+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).
+
+ These arguments can be preserved in the program, since they can
+ be translated to input ports later on. However, since we can
+ only connect signals to input ports, these arguments must be
+ reduced to simple variables (for which signals will be
+ produced). This is taken care of by the argument extraction
+ transform.
+ \item Non-runtime representable typed arguments.
+
+ These arguments cannot be preserved in the program, since we
+ cannot represent them as input or output ports in the resulting
+ VHDL. To remove them, we create a specialized version of the
+ called function with these arguments filled in. This is done by
+ the argument propagation transform.
+\stopitemize
+
+When looking at the arguments of a builtin function, we can divide them
+into categories:
+
+\startitemize
+ \item Arguments with a runtime representable type.
+
+ As we have seen with user-defined functions, these arguments can
+ always be reduced to a simple variable reference, by the
+ argument extraction transform. Performing this transform for
+ builtin functions as well, means that the translation of builtin
+ functions can be limited to signal references, instead of
+ needing to support all possible expressions.
+
+ \item Arguments with a function type.
+
+ These arguments are functions passed to higher order builtins,
+ like \lam{map} and \lam{foldl}. Since implementing these
+ functions for arbitrary function-typed expressions (\eg, lambda
+ expressions) is rather comlex, we reduce these arguments to
+ (partial applications of) global functions.
+
+ We can still support arbitrary expressions from the user code,
+ by creating a new global function containing that expression.
+ This way, we can simply replace the argument with a reference to
+ that new function. However, since the expression can contain any
+ number of free variables we also have to include partial
+ applications in our normal form.
+
+ This category of arguments is handled by the function extraction
+ transform.
+ \item Other unrepresentable arguments.
+
+ These arguments can take a few different forms:
+ \startdesc{Type arguments}
+ In the core language, type arguments can only take a single
+ form: A type wrapped in the Type constructor. Also, there is
+ nothing that can be done with type expressions, except for
+ applying functions to them, so we can simply leave type
+ arguments as they are.
+ \stopdesc
+ \startdesc{Dictionary arguments}
+ In the core language, dictionary arguments are used to find
+ operations operating on one of the type arguments (mostly for
+ finding class methods). Since we will not actually evaluatie
+ the function body for builtin functions and can generate
+ code for builtin functions by just looking at the type
+ arguments, these arguments can be ignored and left as they
+ are.
+ \stopdesc
+ \startdesc{Type level arguments}
+ Sometimes, we want to pass a value to a builtin function, but
+ we need to know the value at compile time. Additionally, the
+ value has an impact on the type of the function. This is
+ encoded using type-level values, where the actual value of the
+ argument is not important, but the type encodes some integer,
+ for example. Since the value is not important, the actual form
+ of the expression does not matter either and we can leave
+ these arguments as they are.
+ \stopdesc
+ \startdesc{Other arguments}
+ Technically, there is still a wide array of arguments that can
+ be passed, but does not fall into any of the above categories.
+ However, none of the supported builtin functions requires such
+ an argument. This leaves use with passing unsupported types to
+ a function, such as calling \lam{head} on a list of functions.
+
+ In these cases, it would be impossible to generate hardware
+ for such a function call anyway, so we can ignore these
+ arguments.
+
+ The only way to generate hardware for builtin functions with
+ arguments like these, is to expand the function call into an
+ equivalent core expression (\eg, expand map into a series of
+ function applications). But for now, we choose to simply not
+ support expressions like these.
+ \stopdesc
+
+ From the above, we can conclude that we can simply ignore these
+ other unrepresentable arguments and focus on the first two
+ categories instead.
+\stopitemize
+
+\subsubsection{Argument extraction}
+This transform deals with arguments to functions that
+are of a runtime representable type.
+
+TODO: It seems we can map an expression to a port, not only a signal.
+Perhaps this makes this transformation not needed?
+TODO: Say something about dataconstructors (without arguments, like True
+or False), which are variable references of a runtime representable
+type, but do not result in a signal.
+
+To reduce a complex expression to a simple variable reference, we create
+a new let expression around the application, which binds the complex
+expression to a new variable. The original function is then applied to
+this variable.
+
+\transform{Argument extract}
+{
+\lam{Y} is of a hardware representable type
+
+\lam{Y} is not a variable referene
+
+\conclusion
+
+\trans{X Y}{let z = Y in X z}
+}
+
+\subsubsection{Function extraction}
+This transform deals with function-typed arguments to builtin functions.
+Since these arguments cannot be propagated, we choose to extract them
+into a new global function instead.
+
+Any free variables occuring in the extracted arguments will become
+parameters to the new global function. The original argument is replaced
+with a reference to the new function, applied to any free variables from
+the original argument.
+
+\transform{Function extraction}
+{
+\lam{X} is a (partial application of) a builtin function
+
+\lam{Y} is not an application
+
+\lam{Y} is not a variable reference
+
+\conclusion