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\fnote{\copyright{} Christian Urban, King's College London, 2019, 2023}


% CPS translations
% https://felleisen.org/matthias/4400-s20/lecture17.html

%% pattern matching resources
%% https://www.reddit.com/r/ProgrammingLanguages/comments/g1vno3/beginner_resources_for_compiling_pattern_matching/

\section*{Handout 9 (LLVM, SSA and CPS)}

Reflecting on our two tiny compilers targetting the JVM, the code
generation part was actually not so hard, no? Pretty much just some
post-traversal of the abstract syntax tree, yes? One of the reasons
for this ease is that the JVM is a stack-based virtual machine and it
is therefore not hard to translate deeply-nested arithmetic
expressions into a sequence of instructions manipulating the
stack. The problem is that ``real'' CPUs, although supporting stack
operations, are not really designed to be \emph{stack machines}.  The
design of CPUs is more like: Here are some operations and a chunk of
memory---compiler, or better compiler writers, do something with
them. Consequently, modern compilers need to go the extra mile in
order to generate code that is much easier and faster to process by
actual CPUs, rather than running code on virtual machines that
introduce an additional layer of indirection. To make this all
tractable for this module, we target the LLVM Intermediate
Language. In this way we can take advantage of the tools coming with
LLVM.\footnote{\url{http://llvm.org}} For example we do not have to
worry about things like register allocations. By using LLVM, however,
we also have to pay price in the sense that generating code gets
harder\ldots{}unfortunately.

\subsection*{LLVM and LLVM-IR}

\noindent LLVM is a beautiful example
that projects from Academia can make a difference in the World. LLVM
started in 2000 as a project by two researchers at the  University of
Illinois at Urbana-Champaign. At the time the behemoth of compilers was
gcc with its myriad of front-ends for different programming languages (C++, Fortran,
Ada, Go, Objective-C, Pascal etc). The problem was that gcc morphed over
time into a monolithic gigantic piece of m\ldots ehm complicated
software, which you
could not mess about in an afternoon. In contrast, LLVM is designed to
be a modular suite of tools with which you can play around easily and
try out something new. LLVM became a big player once Apple hired one of
the original developers (I cannot remember the reason why Apple did not
want to use gcc, but maybe they were also just disgusted by gcc's big
monolithic codebase). Anyway, LLVM is now the big player and gcc is more
or less legacy. This does not mean that programming languages like C and
C++ are dying out any time soon---they are nicely supported by LLVM.

We will target the LLVM Intermediate Language, or LLVM Intermediate
Representation (short LLVM-IR). The LLVM-IR looks very similar to the
assembly language of Jasmin and Krakatau. Targetting LLVM-IR will also
allow us to benefit from the modular structure of the LLVM compiler
and we can let, for example, the compiler generate code for different
CPUs, say X86 or ARM.  That means we can be agnostic about where our
code is actually going to run.\footnote{Anybody want to try to run our
  programs on Android phones?}  We can also be somewhat ignorant about
optimising our code and about allocating memory efficiently: the LLVM
tools will take care of all this.

However, what we have to do in order to make LLVM to play ball is to
generate code in \emph{Static Single-Assignment} format (short SSA),
because that is what the LLVM-IR expects from us. A reason why LLVM
uses the SSA format, rather than JVM-like stack instructions, is that
stack instructions are difficult to optimise---you cannot just
re-arrange instructions without messing about with what is calculated
on the stack. Also it is hard to find out if all the calculations on
the stack are actually necessary and not by chance dead code. The JVM
has for all these obstacles sophisticated machinery to make such
``high-level'' code still run fast, but let's say that for the sake of
argument we do not want to rely on it. We want to generate fast code
ourselves. This means we have to work around the intricacies of what
instructions CPUs can actually process fast. This is what the SSA
format is designed for.


The main idea behind the SSA format is to use very simple variable
assignments where every tmp-variable is assigned only once. The
assignments also need to be primitive in the sense that they can be
just simple operations like addition, multiplication, jumps,
comparisons, function calls and so on.  Say, we have an expression
$((1 + a) + (foo(3 + 2) + (b * 5)))$, then the corresponding SSA format is
 
\begin{lstlisting}[language=LLVMIR,numbers=left]
let tmp0 = add 1 a in
let tmp1 = add 3 2 in  
let tmp2 = call foo(tmp1)  
let tmp3 = mul b 5 in 
let tmp4 = add tmp2 tmp3 in 
let tmp5 = add tmp0 tmp4 in
  return tmp5 
\end{lstlisting}

\noindent where every tmp-variable is used only once (we could, for
example, not write \texttt{tmp1 = add tmp2 tmp3} in Line 5 even if this
would not change the overall result).

There are sophisticated algorithms for imperative languages, like C,
that efficiently transform a high-level program into SSA format. But
we can ignore them here. We want to compile a functional language and
there things get much more interesting than just sophisticated. We
will need to have a look at CPS translations, where the CPS stands for
Continuation-Passing-Style---basically black programming art or
abracadabra programming. So sit tight.

\subsection*{LLVM-IR}

Before we start, let's however first have a look at the \emph{LLVM Intermediate
Representation} in more detail. The LLVM-IR is in between the frontends
and backends of the LLVM framework. It allows compilation of multiple
source languages to multiple targets. It is also the place where most of
the target independent optimisations are performed. 

What is good about our toy Fun-language is that it basically only
contains expressions (be they arithmetic expressions, boolean
expressions or if-expressions). The exception are function definitions.
Luckily, for them we can use the mechanism of defining functions in the
LLVM-IR (this is similar to using JVM methods for functions in our
earlier compiler). For example the simple Fun-program 


\begin{lstlisting}[language=Scala,numbers=none]
def sqr(x) = x * x
\end{lstlisting}

\noindent
can be compiled to the following LLVM-IR function:

\begin{lstlisting}[language=LLVM]
define i32 @sqr(i32 %x) {
   %tmp = mul i32 %x, %x
   ret i32 %tmp
}    
\end{lstlisting}

\noindent First notice that all ``local'' variable names, in this case
\texttt{x} and \texttt{tmp}, are prefixed with \texttt{\%} in the
LLVM-IR.  Temporary variables can be named with an identifier, such as
\texttt{tmp}, or numbers. In contrast, function names, since they are
``global'', need to be prefixed with an @-symbol. Also, the LLVM-IR is
a fully typed language. The \texttt{i32} type stands for 32-bit
integers. There are also types for 64-bit integers (\texttt{i64}),
chars (\texttt{i8}), floats, arrays and even pointer types. In the
code above, \texttt{sqr} takes an argument of type \texttt{i32} and
produces a result of type \texttt{i32} (the result type is in front of
the function name, like in C). Each arithmetic operation, for example
addition and multiplication, are also prefixed with the type they
operate on. Obviously these types need to match up\ldots{} but since
we have in our programs only integers, for the moment \texttt{i32}
everywhere will do. We do not have to generate any other types, but
obviously this is a limitation in our Fun-language (and which we
lift in the final CW).
 
There are a few interesting instructions in the LLVM-IR which are quite
different than what we have seen in the JVM. Can you remember the
kerfuffle we had to go through with boolean expressions and negating the
condition? In the LLVM-IR, branching  if-conditions are implemented
differently: there is a separate \texttt{br}-instruction as follows:

\begin{lstlisting}[language=LLVM]
br i1 %var, label %if_br, label %else_br
\end{lstlisting}

\noindent
The type \texttt{i1} stands for booleans. If the variable is true,
then this instruction jumps to the if-branch, which needs an explicit
label; otherwise it jumps to the else-branch, again with its own
label. This allows us to keep the meaning of the boolean expression
``as is'' when compiling if's---thanks god no more negating of
booleans.

A value of type boolean is generated in the LLVM-IR by the
\texttt{icmp}-instruction. This instruction is for integers (hence the
\texttt{i}) and takes the comparison operation as argument. For example

\begin{lstlisting}[language=LLVM]
icmp eq  i32 %x, %y     ; for equal
icmp sle i32 %x, %y     ; signed less or equal
icmp slt i32 %x, %y     ; signed less than
icmp ult i32 %x, %y     ; unsigned less than 
\end{lstlisting}

\noindent
Note that in some operations the LLVM-IR distinguishes between signed and 
unsigned representations of integers. I assume you know what this means,
otherwise just ask.

It is also easy to call another function in LLVM-IR: as can be 
seen from Figure~\ref{lli} we can just call a function with the 
instruction \texttt{call} and can also assign the result to 
a variable. The syntax is as follows

\begin{lstlisting}[language=LLVM]
%var = call i32 @foo(...args...)
\end{lstlisting}

\noindent
where the arguments can only be simple variables and numbers, but not compound
expressions.

Conveniently, you can use the program \texttt{lli}, which comes with
LLVM, to interpret programs written in the LLVM-IR. So you can easily
check whether the code you produced actually works. To get a running
program that does something interesting you need to add some boilerplate
about printing out numbers and a main-function that is the entry point
for the program (see Figure~\ref{lli} for a complete listing of the square function). Again
this is very similar to the boilerplate we needed to add in our JVM
compiler. 

You can generate a binary for the program in Figure~\ref{lli} by using
the \texttt{llc}-compiler and then \texttt{gcc}/\texttt{clang}, whereby \texttt{llc} generates
an object file and \texttt{gcc} (that is actually \texttt{clang} on my Mac) generates the
executable binary:

\begin{lstlisting}[language=bash,numbers=none]
llc -filetype=obj sqr.ll
gcc sqr.o -o a.out
./a.out
> 25
\end{lstlisting}

\begin{figure}[t]\small 
\lstinputlisting[language=LLVM,numbers=left]{../progs/sqr.ll}
\caption{An LLVM-IR program for calculating the square function. It
  calls the sqr-function in \texttt{@main} with the argument \texttt{5}
  (Line 20). The
  code for the \texttt{sqr} function is in Lines 13 -- 16. It stores
  the result of sqr in the variable called \texttt{\%i} and then
  prints out the contents of this variable in Line 21. The other
  code is boilerplate for printing out integers.\label{lli}}
\end{figure}   


    
\subsection*{Our Own Intermediate Language}

Let's get back to our compiler: Remember compilers have to solve the
problem of bridging the gap between ``high-level'' programs and
``low-level'' hardware. If the gap is too wide for one step, then a
good strategy is to lay a stepping stone somewhere in between. The
LLVM-IR itself is such a stepping stone to make the task of generating
and optimising code easier. Like a real compiler we will use our own
stepping stone which I call the \emph{K-language}.

\begin{center}
  \begin{tikzpicture}[scale=0.9,font=\bf,
                      node/.style={
                      rectangle,rounded corners=3mm,
                      ultra thick,draw=black!50,minimum height=16mm, 
                      minimum width=20mm,
                      top color=white,bottom color=black!20}]

  %\node (0) at (-3,0) {};  
  \node (A) at (0.0,0) [node,text width=1.6cm,text centered,label=above:{\small\it{}source}] {Fun-Language};
  \node (B) at (3.5,0) [node,text width=1.6cm,text centered,label=above:{\small\it{}stepping stone}] {K-Language};
  \node (C) at (7.0,0) [node,text width=1.6cm,text centered,label=above:{\small\it{}target}] {LLVM-IR};
 
  \draw [->,line width=2.5mm] (A) -- node [above,pos=0.35] {} (B); 
  \draw [->,line width=2.5mm] (B) -- node [above,pos=0.35] {} (C); 
  \end{tikzpicture}
  \end{center}

  \noindent
  To see why we need a stepping stone for generating code in SSA-format,
  considder again arithmetic expressions such as
  $((1 + a) + (3 + (b * 5)))$. They need to be broken up into smaller
  ``atomic'' steps, like so
 
\begin{lstlisting}[language=LLVMIR,numbers=none]
let tmp0 = add 1 a in   
let tmp1 = mul b 5 in 
let tmp2 = add 3 tmp1 in 
let tmp3 = add tmp0 tmp2 in
  return tmp3 
\end{lstlisting}

\noindent
Here \texttt{tmp3} will contain the result of what the whole
expression stands for. In each individual step we can only perform an
``atomic'' or ``trival'' operation, like addition or multiplication of
a number or a variable.  We are not allowed to have for example a
nested addition or an if-condition on the right-hand side of an
assignment. Such constraints are forced upon us because of how the SSA
format works in the LLVM-IR. To simplify matters we represent
assignments with two kinds of syntactic entities, namely
\emph{K-values} and \emph{K-expressions}. K-values are all ``things''
that can appear on the right-hand side of an equal. Say we have
the following definition for K-values:

\begin{lstlisting}[language=Scala,numbers=none]
enum KVal {
  case KVar(s: String)
  case KNum(n: Int)
  case KAop(op: String, v1: KVal, v2: KVal)
  case KCall(fname: String, args: List[KVal])
}
\end{lstlisting}

\noindent
where a K-value can be a variable, a number or a ``trivial'' binary
operation, such as addition or multiplication. The two arguments of a
binary operation need to be K-values as well. Finally, we have
function calls, but again each argument of the function call needs to
be a K-value.  One point we need to be careful, however, is that the
arguments of the binary operations and function calls are in fact only
variables or numbers. To encode this constraint into the type-system
of Scala would make things too complicated---to satisfy this
constraint is therefore on us. For our Fun-language, we will later on
consider some further K-values.


Our K-expressions will encode the \texttt{let} and the \texttt{return}
from the SSA-format (again for the moment we only consider these two
constructors---later on we will have if-branches as well).

\begin{lstlisting}[language=Scala,numbers=none]
enum KExp {
  case KLet(x: String, v: KVal, e: KExp)
  case KReturn(v: KVal)
}
\end{lstlisting}

\noindent
By having in \texttt{KLet} taking first a string (standing for an
intermediate variable) and second a value, we can fulfil the SSA
constraint in assignments ``by construction''---there is no way we
could write anything else than a K-value.  Note that the third
argument of a \texttt{KLet} is again a K-expression, meaning either
another \texttt{KLet} or a \texttt{KReturn}. In this way we can
construct a sequence of computations and return a final result.  Of
course we also have to ensure that all intermediary computations are
given (fresh) names---we will use an (ugly) counter for this.

To sum up, K-values are the atomic operations that can be on the
right-hand side of assignemnts. The K-language is restricted such that
it is easy to generate the SSA format for the LLVM-IR. What remains to
be done is a translation of our Fun-language into the K-language. The
main difficulty is that we need to order the computationa---in teh
K-language we only have linear sequence of instructions to need to be
followed. Before we explain this, we have a look at some
CPS-translation.




\subsection*{CPS-Translations}

CPS stands for Continuation-Passing-Style. It is a kind of programming
technique often used in advanced functional programming. Before we delve
into the CPS-translation for our Fun-language, let us look at 
CPS-versions of some well-known functions. Consider

\begin{lstlisting}[language=Scala, numbers=none]
def fact(n: Int) : Int = 
  if (n == 0) 1 else n * fact(n - 1) 
\end{lstlisting}

\noindent 
This is clearly the usual factorial function. But now consider the
following version of the factorial function:

\begin{lstlisting}[language=Scala, numbers=none]
def factC(n: Int, ret: Int => Int) : Int = 
  if (n == 0) ret(1) 
  else factC(n - 1, x => ret(n * x)) 

factC(3, identity)
\end{lstlisting}

\noindent
This function is called with the number, in this case 3, and the
identity-function (which returns just its input). The recursive
calls are:

\begin{lstlisting}[language=Scala, numbers=none]
factC(2, x => identity(3 * x))
factC(1, x => identity(3 * (2 * x)))
factC(0, x => identity(3 * (2 * (1 * x))))
\end{lstlisting}

\noindent
Having reached 0, we get out of the recursion and apply 1 to the
continuation (see if-branch above). This gives

\begin{lstlisting}[language=Scala, numbers=none]
identity(3 * (2 * (1 * 1)))
= 3 * (2 * (1 * 1))
= 6
\end{lstlisting}

\noindent
which is the expected result. If this looks somewhat familiar to you,
than this is because functions with continuations can be
seen as a kind of generalisation of tail-recursive functions.
So far we did not look at this generalisation in earnest.
Anyway
notice how the continuations is ``stacked up'' during the recursion and
then ``unrolled'' when we apply 1 to the continuation. Interestingly, we
can do something similar to the Fibonacci function where in the traditional
version we have two recursive calls. Consider the following function

\begin{lstlisting}[language=Scala, numbers=none]
def fibC(n: Int, ret: Int => Int) : Int = 
  if (n == 0 || n == 1) ret(1) 
  else fibC(n - 1,
             r1 => fibC(n - 2,
               r2 => ret(r1 + r2)))
\end{lstlisting}

\noindent
Here the continuation is a nested function essentially wrapping up 
the second recursive call. Let us check how the recursion unfolds
when called with 3 and the identity function:

\begin{lstlisting}[language=Scala, numbers=none]
fibC(3, id)
fibC(2, r1 => fibC(1, r2 => id(r1 + r2)))
fibC(1, r1 => 
   fibC(0, r2 => fibC(1, r2a => id((r1 + r2) + r2a))))
fibC(0, r2 => fibC(1, r2a => id((1 + r2) + r2a)))
fibC(1, r2a => id((1 + 1) + r2a))
id((1 + 1) + 1)
(1 + 1) + 1
3
\end{lstlisting}

\noindent
The point of this section is that you are playing around with these
simple definitions and make sure they calculate the expected result.
In the next step, you should understand \emph{how} these functions
calculate the result with the continuations. 


\section*{Worked Example}

Let us now come back to the CPS-translations for the Fun-language. The
main difficulty of generating instructions in SSA format is that large
compound expressions need to be broken up into smaller pieces and
intermediate results need to be chained into later instructions. To do
this conveniently, we use the CPS-translation mechanism. What continuations
essentially solve is the following problem: Given an expressions

\begin{equation}\label{exp}
(1 + 2) * (3 + 4)
\end{equation}  

\noindent
which of the subexpressions should be calculated first? We just arbitrarily
going to decide that it will be from left to right. This means we have
to tear the expression shown in \eqref{exp} apart as follows:

\[
(1 + 2) \qquad\text{and}\qquad  \Box * (3 + 4)
\]  

\noindent
The first one will give us a result, but the second one is not
really an expression that makes sense. It will only make sense
as the next step in the computation when we fill-in the result of
$1+2$ into the ``hole'' indicated by $\Box$. Such incomplete
expressions can be represented in Scala as functions

\begin{lstlisting}[language=Scala, numbers=none]
r => r * (3 + 4)
\end{lstlisting}  

\noindent
where \texttt{r} is any result that has been computed earlier. By the
way in lambda-calculus notation we would write
$\lambda r.\, r * (3 + 4)$ for the same function. To sum up, we use
functions (``continuations'') to represent what is coming next in a
sequence of instructions. In our case, continuations are functions of
type \code{KVal} to \code{KExp}. They can be seen as a sequence of
\code{KLet}s where there is a ``hole'' that needs to be
filled. Consider for example

\begin{lstlisting}[language=LLVMIR,numbers=left,escapeinside={(*@}{@*)}]
let tmp0 = add 1 a in   
let tmp1 = mul (*@$\Box$@*) 5 in 
let tmp2 = add 3 tmp1 in 
let tmp3 = add tmp0 tmp2 in
  retrun tmp3 
\end{lstlisting}

\noindent
where in the second line is a $\Box$ which still expects a \code{KVal}
to be filled in before it becomes a ``proper'' \code{KExp}. When we 
apply an argument to the continuation (remember they are functions)
we essentially fill something into the corresponding hole.

Lets look at concrete code. To simplify matters first, 
suppose our source language consists just of three kinds of expressions

\begin{lstlisting}[language=Scala,numbers=none]
enum Expr {
    case Num(n: Int)
    case Bop(op: String, e1: Expr, e2: Expr)
    case Call(fname: String, args: List[Expr])
}
\end{lstlisting}

\noindent
The code of the CPS-translation is then of the form

\begin{lstlisting}[language=Scala,numbers=none]
def CPS(e: Exp)(k: KVal => KExp) : KExp = 
  e match { ... }
\end{lstlisting}

\noindent 
where \code{k} is the continuation and \code{e} is the expression to
be compiled. In case we have numbers, we can just pass them to the
continuations because numbers need not be further teared apart as they
are already atomic. Passing the number to the continuation means we 
apply the continuation like 

\begin{lstlisting}[language=Scala,numbers=none]
  case Num(i) => k(KNum(i))
\end{lstlisting}

\noindent This would just fill in the $\Box$ in a \code{KLet}-expression.
There is no need to create a temporary variable for simple numbers.
More interesting is the case for arithmetic operations.

\begin{lstlisting}[language=Scala,numbers=none,xleftmargin=0mm]
case Aop(op, e1, e2) => {
  val z = Fresh("tmp")
  CPS(e1)(r1 => 
    CPS(e2)(r2 => KLet(z, KAop(op, r1, r2), k(KVar(z)))))
}
\end{lstlisting}

\noindent
What we essentially have to do in this case is the following: compile
the subexpressions \texttt{e1} and \texttt{e2}. They will produce some
result that is stored in two temporary variables (assuming they are more
complicated than just numbers). We need to use these two temporary
variables and feed them into a new assignment of the form

\begin{lstlisting}[language=LLVMIR,numbers=none,escapeinside={(*@}{@*)}]
let z = op (*@$\Box_\texttt{r1}$@*) (*@$\Box_\texttt{r2}$@*) in
...
\end{lstlisting}

\noindent
Note that this assignment has two ``holes'', one for the left
subexpression and one for the right subexpression. We can fill both
holes by calling the CPS function twice---this is a bit of the black
art in CPS. We can use the second call of CPS as the continuation
of the first call: The reason is that

\begin{lstlisting}[language=Scala,numbers=none,xleftmargin=0mm]
r1 => CPS(e2)(r2 => KLet(z, KAop(op, r1, r2), k(KVar(z))))
\end{lstlisting}

\noindent is a function from \code{KVal} to \code{KExp}. Once
we created the assignment with the fresh temporary variable
\texttt{z}, we need to ``communicate'' that the result of the
computation of the arithmetic expression is stored in \texttt{z} to the
computations that follow. In this way we apply the continuation \texttt{k} with this
new variable (essentially we are plugging in a hole further down the line).

The last case we need to consider in our small expression language are
function calls.

\begin{lstlisting}[language=Scala,numbers=none,xleftmargin=0mm]
case Call(fname, args) => {
  def aux(args: List[Expr], vs: List[KVal]): KExp = args match {
       case Nil => {
         val z = Fresh("tmp")
         KLet(z, KCall(fname, vs), k(KVar(z)))
       }
       case a::as => CPS(a)(r => aux(as, vs ::: List(r)))
  }
  aux(args, Nil)  
}
\end{lstlisting}

\noindent
For them we introduce an auxiliary function that compiles first all
function arguments---remember in our source language we can have calls
such as $foo(3 + 4, g(3))$ where we first have to create temporary
variables (and computations) for each argument. Therefore \texttt{aux}
analyses the arguments from left to right. In case there is an argument
\texttt{a} on the front of the list (the case \texttt{a::as}), we call
CPS recursively for the corresponding subexpression. The temporary variable
containing the result for this argument we add to the end of the K-values we
have already analysed before. Again very conveniently we can use the
recursive call to \texttt{aux} as the continuation for the computations
that follow. If we reach the end of the argument list (the \texttt{Nil}-case),
then we collect all the K-values \texttt{v1} to \texttt{vn} and call the
function in the K-language like so


\begin{lstlisting}[language=LLVMIR,numbers=none,escapeinside={(*@}{@*)}]
let z = call foo(v1,...,vn) in
...
\end{lstlisting}

\noindent
Again we need to communicate the result of the function call, namely the
fresh temporary variable \texttt{z}, to the rest of the computation. Therefore
we apply the continuation \texttt{k} with this variable.



\begin{figure}[p]\small
  \lstinputlisting[numbers=left,firstline=1,lastline=24]{../progs/fun/simple-cps.sc}
  \hspace{10mm}...
  \lstinputlisting[numbers=left,firstline=32,firstnumber=32,lastline=51]{../progs/fun/simple-cps.sc}
\caption{CPS-translation for a simple expression language.\label{cps}}
\end{figure}

The last question we need to answer in the code (see Figure~\ref{cps}) is how we
start the CPS-translation function, or more precisely with which continuation we
should start CPS? It turns out that this initial continuation will be the last one that is
called. What should be the last step in the computation? Yes, we need to return the
temporary variable where the last result is stored (if it is a simple number, then we can
just return this number). Therefore we cal CPS with the code

\begin{lstlisting}[language=Scala,numbers=none]
def CPSi(e: Expr) : KExp = CPS(e)(KReturn(_))
\end{lstlisting}

\noindent
where we give the function \code{KReturn(_)}. 


\begin{figure}[p]\small
\begin{lstlisting}[language=Scala,numbers=none]
// Fun language (expressions)
abstract class Exp 
abstract class BExp 

case class Call(name: String, args: List[Exp]) extends Exp
case class If(a: BExp, e1: Exp, e2: Exp) extends Exp
case class Write(e: Exp) extends Exp
case class Var(s: String) extends Exp
case class Num(i: Int) extends Exp
case class Aop(o: String, a1: Exp, a2: Exp) extends Exp
case class Sequence(e1: Exp, e2: Exp) extends Exp
case class Bop(o: String, a1: Exp, a2: Exp) extends BExp  



// K-language (K-expressions, K-values)
abstract class KExp
abstract class KVal

case class KVar(s: String) extends KVal
case class KNum(i: Int) extends KVal
case class Kop(o: String, v1: KVal, v2: KVal) extends KVal
case class KCall(o: String, vrs: List[KVal]) extends KVal
case class KWrite(v: KVal) extends KVal

case class KIf(x1: String, e1: KExp, e2: KExp) extends KExp
case class KLet(x: String, v: KVal, e: KExp) extends KExp
case class KReturn(v: KVal) extends KExp
\end{lstlisting}
\caption{Abstract syntax trees for the Fun-language.\label{absfun}}
\end{figure}



\noindent
\end{document}


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