Remove an unused (and commented) macro.

[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
% will end up on a single line. The strut=no option prevents a bunch of empty
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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]
    \setuptyping[option=none,style=\tttf]
  }
}

\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?