-\chapter{Normalization}
+\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.
% will end up on a single line. The strut=no option prevents a bunch of empty
% space at the start of the frame.
\define[1]\example{
- \framed[offset=1mm,align=right,strut=no]{
+ \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]
TODO: Describe core properties not supported in VHDL, and describe how the
VHDL we want to generate should look like.
-\section{Goal}
+\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.
+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:
-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.
+\placedefinition{}{A program is in \emph{normal form} if none of the
+transformations from this chapter apply.}
-An example of a program in canonical form would be:
+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 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 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.
+
+\startbuffer[NormalComplete]
+ regbank :: Bit
+ -> Word
+ -> State (Word, Word)
+ -> (State (Word, Word), Word)
-\startlambda
-- All arguments are an inital lambda
- λa.λd.λsp.
+ regbank = λa.λd.λsp.
-- There are nested let expressions at top level
let
- -- Unpack the state by coercion
+ -- 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
High -> r1
Low -> r2
r1' = case a of
- High -> d
+ High -> d'
Low -> r1
r2' = case a of
High -> r2
- Low -> d
+ Low -> d'
-- Packing a tuple
s' = (,) r1' r2'
- -- Packing the state by coercion
+ -- 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
-\stoplambda
+\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{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(typeof(var)))
+\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(typeof(var)), fvar(var))
-\italic{binding} = var = \italic{rhs} (representable(typeof(rhs)))
+ | var (representable(varvar))
+\italic{binding} = var = \italic{rhs} (representable(rhs))
-- State packing and unpacking by coercion
- | var0 = var1 :: State ty (fvar(var1))
- | var0 = var1 :: ty (var0 :: State ty) (fvar(var1))
+ | 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 (fvar(var))
+ | case var of C a0 ... an -> ai (lvar(var))
-- Selector case
- | case var of (fvar(var))
- DEFAULT -> var0 (fvar(var0))
- C w0 ... wn -> resvar (\forall{}i, wi \neq resvar, fvar(resvar))
+ | 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 (tvar(var))
-\italic{userarg} = var (fvar(var))
+\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: Define tvar, fvar, typeof, representable
-- TODO: Limit builtinarg further
-- TODO: There can still be other casts around (which the code can handle,
(\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
+\subsection{Definitions}
+In the following sections, we will be using a number of functions and
+notations, which we will define here.
+
+\subsubsection{Transformations}
+The most important notation is the one for transformation, which looks like
+the following:
+
+\starttrans
+context conditions
+~
+from
+------------------------ expression conditions
+to
+~
+context additions
+\stoptrans
-TODO: Say something about uniqueness of identifiers
+Here, we describe a transformation. The most import parts are \lam{from} and
+\lam{to}, which describe the Core expresssion that should be matched and the
+expression that it should be replaced with. This matching can occur anywhere
+in function that is being normalized, so it applies to any subexpression as
+well.
-\subsection{Builtin expressions}
-A $builtinexpr$, as defined at \in[builtinexpr] can have any of the following forms.
+The \lam{expression conditions} list a number of conditions on the \lam{from}
+expression that must hold for the transformation to apply.
-\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
+Furthermore, there is some way to look into the environment (\eg, other top
+level bindings). The \lam{context conditions} part specifies any number of
+top level bindings that must be present for the transformation to apply.
+Usually, this lists a top level binding that binds an identfier that is also
+used in the \lam{from} expression, allowing us to "access" the value of a top
+level binding in the \lam{to} expression (\eg, for inlining).
-\section{Transform passes}
+Finally, there is a way to influence the environment. The \lam{context
+additions} part lists any number of new top level bindings that should be
+added.
+
+If there are no \lam{context conditions} or \lam{context additions}, they can
+be left out alltogether, along with the separator \lam{~}.
+
+TODO: Example
+
+\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 local variable is any other
+variable (\eg, variables local to a function, which can be bound by lambda
+abstractions, let expressions and case expressions).
+
+A \emph{hardware representable} type is 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.
+
+A \emph{builtin function} is a function for which a builtin
+hardware translation is available, because its actual definition is not
+translatable. A user-defined function is any other function.
+
+\subsubsection{Functions}
+Here, we define a number of functions 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 inverse of \emph{gvar}; it is true when \emph{expr}
+references a local variable, false when it references a global variable.
+
+\emph{representable(expr)} or \emph{representable(var)} is true when
+\emph{expr} or \emph{var} has a type that is representable at runtime.
+
+\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).
+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
\stoptrans
\startbuffer[from]
-foo = λa -> case a of
+foo = λa.case a of
True -> λb.mul b b
False -> id
\stopbuffer
\startbuffer[to]
-foo = λa.λx -> (case a of
+foo = λa.λx.(case a of
True -> λb.mul b b
False -> λy.id y) x
\stopbuffer
When looking at the arguments of a user-defined function, we can
divide them into two categories:
\startitemize
- \item Arguments with a runtime representable type (\eg bits or vectors).
+ \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
into categories:
\startitemize
- \item Arguments with a runtime representable type.
+ \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
functions can be limited to signal references, instead of
needing to support all possible expressions.
- \item Arguments with a function type.
+ \item Arguments of a function type.
These arguments are functions passed to higher order builtins,
like \lam{map} and \lam{foldl}. Since implementing these
inlining transformation to replace such a variable with an expression we
can propagate again.
-TODO: Move these definitions somewhere sensible.
-
-Definition: A global variable is any variable that is bound at the
-top level of a program. A local variable is any other variable.
-
-Definition: A hardware representable type is 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.
-
-Definition: A builtin function is a function for which a builtin
-hardware translation is available, because its actual definition is not
-translatable. A user-defined function is any other function.
-
\starttrans
x = E
~
\stopbuffer
\transexample{Return value simplification}{from}{to}
-
-\subsection{Example sequence}
-
-This section lists an example expression, with a sequence of transforms
-applied to it. The exact transforms given here probably don't exactly
-match the transforms given above anymore, but perhaps this can clarify
-the big picture a bit.
-
-TODO: Update or remove this section.
-
-\startlambda
- λx.
- let s = foo x
- in
- case s of
- (a, b) ->
- case a of
- High -> add
- Low -> let
- op' = case b of
- High -> sub
- Low -> λc.λd.c
- in
- λc.λd.op' d c
-\stoplambda
-
-After top-level η-abstraction:
-
-\startlambda
- λx.λc.λd.
- (let s = foo x
- in
- case s of
- (a, b) ->
- case a of
- High -> add
- Low -> let
- op' = case b of
- High -> sub
- Low -> λc.λd.c
- in
- λc.λd.op' d c
- ) c d
-\stoplambda
-
-After (extended) β-reduction:
-
-\startlambda
- λx.λc.λd.
- let s = foo x
- in
- case s of
- (a, b) ->
- case a of
- High -> add c d
- Low -> let
- op' = case b of
- High -> sub
- Low -> λc.λd.c
- in
- op' d c
-\stoplambda
-
-After return value extraction:
-
-\startlambda
- λx.λc.λd.
- let s = foo x
- r = case s of
- (a, b) ->
- case a of
- High -> add c d
- Low -> let
- op' = case b of
- High -> sub
- Low -> λc.λd.c
- in
- op' d c
- in
- r
-\stoplambda
-
-Scrutinee simplification does not apply.
-
-After case binder wildening:
-
-\startlambda
- λx.λc.λd.
- let s = foo x
- a = case s of (a, _) -> a
- b = case s of (_, b) -> b
- r = case s of (_, _) ->
- case a of
- High -> add c d
- Low -> let op' = case b of
- High -> sub
- Low -> λc.λd.c
- in
- op' d c
- in
- r
-\stoplambda
-
-After case value simplification
-
-\startlambda
- λx.λc.λd.
- let s = foo x
- a = case s of (a, _) -> a
- b = case s of (_, b) -> b
- r = case s of (_, _) -> r'
- rh = add c d
- rl = let rll = λc.λd.c
- op' = case b of
- High -> sub
- Low -> rll
- in
- op' d c
- r' = case a of
- High -> rh
- Low -> rl
- in
- r
-\stoplambda
-
-After let flattening:
-
-\startlambda
- λx.λc.λd.
- let s = foo x
- a = case s of (a, _) -> a
- b = case s of (_, b) -> b
- r = case s of (_, _) -> r'
- rh = add c d
- rl = op' d c
- rll = λc.λd.c
- op' = case b of
- High -> sub
- Low -> rll
- r' = case a of
- High -> rh
- Low -> rl
- in
- r
-\stoplambda
-
-After function inlining:
-
-\startlambda
- λx.λc.λd.
- let s = foo x
- a = case s of (a, _) -> a
- b = case s of (_, b) -> b
- r = case s of (_, _) -> r'
- rh = add c d
- rl = (case b of
- High -> sub
- Low -> λc.λd.c) d c
- r' = case a of
- High -> rh
- Low -> rl
- in
- r
-\stoplambda
-
-After (extended) β-reduction again:
-
-\startlambda
- λx.λc.λd.
- let s = foo x
- a = case s of (a, _) -> a
- b = case s of (_, b) -> b
- r = case s of (_, _) -> r'
- rh = add c d
- rl = case b of
- High -> sub d c
- Low -> d
- r' = case a of
- High -> rh
- Low -> rl
- in
- r
-\stoplambda
-
-After case value simplification again:
-
-\startlambda
- λx.λc.λd.
- let s = foo x
- a = case s of (a, _) -> a
- b = case s of (_, b) -> b
- r = case s of (_, _) -> r'
- rh = add c d
- rlh = sub d c
- rl = case b of
- High -> rlh
- Low -> d
- r' = case a of
- High -> rh
- Low -> rl
- in
- r
-\stoplambda
-
-After case removal:
-
-\startlambda
- λx.λc.λd.
- let s = foo x
- a = case s of (a, _) -> a
- b = case s of (_, b) -> b
- r = r'
- rh = add c d
- rlh = sub d c
- rl = case b of
- High -> rlh
- Low -> d
- r' = case a of
- High -> rh
- Low -> rl
- in
- r
-\stoplambda
-
-After let bind removal:
-
-\startlambda
- λx.λc.λd.
- let s = foo x
- a = case s of (a, _) -> a
- b = case s of (_, b) -> b
- rh = add c d
- rlh = sub d c
- rl = case b of
- High -> rlh
- Low -> d
- r' = case a of
- High -> rh
- Low -> rl
- in
- r'
-\stoplambda
-
-Application simplification is not applicable.
projects / matthijs / master-project / report.git / blobdiff