reduce all higher order values to one of the above items, a number
        of transformations we've already seen are used. The η-abstraction
        transformation from \in{section}[sec:normalization:eta] ensures all
        function arguments are introduced by lambda abstraction on the highest
        level of a function. These lambda arguments are allowed because of
        \in{item}[item:toplambda] above. After η-abstraction, our example
        becomes a bit bigger:

        \startlambda
        λy.λq.(let double = λx. x + x in
                 case y of
                   Low -> map double
                   High -> λz. z
              ) q
        \stoplambda

        η-abstraction also introduces extra applications (the application of
        the let expression to \lam{q} in the above example). These
        applications can then propagated down by the application propagation
        transformation (\in{section}[sec:normalization:appprop]). In our
        example, the \lam{q} and \lam{r} variable will be propagated into the
        let expression and then into the case expression:

        \startlambda
        λy.λq.let double = λx. x + x in
                case y of
                  Low -> map double q
                  High -> (λz. z) q
        \stoplambda
        
        This propagation makes higher order values become applied (in
        particular both of the alternatives of the case now have a
        representable type. Completely applied top level functions (like the
        first alternative) are now no longer invalid (they fall under
        \in{item}[item:completeapp] above). (Completely) applied lambda
        abstractions can be removed by β-abstraction. For our example,
        applying β-abstraction results in the following:

        \startlambda
        λy.λq.let double = λx. x + x in
                case y of
                  Low -> map double q
                  High -> q
        \stoplambda

        As you can see in our example, all of this moves applications towards
        the higher order values, but misses higher order functions bound by
        let expressions. The applications cannot be moved towards these values
        (since they can be used in multiple places), so the values will have
        to be moved towards the applications. This is achieved by inlining all
        higher order values bound by let applications, by the
        non-representable binding inlining transformation below. When applying
        it to our example, we get the following:
        
        \startlambda
        λy.λq.case y of
                Low -> map (λx. x + x) q
                High -> q
        \stoplambda

        We've nearly eliminated all unsupported higher order values from this
        expressions. The one that's remaining is the first argument to the
        \lam{map} function. Having higher order arguments to a builtin
        function like \lam{map} is allowed in the intended normal form, but
        only if the argument is a (partial application) of a top level
        function. This is easily done by introducing a new top level function
        and put the lambda abstraction inside. This is done by the function
        extraction transformation from
        \in{section}[sec:normalization:funextract].

        \startlambda
        λy.λq.case y of
                Low -> map func q
                High -> q
        \stoplambda

        This also introduces a new function, that we have called \lam{func}:

        \startlambda
        func = λx. x + x
        \stoplambda

        Note that this does not actually remove the lambda, but now it is a
        lambda at the highest level of a function, which is allowed in the
        intended normal form.

        There is one case that has not been discussed yet. What if the
        \lam{map} function in the example above was not a builtin function
        but a user-defined function? Then extracting the lambda expression
        into a new function would not be enough, since user-defined functions
        can never have higher order arguments. For example, the following
        expression shows an example:

        \startlambda
        twice :: (Word -> Word) -> Word -> Word
        twice = λf.λa.f (f a)

        main = λa.app (λx. x + x) a
        \stoplambda

        This example shows a function \lam{twice} that takes a function as a
        first argument and applies that function twice to the second argument.
        Again, we've made the function monomorphic for clarity, even though
        this function would be a lot more useful if it was polymorphic. The
        function \lam{main} uses \lam{twice} to apply a lambda epression twice.

        When faced with a user defined function, a body is available for that
        function. This means we could create a specialized version of the
        function that only works for this particular higher order argument
        (\ie, we can just remove the argument and call the specialized
        function without the argument). This transformation is detailed below.
        Applying this transformation to the example gives:

        \startlambda
        twice' :: Word -> Word
        twice' = λb.(λf.λa.f (f a)) (λx. x + x) b

        main = λa.app' a
        \stoplambda

        The \lam{main} function is now in normal form, since the only higher
        order value there is the top level lambda expression. The new
        \lam{twice'} function is a bit complex, but the entire original body of
        the original \lam{twice} function is wrapped in a lambda abstraction
        and applied to the argument we've specialized for (\lam{λx. x + x})
        and the other arguments. This complex expression can fortunately be
        effectively reduced by repeatedly applying β-reduction: 

        \startlambda
        twice' :: Word -> Word
        twice' = λb.(b + b) + (b + b)
        \stoplambda

        This example also shows that the resulting normal form might not be as
        efficient as we might hope it to be (it is calculating \lam{(b + b)}
        twice). This is discussed in more detail in
        \in{section}[sec:normalization:duplicatework].

      \subsubsection{Literals}
        There are a limited number of literals available in Haskell and Core.
        \refdef{enumerated types} When using (enumerating) algebraic
        datatypes, a literal is just a reference to the corresponding data
        constructor, which has a representable type (the algebraic datatype)
        and can be translated directly. This also holds for literals of the
        \hs{Bool} Haskell type, which is just an enumerated type.

        There is, however, a second type of literal that does not have a
        representable type: Integer literals. Cλash supports using integer
        literals for all three integer types supported (\hs{SizedWord},
        \hs{SizedInt} and \hs{RangedWord}). This is implemented using
        Haskell's \hs{Num} typeclass, which offers a \hs{fromInteger} method
        that converts any \hs{Integer} to the Cλash datatypes.

        When \GHC sees integer literals, it will automatically insert calls to
        the \hs{fromInteger} method in the resulting Core expression. For
        example, the following expression in Haskell creates a 32 bit unsigned
        word with the value 1. The explicit type signature is needed, since
        there is no context for \GHC to determine the type from otherwise.

        \starthaskell
        1 :: SizedWord D32
        \stophaskell

        This Haskell code results in the following Core expression:

        \startlambda
        fromInteger @(SizedWord D32) \$dNum (smallInteger 10)
        \stoplambda

        The literal 10 will have the type \lam{GHC.Prim.Int\#}, which is
        converted into an \lam{Integer} by \lam{smallInteger}. Finally, the
        \lam{fromInteger} function will finally convert this into a
        \lam{SizedWord D32}.

        Both the \lam{GHC.Prim.Int\#} and \lam{Integer} types are not
        representable, and cannot be translated directly. Fortunately, there
        is no need to translate them, since \lam{fromInteger} is a builtin
        function that knows how to handle these values. However, this does
        require that the \lam{fromInteger} function is directly applied to
        these non-representable literal values, otherwise errors will occur.
        For example, the following expression is not in the intended normal
        form, since one of the let bindings has an unrepresentable type
        (\lam{Integer}):

        \startlambda
        let l = smallInteger 10 in fromInteger @(SizedWord D32) \$dNum l
        \stoplambda

        By inlining these let-bindings, we can ensure that unrepresentable
        literals bound by a let binding end up in an application of the
        appropriate builtin function, where they are allowed. Since it is
        possible that the application of that function is in a different
        function than the definition of the literal value, we will always need
        to specialize away any unrepresentable literals that are used as
        function arguments. The following two transformations do exactly this.

      \subsubsection{Non-representable binding inlining}
        This transform inlines let bindings that are bound to a
        non-representable value. Since we can never generate a signal
        assignment for these bindings (we cannot declare a signal assignment
        with a non-representable type, for obvious reasons), we have no choice
        but to inline the binding to remove it.

        As we have seen in the previous sections, inlining these bindings
        solves (part of) the polymorphism, higher order values and
        unrepresentable literals in an expression.

        \starttrans
        letrec 
          a0 = E0
          \vdots
          ai = Ei
          \vdots
          an = En
        in
          M
        --------------------------    \lam{Ei} has a non-representable type.
        letrec
          a0 = E0 [ai=>Ei] \vdots
          ai-1 = Ei-1 [ai=>Ei]
          ai+1 = Ei+1 [ai=>Ei]
          \vdots
          an = En [ai=>Ei]
        in
          M[ai=>Ei]
        \stoptrans

        \startbuffer[from]
        letrec
          a = smallInteger 10
          inc = λb -> add b 1
          inc' = add 1
          x = fromInteger a 
        in
          inc (inc' x)
        \stopbuffer

        \startbuffer[to]
        letrec
          x = fromInteger (smallInteger 10)
        in
          (λb -> add b 1) (add 1 x)
        \stopbuffer

        \transexample{nonrepinline}{Nonrepresentable binding inlining}{from}{to}

      \subsubsection[sec:normalization:specialize]{Function specialization}
        This transform removes arguments to user-defined functions that are
        not representable at runtime. This is done by creating a
        \emph{specialized} version of the function that only works for one
        particular value of that argument (in other words, the argument can be
        removed).

        Specialization means to create a specialized version of the called
        function, with one argument already filled in. As a simple example, in
        the following program (this is not actual Core, since it directly uses
        a literal with the unrepresentable type \lam{GHC.Prim.Int\#}).

        \startlambda
        f = λa.λb.a + b
        inc = λa.f a 1
        \stoplambda

        We could specialize the function \lam{f} against the literal argument
        1, with the following result:

        \startlambda
        f' = λa.a + 1
        inc = λa.f' a
        \stoplambda

        In some way, this transformation is similar to β-reduction, but it
        operates across function boundaries. It is also similar to
        non-representable let binding inlining above, since it sort of
        \quote{inlines} an expression into a called function.

        Special care must be taken when the argument has any free variables.
        If this is the case, the original argument should not be removed
        completely, but replaced by all the free variables of the expression.
        In this way, the original expression can still be evaluated inside the
        new function.

        To prevent us from propagating the same argument over and over, a
        simple local variable reference is not propagated (since is has
        exactly one free variable, itself, we would only replace that argument
        with itself).

        This shows that any free local variables that are not runtime
        representable cannot be brought into normal form by this transform. We
        rely on an inlining or β-reduction transformation to replace such a
        variable with an expression we can propagate again.

        \starttrans
        x = E
        ~
        x Y0 ... Yi ... Yn                               \lam{Yi} is not representable
        ---------------------------------------------    \lam{Yi} is not a local variable reference
        x' y0 ... yi-1 f0 ...  fm Yi+1 ... Yn            \lam{f0 ... fm} are all free local vars of \lam{Yi}
        ~                                                \lam{T0 ... Tn} are the types of \lam{Y0 ... Yn}
        x' = λ(y0 :: T0) ... λ(yi-1 :: Ty-1). λf0 ... λfm. λ(yi+1 :: Ty+1) ...  λ(yn :: Tn).       
              E y0 ... yi-1 Yi yi+1 ... yn   
        \stoptrans

        This is a bit of a complex transformation. It transforms an
        application of the function \lam{x}, where one of the arguments
        (\lam{Y_i}) is not representable. A new
        function \lam{x'} is created that wraps the body of the old function.
        The body of the new function becomes a number of nested lambda
        abstractions, one for each of the original arguments that are left
        unchanged.
        
        The ith argument is replaced with the free variables of
        \lam{Y_i}. Note that we reuse the same binders as those used in
        \lam{Y_i}, since we can then just use \lam{Y_i} inside the new
        function body and have all of the variables it uses be in scope.

        The argument that we are specializing for, \lam{Y_i}, is put inside
        the new function body. The old function body is applied to it. Since
        we use this new function only in place of an application with that
        particular argument \lam{Y_i}, behaviour should not change.
        
        Note that the types of the arguments of our new function are taken
        from the types of the \emph{actual} arguments (\lam{T0 ... Tn}). This
        means that any polymorphism in the arguments is removed, even when the
        corresponding explicit type lambda is not removed
        yet.\refdef{type lambda}

        \todo{Examples. Perhaps reference the previous sections}


  \section{Unsolved problems}
    The above system of transformations has been implemented in the prototype
    and seems to work well to compile simple and more complex examples of
    hardware descriptions. \todo{Ref christiaan?} However, this normalization
    system has not seen enough review and work to be complete and work for
    every Core expression that is supplied to it. A number of problems
    have already been identified and are discussed in this section.

    \subsection[sec:normalization:duplicatework]{Work duplication}
        A possible problem of β-reduction is that it could duplicate work.
        When the expression applied is not a simple variable reference, but
        requires calculation and the binder the lambda abstraction binds to
        is used more than once, more hardware might be generated than strictly
        needed. 

        As an example, consider the expression:

        \startlambda
        (λx. x + x) (a * b)
        \stoplambda

        When applying β-reduction to this expression, we get:

        \startlambda
        (a * b) + (a * b)
        \stoplambda

        which of course calculates \lam{(a * b)} twice.
        
        A possible solution to this would be to use the following alternative
        transformation, which is of course no longer normal β-reduction. The
        followin transformation has not been tested in the prototype, but is
        given here for future reference:

        \starttrans
        (λx.E) M
        -----------------
        letrec x = M in E
        \stoptrans
        
        This doesn't seem like much of an improvement, but it does get rid of
        the lambda expression (and the associated higher order value), while
        at the same time introducing a new let binding. Since the result of
        every application or case expression must be bound by a let expression
        in the intended normal form anyway, this is probably not a problem. If
        the argument happens to be a variable reference, then simple let
        binding removal (\in{section}[sec:normalization:simplelet]) will
        remove it, making the result identical to that of the original
        β-reduction transformation.

        When also applying argument simplification to the above example, we
        get the following expression:

        \startlambda
        let y = (a * b)
            z = (a * b)
        in y + z
        \stoplambda

        Looking at this, we could imagine an alternative approach: Create a
        transformation that removes let bindings that bind identical values.
        In the above expression, the \lam{y} and \lam{z} variables could be
        merged together, resulting in the more efficient expression:

        \startlambda
        let y = (a * b) in y + y
        \stoplambda

      \subsection{Non-determinism}
        As an example, again consider the following expression:

        \startlambda
        (λx. x + x) (a * b)
        \stoplambda

        We can apply both β-reduction (\in{section}[sec:normalization:beta])
        as well as argument simplification
        (\in{section}[sec:normalization:argsimpl]) to this expression.

        When applying argument simplification first and then β-reduction, we
        get the following expression:

        \startlambda
        let y = (a * b) in y + y
        \stoplambda

        When applying β-reduction first and then argument simplification, we
        get the following expression:

        \startlambda
        let y = (a * b)
            z = (a * b)
        in y + z
        \stoplambda

        As you can see, this is a different expression. This means that the
        order of expressions, does in fact change the resulting normal form,
        which is something that we would like to avoid. In this particular
        case one of the alternatives is even clearly more efficient, so we
        would of course like the more efficient form to be the normal form.

        For this particular problem, the solutions for duplication of work
        seem from the previous section seem to fix the determinism of our
        transformation system as well. However, it is likely that there are
        other occurences of this problem.

      \subsection{Casts}
        We do not fully understand the use of cast expressions in Core, so
        there are probably expressions involving cast expressions that cannot
        be brought into intended normal form by this transformation system.

        The uses of casts in the core system should be investigated more and
        transformations will probably need updating to handle them in all
        cases.

  \section[sec:normalization:properties]{Provable properties}
    When looking at the system of transformations outlined above, there are a
    number of questions that we can ask ourselves. The main question is of course:
    \quote{Does our system work as intended?}. We can split this question into a
    number of subquestions:

    \startitemize[KR]
    \item[q:termination] Does our system \emph{terminate}? Since our system will
    keep running as long as transformations apply, there is an obvious risk that
    it will keep running indefinitely. This typically happens when one
    transformation produces a result that is transformed back to the original
    by another transformation, or when one or more transformations keep
    expanding some expression.
    \item[q:soundness] Is our system \emph{sound}? Since our transformations
    continuously modify the expression, there is an obvious risk that the final
    normal form will not be equivalent to the original program: Its meaning could
    have changed.
    \item[q:completeness] Is our system \emph{complete}? Since we have a complex
    system of transformations, there is an obvious risk that some expressions will
    not end up in our intended normal form, because we forgot some transformation.
    In other words: Does our transformation system result in our intended normal
    form for all possible inputs?
    \item[q:determinism] Is our system \emph{deterministic}? Since we have defined
    no particular order in which the transformation should be applied, there is an
    obvious risk that different transformation orderings will result in
    \emph{different} normal forms. They might still both be intended normal forms
    (if our system is \emph{complete}) and describe correct hardware (if our
    system is \emph{sound}), so this property is less important than the previous
    three: The translator would still function properly without it.
    \stopitemize

    Unfortunately, the final transformation system has only been
    developed in the final part of the research, leaving no more time
    for verifying these properties. In fact, it is likely that the
    current transformation system still violates some of these
    properties in some cases and should be improved (or extra conditions
    on the input hardware descriptions should be formulated).

    This is most likely the case with the completeness and determinism
    properties, perhaps als the termination property. The soundness
    property probably holds, since it is easier to manually verify (each
    transformation can be reviewed separately).

    Even though no complete proofs have been made, some ideas for
    possible proof strategies are shown below.

    \subsection{Graph representation}
      Before looking into how to prove these properties, we'll look at our
      transformation system from a graph perspective. The nodes of the graph are
      all possible Core expressions. The (directed) edges of the graph are
      transformations. When a transformation α applies to an expression \lam{A} to
      produce an expression \lam{B}, we add an edge from the node for \lam{A} to the
      node for \lam{B}, labeled α.

      \startuseMPgraphic{TransformGraph}
        save a, b, c, d;

        % Nodes
        newCircle.a(btex \lam{(λx.λy. (+) x y) 1} etex);
        newCircle.b(btex \lam{λy. (+) 1 y} etex);
        newCircle.c(btex \lam{(λx.(+) x) 1} etex);
        newCircle.d(btex \lam{(+) 1} etex);

        b.c = origin;
        c.c = b.c + (4cm, 0cm);
        a.c = midpoint(b.c, c.c) + (0cm, 4cm);
        d.c = midpoint(b.c, c.c) - (0cm, 3cm);

        % β-conversion between a and b
        ncarc.a(a)(b) "name(bred)";
        ObjLabel.a(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)";
        ncarc.b(b)(a) "name(bexp)", "linestyle(dashed withdots)";
        ObjLabel.b(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)";

        % η-conversion between a and c
        ncarc.a(a)(c) "name(ered)";
        ObjLabel.a(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)";
        ncarc.c(c)(a) "name(eexp)", "linestyle(dashed withdots)";
        ObjLabel.c(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)";

        % η-conversion between b and d
        ncarc.b(b)(d) "name(ered)";
        ObjLabel.b(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)";
        ncarc.d(d)(b) "name(eexp)", "linestyle(dashed withdots)";
        ObjLabel.d(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)";

        % β-conversion between c and d
        ncarc.c(c)(d) "name(bred)";
        ObjLabel.c(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)";
        ncarc.d(d)(c) "name(bexp)", "linestyle(dashed withdots)";
        ObjLabel.d(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)";

        % Draw objects and lines
        drawObj(a, b, c, d);
      \stopuseMPgraphic

      \placeexample[right][ex:TransformGraph]{Partial graph of a lambda calculus
      system with β and η reduction (solid lines) and expansion (dotted lines).}
          \boxedgraphic{TransformGraph}

      Of course our graph is unbounded, since we can construct an infinite amount of
      Core expressions. Also, there might potentially be multiple edges between two
      given nodes (with different labels), though seems unlikely to actually happen
      in our system.

      See \in{example}[ex:TransformGraph] for the graph representation of a very
      simple lambda calculus that contains just the expressions \lam{(λx.λy. (+) x
      y) 1}, \lam{λy. (+) 1 y}, \lam{(λx.(+) x) 1} and \lam{(+) 1}. The
      transformation system consists of β-reduction and η-reduction (solid edges) or
      β-expansion and η-expansion (dotted edges).

      \todo{Define β-reduction and η-reduction?}

      Note that the normal form of such a system consists of the set of nodes
      (expressions) without outgoing edges, since those are the expression to which
      no transformation applies anymore. We call this set of nodes the \emph{normal
      set}. The set of nodes containing expressions in intended normal
      form \refdef{intended normal form} is called the \emph{intended
      normal set}.

      From such a graph, we can derive some properties easily:
      \startitemize[KR]
        \item A system will \emph{terminate} if there is no path of infinite length
        in the graph (this includes cycles, but can also happen without cycles).
        \item Soundness is not easily represented in the graph.
        \item A system is \emph{complete} if all of the nodes in the normal set have
        the intended normal form. The inverse (that all of the nodes outside of
        the normal set are \emph{not} in the intended normal form) is not
        strictly required. In other words, our normal set must be a
        subset of the intended normal form, but they do not need to be
        the same set.
        form.
        \item A system is deterministic if all paths starting at a particular
        node, which end in a node in the normal set, end at the same node.
      \stopitemize

      When looking at the \in{example}[ex:TransformGraph], we see that the system
      terminates for both the reduction and expansion systems (but note that, for
      expansion, this is only true because we've limited the possible
      expressions.  In comlete lambda calculus, there would be a path from
      \lam{(λx.λy. (+) x y) 1} to \lam{(λx.λy.(λz.(+) z) x y) 1} to
      \lam{(λx.λy.(λz.(λq.(+) q) z) x y) 1} etc.)

      If we would consider the system with both expansion and reduction, there
      would no longer be termination either, since there would be cycles all
      over the place.

      The reduction and expansion systems have a normal set of containing just
      \lam{(+) 1} or \lam{(λx.λy. (+) x y) 1} respectively. Since all paths in
      either system end up in these normal forms, both systems are \emph{complete}.
      Also, since there is only one node in the normal set, it must obviously be
      \emph{deterministic} as well.

    \todo{Add content to these sections}
    \subsection{Termination}
      In general, proving termination of an arbitrary program is a very
      hard problem. \todo{Ref about arbitrary termination} Fortunately,
      we only have to prove termination for our specific transformation
      system.

      A common approach for these kinds of proofs is to associate a
      measure with each possible expression in our system. If we can
      show that each transformation strictly decreases this measure
      (\ie, the expression transformed to has a lower measure than the
      expression transformed from).  \todo{ref about measure-based
      termination proofs / analysis}
      
      A good measure for a system consisting of just β-reduction would
      be the number of lambda expressions in the expression. Since every
      application of β-reduction removes a lambda abstraction (and there
      is always a bounded number of lambda abstractions in every
      expression) we can easily see that a transformation system with
      just β-reduction will always terminate.

      For our complete system, this measure would be fairly complex
      (probably the sum of a lot of things). Since the (conditions on)
      our transformations are pretty complex, we would need to include
      both simple things like the number of let expressions as well as
      more complex things like the number of case expressions that are
      not yet in normal form.

      No real attempt has been made at finding a suitable measure for
      our system yet.

    \subsection{Soundness}
      Soundness is a property that can be proven for each transformation
      separately. Since our system only runs separate transformations
      sequentially, if each of our transformations leaves the
      \emph{meaning} of the expression unchanged, then the entire system
      will of course leave the meaning unchanged and is thus
      \emph{sound}.

      The current prototype has only been verified in an ad-hoc fashion
      by inspecting (the code for) each transformation. A more formal
      verification would be more appropriate.

      To be able to formally show that each transformation properly
      preserves the meaning of every expression, we require an exact
      definition of the \emph{meaning} of every expression, so we can
      compare them. Currently there seems to be no formal definition of
      the meaning or semantics of \GHC's core language, only informal
      descriptions are available.

      It should be possible to have a single formal definition of
      meaning for Core for both normal Core compilation by \GHC and for
      our compilation to \VHDL. The main difference seems to be that in
      hardware every expression is always evaluated, while in software
      it is only evaluated if needed, but it should be possible to
      assign a meaning to core expressions that assumes neither.
      
      Since each of the transformations can be applied to any
      subexpression as well, there is a constraint on our meaning
      definition: The meaning of an expression should depend only on the
      meaning of subexpressions, not on the expressions themselves. For
      example, the meaning of the application in \lam{f (let x = 4 in
      x)} should be the same as the meaning of the application in \lam{f
      4}, since the argument subexpression has the same meaning (though
      the actual expression is different).
      
    \subsection{Completeness}
      Proving completeness is probably not hard, but it could be a lot
      of work. We have seen above that to prove completeness, we must
      show that the normal set of our graph representation is a subset
      of the intended normal set.

      However, it is hard to systematically generate or reason about the
      normal set, since it is defined as any nodes to which no
      transformation applies. To determine this set, each transformation
      must be considered and when a transformation is added, the entire
      set should be re-evaluated. This means it is hard to show that
      each node in the normal set is also in the intended normal set.
      Reasoning about our intended normal set is easier, since we know
      how to generate it from its definition. \refdef{intended normal
      form definition}.

      Fortunately, we can also prove the complement (which is
      equivalent, since $A \subseteq B \Leftrightarrow \overline{B}
      \subseteq \overline{A}$): Show that the set of nodes not in
      intended normal form is a subset of the set of nodes not in normal
      form. In other words, show that for every expression that is not
      in intended normal form, that there is at least one transformation
      that applies to it (since that means it is not in normal form
      either and since $A \subseteq C \Leftrightarrow \forall x (x \in A
      \rightarrow x \in C)$).

      By systematically reviewing the entire Core language definition
      along with the intended normal form definition (both of which have
      a similar structure), it should be possible to identify all
      possible (sets of) core expressions that are not in intended
      normal form and identify a transformation that applies to it.
      
      This approach is especially useful for proving completeness of our
      system, since if expressions exist to which none of the
      transformations apply (\ie if the system is not yet complete), it
      is immediately clear which expressions these are and adding
      (or modifying) transformations to fix this should be relatively
      easy.

      As observed above, applying this approach is a lot of work, since
      we need to check every (set of) transformation(s) separately.

      \todo{Perhaps do a few steps of the proofs as proof-of-concept}

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