PL: Lecture #7  Tuesday, January 29th
(text file)

Implementing First Class Functions

PLAI §6 (uses some stuff from PLAI §5, which we do later)

This is a simple plan, but it is directly related to how functions are going to be used in our language. We know that {call {fun {x} E1} E2} is equivalent to a with expression, but the new thing here is that we do allow writing just the {fun ...} expression by itself, and therefore we need to have some meaning for it. The meaning, or the value of this expression, should roughly be “an expression that needs a value to be plugged in for x”. In other words, our language will have these new kinds of values that contain an expression to be evaluated later on.

There are three basic approaches that classify programming languages in relation to how the deal with functions:

  1. First order: functions are not real values. They cannot be used or returned as values by other functions. This means that they cannot be stored in data structures. This is what most “conventional” languages used to have in the past. (You will be implementing such a language in homework 4.)

    An example of such a language is the Beginner Student language that is used in HtDP, where the language is intentionally first-order to help students write correct code (at the early stages where using a function as a value is usually an error). It’s hard to find practical modern languages that fall in this category.

  2. Higher order: functions can receive and return other functions as values. This is what you get with C and modern Fortran.

  3. First class: functions are values with all the rights of other values. In particular, they can be supplied to other functions, returned from functions, stored in data structures, and new functions can be created at run-time. (And most modern languages have first class functions.)

The last category is the most interesting one. Back in the old days, complex expressions were not first-class in that they could not be freely composed. This is still the case in machine-code: as we’ve seen earlier, to compute an expression such as

(-b + sqrt(b^2 - 4*a*c)) / 2a

you have to do something like this:

x = b * b
y = 4 * a
y = y * c
x = x - y
x = sqrt(x)
y = -b
x = y + x
y = 2 * a
s = x / y

In other words, every intermediate value needs to have its own name. But with proper (“high-level”) programming languages (at least most of them…) you can just write the original expression, with no names for these values.

With first-class functions something similar happens — it is possible to have complex expressions that consume and return functions, and they do not need to be named.

What we get with our fun expression (if we can make it work) is exactly this: it generates a function, and you can choose to either bind it to a name, or not. The important thing is that the value exists independently of a name.

This has a major effect on the “personality” of a programming language as we will see. In fact, just adding this feature will make our language much more advanced than languages with just higher-order or first-order functions.


Quick Example: the following is working JavaScript code, that uses first class functions.

function foo(x) {
  function bar(y) { return x + y; }
  return bar;
}
function main() {
  var f = foo(1);
  var g = foo(10);
  return [f(2), g(2)];
}

Note that the above definition of foo does not use an anonymous “lambda expression” — in Racket terms, it’s translated to

(define (foo x)
  (define (bar y) (+ x y))
  bar)

The returned function is not anonymous, but it’s not really named either: the bar name is bound only inside the body of foo, and outside of it that name no longer exists since it’s not its scope. It gets used in the printed form if the function value is displayed, but this is merely a debugging aid. The anonymous lambda version that is common in Racket can be used in JavaScript too:

function foo(x) {
  return function(y) { return x + y; }
}

Side-note: GCC includes extensions that allow internal function definitions, but it still does not have first class functions — trying to do the above is broken:

#include <stdio.h>
typedef int(*int2int)(int);
int2int foo(int x) {
  int bar(int y) { return x + y; }
  return bar;
}
int main() {
  int2int f = foo(1);
  int2int g = foo(10);
  printf(">> %d, %d\n", f(2), g(2));
}

The FLANG Language

Now for the implementation — we call this new language FLANG.

First, the BNF:

<FLANG> ::= <num>
          | { + <FLANG> <FLANG> }
          | { - <FLANG> <FLANG> }
          | { * <FLANG> <FLANG> }
          | { / <FLANG> <FLANG> }
          | { with { <id> <FLANG> } <FLANG> }
          | <id>
          | { fun { <id> } <FLANG> }
          | { call <FLANG> <FLANG> }

And the matching type definition:

(define-type FLANG
  [Num  Number]
  [Add  FLANG FLANG]
  [Sub  FLANG FLANG]
  [Mul  FLANG FLANG]
  [Div  FLANG FLANG]
  [Id  Symbol]
  [With Symbol FLANG FLANG]
  [Fun  Symbol      FLANG] ; No named-expression
  [Call FLANG FLANG])

The parser for this grammar is, as usual, straightforward:

(: parse-sexpr : Sexpr -> FLANG)
;; parses s-expressions into FLANGs
(define (parse-sexpr sexpr)
  (match sexpr
    [(number: n)    (Num n)]
    [(symbol: name) (Id name)]
    [(cons 'with more)
    (match sexpr
      [(list 'with (list (symbol: name) named) body)
        (With name (parse-sexpr named) (parse-sexpr body))]
      [else (error 'parse-sexpr "bad `with' syntax in ~s" sexpr)])]
    [(cons 'fun more)
    (match sexpr
      [(list 'fun (list (symbol: name)) body)
        (Fun name (parse-sexpr body))]
      [else (error 'parse-sexpr "bad `fun' syntax in ~s" sexpr)])]
    [(list '+ lhs rhs) (Add (parse-sexpr lhs) (parse-sexpr rhs))]
    [(list '- lhs rhs) (Sub (parse-sexpr lhs) (parse-sexpr rhs))]
    [(list '* lhs rhs) (Mul (parse-sexpr lhs) (parse-sexpr rhs))]
    [(list '/ lhs rhs) (Div (parse-sexpr lhs) (parse-sexpr rhs))]
    [(list 'call fun arg)
                      (Call (parse-sexpr fun) (parse-sexpr arg))]
    [else (error 'parse-sexpr "bad syntax in ~s" sexpr)]))

We also need to patch up the substitution function to deal with these things. The scoping rule for the new function form is, unsurprisingly, similar to the rule of with, except that there is no extra expression now, and the scoping rule for call is the same as for the arithmetic operators:

N[v/x]                = N

{+ E1 E2}[v/x]        = {+ E1[v/x] E2[v/x]}

{- E1 E2}[v/x]        = {- E1[v/x] E2[v/x]}

{* E1 E2}[v/x]        = {* E1[v/x] E2[v/x]}

{/ E1 E2}[v/x]        = {/ E1[v/x] E2[v/x]}

y[v/x]                = y
x[v/x]                = v

{with {y E1} E2}[v/x] = {with {y E1[v/x]} E2[v/x]}
{with {x E1} E2}[v/x] = {with {x E1[v/x]} E2}

{call E1 E2}[v/x]    = {call E1[v/x] E2[v/x]}

{fun {y} E}[v/x]      = {fun {y} E[v/x]}
{fun {x} E}[v/x]      = {fun {x} E}

And the matching code:

(: subst : FLANG Symbol FLANG -> FLANG)
;; substitutes the second argument with the third argument in the
;; first argument, as per the rules of substitution; the resulting
;; expression contains no free instances of the second argument
(define (subst expr from to)
  (cases expr
    [(Num n) expr]
    [(Add l r) (Add (subst l from to) (subst r from to))]
    [(Sub l r) (Sub (subst l from to) (subst r from to))]
    [(Mul l r) (Mul (subst l from to) (subst r from to))]
    [(Div l r) (Div (subst l from to) (subst r from to))]
    [(Id name) (if (eq? name from) to expr)]
    [(With bound-id named-expr bound-body)
    (With bound-id
          (subst named-expr from to)
          (if (eq? bound-id from)
            bound-body
            (subst bound-body from to)))]
    [(Call l r) (Call (subst l from to) (subst r from to))]
    [(Fun bound-id bound-body)
    (if (eq? bound-id from)
      expr
      (Fun bound-id (subst bound-body from to)))]))

Now, before we start working on an evaluator, we need to decide on what exactly do we use to represent values of this language. Before we had functions, we had only number values and we used Racket numbers to represent them. Now we have two kinds of values — numbers and functions. It seems easy enough to continue using Racket numbers to represent numbers, but what about functions? What should be the result of evaluating

{fun {x} {+ x 1}}

? Well, this is the new toy we have: it should be a function value, which is something that can be used just like numbers, but instead of arithmetic operations, we can call these things. What we need is a way to avoid evaluating the body expression of the function — delay it — and instead use some value that will contain this delayed expression in a way that can be used later.

To accommodate this, we will change our implementation strategy a little: we will use our syntax objects for numbers ((Num n) instead of just n), which will be a little inconvenient when we do the arithmetic operations, but it will simplify life by making it possible to evaluate functions in a similar way: simply return their own syntax object as their values. The syntax object has what we need: the body expression that needs to be evaluated later when the function is called, and it also has the identifier name that should be replaced with the actual input to the function call. This means that evaluating:

(Add (Num 1) (Num 2))

now yields

(Num 3)

and a number (Num 5) evaluates to (Num 5).

In a similar way, (Fun 'x (Num 2)) evaluates to (Fun 'x (Num 2)).

Why would this work? Well, because call will be very similar to with — the only difference is that its arguments are ordered a little differently, being retrieved from the function that is applied and the argument.

The formal evaluation rules are therefore treating functions like numbers, and use the syntax object to represent both values:

eval(N)        = N

eval({+ E1 E2}) = eval(E1) + eval(E2)

eval({- E1 E2}) = eval(E1) - eval(E2)

eval({* E1 E2}) = eval(E1) * eval(E2)

eval({/ E1 E2}) = eval(E1) / eval(E2)

eval(id)        = error!

eval({with {x E1} E2}) = eval(E2[eval(E1)/x])

eval(FUN)      = FUN ; assuming FUN is a function expression

eval({call E1 E2})
                = eval(Ef[eval(E2)/x])  if eval(E1) = {fun {x} Ef}
                = error!                otherwise

Note that the last rule could be written using a translation to a with expression:

eval({call E1 E2})
                = eval({with {x E2} Ef}) if eval(E1) = {fun {x} Ef}
                = error!                otherwise

And alternatively, we could specify with using call and fun:

eval({with {x E1} E2}) = eval({call {fun {x} E2} E1})

There is a small problem in these rules which is intuitively seen by the fact that the evaluation rule for a call is expected to be very similar to the one for arithmetic operations. We now have two kinds of values, so we need to check the arithmetic operation’s arguments too:

eval({+ E1 E2}) = N1 + N2
                    if eval(E1), eval(E2) evaluate to numbers N1, N2
                    otherwise error!
...

The corresponding code is:

(: eval : FLANG -> FLANG)                      ;*** note return type
;; evaluates FLANG expressions by reducing them to *expressions* but
;; only expressions that stand for values: only `Fun`s and `Num`s
(define (eval expr)
  (cases expr
    [(Num n) expr]                            ;*** change here
    [(Add l r) (arith-op + (eval l) (eval r))]
    [(Sub l r) (arith-op - (eval l) (eval r))]
    [(Mul l r) (arith-op * (eval l) (eval r))]
    [(Div l r) (arith-op / (eval l) (eval r))]
    [(With bound-id named-expr bound-body)
    (eval (subst bound-body
                  bound-id
                  (eval named-expr)))]        ;*** no `(Num ...)'
    [(Id name) (error 'eval "free identifier: ~s" name)]
    [(Fun bound-id bound-body) expr]          ;*** similar to `Num'
    [(Call (Fun bound-id bound-body) arg-expr) ;*** nested pattern
    (eval (subst bound-body                  ;*** just like `with'
                  bound-id
                  (eval arg-expr)))]
    [(Call something arg-expr)
    (error 'eval "`call' expects a function, got: ~s" something)]))

Note that the Call case is doing the same thing we do in the With case. In fact, we could have just generated a With expression and evaluate that instead:

    ...
    [(Call (Fun bound-id bound-body) arg-expr)
    (eval (With bound-id arg-expr bound-body))]
    ...

The arith-op function is in charge of checking that the input values are numbers (represented as FLANG numbers), translating them to plain numbers, performing the Racket operation, then re-wrapping the result in a Num. Note how its type indicates that it is a higher-order function.

(: arith-op : (Number Number -> Number) FLANG FLANG -> FLANG)
;; gets a Racket numeric binary operator, and uses it within a FLANG
;; `Num' wrapper (note the H.O. type, and note the hack of the `val`
;; name which is actually an AST that represents a runtime value)
(define (arith-op op val1 val2)
  (Num (op (Num->number val1) (Num->number val2))))

It uses the following function to convert FLANG numbers to Racket numbers. (Note that else is almost always a bad idea since it can prevent the compiler from showing you places to edit code — but this case is an exception since we never want to deal with anything other than Nums.) The reason that this function is relatively trivial is that we chose the easy way and represented numbers using Racket numbers, but we could have used strings or anything else.

(: Num->number : FLANG -> Number)
;; convert a FLANG number to a Racket one
(define (Num->number e)
  (cases e
    [(Num n) n]
    [else (error 'arith-op "expected a number, got: ~s" e)]))

We can also make things a little easier to use if we make run convert the result to a number:

(: run : String -> Number)
;; evaluate a FLANG program contained in a string
(define (run str)
  (let ([result (eval (parse str))])
    (cases result
      [(Num n) n]
      [else (error 'run "evaluation returned a non-number: ~s"
                  result)])))

Adding few simple tests we get:

;; The Flang interpreter

#lang pl

#|
The grammar:
  <FLANG> ::= <num>
            | { + <FLANG> <FLANG> }
            | { - <FLANG> <FLANG> }
            | { * <FLANG> <FLANG> }
            | { / <FLANG> <FLANG> }
            | { with { <id> <FLANG> } <FLANG> }
            | <id>
            | { fun { <id> } <FLANG> }
            | { call <FLANG> <FLANG> }

Evaluation rules:

  subst:
    N[v/x]                = N
    {+ E1 E2}[v/x]        = {+ E1[v/x] E2[v/x]}
    {- E1 E2}[v/x]        = {- E1[v/x] E2[v/x]}
    {* E1 E2}[v/x]        = {* E1[v/x] E2[v/x]}
    {/ E1 E2}[v/x]        = {/ E1[v/x] E2[v/x]}
    y[v/x]                = y
    x[v/x]                = v
    {with {y E1} E2}[v/x] = {with {y E1[v/x]} E2[v/x]} ; if y =/= x
    {with {x E1} E2}[v/x] = {with {x E1[v/x]} E2}
    {call E1 E2}[v/x]    = {call E1[v/x] E2[v/x]}
    {fun {y} E}[v/x]      = {fun {y} E[v/x]}          ; if y =/= x
    {fun {x} E}[v/x]      = {fun {x} E}

  eval:
    eval(N)            = N
    eval({+ E1 E2})    = eval(E1) + eval(E2)  \ if both E1 and E2
    eval({- E1 E2})    = eval(E1) - eval(E2)  \ evaluate to numbers
    eval({* E1 E2})    = eval(E1) * eval(E2)  / otherwise error!
    eval({/ E1 E2})    = eval(E1) / eval(E2)  /
    eval(id)          = error!
    eval({with {x E1} E2}) = eval(E2[eval(E1)/x])
    eval(FUN)          = FUN ; assuming FUN is a function expression
    eval({call E1 E2}) = eval(Ef[eval(E2)/x])
                          if eval(E1)={fun {x} Ef}, otherwise error!
|#

(define-type FLANG
  [Num  Number]
  [Add  FLANG FLANG]
  [Sub  FLANG FLANG]
  [Mul  FLANG FLANG]
  [Div  FLANG FLANG]
  [Id  Symbol]
  [With Symbol FLANG FLANG]
  [Fun  Symbol FLANG]
  [Call FLANG FLANG])

(: parse-sexpr : Sexpr -> FLANG)
;; parses s-expressions into FLANGs
(define (parse-sexpr sexpr)
  (match sexpr
    [(number: n)    (Num n)]
    [(symbol: name) (Id name)]
    [(cons 'with more)
    (match sexpr
      [(list 'with (list (symbol: name) named) body)
        (With name (parse-sexpr named) (parse-sexpr body))]
      [else (error 'parse-sexpr "bad `with' syntax in ~s" sexpr)])]
    [(cons 'fun more)
    (match sexpr
      [(list 'fun (list (symbol: name)) body)
        (Fun name (parse-sexpr body))]
      [else (error 'parse-sexpr "bad `fun' syntax in ~s" sexpr)])]
    [(list '+ lhs rhs) (Add (parse-sexpr lhs) (parse-sexpr rhs))]
    [(list '- lhs rhs) (Sub (parse-sexpr lhs) (parse-sexpr rhs))]
    [(list '* lhs rhs) (Mul (parse-sexpr lhs) (parse-sexpr rhs))]
    [(list '/ lhs rhs) (Div (parse-sexpr lhs) (parse-sexpr rhs))]
    [(list 'call fun arg)
                      (Call (parse-sexpr fun) (parse-sexpr arg))]
    [else (error 'parse-sexpr "bad syntax in ~s" sexpr)]))

(: parse : String -> FLANG)
;; parses a string containing a FLANG expression to a FLANG AST
(define (parse str)
  (parse-sexpr (string->sexpr str)))

(: subst : FLANG Symbol FLANG -> FLANG)
;; substitutes the second argument with the third argument in the
;; first argument, as per the rules of substitution; the resulting
;; expression contains no free instances of the second argument
(define (subst expr from to)
  (cases expr
    [(Num n) expr]
    [(Add l r) (Add (subst l from to) (subst r from to))]
    [(Sub l r) (Sub (subst l from to) (subst r from to))]
    [(Mul l r) (Mul (subst l from to) (subst r from to))]
    [(Div l r) (Div (subst l from to) (subst r from to))]
    [(Id name) (if (eq? name from) to expr)]
    [(With bound-id named-expr bound-body)
    (With bound-id
          (subst named-expr from to)
          (if (eq? bound-id from)
            bound-body
            (subst bound-body from to)))]
    [(Call l r) (Call (subst l from to) (subst r from to))]
    [(Fun bound-id bound-body)
    (if (eq? bound-id from)
      expr
      (Fun bound-id (subst bound-body from to)))]))

(: Num->number : FLANG -> Number)
;; convert a FLANG number to a Racket one
(define (Num->number e)
  (cases e
    [(Num n) n]
    [else (error 'arith-op "expected a number, got: ~s" e)]))

(: arith-op : (Number Number -> Number) FLANG FLANG -> FLANG)
;; gets a Racket numeric binary operator, and uses it within a FLANG
;; `Num' wrapper
(define (arith-op op val1 val2)
  (Num (op (Num->number val1) (Num->number val2))))

(: eval : FLANG -> FLANG)
;; evaluates FLANG expressions by reducing them to *expressions*
(define (eval expr)
  (cases expr
    [(Num n) expr]
    [(Add l r) (arith-op + (eval l) (eval r))]
    [(Sub l r) (arith-op - (eval l) (eval r))]
    [(Mul l r) (arith-op * (eval l) (eval r))]
    [(Div l r) (arith-op / (eval l) (eval r))]
    [(With bound-id named-expr bound-body)
    (eval (subst bound-body
                  bound-id
                  (eval named-expr)))]
    [(Id name) (error 'eval "free identifier: ~s" name)]
    [(Fun bound-id bound-body) expr]
    [(Call (Fun bound-id bound-body) arg-expr)
    (eval (subst bound-body
                  bound-id
                  (eval arg-expr)))]
    [(Call something arg-expr)
    (error 'eval "`call' expects a function, got: ~s" something)]))

(: run : String -> Number)
;; evaluate a FLANG program contained in a string
(define (run str)
  (let ([result (eval (parse str))])
    (cases result
      [(Num n) n]
      [else (error 'run "evaluation returned a non-number: ~s"
                  result)])))

;; tests
(test (run "{call {fun {x} {+ x 1}} 4}")
      => 5)
(test (run "{with {add3 {fun {x} {+ x 3}}}
              {call add3 1}}")
      => 4)
(test (run "{with {add3 {fun {x} {+ x 3}}}
              {with {add1 {fun {x} {+ x 1}}}
                {with {x 3}
                  {call add1 {call add3 x}}}}}")
      => 7)

There is still a problem with this version. First a question — if call is similar to arithmetic operations (and to with in what it actually does), then how come the code is different enough that it doesn’t even need an auxiliary function?

Second question: what should happen if we evaluate these code snippets:

(run "{with {add {fun {x}
                  {fun {y}
                    {+ x y}}}}
      {call {call add 8} 9}}")
(run "{with {identity {fun {x} x}}
        {with {foo {fun {x} {+ x 1}}}
          {call {call identity foo} 123}}}")
(run "{call {call {fun {x} {call x 1}}
                  {fun {x} {fun {y} {+ x y}}}}
            123}")

Third question, what will happen if we do the above?

What we’re missing is an evaluation of the function expression, in case it’s not a literal fun form. The following fixes this:

(: eval : FLANG -> FLANG)
;; evaluates FLANG expressions by reducing them to *expressions*
(define (eval expr)
  (cases expr
    [(Num n) expr]
    [(Add l r) (arith-op + (eval l) (eval r))]
    [(Sub l r) (arith-op - (eval l) (eval r))]
    [(Mul l r) (arith-op * (eval l) (eval r))]
    [(Div l r) (arith-op / (eval l) (eval r))]
    [(With bound-id named-expr bound-body)
    (eval (subst bound-body
                  bound-id
                  (eval named-expr)))]
    [(Id name) (error 'eval "free identifier: ~s" name)]
    [(Fun bound-id bound-body) expr]
    [(Call fun-expr arg-expr)
    (let ([fval (eval fun-expr)])      ;*** need to evaluate this!
      (cases fval
        [(Fun bound-id bound-body)
          (eval (subst bound-body
                      bound-id
                      (eval arg-expr)))]
        [else (error 'eval "`call' expects a function, got: ~s"
                            fval)]))]))

The complete code is:

;; The Flang interpreter

#lang pl

#|
The grammar:
  <FLANG> ::= <num>
            | { + <FLANG> <FLANG> }
            | { - <FLANG> <FLANG> }
            | { * <FLANG> <FLANG> }
            | { / <FLANG> <FLANG> }
            | { with { <id> <FLANG> } <FLANG> }
            | <id>
            | { fun { <id> } <FLANG> }
            | { call <FLANG> <FLANG> }

Evaluation rules:

  subst:
    N[v/x]                = N
    {+ E1 E2}[v/x]        = {+ E1[v/x] E2[v/x]}
    {- E1 E2}[v/x]        = {- E1[v/x] E2[v/x]}
    {* E1 E2}[v/x]        = {* E1[v/x] E2[v/x]}
    {/ E1 E2}[v/x]        = {/ E1[v/x] E2[v/x]}
    y[v/x]                = y
    x[v/x]                = v
    {with {y E1} E2}[v/x] = {with {y E1[v/x]} E2[v/x]} ; if y =/= x
    {with {x E1} E2}[v/x] = {with {x E1[v/x]} E2}
    {call E1 E2}[v/x]    = {call E1[v/x] E2[v/x]}
    {fun {y} E}[v/x]      = {fun {y} E[v/x]}          ; if y =/= x
    {fun {x} E}[v/x]      = {fun {x} E}

  eval:
    eval(N)            = N
    eval({+ E1 E2})    = eval(E1) + eval(E2)  \ if both E1 and E2
    eval({- E1 E2})    = eval(E1) - eval(E2)  \ evaluate to numbers
    eval({* E1 E2})    = eval(E1) * eval(E2)  / otherwise error!
    eval({/ E1 E2})    = eval(E1) / eval(E2)  /
    eval(id)          = error!
    eval({with {x E1} E2}) = eval(E2[eval(E1)/x])
    eval(FUN)          = FUN ; assuming FUN is a function expression
    eval({call E1 E2}) = eval(Ef[eval(E2)/x])
                          if eval(E1)={fun {x} Ef}, otherwise error!
|#

(define-type FLANG
  [Num  Number]
  [Add  FLANG FLANG]
  [Sub  FLANG FLANG]
  [Mul  FLANG FLANG]
  [Div  FLANG FLANG]
  [Id  Symbol]
  [With Symbol FLANG FLANG]
  [Fun  Symbol FLANG]
  [Call FLANG FLANG])

(: parse-sexpr : Sexpr -> FLANG)
;; parses s-expressions into FLANGs
(define (parse-sexpr sexpr)
  (match sexpr
    [(number: n)    (Num n)]
    [(symbol: name) (Id name)]
    [(cons 'with more)
    (match sexpr
      [(list 'with (list (symbol: name) named) body)
        (With name (parse-sexpr named) (parse-sexpr body))]
      [else (error 'parse-sexpr "bad `with' syntax in ~s" sexpr)])]
    [(cons 'fun more)
    (match sexpr
      [(list 'fun (list (symbol: name)) body)
        (Fun name (parse-sexpr body))]
      [else (error 'parse-sexpr "bad `fun' syntax in ~s" sexpr)])]
    [(list '+ lhs rhs) (Add  (parse-sexpr lhs) (parse-sexpr rhs))]
    [(list '- lhs rhs) (Sub  (parse-sexpr lhs) (parse-sexpr rhs))]
    [(list '* lhs rhs) (Mul  (parse-sexpr lhs) (parse-sexpr rhs))]
    [(list '/ lhs rhs) (Div  (parse-sexpr lhs) (parse-sexpr rhs))]
    [(list 'call fun arg)
                      (Call (parse-sexpr fun) (parse-sexpr arg))]
    [else (error 'parse-sexpr "bad syntax in ~s" sexpr)]))

(: parse : String -> FLANG)
;; parses a string containing a FLANG expression to a FLANG AST
(define (parse str)
  (parse-sexpr (string->sexpr str)))

(: subst : FLANG Symbol FLANG -> FLANG)
;; substitutes the second argument with the third argument in the
;; first argument, as per the rules of substitution; the resulting
;; expression contains no free instances of the second argument
(define (subst expr from to)
  (cases expr
    [(Num n) expr]
    [(Add l r) (Add (subst l from to) (subst r from to))]
    [(Sub l r) (Sub (subst l from to) (subst r from to))]
    [(Mul l r) (Mul (subst l from to) (subst r from to))]
    [(Div l r) (Div (subst l from to) (subst r from to))]
    [(Id name) (if (eq? name from) to expr)]
    [(With bound-id named-expr bound-body)
    (With bound-id
          (subst named-expr from to)
          (if (eq? bound-id from)
            bound-body
            (subst bound-body from to)))]
    [(Call l r) (Call (subst l from to) (subst r from to))]
    [(Fun bound-id bound-body)
    (if (eq? bound-id from)
      expr
      (Fun bound-id (subst bound-body from to)))]))

(: Num->number : FLANG -> Number)
;; convert a FLANG number to a Racket one
(define (Num->number e)
  (cases e
    [(Num n) n]
    [else (error 'arith-op "expected a number, got: ~s" e)]))

(: arith-op : (Number Number -> Number) FLANG FLANG -> FLANG)
;; gets a Racket numeric binary operator, and uses it within a FLANG
;; `Num' wrapper
(define (arith-op op val1 val2)
  (Num (op (Num->number val1) (Num->number val2))))

(: eval : FLANG -> FLANG)
;; evaluates FLANG expressions by reducing them to *expressions*
(define (eval expr)
  (cases expr
    [(Num n) expr]
    [(Add l r) (arith-op + (eval l) (eval r))]
    [(Sub l r) (arith-op - (eval l) (eval r))]
    [(Mul l r) (arith-op * (eval l) (eval r))]
    [(Div l r) (arith-op / (eval l) (eval r))]
    [(With bound-id named-expr bound-body)
    (eval (subst bound-body
                  bound-id
                  (eval named-expr)))]
    [(Id name) (error 'eval "free identifier: ~s" name)]
    [(Fun bound-id bound-body) expr]
    [(Call fun-expr arg-expr)
    (let ([fval (eval fun-expr)])
      (cases fval
        [(Fun bound-id bound-body)
          (eval (subst bound-body
                      bound-id
                      (eval arg-expr)))]
        [else (error 'eval "`call' expects a function, got: ~s"
                            fval)]))]))

(: run : String -> Number)
;; evaluate a FLANG program contained in a string
(define (run str)
  (let ([result (eval (parse str))])
    (cases result
      [(Num n) n]
      [else (error 'run "evaluation returned a non-number: ~s"
                  result)])))

;; tests
(test (run "{call {fun {x} {+ x 1}} 4}")
      => 5)
(test (run "{with {add3 {fun {x} {+ x 3}}}
              {call add3 1}}")
      => 4)
(test (run "{with {add3 {fun {x} {+ x 3}}}
              {with {add1 {fun {x} {+ x 1}}}
                {with {x 3}
                  {call add1 {call add3 x}}}}}")
      => 7)
(test (run "{with {add {fun {x}
                        {fun {y}
                          {+ x y}}}}
              {call {call add 8} 9}}")
      => 17)
(test (run "{with {identity {fun {x} x}}
              {with {foo {fun {x} {+ x 1}}}
                {call {call identity foo} 123}}}")
      => 124)
(test (run "{call {call {fun {x} {call x 1}}
                        {fun {x} {fun {y} {+ x y}}}}
                  123}")
      => 124)