-}
-
-
-% A transformation example
-\definefloat[example][examples]
-\setupcaption[example][location=top] % Put captions on top
-
-\define[3]\transexample{
- \placeexample[here]{#1}
- \startcombination[2*1]
- {\example{#2}}{Original program}
- {\example{#3}}{Transformed program}
- \stopcombination
-}
-%
-%\define[3]\transexampleh{
-%% \placeexample[here]{#1}
-%% \startcombination[1*2]
-%% {\example{#2}}{Original program}
-%% {\example{#3}}{Transformed program}
-%% \stopcombination
-%}
-
-The first step in the core to VHDL translation process, is normalization. We
-aim to bring the core description into a simpler form, which we can
-subsequently translate into VHDL easily. This normal form is needed because
-the full core language is more expressive than VHDL in some areas and because
-core can describe expressions that do not have a direct hardware
-interpretation.
-
-TODO: Describe core properties not supported in VHDL, and describe how the
-VHDL we want to generate should look like.
-
-\section{Goal}
-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.
-
-This {\em normal form} is again a Core program, but with a very specific
-structure. A function in normal form has nested lambda's at the top, which
-produce a let expression. This let expression binds every function application
-in the function and produces a simple identifier. Every bound value in
-the let expression is either a simple function application or a case
-expression to extract a single element from a tuple returned by a
-function.
-
-An example of a program in canonical form would be:
-
-\startlambda
- -- All arguments are an inital lambda
- λa.λd.λsp.
- -- There are nested let expressions at top level
- let
- -- Unpack the state by coercion
- 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'
- -- Packing the state by coercion
- sp' = s' :: State (Word, Word)
- -- Pack our return value
- res = (,) sp' out
- in
- -- The actual result
- res
-\stoplambda
-
-\startlambda
-\italic{normal} = \italic{lambda}
-\italic{lambda} = λvar.\italic{lambda} (representable(typeof(var)))
- | \italic{toplet}
-\italic{toplet} = let \italic{binding} in \italic{toplet}
- | letrec [\italic{binding}] in \italic{toplet}
- | var (representable(typeof(var)), fvar(var))
-\italic{binding} = var = \italic{rhs} (representable(typeof(rhs)))
- -- State packing and unpacking by coercion
- | var0 = var1 :: State ty (fvar(var1))
- | var0 = var1 :: ty (var0 :: State ty) (fvar(var1))
-\italic{rhs} = userapp
- | builtinapp
- -- Extractor case
- | case var of C a0 ... an -> ai (fvar(var))
- -- Selector case
- | case var of (fvar(var))
- DEFAULT -> var0 (fvar(var0))
- C w0 ... wn -> resvar (\forall{}i, wi \neq resvar, fvar(resvar))
-\italic{userapp} = \italic{userfunc}
- | \italic{userapp} {userarg}
-\italic{userfunc} = var (tvar(var))
-\italic{userarg} = var (fvar(var))
-\italic{builtinapp} = \italic{builtinfunc}
- | \italic{builtinapp} \italic{builtinarg}
-\italic{builtinfunc} = var (bvar(var))
-\italic{builtinarg} = \italic{coreexpr}
-\stoplambda
-
--- TODO: Define tvar, fvar, typeof, representable
--- 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 VHDL translation is
-available.
-
-\subsection{Normal definition}
-Formally, the normal form is a core program obeying the following
-constraints. TODO: Update this section, this is probably not completely
-accurate or relevant anymore.
-
-\startitemize[R,inmargin]
-%\item All top level binds must have the form $\expr{\bind{fun}{lamexpr}}$.
-%$fun$ is an identifier that will be bound as a global identifier.
-%\item A $lamexpr$ has the form $\expr{\lam{arg}{lamexpr}}$ or
-%$\expr{letexpr}$. $arg$ is an identifier which will be bound as an $argument$.
-%\item[letexpr] A $letexpr$ has the form $\expr{\letexpr{letbinds}{retexpr}}$.
-%\item $letbinds$ is a list with elements of the form
-%$\expr{\bind{res}{appexpr}}$ or $\expr{\bind{res}{builtinexpr}}$, where $res$ is
-%an identifier that will be bound as local identifier. The type of the bound
-%value must be a $hardware\;type$.
-%\item[builtinexpr] A $builtinexpr$ is an expression that can be mapped to an
-%equivalent VHDL expression. Since there are many supported forms for this,
-%these are defined in a separate table.
-%\item An $appexpr$ has the form $\expr{fun}$ or $\expr{\app{appexpr}{x}}$,
-%where $fun$ is a global identifier and $x$ is a local identifier.
-%\item[retexpr] A $retexpr$ has the form $\expr{x}$ or $\expr{tupexpr}$, where $x$ is a local identifier that is bound as an $argument$ or $result$. A $retexpr$ must
-%be of a $hardware\;type$.
-%\item A $tupexpr$ has the form $\expr{con}$ or $\expr{\app{tupexpr}{x}}$,
-%where $con$ is a tuple constructor ({\em e.g.} $(,)$ or $(,,,)$) and $x$ is
-%a local identifier.
-%\item A $hardware\;type$ is a type that can be directly translated to
-%hardware. This includes the types $Bit$, $SizedWord$, tuples containing
-%elements of $hardware\;type$s, and will include others. This explicitely
-%excludes function types.
-\stopitemize
-
-TODO: Say something about uniqueness of identifiers
-
-\subsection{Builtin expressions}
-A $builtinexpr$, as defined at \in[builtinexpr] can have any of the following forms.
-
-\startitemize[m,inmargin]
-%\item
-%$tuple\_extract=\expr{\case{t}{\alt{\app{con}{x_0\;x_1\;..\;x_n}}{x_i}}}$,
-%where $t$ can be any local identifier, $con$ is a tuple constructor ({\em
-%e.g.} $(,)$ or $(,,,)$), $x_0$ to $x_n$ can be any identifier, and $x_i$ can
-%be any of $x_0$ to $x_n$. A case expression must have a $hardware\;type$.
-%\item TODO: Many more!
-\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
-expressions can be applied anymore, the program is in normal form. 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.
-
-TODO: Formally describe the "apply to every (sub)expression" in terms of
-rules with full transformations in the conditions.
-
-\subsection{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 :: * -> *}
--------------- \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}
-
-\subsection{Extended β-reduction}
-This transformation is meant to propagate application expressions downwards
-into expressions as far as possible. In lambda calculus, this reduction
-is known as β-reduction, but it is of course only defined for
-applications of lambda abstractions. We extend this reduction to also
-work for the rest of core (case and let expressions).
-
-For let expressions:
-\starttrans
-let binds in E) M
------------------
-let binds in E M
-\stoptrans
-
-For case statements:
-\starttrans
-(case x of
- p1 -> E1
- \vdots
- pn -> En) M
------------------
-case x of
- p1 -> E1 M
- \vdots
- pn -> En M
-\stoptrans
-
-For lambda expressions:
-\starttrans
-(λx.E) M
------------------
-E[M/x]
-\stoptrans
-
-% And an example
-\startbuffer[from]
-( let a = (case x of
+
+ \define[4]\transexample{
+ \placeexample[here][ex:trans:#1]{#2}
+ \startcombination[2*1]
+ {\example{#3}}{Original program}
+ {\example{#4}}{Transformed program}
+ \stopcombination
+ }
+
+ The first step in the core to \small{VHDL} translation process, is normalization. We
+ aim to bring the core description into a simpler form, which we can
+ subsequently translate into \small{VHDL} easily. This normal form is needed because
+ the full core language is more expressive than \small{VHDL} in some
+ areas (higher-order expressions, limited polymorphism using type
+ classes, etc.) and because core can describe expressions that do not
+ have a direct hardware interpretation.
+
+ \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
+ \VHDL, 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{}{\startboxed A program is in \emph{normal form} if none of the
+ transformations from this chapter apply.\stopboxed}
+
+ 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 us try to get a feeling for it first. The easiest way to do this
+ is by describing the things that are unwanted in the intended normal form.
+
+ \startitemize
+ \item Any \emph{polymorphism} must be removed. When laying down hardware, we
+ cannot generate any signals that can have multiple types. All types must be
+ completely known to generate hardware.
+
+ \item All \emph{higher-order} constructions must be removed. We cannot
+ generate a hardware signal that contains a function, so all values,
+ arguments and return values used must be first order.
+
+ \item All complex \emph{nested scopes} must be removed. In the \small{VHDL}
+ description, every signal is in a single scope. Also, full expressions are
+ not supported everywhere (in particular port maps can only map signal
+ names and constants, not complete expressions). To make the \small{VHDL}
+ generation easy, a separate binder must be bound to ever application or
+ other expression.
+ \stopitemize
+
+ \todo{Intermezzo: functions vs plain values}
+
+ A very simple example of a program in normal form is given in
+ \in{example}[ex:MulSum]. As you can see, all arguments to the function (which
+ will become input ports in the generated \VHDL) are at the outer level.
+ This means that the body of the inner lambda abstraction is never a
+ function, but always a plain value.
+
+ As the body of the inner lambda abstraction, we see a single (recursive)
+ let expression, that binds two variables (\lam{mul} and \lam{sum}). These
+ variables will be signals in the generated \VHDL, bound to the output port
+ of the \lam{*} and \lam{+} components.
+
+ The final line (the \quote{return value} of the function) selects the
+ \lam{sum} signal to be the output port of the function. This \quote{return
+ value} can always only be a variable reference, never a more complex
+ expression.
+
+ \todo{Add generated VHDL}
+
+ \startbuffer[MulSum]
+ alu :: Bit -> Word -> Word -> Word
+ alu = λa.λb.λc.
+ let
+ mul = (*) a b
+ sum = (+) mul c
+ in
+ sum
+ \stopbuffer
+
+ \startuseMPgraphic{MulSum}
+ save a, b, c, mul, add, sum;
+
+ % I/O ports
+ newCircle.a(btex $a$ etex) "framed(false)";
+ newCircle.b(btex $b$ etex) "framed(false)";
+ newCircle.c(btex $c$ etex) "framed(false)";
+ newCircle.sum(btex $sum$ etex) "framed(false)";
+
+ % Components
+ newCircle.mul(btex * etex);
+ newCircle.add(btex + etex);
+
+ a.c - b.c = (0cm, 2cm);
+ b.c - c.c = (0cm, 2cm);
+ add.c = c.c + (2cm, 0cm);
+ mul.c = midpoint(a.c, b.c) + (2cm, 0cm);
+ sum.c = add.c + (2cm, 0cm);
+ c.c = origin;
+
+ % Draw objects and lines
+ drawObj(a, b, c, mul, add, sum);
+
+ ncarc(a)(mul) "arcangle(15)";
+ ncarc(b)(mul) "arcangle(-15)";
+ ncline(c)(add);
+ ncline(mul)(add);
+ ncline(add)(sum);
+ \stopuseMPgraphic
+
+ \placeexample[here][ex:MulSum]{Simple architecture consisting of a
+ multiplier and a subtractor.}
+ \startcombination[2*1]
+ {\typebufferlam{MulSum}}{Core description in normal form.}
+ {\boxedgraphic{MulSum}}{The architecture described by the normal form.}
+ \stopcombination
+
+ \in{Example}[ex:MulSum] showed a function that just applied two
+ other functions (multiplication and addition), resulting in a simple
+ architecture with two components and some connections. There is of
+ course also some mechanism for choice in the normal form. In a
+ normal Core program, the \emph{case} expression can be used in a few
+ different ways to describe choice. In normal form, this is limited
+ to a very specific form.
+
+ \in{Example}[ex:AddSubAlu] shows an example describing a
+ simple \small{ALU}, which chooses between two operations based on an opcode
+ bit. The main structure is similar to \in{example}[ex:MulSum], but this
+ time the \lam{res} variable is bound to a case expression. This case
+ expression scrutinizes the variable \lam{opcode} (and scrutinizing more
+ complex expressions is not supported). The case expression can select a
+ different variable based on the constructor of \lam{opcode}.
+ \refdef{case expression}
+
+ \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 shows everything that
+ is allowed in normal form, except for built-in higher-order functions
+ (like \lam{map}). The graphical version of the architecture contains
+ a slightly simplified version, since the state tuple packing and
+ unpacking have been left out. Instead, two separate registers are
+ drawn. Also note that most synthesis tools will further optimize
+ this architecture by removing the multiplexers at the register input
+ and instead put some gates in front of the register's clock input,
+ but we want to show the architecture as close to the description as
+ possible.
+
+ As you can see from the previous examples, the generation of the final
+ architecture from the normal form is straightforward. In each of the
+ examples, there is a direct match between the normal form structure,
+ the generated VHDL and the architecture shown in the images.
+
+ \startbuffer[NormalComplete]
+ regbank :: Bit
+ -> Word
+ -> State (Word, Word)
+ -> (State (Word, Word), Word)
+
+ -- All arguments are an inital lambda (address, data, packed state)
+ regbank = λa.λd.λsp.
+ -- There are nested let expressions at top level
+ let
+ -- Unpack the state by coercion (\eg, cast from
+ -- State (Word, Word) to (Word, Word))
+ s = sp ▶ (Word, Word)
+ -- Extract both registers from the state
+ r1 = case s of (a, b) -> a
+ r2 = case s of (a, b) -> b
+ -- Calling some other user-defined function.
+ d' = foo d
+ -- Conditional connections
+ out = case a of
+ High -> r1
+ Low -> r2
+ r1' = case a of
+ High -> d'
+ Low -> r1
+ r2' = case a of
+ High -> r2
+ Low -> d'
+ -- Packing a tuple
+ s' = (,) r1' r2'
+ -- pack the state by coercion (\eg, cast from
+ -- (Word, Word) to State (Word, Word))
+ sp' = s' ▶ State (Word, Word)
+ -- Pack our return value
+ res = (,) sp' out
+ in
+ -- The actual result
+ res
+ \stopbuffer
+
+ \startuseMPgraphic{NormalComplete}
+ save a, d, r, foo, muxr, muxout, out;
+
+ % I/O ports
+ newCircle.a(btex \lam{a} etex) "framed(false)";
+ newCircle.d(btex \lam{d} etex) "framed(false)";
+ newCircle.out(btex \lam{out} etex) "framed(false)";
+ % Components
+ %newCircle.add(btex + etex);
+ newBox.foo(btex \lam{foo} etex);
+ newReg.r1(btex $\lam{r1}$ etex) "dx(4mm)", "dy(6mm)";
+ newReg.r2(btex $\lam{r2}$ etex) "dx(4mm)", "dy(6mm)", "reflect(true)";
+ newMux.muxr1;
+ % Reflect over the vertical axis
+ reflectObj(muxr1)((0,0), (0,1));
+ newMux.muxr2;
+ newMux.muxout;
+ rotateObj(muxout)(-90);
+
+ d.c = foo.c + (0cm, 1.5cm);
+ a.c = (xpart r2.c + 2cm, ypart d.c - 0.5cm);
+ foo.c = midpoint(muxr1.c, muxr2.c) + (0cm, 2cm);
+ muxr1.c = r1.c + (0cm, 2cm);
+ muxr2.c = r2.c + (0cm, 2cm);
+ r2.c = r1.c + (4cm, 0cm);
+ r1.c = origin;
+ muxout.c = midpoint(r1.c, r2.c) - (0cm, 2cm);
+ out.c = muxout.c - (0cm, 1.5cm);
+
+ % % Draw objects and lines
+ drawObj(a, d, foo, r1, r2, muxr1, muxr2, muxout, out);
+
+ ncline(d)(foo);
+ nccurve(foo)(muxr1) "angleA(-90)", "posB(inpa)", "angleB(180)";
+ nccurve(foo)(muxr2) "angleA(-90)", "posB(inpb)", "angleB(0)";
+ nccurve(muxr1)(r1) "posA(out)", "angleA(180)", "posB(d)", "angleB(0)";
+ nccurve(r1)(muxr1) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(180)";
+ nccurve(muxr2)(r2) "posA(out)", "angleA(0)", "posB(d)", "angleB(180)";
+ nccurve(r2)(muxr2) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(0)";
+ nccurve(r1)(muxout) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(-90)";
+ nccurve(r2)(muxout) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(-90)";
+ % Connect port a
+ nccurve(a)(muxout) "angleA(-90)", "angleB(180)", "posB(sel)";
+ nccurve(a)(muxr1) "angleA(180)", "angleB(-90)", "posB(sel)";
+ nccurve(a)(muxr2) "angleA(180)", "angleB(-90)", "posB(sel)";
+ ncline(muxout)(out) "posA(out)";
+ \stopuseMPgraphic
+
+ \todo{Don't split registers in this image?}
+ \placeexample[here][ex:NormalComplete]{Simple architecture consisting of an adder and a
+ subtractor.}
+ \startcombination[2*1]
+ {\typebufferlam{NormalComplete}}{Core description in normal form.}
+ {\boxedgraphic{NormalComplete}}{The architecture described by the normal form.}
+ \stopcombination
+
+
+
+ \subsection[sec:normalization:intendednormalform]{Intended normal form definition}
+ Now we have some intuition for the normal form, we can describe how we want
+ the normal form to look like in a slightly more formal manner. The following
+ EBNF-like description captures most of the intended structure (and
+ generates a subset of \GHC's core format).
+
+ There are two things missing: Cast expressions are sometimes
+ allowed by the prototype, but not specified here and the below
+ definition allows uses of state that cannot be translated to \VHDL
+ properly. These two problems are discussed in
+ \in{section}[sec:normalization:castproblems] and
+ \in{section}[sec:normalization:stateproblems] respectively.
+
+ Some clauses have an expression listed behind them in parentheses.
+ These are conditions that need to apply to the clause. The
+ predicates used there (\lam{lvar()}, \lam{representable()},
+ \lam{gvar()}) will be defined in
+ \in{section}[sec:normalization:predicates].
+
+ An expression is in normal form if it matches the first
+ definition, \emph{normal}.
+
+ \todo{Fix indentation}
+ \startbuffer[IntendedNormal]
+ \italic{normal} := \italic{lambda}
+ \italic{lambda} := λvar.\italic{lambda} (representable(var))
+ | \italic{toplet}
+ \italic{toplet} := letrec [\italic{binding}...] in var (representable(var))
+ \italic{binding} := var = \italic{rhs} (representable(rhs))
+ -- State packing and unpacking by coercion
+ | var0 = var1 ▶ State ty (lvar(var1))
+ | var0 = var1 ▶ ty (var1 :: State ty ∧ lvar(var1))
+ \italic{rhs} := \italic{userapp}
+ | \italic{builtinapp}
+ -- Extractor case
+ | case var of C a0 ... an -> ai (lvar(var))
+ -- Selector case
+ | case var of (lvar(var))
+ [ DEFAULT -> var ] (lvar(var))
+ C0 w0,0 ... w0,n -> var0
+ \vdots
+ Cm wm,0 ... wm,n -> varm (\forall{}i \forall{}j, wi,j \neq vari, lvar(vari))
+ \italic{userapp} := \italic{userfunc}
+ | \italic{userapp} {userarg}
+ \italic{userfunc} := var (gvar(var))
+ \italic{userarg} := var (lvar(var))
+ \italic{builtinapp} := \italic{builtinfunc}
+ | \italic{builtinapp} \italic{builtinarg}
+ \italic{built-infunc} := var (bvar(var))
+ \italic{built-inarg} := var (representable(var) ∧ lvar(var))
+ | \italic{partapp} (partapp :: a -> b)
+ | \italic{coreexpr} (¬representable(coreexpr) ∧ ¬(coreexpr :: a -> b))
+ \italic{partapp} := \italic{userapp}
+ | \italic{builtinapp}
+ \stopbuffer
+
+ \placedefinition[][def:IntendedNormal]{Definition of the intended nnormal form using an \small{EBNF}-like syntax.}
+ {\defref{intended normal form definition}
+ \typebufferlam{IntendedNormal}}
+
+ When looking at such a program from a hardware perspective, the
+ top level lambda abstractions define the input ports. Lambda
+ abstractions cannot appear anywhere else. The variable reference
+ in the body of the recursive let expression is the output port.
+ Most function applications bound by the let expression define a
+ component instantiation, where the input and output ports are
+ mapped to local signals or arguments. Some of the others use a
+ built-in construction (\eg the \lam{case} expression) or call a
+ built-in function (\eg \lam{+} or \lam{map}). For these, a
+ hardcoded \small{VHDL} translation is available.
+
+ \section[sec:normalization:transformation]{Transformation notation}
+ To be able to concisely present transformations, we use a specific format
+ for them. It is a simple format, similar to one used in logic reasoning.
+
+ Such a transformation description looks like the following.
+
+ \starttrans
+ <context conditions>
+ ~
+ <original expression>
+ -------------------------- <expression conditions>
+ <transformed expression>
+ ~
+ <context additions>
+ \stoptrans
+
+ This format describes a transformation that applies to \lam{<original
+ expression>} and transforms it into \lam{<transformed expression>}, assuming
+ that all conditions are satisfied. In this format, there are a number of placeholders
+ in pointy brackets, most of which should be rather obvious in their meaning.
+ Nevertheless, we will more precisely specify their meaning below:
+
+ \startdesc{<original expression>} The expression pattern that will be matched
+ against (subexpressions of) the expression to be transformed. We call this a
+ pattern, because it can contain \emph{placeholders} (variables), which match
+ any expression or binder. Any such placeholder is said to be \emph{bound} to
+ the expression it matches. It is convention to use an uppercase letter (\eg
+ \lam{M} or \lam{E}) to refer to any expression (including a simple variable
+ reference) and lowercase letters (\eg \lam{v} or \lam{b}) to refer to
+ (references to) binders.
+
+ For example, the pattern \lam{a + B} will match the expression
+ \lam{v + (2 * w)} (binding \lam{a} to \lam{v} and \lam{B} to
+ \lam{(2 * w)}), but not \lam{(2 * w) + v}.
+ \stopdesc
+
+ \startdesc{<expression conditions>}
+ These are extra conditions on the expression that is matched. These
+ conditions can be used to further limit the cases in which the
+ transformation applies, commonly to prevent a transformation from
+ causing a loop with itself or another transformation.
+
+ Only if these conditions are \emph{all} satisfied, the transformation
+ applies.
+ \stopdesc
+
+ \startdesc{<context conditions>}
+ These are a number of extra conditions on the context of the function. In
+ particular, these conditions can require some (other) top level function to be
+ present, whose value matches the pattern given here. The format of each of
+ these conditions is: \lam{binder = <pattern>}.
+
+ Typically, the binder is some placeholder bound in the \lam{<original
+ expression>}, while the pattern contains some placeholders that are used in
+ the \lam{transformed expression}.
+
+ Only if a top level binder exists that matches each binder and pattern,
+ the transformation applies.
+ \stopdesc
+
+ \startdesc{<transformed expression>}
+ This is the expression template that is the result of the transformation. If, looking
+ at the above three items, the transformation applies, the \lam{<original
+ expression>} is completely replaced by 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 be added to the context.
+ This is a way to let a transformation create new top level
+ functions.
+
+ Each addition has the form \lam{binder = template}. As above, any
+ placeholder in the addition is replaced with the value bound to it, and any
+ binder placeholder that has no value bound to it yet will be bound to (and
+ replaced with) a fresh binder.
+ \stopdesc
+
+ To understand this notation better, the step by step application of
+ the η-abstraction transformation to a simple \small{ALU} will be
+ shown. Consider η-abstraction, which is a common transformation from
+ labmda calculus, described using above notation as follows:
+
+ \starttrans
+ E \lam{E :: a -> b}
+ -------------- \lam{E} does not occur on a function position in an application
+ λx.E x \lam{E} is not a lambda abstraction.
+ \stoptrans
+
+ η-abstraction is a well known transformation from lambda calculus. What
+ this transformation does, is take any expression that has a function type
+ and turn it into a lambda expression (giving an explicit name to the
+ argument). There are some extra conditions that ensure that this
+ transformation does not apply infinitely (which are not necessarily part
+ of the conventional definition of η-abstraction).
+
+ Consider the following function, in Core notation, which is a fairly obvious way to specify a
+ simple \small{ALU} (Note that it is not yet in normal form, but
+ \in{example}[ex:AddSubAlu] shows the normal form of this function).
+ The parentheses around the \lam{+} and \lam{-} operators are
+ commonly used in Haskell to show that the operators are used as
+ normal functions, instead of \emph{infix} operators (\eg, the
+ operators appear before their arguments, instead of in between).
+
+ \startlambda
+ alu :: Bit -> Word -> Word -> Word
+ alu = λopcode. case opcode of
+ Low -> (+)
+ High -> (-)
+ \stoplambda
+
+ There are a few subexpressions in this function to which we could possibly
+ apply the transformation. Since the pattern of the transformation is only
+ the placeholder \lam{E}, any expression will match that. Whether the
+ transformation applies to an expression is thus solely decided by the
+ conditions to the right of the transformation.
+
+ We will look at each expression in the function in a top down manner. The
+ first expression is the entire expression the function is bound to.
+
+ \startlambda
+ λopcode. case opcode of
+ Low -> (+)
+ High -> (-)
+ \stoplambda
+
+ As said, the expression pattern matches this. The type of this expression is
+ \lam{Bit -> Word -> Word -> Word}, which matches \lam{a -> b} (Note that in
+ this case \lam{a = Bit} and \lam{b = Word -> Word -> Word}).
+
+ Since this expression is at top level, it does not occur at a function
+ position of an application. However, The expression is a lambda abstraction,
+ so this transformation does not apply.
+
+ The next expression we could apply this transformation to, is the body of
+ the lambda abstraction:
+
+ \startlambda
+ case opcode of
+ Low -> (+)
+ High -> (-)
+ \stoplambda
+
+ The type of this expression is \lam{Word -> Word -> Word}, which again
+ matches \lam{a -> b}. The expression is the body of a lambda expression, so
+ it does not occur at a function position of an application. Finally, the
+ expression is not a lambda abstraction but a case expression, so all the
+ conditions match. There are no context conditions to match, so the
+ transformation applies.
+
+ By now, the placeholder \lam{E} is bound to the entire expression. The
+ placeholder \lam{x}, which occurs in the replacement template, is not bound
+ yet, so we need to generate a fresh binder for that. Let us use the binder
+ \lam{a}. This results in the following replacement expression:
+
+ \startlambda
+ λa.(case opcode of
+ Low -> (+)
+ High -> (-)) a
+ \stoplambda
+
+ Continuing with this expression, we see that the transformation does not
+ apply again (it is a lambda expression). Next we look at the body of this
+ lambda abstraction:
+
+ \startlambda
+ (case opcode of
+ Low -> (+)
+ High -> (-)) a
+ \stoplambda
+
+ Here, the transformation does apply, binding \lam{E} to the entire
+ expression (which has type \lam{Word -> Word}) and binding \lam{x}
+ to the fresh binder \lam{b}, resulting in the replacement:
+
+ \startlambda
+ λb.(case opcode of
+ Low -> (+)
+ High -> (-)) a b
+ \stoplambda
+
+ The transformation does not apply to this lambda abstraction, so we
+ look at its body. For brevity, we will put the case expression on one line from
+ now on.
+
+ \startlambda
+ (case opcode of Low -> (+); High -> (-)) a b
+ \stoplambda
+
+ The type of this expression is \lam{Word}, so it does not match \lam{a -> b}
+ and the transformation does not apply. Next, we have two options for the
+ next expression to look at: The function position and argument position of
+ the application. The expression in the argument position is \lam{b}, which
+ has type \lam{Word}, so the transformation does not apply. The expression in
+ the function position is:
+
+ \startlambda
+ (case opcode of Low -> (+); High -> (-)) a
+ \stoplambda
+
+ Obviously, the transformation does not apply here, since it occurs in
+ function position (which makes the second condition false). In the same
+ way the transformation does not apply to both components of this
+ expression (\lam{case opcode of Low -> (+); High -> (-)} and \lam{a}), so
+ we will skip to the components of the case expression: The scrutinee and
+ both alternatives. Since the opcode is not a function, it does not apply
+ here.
+
+ The first alternative is \lam{(+)}. This expression has a function type
+ (the operator still needs two arguments). It does not occur in function
+ position of an application and it is not a lambda expression, so the
+ transformation applies.
+
+ We look at the \lam{<original expression>} pattern, which is \lam{E}.
+ This means we bind \lam{E} to \lam{(+)}. We then replace the expression
+ with the \lam{<transformed expression>}, replacing all occurences of
+ \lam{E} with \lam{(+)}. In the \lam{<transformed expression>}, the This gives us the replacement expression:
+ \lam{λx.(+) x} (A lambda expression binding \lam{x}, with a body that
+ applies the addition operator to \lam{x}).
+
+ The complete function then becomes:
+ \startlambda
+ (case opcode of Low -> λa1.(+) a1; High -> (-)) a
+ \stoplambda
+
+ Now the transformation no longer applies to the complete first alternative
+ (since it is a lambda expression). It does not apply to the addition
+ operator again, since it is now in function position in an application. It
+ does, however, apply to the application of the addition operator, since
+ that is neither a lambda expression nor does it occur in function
+ position. This means after one more application of the transformation, the
+ function becomes:
+
+ \startlambda
+ (case opcode of Low -> λa1.λb1.(+) a1 b1; High -> (-)) a
+ \stoplambda
+
+ The other alternative is left as an exercise to the reader. The final
+ function, after applying η-abstraction until it does no longer apply is:
+
+ \startlambda
+ alu :: Bit -> Word -> Word -> Word
+ alu = λopcode.λa.b. (case opcode of
+ Low -> λa1.λb1 (+) a1 b1
+ High -> λa2.λb2 (-) a2 b2) a b
+ \stoplambda
+
+ \subsection{Transformation application}
+ In this chapter we define a number of transformations, but how will we apply
+ these? As stated before, our normal form is reached as soon as no
+ transformation applies anymore. This means our application strategy is to
+ simply apply any transformation that applies, and continuing to do that with
+ the result of each transformation.
+
+ In particular, we define no particular order of transformations. Since
+ transformation order should not influence the resulting normal form,
+ this leaves the implementation free to choose any application order that
+ results in an efficient implementation. Unfortunately this is not
+ entirely true for the current set of transformations. See
+ \in{section}[sec:normalization:non-determinism] for a discussion of this
+ problem.
+
+ When applying a single transformation, we try to apply it to every (sub)expression
+ in a function, not just the top level function body. This allows us to
+ keep the transformation descriptions concise and powerful.
+
+ \subsection{Definitions}
+ A \emph{global variable} is any variable (binder) that is bound at the
+ top level of a program, or an external module. A \emph{local variable} is any
+ other variable (\eg, variables local to a function, which can be bound by
+ lambda abstractions, let expressions and pattern matches of case
+ alternatives). This is a slightly different notion of global versus
+ local than what \small{GHC} uses internally, but for our purposes
+ the distinction \GHC makes is not useful.
+ \defref{global variable} \defref{local variable}
+
+ A \emph{hardware representable} (or just \emph{representable}) type or value
+ is (a value of) a type that we can generate a signal for in hardware. For
+ example, a bit, a vector of bits, a 32 bit unsigned word, etc. Values that are
+ not runtime representable notably include (but are not limited to): Types,
+ dictionaries, functions.
+ \defref{representable}
+
+ A \emph{built-in 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{built-in function} \defref{user-defined function}
+
+ For these functions, Cλash has a \emph{built-in 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[sec:normalization:predicates]{Predicates}
+ Here, we define a number of predicates that can be used below to concisely
+ specify conditions.
+
+ \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.
+
+ \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.
+
+ \emph{representable(expr)} is true when \emph{expr} is \emph{representable}.
+
+ \subsection[sec:normalization:uniq]{Binder uniqueness}
+ A common problem in transformation systems, is binder uniqueness. When not
+ considering this problem, it is easy to create transformations that mix up
+ bindings and cause name collisions. Take for example, the following core
+ expression:
+
+ \startlambda
+ (λa.λb.λc. a * b * c) x c
+ \stoplambda
+
+ By applying β-reduction (see \in{section}[sec:normalization:beta]) once,
+ we can simplify this expression to:
+
+ \startlambda
+ (λb.λc. x * b * c) c
+ \stoplambda
+
+ Now, we have replaced the \lam{a} binder with a reference to the \lam{x}
+ binder. No harm done here. But note that we see multiple occurences of the
+ \lam{c} binder. The first is a binding occurence, to which the second refers.
+ The last, however refers to \emph{another} instance of \lam{c}, which is
+ bound somewhere outside of this expression. Now, if we would apply beta
+ reduction without taking heed of binder uniqueness, we would get:
+
+ \startlambda
+ λc. x * c * c
+ \stoplambda
+
+ This is obviously not what was supposed to happen! The root of this problem is
+ the reuse of binders: Identical binders can be bound in different,
+ but overlapping scopes. Any variable reference in those
+ overlapping scopes then refers to the variable bound in the inner
+ (smallest) scope. There is not way to refer to the variable in the
+ outer scope. This effect is usually referred to as
+ \emph{shadowing}: When a binder is bound in a scope where the
+ binder already had a value, the inner binding is said to
+ \emph{shadow} the outer binding. 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}
+ can be accessed.
+
+ 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
+ does not use \small{GHC}'s subsitution code). Also, the following points
+ can be observed.
+
+ \startitemize
+ \item Deshadowing does not guarantee overall uniqueness. For example, the
+ following (slightly contrived) expression shows the identifier \lam{x} bound in
+ two seperate places (and to different values), even though no shadowing
+ occurs.
+
+ \startlambda
+ (let x = 1 in x) + (let x = 2 in x)
+ \stoplambda
+
+ \item In our normal form (and the resulting \small{VHDL}), all binders
+ (signals) within the same function (entity) will end up in the same
+ scope. To allow this, all binders within the same function should be
+ unique.
+
+ \item When we know that all binders in an expression are unique, moving around
+ or removing a subexpression will never cause any binder conflicts. If we have
+ some way to generate fresh binders, introducing new subexpressions will not
+ cause any problems either. The only way to cause conflicts is thus to
+ duplicate an existing subexpression.
+ \stopitemize
+
+ Given the above, our prototype maintains a unique binder invariant. This
+ means that in any given moment during normalization, all binders \emph{within
+ a single function} must be unique. To achieve this, we apply the following
+ technique.
+
+ \todo{Define fresh binders and unique supplies}
+
+ \startitemize
+ \item Before starting normalization, all binders in the function are made
+ unique. This is done by generating a fresh binder for every binder used. This
+ also replaces binders that did not cause any conflict, but it does ensure that
+ all binders within the function are generated by the same unique supply.
+ \refdef{fresh binder}
+ \item Whenever a new binder must be generated, we generate a fresh binder that
+ is guaranteed to be different from \emph{all binders generated so far}. This
+ can thus never introduce duplication and will maintain the invariant.
+ \item Whenever (a part of) an expression is duplicated (for example when
+ inlining), all binders in the expression are replaced with fresh binders
+ (using the same method as at the start of normalization). These fresh binders
+ can never introduce duplication, so this will maintain the invariant.
+ \item Whenever we move part of an expression around within the function, there
+ is no need to do anything special. There is obviously no way to introduce
+ duplication by moving expressions around. Since we know that each of the
+ binders is already unique, there is no way to introduce (incorrect) shadowing
+ either.
+ \stopitemize
+
+ \section{Transform passes}
+ In this section we describe the actual transforms.
+
+ Each transformation will be described informally first, explaining
+ the need for and goal of the transformation. Then, we will formally define
+ the transformation using the syntax introduced in
+ \in{section}[sec:normalization:transformation].
+
+ \subsection{General cleanup}
+ These transformations are general cleanup transformations, that aim to
+ make expressions simpler. These transformations usually clean up the
+ mess left behind by other transformations or clean up expressions to
+ expose new transformation opportunities for other transformations.
+
+ Most of these transformations are standard optimizations in other
+ compilers as well. However, in our compiler, most of these are not just
+ optimizations, but they are required to get our program into intended
+ normal form.
+
+ \placeintermezzo{}{
+ \defref{substitution notation}
+ \startframedtext[width=8cm,background=box,frame=no]
+ \startalignment[center]
+ {\tfa Substitution notation}
+ \stopalignment
+ \blank[medium]
+
+ In some of the transformations in this chapter, we need to perform
+ substitution on an expression. Substitution means replacing every
+ occurence of some expression (usually a variable reference) with
+ another expression.
+
+ There have been a lot of different notations used in literature for
+ specifying substitution. The notation that will be used in this report
+ is the following:
+
+ \startlambda
+ E[A=>B]
+ \stoplambda
+
+ This means expression \lam{E} with all occurences of \lam{A} replaced
+ with \lam{B}.
+ \stopframedtext
+ }
+
+ \subsubsection[sec:normalization:beta]{β-reduction}
+ β-reduction is a well known transformation from lambda calculus, where it is
+ the main reduction step. It reduces applications of lambda abstractions,
+ removing both the lambda abstraction and the application.
+
+ In our transformation system, this step helps to remove unwanted lambda
+ abstractions (basically all but the ones at the top level). Other
+ transformations (application propagation, non-representable inlining) make
+ sure that most lambda abstractions will eventually be reducable by
+ β-reduction.
+
+ Note that β-reduction also works on type lambda abstractions and type
+ applications as well. This means the substitution below also works on
+ type variables, in the case that the binder is a type variable and teh
+ expression applied to is a type.
+
+ \starttrans
+ (λx.E) M
+ -----------------
+ E[x=>M]
+ \stoptrans
+
+ % And an example
+ \startbuffer[from]
+ (λa. 2 * a) (2 * b)
+ \stopbuffer
+
+ \startbuffer[to]
+ 2 * (2 * b)
+ \stopbuffer
+
+ \transexample{beta}{β-reduction}{from}{to}
+
+ \startbuffer[from]
+ (λt.λa::t. a) @Int
+ \stopbuffer
+
+ \startbuffer[to]
+ (λa::Int. a)
+ \stopbuffer
+
+ \transexample{beta-type}{β-reduction for type abstractions}{from}{to}
+
+ \subsubsection{Unused let binding removal}
+ This transformation removes let bindings that are never used.
+ Occasionally, \GHC's desugarer introduces some unused let bindings.
+
+ This normalization pass should really be not be necessary to get
+ into intended normal form (since the intended normal form
+ definition \refdef{intended normal form definition} does not
+ require that every binding is used), but in practice the
+ desugarer or simplifier emits some bindings that cannot be
+ normalized (e.g., calls to a
+ \hs{Control.Exception.Base.patError}) but are not used anywhere
+ either. To prevent the \VHDL generation from breaking on these
+ artifacts, this transformation removes them.
+
+ \todo{Do not use old-style numerals in transformations}
+ \starttrans
+ letrec
+ a0 = E0
+ \vdots
+ ai = Ei
+ \vdots
+ an = En
+ in
+ M \lam{ai} does not occur free in \lam{M}
+ ---------------------------- \lam{\forall j, 0 ≤ j ≤ n, j ≠ i} (\lam{ai} does not occur free in \lam{Ej})
+ letrec
+ a0 = E0
+ \vdots
+ ai-1 = Ei-1
+ ai+1 = Ei+1
+ \vdots
+ an = En
+ in
+ M
+ \stoptrans
+
+ % And an example
+ \startbuffer[from]
+ let
+ x = 1
+ in
+ 2
+ \stopbuffer
+
+ \startbuffer[to]
+ let
+ in
+ 2
+ \stopbuffer
+
+ \transexample{unusedlet}{Unused let binding removal}{from}{to}
+
+ \subsubsection{Empty let removal}
+ This transformation is simple: It removes recursive lets that have no bindings
+ (which usually occurs when unused let binding removal removes the last
+ binding from it).
+
+ Note that there is no need to define this transformation for
+ non-recursive lets, since they always contain exactly one binding.
+
+ \starttrans
+ letrec in M
+ --------------
+ M
+ \stoptrans
+
+ % And an example
+ \startbuffer[from]
+ let
+ in
+ 2
+ \stopbuffer
+
+ \startbuffer[to]
+ 2
+ \stopbuffer
+
+ \transexample{emptylet}{Empty let removal}{from}{to}
+
+ \subsubsection[sec:normalization:simplelet]{Simple let binding removal}
+ This transformation inlines simple let bindings, that bind some
+ binder to some other binder instead of a more complex expression (\ie
+ a = b).
+
+ This transformation is not needed to get an expression into intended
+ normal form (since these bindings are part of the intended normal
+ form), but makes the resulting \small{VHDL} a lot shorter.
+
+ \refdef{substitution notation}
+ \starttrans
+ letrec
+ a0 = E0
+ \vdots
+ ai = b
+ \vdots
+ an = En
+ in
+ M
+ ----------------------------- \lam{b} is a variable reference
+ letrec \lam{ai} ≠ \lam{b}
+ a0 = E0 [ai=>b]
+ \vdots
+ ai-1 = Ei-1 [ai=>b]
+ ai+1 = Ei+1 [ai=>b]
+ \vdots
+ an = En [ai=>b]
+ in
+ M[ai=>b]
+ \stoptrans
+
+ \todo{example}
+
+ \subsubsection{Cast propagation / simplification}
+ This transform pushes casts down into the expression as far as
+ possible. This transformation has been added to make a few
+ specific corner cases work, but it is not clear yet if this
+ transformation handles cast expressions completely or in the
+ right way. See \in{section}[sec:normalization:castproblems].
+
+ \starttrans
+ (let binds in E) ▶ T
+ -------------------------
+ let binds in (E ▶ T)
+ \stoptrans
+
+ \starttrans
+ (case S of
+ p0 -> E0
+ \vdots
+ pn -> En
+ ) ▶ T
+ -------------------------
+ case S of
+ p0 -> E0 ▶ T
+ \vdots
+ pn -> En ▶ T
+ \stoptrans
+
+ \subsubsection{Top level binding inlining}
+ \refdef{top level binding}
+ This transform takes simple top level bindings generated by the
+ \small{GHC} compiler. \small{GHC} sometimes generates very simple
+ \quote{wrapper} bindings, which are bound to just a variable
+ reference, or contain just a (partial) function appliation with
+ the type and dictionary arguments filled in (such as the
+ \lam{(+)} in the example below).
+
+ Note that this transformation is completely optional. It is not
+ required to get any function into intended normal form, but it does help making
+ the resulting VHDL output easier to read (since it removes components
+ that do not add any real structure, but do hide away operations and
+ cause extra clutter).
+
+ This transform takes any top level binding generated by \GHC,
+ whose normalized form contains only a single let binding.
+
+ \starttrans
+ x = λa0 ... λan.let y = E in y
+ ~
+ x
+ -------------------------------------- \lam{x} is generated by the compiler
+ λa0 ... λan.let y = E in y
+ \stoptrans
+
+ \startbuffer[from]
+ (+) :: Word -> Word -> Word
+ (+) = GHC.Num.(+) @Word \$dNum
+ ~
+ (+) a b
+ \stopbuffer
+ \startbuffer[to]
+ GHC.Num.(+) @ Alu.Word \$dNum a b
+ \stopbuffer
+
+ \transexample{toplevelinline}{Top level binding inlining}{from}{to}
+
+ \in{Example}[ex:trans:toplevelinline] shows a typical application of
+ the addition operator generated by \GHC. The type and dictionary
+ arguments used here are described in
+ \in{Section}[section:prototype:polymorphism].
+
+ Without this transformation, there would be a \lam{(+)} entity
+ in the \VHDL which would just add its inputs. This generates a
+ lot of overhead in the \VHDL, which is particularly annoying
+ when browsing the generated RTL schematic (especially since most
+ non-alphanumerics, like all characters in \lam{(+)}, are not
+ allowed in \VHDL architecture names\footnote{Technically, it is
+ allowed to use non-alphanumerics when using extended
+ identifiers, but it seems that none of the tooling likes
+ extended identifiers in filenames, so it effectively does not
+ work.}, so the entity would be called \quote{w7aA7f} or
+ something similarly meaningless and autogenerated).
+
+ \subsection{Program structure}
+ These transformations are aimed at normalizing the overall structure
+ into the intended form. This means ensuring there is a lambda abstraction
+ at the top for every argument (input port or current state), putting all
+ of the other value definitions in let bindings and making the final
+ return value a simple variable reference.
+
+ \subsubsection[sec:normalization:eta]{η-abstraction}
+ This transformation makes sure that all arguments of a function-typed
+ expression are named, by introducing lambda expressions. When combined with
+ β-reduction and non-representable binding inlining, all function-typed
+ expressions should be lambda abstractions or global identifiers.
+
+ \starttrans
+ E \lam{E :: a -> b}
+ -------------- \lam{E} does not occur on a function position in an application
+ λx.E x \lam{E} is not a lambda abstraction.
+ \stoptrans
+
+ \startbuffer[from]
+ foo = λa.case a of
+ True -> λb.mul b b
+ False -> id
+ \stopbuffer
+
+ \startbuffer[to]
+ foo = λa.λx.(case a of
+ True -> λb.mul b b
+ False -> λy.id y) x
+ \stopbuffer
+
+ \transexample{eta}{η-abstraction}{from}{to}
+
+ \subsubsection[sec:normalization:appprop]{Application propagation}
+ This transformation is meant to propagate application expressions downwards
+ into expressions as far as possible. This allows partial applications inside
+ expressions to become fully applied and exposes new transformation
+ opportunities for other transformations (like β-reduction and
+ specialization).
+
+ Since all binders in our expression are unique (see
+ \in{section}[sec:normalization:uniq]), there is no risk that we will
+ introduce unintended shadowing by moving an expression into a lower
+ scope. Also, since only move expression into smaller scopes (down into
+ our expression), there is no risk of moving a variable reference out
+ of the scope in which it is defined.
+
+ \starttrans
+ (letrec binds in E) M
+ ------------------------
+ letrec binds in E M
+ \stoptrans
+
+ % And an example
+ \startbuffer[from]
+ ( letrec
+ val = 1
+ in
+ add val
+ ) 3
+ \stopbuffer
+
+ \startbuffer[to]
+ letrec
+ val = 1
+ in
+ add val 3
+ \stopbuffer
+
+ \transexample{appproplet}{Application propagation for a let expression}{from}{to}
+
+ \starttrans
+ (case x of
+ p0 -> E0
+ \vdots
+ pn -> En) M
+ -----------------
+ case x of
+ p0 -> E0 M
+ \vdots
+ pn -> En M
+ \stoptrans
+
+ % And an example
+ \startbuffer[from]
+ ( case x of