[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
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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=]
      \typebuffer[#1]
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  \define[3]\transexample{
    \placeexample[here]{#1}
    \startcombination[2*1]
      {\example{#2}}{Original program}
      {\example{#3}}{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 \small{VHDL}, and describe how the
  \small{VHDL} we want to generate should look like.

  \section{Normal form}
    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,
      not expressions). To make the \small{VHDL} generation easy, all values must be bound
      on the \quote{top level}.
    \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 top. This means that
    the body of the final lambda abstraction is never a function, but always a
    plain value.

    After the lambda abstractions, we see a single 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.

    \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 an adder 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 component and
    connection. 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 the same as in \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 replace
    them with some logic in the clock inputs, but we want to show the architecture
    as close to the description as possible.

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

      -- All arguments are an inital lambda
      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 (fst, snd) -> fst
        r2 = case s of (fst, snd) -> snd
        -- 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

    \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{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.

      \startlambda
      \italic{normal} = \italic{lambda}
      \italic{lambda} = λvar.\italic{lambda} (representable(var))
                      | \italic{toplet} 
      \italic{toplet} = let \italic{binding} in \italic{toplet} 
                      | letrec [\italic{binding}] in \italic{toplet}
                      | var (representable(varvar))
      \italic{binding} = var = \italic{rhs} (representable(rhs))
                       -- State packing and unpacking by coercion
                       | var0 = var1 :: State ty (lvar(var1))
                       | var0 = var1 :: ty (var0 :: 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 -> var0 (lvar(var0))
                      C w0 ... wn -> resvar (\forall{}i, wi \neq resvar, lvar(resvar))
      \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} = \italic{coreexpr}
      \stoplambda

      -- TODO: Limit builtinarg further

      -- TODO: There can still be other casts around (which the code can handle,
      e.g., ignore), which still need to be documented here.

      -- TODO: Note about the selector case. It just supports Bit and Bool
      currently, perhaps it should be generalized in the normal form?

      When looking at such a program from a hardware perspective, the top level
      lambda's define the input ports. The value produced by the 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} statement) or call a builtin function
      (\eg \lam{add} or \lam{sub}). For these, a hardcoded \small{VHDL} translation is
      available.

  \section{Transformation notation}
    To be able to concisely present transformations, we use a specific format to
    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 latter (\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)} (and bind \lam{a} to \lam{v} and \lam{B} to 
      \lam{(2 * 2)}), but not \lam{v + (2 * w)}.
      \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, in particular to prevent a transformation from
      causing a loop with itself or another transformation.

      Only if these if these conditions are \emph{all} true, this 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, this
      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 functiosn.

      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

    Consider the following function, which is a fairly obvious way to specify a
    simple ALU (Note \at{example}[ex:AddSubAlu] is the normal form of this
    function):

    \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
    labmda 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. 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, and we'll leave both alternatives as an exercise to the
    reader. The final function, after all these transformations becomes:

    \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

    In this case, the transformation does not apply anymore, though this might
    not always be the case (e.g., the application of a transformation on a
    subexpression might open up possibilities to apply the transformation
    further up in the expression).

    \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 (see TODO
      ref), 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. 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.

      \subsubsection{Other concepts}
        A \emph{global variable} is any variable 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. Types 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{Functions}
        Here, we define a number of functions 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{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 below) 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, we can observe the following
      points.

      \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) 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
      meanst 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 pose any conflict, but it does ensure that
      all binders within the function are generated by the same unique supply. See
      (TODO: ref 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 (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. Here we're using
    the core language in a notation that resembles lambda calculus.

    Each of these transforms is meant to be applied to every (sub)expression
    in a program, for as long as it applies. Only when none of the
    transformations can be applied anymore, the program is in normal form (by
    definition). We hope to be able to prove that this form will obey all of the
    constraints defined above, but this has yet to happen (though it seems likely
    that it will).

    Each of the transforms will be described informally first, explaining
    the need for and goal of the transform. Then, a formal definition is
    given, using a familiar syntax from the world of logic. Each transform
    is specified as a number of conditions (above the horizontal line) and a
    number of conclusions (below the horizontal line). The details of using
    this notation are still a bit fuzzy, so comments are welcom.

    \subsection{General cleanup}

      \subsubsection{β-reduction}
        β-reduction is a well known transformation from lambda calculus, where it is
        the main reduction step. It reduces applications of labmda 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.

        TODO: Define substitution syntax

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

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

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

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

      \subsubsection{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
        possibilities for other transformations (like β-reduction).

        \starttrans
        let binds in E) M
        -----------------
        let binds in E M
        \stoptrans

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

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

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

      \subsubsection{Empty let removal}
        This transformation is simple: It removes recursive lets that have no bindings
        (which usually occurs when let derecursification removes the last binding from
        it).

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

        \subsubsection{Simple let binding removal}
        This transformation inlines simple let bindings (\eg a = b).

        This transformation is not needed to get into normal form, but makes the
        resulting \small{VHDL} a lot shorter.

        \starttrans
        letnonrec
          a = b
        in
          M
        -----------------
        M[b/a]
        \stoptrans

        \starttrans
        letrec
          \vdots
          a = b
          \vdots
        in
          M
        -----------------
        let
          \vdots [b/a]
          \vdots [b/a]
        in
          M[b/a]
        \stoptrans

      \subsubsection{Unused let binding removal}
        This transformation removes let bindings that are never used. Usually,
        the desugarer introduces some unused let bindings.

        This normalization pass should really be unneeded to get into normal form
        (since ununsed 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.

        \starttrans
        let a = E in M
        ----------------------------    \lam{a} does not occur free in \lam{M}
        M
        \stoptrans

        \starttrans
        letrec
          \vdots
          a = E
          \vdots
        in
          M
        ----------------------------    \lam{a} does not occur free in \lam{M}
        letrec
          \vdots
          \vdots
        in
          M
        \stoptrans

      \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.

      \subsubsection{Compiler generated top level binding inlining}
        TODO

    \section{Program structure}

      \subsubsection{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 function inlining below, 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{η-abstraction}{from}{to}

      \subsubsection{Let derecursification}
        This transformation is meant to make lets non-recursive whenever possible.
        This might allow other optimizations to do their work better. TODO: Why is
        this needed exactly?

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

        Note that this transformation does not try to be smart when faced with
        recursive lets, it will just leave the lets recursive (possibly joining a
        recursive and non-recursive let into a single recursive let). The let
        rederecursification transformation will do this instead.

        \starttrans
        letnonrec x = (let bindings in M) in N
        ------------------------------------------
        let bindings in (letnonrec x = M) in N
        \stoptrans

        \starttrans
        letrec 
          \vdots
          x = (let bindings in M)
          \vdots
        in
          N
        ------------------------------------------
        letrec
          \vdots
          bindings
          x = M
          \vdots
        in
          N
        \stoptrans

        \startbuffer[from]
        let
          a = letrec
            x = 1
            y = 2
          in
            x + y
        in
          letrec
            b = let c = 3 in a + c
            d = 4
          in
            d + b
        \stopbuffer
        \startbuffer[to]
        letrec
          x = 1
          y = 2
        in
          let
            a = x + y
          in
            letrec
              c = 3
              b = a + c
              d = 4
            in
              d + b
        \stopbuffer

        \transexample{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).

        \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
        let 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
        let 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
        ---------------------------
        let x = E in x
        \stoptrans

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

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

        \transexample{Return value simplification}{from}{to}

    \subsection{Argument simplification}
      The transforms in this section deal with simplifying application
      arguments into normal form. The goal here is to:

      \startitemize
       \item Make all arguments of user-defined functions (\eg, of which
       we have a function body) simple variable references of a runtime
       representable type. This is needed, since these applications will be turned
       into component instantiations.
       \item Make all arguments of builtin functions one of:
         \startitemize
          \item A type argument.
          \item A dictionary argument.
          \item A type level expression.
          \item A variable reference of a runtime representable type.
          \item A variable reference or partial application of a function type.
         \stopitemize
      \stopitemize

      When looking at the arguments of a user-defined function, we can
      divide them into two categories:
      \startitemize
        \item Arguments of a runtime representable type (\eg bits or vectors).

              These arguments can be preserved in the program, since they can
              be translated to input ports later on.  However, since we can
              only connect signals to input ports, these arguments must be
              reduced to simple variables (for which signals will be
              produced). This is taken care of by the argument extraction
              transform.
        \item Non-runtime representable typed arguments.
              
              These arguments cannot be preserved in the program, since we
              cannot represent them as input or output ports in the resulting
              \small{VHDL}. To remove them, we create a specialized version of the
              called function with these arguments filled in. This is done by
              the argument propagation transform.

              Typically, these arguments are type and dictionary arguments that are
              used to make functions polymorphic. By propagating these arguments, we
              are essentially doing the same which GHC does when it specializes
              functions: Creating multiple variants of the same function, one for
              each type for which it is used. Other common non-representable
              arguments are functions, e.g. when calling a higher order function
              with another function or a lambda abstraction as an argument.

              The reason for doing this is similar to the reasoning provided for
              the inlining of non-representable let bindings above. In fact, this
              argument propagation could be viewed as a form of cross-function
              inlining.
      \stopitemize

      TODO: Check the following itemization.

      When looking at the arguments of a builtin function, we can divide them
      into categories: 

      \startitemize
        \item Arguments of a runtime representable type.
              
              As we have seen with user-defined functions, these arguments can
              always be reduced to a simple variable reference, by the
              argument extraction transform. Performing this transform for
              builtin functions as well, means that the translation of builtin
              functions can be limited to signal references, instead of
              needing to support all possible expressions.

        \item Arguments of a function type.
              
              These arguments are functions passed to higher order builtins,
              like \lam{map} and \lam{foldl}. Since implementing these
              functions for arbitrary function-typed expressions (\eg, lambda
              expressions) is rather comlex, we reduce these arguments to
              (partial applications of) global functions.
              
              We can still support arbitrary expressions from the user code,
              by creating a new global function containing that expression.
              This way, we can simply replace the argument with a reference to
              that new function. However, since the expression can contain any
              number of free variables we also have to include partial
              applications in our normal form.

              This category of arguments is handled by the function extraction
              transform.
        \item Other unrepresentable arguments.
              
              These arguments can take a few different forms:
              \startdesc{Type arguments}
                In the core language, type arguments can only take a single
                form: A type wrapped in the Type constructor. Also, there is
                nothing that can be done with type expressions, except for
                applying functions to them, so we can simply leave type
                arguments as they are.
              \stopdesc
              \startdesc{Dictionary arguments}
                In the core language, dictionary arguments are used to find
                operations operating on one of the type arguments (mostly for
                finding class methods). Since we will not actually evaluatie
                the function body for builtin functions and can generate
                code for builtin functions by just looking at the type
                arguments, these arguments can be ignored and left as they
                are.
              \stopdesc
              \startdesc{Type level arguments}
                Sometimes, we want to pass a value to a builtin function, but
                we need to know the value at compile time. Additionally, the
                value has an impact on the type of the function. This is
                encoded using type-level values, where the actual value of the
                argument is not important, but the type encodes some integer,
                for example. Since the value is not important, the actual form
                of the expression does not matter either and we can leave
                these arguments as they are.
              \stopdesc
              \startdesc{Other arguments}
                Technically, there is still a wide array of arguments that can
                be passed, but does not fall into any of the above categories.
                However, none of the supported builtin functions requires such
                an argument. This leaves use with passing unsupported types to
                a function, such as calling \lam{head} on a list of functions.

                In these cases, it would be impossible to generate hardware
                for such a function call anyway, so we can ignore these
                arguments.

                The only way to generate hardware for builtin functions with
                arguments like these, is to expand the function call into an
                equivalent core expression (\eg, expand map into a series of
                function applications). But for now, we choose to simply not
                support expressions like these.
              \stopdesc

              From the above, we can conclude that we can simply ignore these
              other unrepresentable arguments and focus on the first two
              categories instead.
      \stopitemize

      \subsubsection{Argument simplification}
        This transform deals with arguments to functions that
        are of a runtime representable type. It ensures that they will all become
        references to global variables, or local signals in the resulting \small{VHDL}. 

        TODO: It seems we can map an expression to a port, not only a signal.
        Perhaps this makes this transformation not needed?
        TODO: Say something about dataconstructors (without arguments, like True
        or False), which are variable references of a runtime representable
        type, but do not result in a signal.

        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.

        \starttrans
        M N
        --------------------    \lam{N} is of a representable type
        let x = N in M x        \lam{N} is not a local variable reference
        \stoptrans

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

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

        \transexample{Argument extraction}{from}{to}

      \subsubsection{Function extraction}
        This transform deals with function-typed arguments to builtin functions.
        Since these arguments cannot be propagated, we choose to extract them
        into a new global function instead.

        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} = 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]
        map (λa . add a b) xs

        map (add b) ys
        \stopbuffer

        \startbuffer[to]
        x0 = λb.λa.add a b
        ~
        map x0 xs

        x1 = λb.add b
        map x1 ys
        \stopbuffer

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

      \subsubsection{Argument propagation}
        This transform deals with arguments to user-defined functions that are
        not representable at runtime. This means these arguments cannot be
        preserved in the final form and most be {\em propagated}.

        Propagation means to create a specialized version of the called
        function, with the propagated argument already filled in. As a simple
        example, in the following program:

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

        we could {\em propagate} the constant argument 1, with the following
        result:

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

        Special care must be taken when the to-be-propagated expression has any
        free variables. If this is the case, the original argument should not be
        removed alltogether, but replaced by all the free variables of the
        expression. In this way, the original expression can still be evaluated
        inside the new function. Also, this brings us closer to our goal: All
        these free variables will be simple variable references.

        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 transformation to replace such a variable with an expression we
        can propagate again.

        \starttrans
        x = E
        ~
        x Y0 ... Yi ... Yn                               \lam{Yi} is not of a runtime representable type
        ---------------------------------------------    \lam{Yi} is not a local variable reference
        x' y0 ... yi-1 f0 ...  fm Yi+1 ... Yn            \lam{f0 ... fm} = free local vars of \lam{Yi}
        ~
        x' = λy0 ... yi-1 f0 ... fm yi+1 ... yn .       
              E y0 ... yi-1 Yi yi+1 ... yn   

        \stoptrans

        TODO: Example

    \subsection{Case simplification}
      \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
        let x = E in 
          case E of
            alts
        \stoptrans

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

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

        \transexample{Let flattening}{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

        \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)
        letnonrec
          v0,0 = case x of C0 v0,0 .. v0,m -> v0,0
          \vdots
          v0,m = case x of C0 v0,0 .. v0,m -> v0,m
          x0 = E0
          \dots
          vn,m = case x of Cn vn,0 .. vn,m -> vn,m
          xn = En
        in
          case E of
            C0 w0,0 ... w0,m -> x0
            \vdots
            Cn wn,0 ... wn,m -> xn
        \stoptrans

        TODO: This transformation specified like this is complicated and misses
        conditions to prevent looping with itself. Perhaps we should split it 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{Selector case simplification}{from}{to}

        \startbuffer[from]
        case a of
          (,) b c -> add b c
        \stopbuffer
        \startbuffer[to]
        letnonrec
          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{Extractor case simplification}{from}{to}

      \subsubsection{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{Case removal}{from}{to}

  \subsection{Monomorphisation}
    TODO: Better name for this section

    Reference type-specialization (== argument propagation)

  \subsubsection{Defunctionalization}
    Reference higher-order-specialization (== argument propagation)

      \subsubsection{Non-representable binding inlining}
        This transform inlines let bindings that have a non-representable type. 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.

        If the binding is non-representable because it is a lambda abstraction, it is
        likely that it will inlined into an application and β-reduction will remove
        the lambda abstraction and turn it into a representable expression at the
        inline site. The same holds for partial applications, which can be turned into
        full applications by inlining.

        Other cases of non-representable bindings we see in practice are primitive
        Haskell types. In most cases, these will not result in a valid normalized
        output, but then the input would have been invalid to start with. There is one
        exception to this: When a builtin function is applied to a non-representable
        expression, things might work out in some cases. For example, when you write a
        literal \hs{SizedInt} in Haskell, like \hs{1 :: SizedInt D8}, this results in
        the following core: \lam{fromInteger (smallInteger 10)}, where for example
        \lam{10 :: GHC.Prim.Int\#} and \lam{smallInteger 10 :: Integer} have
        non-representable types. TODO: This/these paragraph(s) should probably become a
        separate discussion somewhere else.

        \starttrans
        letnonrec a = E in M
        --------------------------    \lam{E} has a non-representable type.
        M[E/a]
        \stoptrans

        \starttrans
        letrec 
          \vdots
          a = E
          \vdots
        in
          M
        --------------------------    \lam{E} has a non-representable type.
        letrec
          \vdots [E/a]
          \vdots [E/a]
        in
          M[E/a]
        \stoptrans

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

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

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


  \section{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. One transformation produces a result that
    is transformed back to the original by another transformation, for example.
    \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

    \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 labmda 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-reduction and η-reduction (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}.

      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).
        \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.
        \item A system is deterministic if all paths from a 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, 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 normal form, it must obviously be
      \emph{deterministic} as well.

    \subsection{Termination}
      Approach: Counting.

      Church-Rosser?

    \subsection{Soundness}
      Needs formal definition of semantics.
      Prove for each transformation seperately, implies soundness of the system.
     
    \subsection{Completeness}
      Show that any transformation applies to every Core expression that is not
      in normal form. To prove: no transformation applies => in intended form.
      Show the reverse: Not in intended form => transformation applies.

    \subsection{Determinism}
      How to prove this?