1 \chapter[chap:normalization]{Normalization}
3 % A helper to print a single example in the half the page width. The example
4 % text should be in a buffer whose name is given in an argument.
6 % The align=right option really does left-alignment, but without the program
7 % will end up on a single line. The strut=no option prevents a bunch of empty
8 % space at the start of the frame.
10 \framed[offset=1mm,align=right,strut=no,background=box,frame=off]{
11 \setuptyping[option=LAM,style=sans,before=,after=]
13 \setuptyping[option=none,style=\tttf]
17 \define[3]\transexample{
18 \placeexample[here]{#1}
19 \startcombination[2*1]
20 {\example{#2}}{Original program}
21 {\example{#3}}{Transformed program}
25 The first step in the core to \small{VHDL} translation process, is normalization. We
26 aim to bring the core description into a simpler form, which we can
27 subsequently translate into \small{VHDL} easily. This normal form is needed because
28 the full core language is more expressive than \small{VHDL} in some areas and because
29 core can describe expressions that do not have a direct hardware
32 TODO: Describe core properties not supported in \small{VHDL}, and describe how the
33 \small{VHDL} we want to generate should look like.
36 The transformations described here have a well-defined goal: To bring the
37 program in a well-defined form that is directly translatable to hardware,
38 while fully preserving the semantics of the program. We refer to this form as
39 the \emph{normal form} of the program. The formal definition of this normal
42 \placedefinition{}{A program is in \emph{normal form} if none of the
43 transformations from this chapter apply.}
45 Of course, this is an \quote{easy} definition of the normal form, since our
46 program will end up in normal form automatically. The more interesting part is
47 to see if this normal form actually has the properties we would like it to
50 But, before getting into more definitions and details about this normal form,
51 let's try to get a feeling for it first. The easiest way to do this is by
52 describing the things we want to not have in a normal form.
55 \item Any \emph{polymorphism} must be removed. When laying down hardware, we
56 can't generate any signals that can have multiple types. All types must be
57 completely known to generate hardware.
59 \item Any \emph{higher order} constructions must be removed. We can't
60 generate a hardware signal that contains a function, so all values,
61 arguments and returns values used must be first order.
63 \item Any complex \emph{nested scopes} must be removed. In the \small{VHDL}
64 description, every signal is in a single scope. Also, full expressions are
65 not supported everywhere (in particular port maps can only map signal names,
66 not expressions). To make the \small{VHDL} generation easy, all values must be bound
67 on the \quote{top level}.
70 TODO: Intermezzo: functions vs plain values
72 A very simple example of a program in normal form is given in
73 \in{example}[ex:MulSum]. As you can see, all arguments to the function (which
74 will become input ports in the final hardware) are at the top. This means that
75 the body of the final lambda abstraction is never a function, but always a
78 After the lambda abstractions, we see a single let expression, that binds two
79 variables (\lam{mul} and \lam{sum}). These variables will be signals in the
80 final hardware, bound to the output port of the \lam{*} and \lam{+}
83 The final line (the \quote{return value} of the function) selects the
84 \lam{sum} signal to be the output port of the function. This \quote{return
85 value} can always only be a variable reference, never a more complex
89 alu :: Bit -> Word -> Word -> Word
98 \startuseMPgraphic{MulSum}
99 save a, b, c, mul, add, sum;
102 newCircle.a(btex $a$ etex) "framed(false)";
103 newCircle.b(btex $b$ etex) "framed(false)";
104 newCircle.c(btex $c$ etex) "framed(false)";
105 newCircle.sum(btex $res$ etex) "framed(false)";
108 newCircle.mul(btex - etex);
109 newCircle.add(btex + etex);
111 a.c - b.c = (0cm, 2cm);
112 b.c - c.c = (0cm, 2cm);
113 add.c = c.c + (2cm, 0cm);
114 mul.c = midpoint(a.c, b.c) + (2cm, 0cm);
115 sum.c = add.c + (2cm, 0cm);
118 % Draw objects and lines
119 drawObj(a, b, c, mul, add, sum);
121 ncarc(a)(mul) "arcangle(15)";
122 ncarc(b)(mul) "arcangle(-15)";
128 \placeexample[here][ex:MulSum]{Simple architecture consisting of an adder and a
130 \startcombination[2*1]
131 {\typebufferlam{MulSum}}{Core description in normal form.}
132 {\boxedgraphic{MulSum}}{The architecture described by the normal form.}
135 The previous example described composing an architecture by calling other
136 functions (operators), resulting in a simple architecture with component and
137 connection. There is of course also some mechanism for choice in the normal
138 form. In a normal Core program, the \emph{case} expression can be used in a
139 few different ways to describe choice. In normal form, this is limited to a
142 \in{Example}[ex:AddSubAlu] shows an example describing a
143 simple \small{ALU}, which chooses between two operations based on an opcode
144 bit. The main structure is the same as in \in{example}[ex:MulSum], but this
145 time the \lam{res} variable is bound to a case expression. This case
146 expression scrutinizes the variable \lam{opcode} (and scrutinizing more
147 complex expressions is not supported). The case expression can select a
148 different variable based on the constructor of \lam{opcode}.
150 \startbuffer[AddSubAlu]
151 alu :: Bit -> Word -> Word -> Word
163 \startuseMPgraphic{AddSubAlu}
164 save opcode, a, b, add, sub, mux, res;
167 newCircle.opcode(btex $opcode$ etex) "framed(false)";
168 newCircle.a(btex $a$ etex) "framed(false)";
169 newCircle.b(btex $b$ etex) "framed(false)";
170 newCircle.res(btex $res$ etex) "framed(false)";
172 newCircle.add(btex + etex);
173 newCircle.sub(btex - etex);
176 opcode.c - a.c = (0cm, 2cm);
177 add.c - a.c = (4cm, 0cm);
178 sub.c - b.c = (4cm, 0cm);
179 a.c - b.c = (0cm, 3cm);
180 mux.c = midpoint(add.c, sub.c) + (1.5cm, 0cm);
181 res.c - mux.c = (1.5cm, 0cm);
184 % Draw objects and lines
185 drawObj(opcode, a, b, res, add, sub, mux);
187 ncline(a)(add) "posA(e)";
188 ncline(b)(sub) "posA(e)";
189 nccurve(a)(sub) "posA(e)", "angleA(0)";
190 nccurve(b)(add) "posA(e)", "angleA(0)";
191 nccurve(add)(mux) "posB(inpa)", "angleB(0)";
192 nccurve(sub)(mux) "posB(inpb)", "angleB(0)";
193 nccurve(opcode)(mux) "posB(n)", "angleA(0)", "angleB(-90)";
194 ncline(mux)(res) "posA(out)";
197 \placeexample[here][ex:AddSubAlu]{Simple \small{ALU} supporting two operations.}
198 \startcombination[2*1]
199 {\typebufferlam{AddSubAlu}}{Core description in normal form.}
200 {\boxedgraphic{AddSubAlu}}{The architecture described by the normal form.}
203 As a more complete example, consider \in{example}[ex:NormalComplete]. This
204 example contains everything that is supported in normal form, with the
205 exception of builtin higher order functions. The graphical version of the
206 architecture contains a slightly simplified version, since the state tuple
207 packing and unpacking have been left out. Instead, two seperate registers are
208 drawn. Also note that most synthesis tools will further optimize this
209 architecture by removing the multiplexers at the register input and replace
210 them with some logic in the clock inputs, but we want to show the architecture
211 as close to the description as possible.
213 \startbuffer[NormalComplete]
216 -> State (Word, Word)
217 -> (State (Word, Word), Word)
219 -- All arguments are an inital lambda
221 -- There are nested let expressions at top level
223 -- Unpack the state by coercion (\eg, cast from
224 -- State (Word, Word) to (Word, Word))
225 s = sp :: (Word, Word)
226 -- Extract both registers from the state
227 r1 = case s of (fst, snd) -> fst
228 r2 = case s of (fst, snd) -> snd
229 -- Calling some other user-defined function.
231 -- Conditional connections
243 -- pack the state by coercion (\eg, cast from
244 -- (Word, Word) to State (Word, Word))
245 sp' = s' :: State (Word, Word)
246 -- Pack our return value
253 \startuseMPgraphic{NormalComplete}
254 save a, d, r, foo, muxr, muxout, out;
257 newCircle.a(btex \lam{a} etex) "framed(false)";
258 newCircle.d(btex \lam{d} etex) "framed(false)";
259 newCircle.out(btex \lam{out} etex) "framed(false)";
261 %newCircle.add(btex + etex);
262 newBox.foo(btex \lam{foo} etex);
263 newReg.r1(btex $\lam{r1}$ etex) "dx(4mm)", "dy(6mm)";
264 newReg.r2(btex $\lam{r2}$ etex) "dx(4mm)", "dy(6mm)", "reflect(true)";
266 % Reflect over the vertical axis
267 reflectObj(muxr1)((0,0), (0,1));
270 rotateObj(muxout)(-90);
272 d.c = foo.c + (0cm, 1.5cm);
273 a.c = (xpart r2.c + 2cm, ypart d.c - 0.5cm);
274 foo.c = midpoint(muxr1.c, muxr2.c) + (0cm, 2cm);
275 muxr1.c = r1.c + (0cm, 2cm);
276 muxr2.c = r2.c + (0cm, 2cm);
277 r2.c = r1.c + (4cm, 0cm);
279 muxout.c = midpoint(r1.c, r2.c) - (0cm, 2cm);
280 out.c = muxout.c - (0cm, 1.5cm);
282 % % Draw objects and lines
283 drawObj(a, d, foo, r1, r2, muxr1, muxr2, muxout, out);
286 nccurve(foo)(muxr1) "angleA(-90)", "posB(inpa)", "angleB(180)";
287 nccurve(foo)(muxr2) "angleA(-90)", "posB(inpb)", "angleB(0)";
288 nccurve(muxr1)(r1) "posA(out)", "angleA(180)", "posB(d)", "angleB(0)";
289 nccurve(r1)(muxr1) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(180)";
290 nccurve(muxr2)(r2) "posA(out)", "angleA(0)", "posB(d)", "angleB(180)";
291 nccurve(r2)(muxr2) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(0)";
292 nccurve(r1)(muxout) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(-90)";
293 nccurve(r2)(muxout) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(-90)";
295 nccurve(a)(muxout) "angleA(-90)", "angleB(180)", "posB(sel)";
296 nccurve(a)(muxr1) "angleA(180)", "angleB(-90)", "posB(sel)";
297 nccurve(a)(muxr2) "angleA(180)", "angleB(-90)", "posB(sel)";
298 ncline(muxout)(out) "posA(out)";
301 \placeexample[here][ex:NormalComplete]{Simple architecture consisting of an adder and a
303 \startcombination[2*1]
304 {\typebufferlam{NormalComplete}}{Core description in normal form.}
305 {\boxedgraphic{NormalComplete}}{The architecture described by the normal form.}
308 \subsection{Intended normal form definition}
309 Now we have some intuition for the normal form, we can describe how we want
310 the normal form to look like in a slightly more formal manner. The following
311 EBNF-like description completely captures the intended structure (and
312 generates a subset of GHC's core format).
314 Some clauses have an expression listed in parentheses. These are conditions
315 that need to apply to the clause.
318 \italic{normal} = \italic{lambda}
319 \italic{lambda} = λvar.\italic{lambda} (representable(var))
321 \italic{toplet} = let \italic{binding} in \italic{toplet}
322 | letrec [\italic{binding}] in \italic{toplet}
323 | var (representable(varvar))
324 \italic{binding} = var = \italic{rhs} (representable(rhs))
325 -- State packing and unpacking by coercion
326 | var0 = var1 :: State ty (lvar(var1))
327 | var0 = var1 :: ty (var0 :: State ty) (lvar(var1))
328 \italic{rhs} = userapp
331 | case var of C a0 ... an -> ai (lvar(var))
333 | case var of (lvar(var))
334 DEFAULT -> var0 (lvar(var0))
335 C w0 ... wn -> resvar (\forall{}i, wi \neq resvar, lvar(resvar))
336 \italic{userapp} = \italic{userfunc}
337 | \italic{userapp} {userarg}
338 \italic{userfunc} = var (gvar(var))
339 \italic{userarg} = var (lvar(var))
340 \italic{builtinapp} = \italic{builtinfunc}
341 | \italic{builtinapp} \italic{builtinarg}
342 \italic{builtinfunc} = var (bvar(var))
343 \italic{builtinarg} = \italic{coreexpr}
346 -- TODO: Limit builtinarg further
348 -- TODO: There can still be other casts around (which the code can handle,
349 e.g., ignore), which still need to be documented here.
351 -- TODO: Note about the selector case. It just supports Bit and Bool
352 currently, perhaps it should be generalized in the normal form?
354 When looking at such a program from a hardware perspective, the top level
355 lambda's define the input ports. The value produced by the let expression is
356 the output port. Most function applications bound by the let expression
357 define a component instantiation, where the input and output ports are mapped
358 to local signals or arguments. Some of the others use a builtin
359 construction (\eg the \lam{case} statement) or call a builtin function
360 (\eg \lam{add} or \lam{sub}). For these, a hardcoded \small{VHDL} translation is
363 \section{Transformation notation}
364 To be able to concisely present transformations, we use a specific format to
365 them. It is a simple format, similar to one used in logic reasoning.
367 Such a transformation description looks like the following.
372 <original expression>
373 -------------------------- <expression conditions>
374 <transformed expresssion>
379 This format desribes a transformation that applies to \lam{original
380 expresssion} and transforms it into \lam{transformed expression}, assuming
381 that all conditions apply. In this format, there are a number of placeholders
382 in pointy brackets, most of which should be rather obvious in their meaning.
383 Nevertheless, we will more precisely specify their meaning below:
385 \startdesc{<original expression>} The expression pattern that will be matched
386 against (subexpressions of) the expression to be transformed. We call this a
387 pattern, because it can contain \emph{placeholders} (variables), which match
388 any expression or binder. Any such placeholder is said to be \emph{bound} to
389 the expression it matches. It is convention to use an uppercase latter (\eg
390 \lam{M} or \lam{E} to refer to any expression (including a simple variable
391 reference) and lowercase letters (\eg \lam{v} or \lam{b}) to refer to
392 (references to) binders.
394 For example, the pattern \lam{a + B} will match the expression
395 \lam{v + (2 * w)} (and bind \lam{a} to \lam{v} and \lam{B} to
396 \lam{(2 * 2)}), but not \lam{v + (2 * w)}.
399 \startdesc{<expression conditions>}
400 These are extra conditions on the expression that is matched. These
401 conditions can be used to further limit the cases in which the
402 transformation applies, in particular to prevent a transformation from
403 causing a loop with itself or another transformation.
405 Only if these if these conditions are \emph{all} true, this transformation
409 \startdesc{<context conditions>}
410 These are a number of extra conditions on the context of the function. In
411 particular, these conditions can require some other top level function to be
412 present, whose value matches the pattern given here. The format of each of
413 these conditions is: \lam{binder = <pattern>}.
415 Typically, the binder is some placeholder bound in the \lam{<original
416 expression>}, while the pattern contains some placeholders that are used in
417 the \lam{transformed expression}.
419 Only if a top level binder exists that matches each binder and pattern, this
420 transformation applies.
423 \startdesc{<transformed expression>}
424 This is the expression template that is the result of the transformation. If, looking
425 at the above three items, the transformation applies, the \lam{original
426 expression} is completely replaced with the \lam{<transformed expression>}.
427 We call this a template, because it can contain placeholders, referring to
428 any placeholder bound by the \lam{<original expression>} or the
429 \lam{<context conditions>}. The resulting expression will have those
430 placeholders replaced by the values bound to them.
432 Any binder (lowercase) placeholder that has no value bound to it yet will be
433 bound to (and replaced with) a fresh binder.
436 \startdesc{<context additions>}
437 These are templates for new functions to add to the context. This is a way
438 to have a transformation create new top level functiosn.
440 Each addition has the form \lam{binder = template}. As above, any
441 placeholder in the addition is replaced with the value bound to it, and any
442 binder placeholder that has no value bound to it yet will be bound to (and
443 replaced with) a fresh binder.
446 As an example, we'll look at η-abstraction:
450 -------------- \lam{E} does not occur on a function position in an application
451 λx.E x \lam{E} is not a lambda abstraction.
454 Consider the following function, which is a fairly obvious way to specify a
455 simple ALU (Note \at{example}[ex:AddSubAlu] is the normal form of this
459 alu :: Bit -> Word -> Word -> Word
460 alu = λopcode. case opcode of
465 There are a few subexpressions in this function to which we could possibly
466 apply the transformation. Since the pattern of the transformation is only
467 the placeholder \lam{E}, any expression will match that. Whether the
468 transformation applies to an expression is thus solely decided by the
469 conditions to the right of the transformation.
471 We will look at each expression in the function in a top down manner. The
472 first expression is the entire expression the function is bound to.
475 λopcode. case opcode of
480 As said, the expression pattern matches this. The type of this expression is
481 \lam{Bit -> Word -> Word -> Word}, which matches \lam{a -> b} (Note that in
482 this case \lam{a = Bit} and \lam{b = Word -> Word -> Word}).
484 Since this expression is at top level, it does not occur at a function
485 position of an application. However, The expression is a lambda abstraction,
486 so this transformation does not apply.
488 The next expression we could apply this transformation to, is the body of
489 the lambda abstraction:
497 The type of this expression is \lam{Word -> Word -> Word}, which again
498 matches \lam{a -> b}. The expression is the body of a lambda expression, so
499 it does not occur at a function position of an application. Finally, the
500 expression is not a lambda abstraction but a case expression, so all the
501 conditions match. There are no context conditions to match, so the
502 transformation applies.
504 By now, the placeholder \lam{E} is bound to the entire expression. The
505 placeholder \lam{x}, which occurs in the replacement template, is not bound
506 yet, so we need to generate a fresh binder for that. Let's use the binder
507 \lam{a}. This results in the following replacement expression:
515 Continuing with this expression, we see that the transformation does not
516 apply again (it is a lambda expression). Next we look at the body of this
525 Here, the transformation does apply, binding \lam{E} to the entire
526 expression and \lam{x} to the fresh binder \lam{b}, resulting in the
535 Again, the transformation does not apply to this lambda abstraction, so we
536 look at its body. For brevity, we'll put the case statement on one line from
540 (case opcode of Low -> (+); High -> (-)) a b
543 The type of this expression is \lam{Word}, so it does not match \lam{a -> b}
544 and the transformation does not apply. Next, we have two options for the
545 next expression to look at: The function position and argument position of
546 the application. The expression in the argument position is \lam{b}, which
547 has type \lam{Word}, so the transformation does not apply. The expression in
548 the function position is:
551 (case opcode of Low -> (+); High -> (-)) a
554 Obviously, the transformation does not apply here, since it occurs in
555 function position. In the same way the transformation does not apply to both
556 components of this expression (\lam{case opcode of Low -> (+); High -> (-)}
557 and \lam{a}), so we'll skip to the components of the case expression: The
558 scrutinee and both alternatives. Since the opcode is not a function, it does
559 not apply here, and we'll leave both alternatives as an exercise to the
560 reader. The final function, after all these transformations becomes:
563 alu :: Bit -> Word -> Word -> Word
564 alu = λopcode.λa.b. (case opcode of
565 Low -> λa1.λb1 (+) a1 b1
566 High -> λa2.λb2 (-) a2 b2) a b
569 In this case, the transformation does not apply anymore, though this might
570 not always be the case (e.g., the application of a transformation on a
571 subexpression might open up possibilities to apply the transformation
572 further up in the expression).
574 \subsection{Transformation application}
575 In this chapter we define a number of transformations, but how will we apply
576 these? As stated before, our normal form is reached as soon as no
577 transformation applies anymore. This means our application strategy is to
578 simply apply any transformation that applies, and continuing to do that with
579 the result of each transformation.
581 In particular, we define no particular order of transformations. Since
582 transformation order should not influence the resulting normal form (see TODO
583 ref), this leaves the implementation free to choose any application order that
584 results in an efficient implementation.
586 When applying a single transformation, we try to apply it to every (sub)expression
587 in a function, not just the top level function. This allows us to keep the
588 transformation descriptions concise and powerful.
590 \subsection{Definitions}
591 In the following sections, we will be using a number of functions and
592 notations, which we will define here.
594 \subsubsection{Other concepts}
595 A \emph{global variable} is any variable that is bound at the
596 top level of a program, or an external module. A \emph{local variable} is any
597 other variable (\eg, variables local to a function, which can be bound by
598 lambda abstractions, let expressions and pattern matches of case
599 alternatives). Note that this is a slightly different notion of global versus
600 local than what \small{GHC} uses internally.
601 \defref{global variable} \defref{local variable}
603 A \emph{hardware representable} (or just \emph{representable}) type or value
604 is (a value of) a type that we can generate a signal for in hardware. For
605 example, a bit, a vector of bits, a 32 bit unsigned word, etc. Types that are
606 not runtime representable notably include (but are not limited to): Types,
607 dictionaries, functions.
608 \defref{representable}
610 A \emph{builtin function} is a function supplied by the Cλash framework, whose
611 implementation is not valid Cλash. The implementation is of course valid
612 Haskell, for simulation, but it is not expressable in Cλash.
613 \defref{builtin function} \defref{user-defined function}
615 For these functions, Cλash has a \emph{builtin hardware translation}, so calls
616 to these functions can still be translated. These are functions like
617 \lam{map}, \lam{hwor} and \lam{length}.
619 A \emph{user-defined} function is a function for which we do have a Cλash
620 implementation available.
622 \subsubsection{Functions}
623 Here, we define a number of functions that can be used below to concisely
626 \refdef{global variable}\emph{gvar(expr)} is true when \emph{expr} is a variable that references a
627 global variable. It is false when it references a local variable.
629 \refdef{local variable}\emph{lvar(expr)} is the complement of \emph{gvar}; it is true when \emph{expr}
630 references a local variable, false when it references a global variable.
632 \refdef{representable}\emph{representable(expr)} or \emph{representable(var)} is true when
633 \emph{expr} or \emph{var} is \emph{representable}.
635 \subsection{Binder uniqueness}
636 A common problem in transformation systems, is binder uniqueness. When not
637 considering this problem, it is easy to create transformations that mix up
638 bindings and cause name collisions. Take for example, the following core
642 (λa.λb.λc. a * b * c) x c
645 By applying β-reduction (see below) once, we can simplify this expression to:
651 Now, we have replaced the \lam{a} binder with a reference to the \lam{x}
652 binder. No harm done here. But note that we see multiple occurences of the
653 \lam{c} binder. The first is a binding occurence, to which the second refers.
654 The last, however refers to \emph{another} instance of \lam{c}, which is
655 bound somewhere outside of this expression. Now, if we would apply beta
656 reduction without taking heed of binder uniqueness, we would get:
662 This is obviously not what was supposed to happen! The root of this problem is
663 the reuse of binders: Identical binders can be bound in different scopes, such
664 that only the inner one is \quote{visible} in the inner expression. In the example
665 above, the \lam{c} binder was bound outside of the expression and in the inner
666 lambda expression. Inside that lambda expression, only the inner \lam{c} is
669 There are a number of ways to solve this. \small{GHC} has isolated this
670 problem to their binder substitution code, which performs \emph{deshadowing}
671 during its expression traversal. This means that any binding that shadows
672 another binding on a higher level is replaced by a new binder that does not
673 shadow any other binding. This non-shadowing invariant is enough to prevent
674 binder uniqueness problems in \small{GHC}.
676 In our transformation system, maintaining this non-shadowing invariant is
677 a bit harder to do (mostly due to implementation issues, the prototype doesn't
678 use \small{GHC}'s subsitution code). Also, we can observe the following
682 \item Deshadowing does not guarantee overall uniqueness. For example, the
683 following (slightly contrived) expression shows the identifier \lam{x} bound in
684 two seperate places (and to different values), even though no shadowing
688 (let x = 1 in x) + (let x = 2 in x)
691 \item In our normal form (and the resulting \small{VHDL}), all binders
692 (signals) will end up in the same scope. To allow this, all binders within the
693 same function should be unique.
695 \item When we know that all binders in an expression are unique, moving around
696 or removing a subexpression will never cause any binder conflicts. If we have
697 some way to generate fresh binders, introducing new subexpressions will not
698 cause any problems either. The only way to cause conflicts is thus to
699 duplicate an existing subexpression.
702 Given the above, our prototype maintains a unique binder invariant. This
703 meanst that in any given moment during normalization, all binders \emph{within
704 a single function} must be unique. To achieve this, we apply the following
707 TODO: Define fresh binders and unique supplies
710 \item Before starting normalization, all binders in the function are made
711 unique. This is done by generating a fresh binder for every binder used. This
712 also replaces binders that did not pose any conflict, but it does ensure that
713 all binders within the function are generated by the same unique supply. See
714 (TODO: ref fresh binder).
715 \item Whenever a new binder must be generated, we generate a fresh binder that
716 is guaranteed to be different from \emph{all binders generated so far}. This
717 can thus never introduce duplication and will maintain the invariant.
718 \item Whenever (part of) an expression is duplicated (for example when
719 inlining), all binders in the expression are replaced with fresh binders
720 (using the same method as at the start of normalization). These fresh binders
721 can never introduce duplication, so this will maintain the invariant.
722 \item Whenever we move part of an expression around within the function, there
723 is no need to do anything special. There is obviously no way to introduce
724 duplication by moving expressions around. Since we know that each of the
725 binders is already unique, there is no way to introduce (incorrect) shadowing
729 \section{Transform passes}
730 In this section we describe the actual transforms. Here we're using
731 the core language in a notation that resembles lambda calculus.
733 Each of these transforms is meant to be applied to every (sub)expression
734 in a program, for as long as it applies. Only when none of the
735 transformations can be applied anymore, the program is in normal form (by
736 definition). We hope to be able to prove that this form will obey all of the
737 constraints defined above, but this has yet to happen (though it seems likely
740 Each of the transforms will be described informally first, explaining
741 the need for and goal of the transform. Then, a formal definition is
742 given, using a familiar syntax from the world of logic. Each transform
743 is specified as a number of conditions (above the horizontal line) and a
744 number of conclusions (below the horizontal line). The details of using
745 this notation are still a bit fuzzy, so comments are welcom.
747 \subsection{General cleanup}
749 \subsubsection{β-reduction}
750 β-reduction is a well known transformation from lambda calculus, where it is
751 the main reduction step. It reduces applications of labmda abstractions,
752 removing both the lambda abstraction and the application.
754 In our transformation system, this step helps to remove unwanted lambda
755 abstractions (basically all but the ones at the top level). Other
756 transformations (application propagation, non-representable inlining) make
757 sure that most lambda abstractions will eventually be reducable by
760 TODO: Define substitution syntax
777 \transexample{β-reduction}{from}{to}
779 \subsubsection{Application propagation}
780 This transformation is meant to propagate application expressions downwards
781 into expressions as far as possible. This allows partial applications inside
782 expressions to become fully applied and exposes new transformation
783 possibilities for other transformations (like β-reduction).
807 \transexample{Application propagation for a let expression}{from}{to}
835 \transexample{Application propagation for a case expression}{from}{to}
837 \subsubsection{Empty let removal}
838 This transformation is simple: It removes recursive lets that have no bindings
839 (which usually occurs when let derecursification removes the last binding from
848 \subsubsection{Simple let binding removal}
849 This transformation inlines simple let bindings (\eg a = b).
851 This transformation is not needed to get into normal form, but makes the
852 resulting \small{VHDL} a lot shorter.
878 \subsubsection{Unused let binding removal}
879 This transformation removes let bindings that are never used. Usually,
880 the desugarer introduces some unused let bindings.
882 This normalization pass should really be unneeded to get into normal form
883 (since ununsed bindings are not forbidden by the normal form), but in practice
884 the desugarer or simplifier emits some unused bindings that cannot be
885 normalized (e.g., calls to a \type{PatError} (TODO: Check this name)). Also,
886 this transformation makes the resulting \small{VHDL} a lot shorter.
890 ---------------------------- \lam{a} does not occur free in \lam{M}
901 ---------------------------- \lam{a} does not occur free in \lam{M}
909 \subsubsection{Cast propagation / simplification}
910 This transform pushes casts down into the expression as far as possible.
911 Since its exact role and need is not clear yet, this transformation is
914 \subsubsection{Compiler generated top level binding inlining}
917 \section{Program structure}
919 \subsubsection{η-abstraction}
920 This transformation makes sure that all arguments of a function-typed
921 expression are named, by introducing lambda expressions. When combined with
922 β-reduction and function inlining below, all function-typed expressions should
923 be lambda abstractions or global identifiers.
927 -------------- \lam{E} is not the first argument of an application.
928 λx.E x \lam{E} is not a lambda abstraction.
929 \lam{x} is a variable that does not occur free in \lam{E}.
939 foo = λa.λx.(case a of
944 \transexample{η-abstraction}{from}{to}
946 \subsubsection{Let derecursification}
947 This transformation is meant to make lets non-recursive whenever possible.
948 This might allow other optimizations to do their work better. TODO: Why is
951 \subsubsection{Let flattening}
952 This transformation puts nested lets in the same scope, by lifting the
953 binding(s) of the inner let into a new let around the outer let. Eventually,
954 this will cause all let bindings to appear in the same scope (they will all be
955 in scope for the function return value).
957 Note that this transformation does not try to be smart when faced with
958 recursive lets, it will just leave the lets recursive (possibly joining a
959 recursive and non-recursive let into a single recursive let). The let
960 rederecursification transformation will do this instead.
963 letnonrec x = (let bindings in M) in N
964 ------------------------------------------
965 let bindings in (letnonrec x = M) in N
971 x = (let bindings in M)
975 ------------------------------------------
994 b = let c = 3 in a + c
1015 \transexample{Let flattening}{from}{to}
1017 \subsubsection{Return value simplification}
1018 This transformation ensures that the return value of a function is always a
1019 simple local variable reference.
1021 Currently implemented using lambda simplification, let simplification, and
1022 top simplification. Should change into something like the following, which
1023 works only on the result of a function instead of any subexpression. This is
1024 achieved by the contexts, like \lam{x = E}, though this is strictly not
1025 correct (you could read this as "if there is any function \lam{x} that binds
1026 \lam{E}, any \lam{E} can be transformed, while we only mean the \lam{E} that
1027 is bound by \lam{x}. This might need some extra notes or something).
1030 x = E \lam{E} is representable
1031 ~ \lam{E} is not a lambda abstraction
1032 E \lam{E} is not a let expression
1033 --------------------------- \lam{E} is not a local variable reference
1039 ~ \lam{E} is representable
1040 E \lam{E} is not a let expression
1041 --------------------------- \lam{E} is not a local variable reference
1046 x = λv0 ... λvn.let ... in E
1047 ~ \lam{E} is representable
1048 E \lam{E} is not a local variable reference
1049 ---------------------------
1058 x = let x = add 1 2 in x
1061 \transexample{Return value simplification}{from}{to}
1063 \subsection{Argument simplification}
1064 The transforms in this section deal with simplifying application
1065 arguments into normal form. The goal here is to:
1068 \item Make all arguments of user-defined functions (\eg, of which
1069 we have a function body) simple variable references of a runtime
1070 representable type. This is needed, since these applications will be turned
1071 into component instantiations.
1072 \item Make all arguments of builtin functions one of:
1074 \item A type argument.
1075 \item A dictionary argument.
1076 \item A type level expression.
1077 \item A variable reference of a runtime representable type.
1078 \item A variable reference or partial application of a function type.
1082 When looking at the arguments of a user-defined function, we can
1083 divide them into two categories:
1085 \item Arguments of a runtime representable type (\eg bits or vectors).
1087 These arguments can be preserved in the program, since they can
1088 be translated to input ports later on. However, since we can
1089 only connect signals to input ports, these arguments must be
1090 reduced to simple variables (for which signals will be
1091 produced). This is taken care of by the argument extraction
1093 \item Non-runtime representable typed arguments.
1095 These arguments cannot be preserved in the program, since we
1096 cannot represent them as input or output ports in the resulting
1097 \small{VHDL}. To remove them, we create a specialized version of the
1098 called function with these arguments filled in. This is done by
1099 the argument propagation transform.
1101 Typically, these arguments are type and dictionary arguments that are
1102 used to make functions polymorphic. By propagating these arguments, we
1103 are essentially doing the same which GHC does when it specializes
1104 functions: Creating multiple variants of the same function, one for
1105 each type for which it is used. Other common non-representable
1106 arguments are functions, e.g. when calling a higher order function
1107 with another function or a lambda abstraction as an argument.
1109 The reason for doing this is similar to the reasoning provided for
1110 the inlining of non-representable let bindings above. In fact, this
1111 argument propagation could be viewed as a form of cross-function
1115 TODO: Check the following itemization.
1117 When looking at the arguments of a builtin function, we can divide them
1121 \item Arguments of a runtime representable type.
1123 As we have seen with user-defined functions, these arguments can
1124 always be reduced to a simple variable reference, by the
1125 argument extraction transform. Performing this transform for
1126 builtin functions as well, means that the translation of builtin
1127 functions can be limited to signal references, instead of
1128 needing to support all possible expressions.
1130 \item Arguments of a function type.
1132 These arguments are functions passed to higher order builtins,
1133 like \lam{map} and \lam{foldl}. Since implementing these
1134 functions for arbitrary function-typed expressions (\eg, lambda
1135 expressions) is rather comlex, we reduce these arguments to
1136 (partial applications of) global functions.
1138 We can still support arbitrary expressions from the user code,
1139 by creating a new global function containing that expression.
1140 This way, we can simply replace the argument with a reference to
1141 that new function. However, since the expression can contain any
1142 number of free variables we also have to include partial
1143 applications in our normal form.
1145 This category of arguments is handled by the function extraction
1147 \item Other unrepresentable arguments.
1149 These arguments can take a few different forms:
1150 \startdesc{Type arguments}
1151 In the core language, type arguments can only take a single
1152 form: A type wrapped in the Type constructor. Also, there is
1153 nothing that can be done with type expressions, except for
1154 applying functions to them, so we can simply leave type
1155 arguments as they are.
1157 \startdesc{Dictionary arguments}
1158 In the core language, dictionary arguments are used to find
1159 operations operating on one of the type arguments (mostly for
1160 finding class methods). Since we will not actually evaluatie
1161 the function body for builtin functions and can generate
1162 code for builtin functions by just looking at the type
1163 arguments, these arguments can be ignored and left as they
1166 \startdesc{Type level arguments}
1167 Sometimes, we want to pass a value to a builtin function, but
1168 we need to know the value at compile time. Additionally, the
1169 value has an impact on the type of the function. This is
1170 encoded using type-level values, where the actual value of the
1171 argument is not important, but the type encodes some integer,
1172 for example. Since the value is not important, the actual form
1173 of the expression does not matter either and we can leave
1174 these arguments as they are.
1176 \startdesc{Other arguments}
1177 Technically, there is still a wide array of arguments that can
1178 be passed, but does not fall into any of the above categories.
1179 However, none of the supported builtin functions requires such
1180 an argument. This leaves use with passing unsupported types to
1181 a function, such as calling \lam{head} on a list of functions.
1183 In these cases, it would be impossible to generate hardware
1184 for such a function call anyway, so we can ignore these
1187 The only way to generate hardware for builtin functions with
1188 arguments like these, is to expand the function call into an
1189 equivalent core expression (\eg, expand map into a series of
1190 function applications). But for now, we choose to simply not
1191 support expressions like these.
1194 From the above, we can conclude that we can simply ignore these
1195 other unrepresentable arguments and focus on the first two
1199 \subsubsection{Argument simplification}
1200 This transform deals with arguments to functions that
1201 are of a runtime representable type. It ensures that they will all become
1202 references to global variables, or local signals in the resulting \small{VHDL}.
1204 TODO: It seems we can map an expression to a port, not only a signal.
1205 Perhaps this makes this transformation not needed?
1206 TODO: Say something about dataconstructors (without arguments, like True
1207 or False), which are variable references of a runtime representable
1208 type, but do not result in a signal.
1210 To reduce a complex expression to a simple variable reference, we create
1211 a new let expression around the application, which binds the complex
1212 expression to a new variable. The original function is then applied to
1217 -------------------- \lam{N} is of a representable type
1218 let x = N in M x \lam{N} is not a local variable reference
1226 let x = add a 1 in add x 1
1229 \transexample{Argument extraction}{from}{to}
1231 \subsubsection{Function extraction}
1232 This transform deals with function-typed arguments to builtin functions.
1233 Since these arguments cannot be propagated, we choose to extract them
1234 into a new global function instead.
1236 Any free variables occuring in the extracted arguments will become
1237 parameters to the new global function. The original argument is replaced
1238 with a reference to the new function, applied to any free variables from
1239 the original argument.
1241 This transformation is useful when applying higher order builtin functions
1242 like \hs{map} to a lambda abstraction, for example. In this case, the code
1243 that generates \small{VHDL} for \hs{map} only needs to handle top level functions and
1244 partial applications, not any other expression (such as lambda abstractions or
1245 even more complicated expressions).
1248 M N \lam{M} is a (partial aplication of a) builtin function.
1249 --------------------- \lam{f0 ... fn} = free local variables of \lam{N}
1250 M x f0 ... fn \lam{N :: a -> b}
1251 ~ \lam{N} is not a (partial application of) a top level function
1256 map (λa . add a b) xs
1270 \transexample{Function extraction}{from}{to}
1272 \subsubsection{Argument propagation}
1273 This transform deals with arguments to user-defined functions that are
1274 not representable at runtime. This means these arguments cannot be
1275 preserved in the final form and most be {\em propagated}.
1277 Propagation means to create a specialized version of the called
1278 function, with the propagated argument already filled in. As a simple
1279 example, in the following program:
1286 we could {\em propagate} the constant argument 1, with the following
1294 Special care must be taken when the to-be-propagated expression has any
1295 free variables. If this is the case, the original argument should not be
1296 removed alltogether, but replaced by all the free variables of the
1297 expression. In this way, the original expression can still be evaluated
1298 inside the new function. Also, this brings us closer to our goal: All
1299 these free variables will be simple variable references.
1301 To prevent us from propagating the same argument over and over, a simple
1302 local variable reference is not propagated (since is has exactly one
1303 free variable, itself, we would only replace that argument with itself).
1305 This shows that any free local variables that are not runtime representable
1306 cannot be brought into normal form by this transform. We rely on an
1307 inlining transformation to replace such a variable with an expression we
1308 can propagate again.
1313 x Y0 ... Yi ... Yn \lam{Yi} is not of a runtime representable type
1314 --------------------------------------------- \lam{Yi} is not a local variable reference
1315 x' y0 ... yi-1 f0 ... fm Yi+1 ... Yn \lam{f0 ... fm} = free local vars of \lam{Yi}
1317 x' = λy0 ... yi-1 f0 ... fm yi+1 ... yn .
1318 E y0 ... yi-1 Yi yi+1 ... yn
1324 \subsection{Case simplification}
1325 \subsubsection{Scrutinee simplification}
1326 This transform ensures that the scrutinee of a case expression is always
1327 a simple variable reference.
1332 ----------------- \lam{E} is not a local variable reference
1351 \transexample{Let flattening}{from}{to}
1354 \subsubsection{Case simplification}
1355 This transformation ensures that all case expressions become normal form. This
1356 means they will become one of:
1358 \item An extractor case with a single alternative that picks a single field
1359 from a datatype, \eg \lam{case x of (a, b) -> a}.
1360 \item A selector case with multiple alternatives and only wild binders, that
1361 makes a choice between expressions based on the constructor of another
1362 expression, \eg \lam{case x of Low -> a; High -> b}.
1367 C0 v0,0 ... v0,m -> E0
1369 Cn vn,0 ... vn,m -> En
1370 --------------------------------------------------- \forall i \forall j, 0 <= i <= n, 0 <= i < m (\lam{wi,j} is a wild (unused) binder)
1372 v0,0 = case x of C0 v0,0 .. v0,m -> v0,0
1374 v0,m = case x of C0 v0,0 .. v0,m -> v0,m
1377 vn,m = case x of Cn vn,0 .. vn,m -> vn,m
1381 C0 w0,0 ... w0,m -> x0
1383 Cn wn,0 ... wn,m -> xn
1386 TODO: This transformation specified like this is complicated and misses
1387 conditions to prevent looping with itself. Perhaps we should split it here for
1406 \transexample{Selector case simplification}{from}{to}
1414 b = case a of (,) b c -> b
1415 c = case a of (,) b c -> c
1422 \transexample{Extractor case simplification}{from}{to}
1424 \subsubsection{Case removal}
1425 This transform removes any case statements with a single alternative and
1428 These "useless" case statements are usually leftovers from case simplification
1429 on extractor case (see the previous example).
1434 ---------------------- \lam{\forall i, 0 <= i <= m} (\lam{vi} does not occur free in E)
1447 \transexample{Case removal}{from}{to}
1449 \subsection{Monomorphisation}
1450 TODO: Better name for this section
1452 Reference type-specialization (== argument propagation)
1454 \subsubsection{Defunctionalization}
1455 Reference higher-order-specialization (== argument propagation)
1457 \subsubsection{Non-representable binding inlining}
1458 This transform inlines let bindings that have a non-representable type. Since
1459 we can never generate a signal assignment for these bindings (we cannot
1460 declare a signal assignment with a non-representable type, for obvious
1461 reasons), we have no choice but to inline the binding to remove it.
1463 If the binding is non-representable because it is a lambda abstraction, it is
1464 likely that it will inlined into an application and β-reduction will remove
1465 the lambda abstraction and turn it into a representable expression at the
1466 inline site. The same holds for partial applications, which can be turned into
1467 full applications by inlining.
1469 Other cases of non-representable bindings we see in practice are primitive
1470 Haskell types. In most cases, these will not result in a valid normalized
1471 output, but then the input would have been invalid to start with. There is one
1472 exception to this: When a builtin function is applied to a non-representable
1473 expression, things might work out in some cases. For example, when you write a
1474 literal \hs{SizedInt} in Haskell, like \hs{1 :: SizedInt D8}, this results in
1475 the following core: \lam{fromInteger (smallInteger 10)}, where for example
1476 \lam{10 :: GHC.Prim.Int\#} and \lam{smallInteger 10 :: Integer} have
1477 non-representable types. TODO: This/these paragraph(s) should probably become a
1478 separate discussion somewhere else.
1481 letnonrec a = E in M
1482 -------------------------- \lam{E} has a non-representable type.
1493 -------------------------- \lam{E} has a non-representable type.
1513 x = fromInteger (smallInteger 10)
1515 (λa -> add a 1) (add 1 x)
1518 \transexample{Let flattening}{from}{to}
1521 \section{Provable properties}
1522 When looking at the system of transformations outlined above, there are a
1523 number of questions that we can ask ourselves. The main question is of course:
1524 \quote{Does our system work as intended?}. We can split this question into a
1525 number of subquestions:
1528 \item[q:termination] Does our system \emph{terminate}? Since our system will
1529 keep running as long as transformations apply, there is an obvious risk that
1530 it will keep running indefinitely. One transformation produces a result that
1531 is transformed back to the original by another transformation, for example.
1532 \item[q:soundness] Is our system \emph{sound}? Since our transformations
1533 continuously modify the expression, there is an obvious risk that the final
1534 normal form will not be equivalent to the original program: Its meaning could
1536 \item[q:completeness] Is our system \emph{complete}? Since we have a complex
1537 system of transformations, there is an obvious risk that some expressions will
1538 not end up in our intended normal form, because we forgot some transformation.
1539 In other words: Does our transformation system result in our intended normal
1540 form for all possible inputs?
1541 \item[q:determinism] Is our system \emph{deterministic}? Since we have defined
1542 no particular order in which the transformation should be applied, there is an
1543 obvious risk that different transformation orderings will result in
1544 \emph{different} normal forms. They might still both be intended normal forms
1545 (if our system is \emph{complete}) and describe correct hardware (if our
1546 system is \emph{sound}), so this property is less important than the previous
1547 three: The translator would still function properly without it.
1550 \subsection{Graph representation}
1551 Before looking into how to prove these properties, we'll look at our
1552 transformation system from a graph perspective. The nodes of the graph are
1553 all possible Core expressions. The (directed) edges of the graph are
1554 transformations. When a transformation α applies to an expression \lam{A} to
1555 produce an expression \lam{B}, we add an edge from the node for \lam{A} to the
1556 node for \lam{B}, labeled α.
1558 \startuseMPgraphic{TransformGraph}
1562 newCircle.a(btex \lam{(λx.λy. (+) x y) 1} etex);
1563 newCircle.b(btex \lam{λy. (+) 1 y} etex);
1564 newCircle.c(btex \lam{(λx.(+) x) 1} etex);
1565 newCircle.d(btex \lam{(+) 1} etex);
1568 c.c = b.c + (4cm, 0cm);
1569 a.c = midpoint(b.c, c.c) + (0cm, 4cm);
1570 d.c = midpoint(b.c, c.c) - (0cm, 3cm);
1572 % β-conversion between a and b
1573 ncarc.a(a)(b) "name(bred)";
1574 ObjLabel.a(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)";
1575 ncarc.b(b)(a) "name(bexp)", "linestyle(dashed withdots)";
1576 ObjLabel.b(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)";
1578 % η-conversion between a and c
1579 ncarc.a(a)(c) "name(ered)";
1580 ObjLabel.a(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)";
1581 ncarc.c(c)(a) "name(eexp)", "linestyle(dashed withdots)";
1582 ObjLabel.c(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)";
1584 % η-conversion between b and d
1585 ncarc.b(b)(d) "name(ered)";
1586 ObjLabel.b(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)";
1587 ncarc.d(d)(b) "name(eexp)", "linestyle(dashed withdots)";
1588 ObjLabel.d(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)";
1590 % β-conversion between c and d
1591 ncarc.c(c)(d) "name(bred)";
1592 ObjLabel.c(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)";
1593 ncarc.d(d)(c) "name(bexp)", "linestyle(dashed withdots)";
1594 ObjLabel.d(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)";
1596 % Draw objects and lines
1597 drawObj(a, b, c, d);
1600 \placeexample[right][ex:TransformGraph]{Partial graph of a labmda calculus
1601 system with β and η reduction (solid lines) and expansion (dotted lines).}
1602 \boxedgraphic{TransformGraph}
1604 Of course our graph is unbounded, since we can construct an infinite amount of
1605 Core expressions. Also, there might potentially be multiple edges between two
1606 given nodes (with different labels), though seems unlikely to actually happen
1609 See \in{example}[ex:TransformGraph] for the graph representation of a very
1610 simple lambda calculus that contains just the expressions \lam{(λx.λy. (+) x
1611 y) 1}, \lam{λy. (+) 1 y}, \lam{(λx.(+) x) 1} and \lam{(+) 1}. The
1612 transformation system consists of β-reduction and η-reduction (solid edges) or
1613 β-reduction and η-reduction (dotted edges).
1615 TODO: Define β-reduction and η-reduction?
1617 Note that the normal form of such a system consists of the set of nodes
1618 (expressions) without outgoing edges, since those are the expression to which
1619 no transformation applies anymore. We call this set of nodes the \emph{normal
1622 From such a graph, we can derive some properties easily:
1624 \item A system will \emph{terminate} if there is no path of infinite length
1625 in the graph (this includes cycles).
1626 \item Soundness is not easily represented in the graph.
1627 \item A system is \emph{complete} if all of the nodes in the normal set have
1628 the intended normal form. The inverse (that all of the nodes outside of
1629 the normal set are \emph{not} in the intended normal form) is not
1631 \item A system is deterministic if all paths from a node, which end in a node
1632 in the normal set, end at the same node.
1635 When looking at the \in{example}[ex:TransformGraph], we see that the system
1636 terminates for both the reduction and expansion systems (but note that, for
1637 expansion, this is only true because we've limited the possible expressions!
1638 In comlete lambda calculus, there would be a path from \lam{(λx.λy. (+) x y)
1639 1} to \lam{(λx.λy.(λz.(+) z) x y) 1} to \lam{(λx.λy.(λz.(λq.(+) q) z) x y) 1}
1642 If we would consider the system with both expansion and reduction, there would
1643 no longer be termination, since there would be cycles all over the place.
1645 The reduction and expansion systems have a normal set of containing just
1646 \lam{(+) 1} or \lam{(λx.λy. (+) x y) 1} respectively. Since all paths in
1647 either system end up in these normal forms, both systems are \emph{complete}.
1648 Also, since there is only one normal form, it must obviously be
1649 \emph{deterministic} as well.
1651 \subsection{Termination}
1656 \subsection{Soundness}
1657 Needs formal definition of semantics.
1658 Prove for each transformation seperately, implies soundness of the system.
1660 \subsection{Completeness}
1661 Show that any transformation applies to every Core expression that is not
1662 in normal form. To prove: no transformation applies => in intended form.
1663 Show the reverse: Not in intended form => transformation applies.
1665 \subsection{Determinism}