ML Expression, help line by line

val y=2;
fun f(x) = x*y;
fun g(h) = let val y=5 in 3+h(y) end;
let val y=3 in g(f) end;

I'm looking for a line by line explanation. I'm new to ML and trying to decipher some online code. Also, a description of the "let/in" commands would be very helpful.

I'm more familiar with ocaml but it all looks the same to me.

val y=2;
fun f(x) = x*y;

The first two lines bind variables y and f. y to an integer 2 and f to a function which takes an integer x and multiplies it by what's bound to y, 2. So you can think of the function f takes some integer and multiplies it by 2. (f(x) = x*2)

fun g(h) = let val y=5
           in
               3+h(y)
           end;

The next line defines a function g which takes some h (which turns out to be a function which takes an integer and returns an integer) and does the following:

1. Binds the integer 5 to a temporary variable y.
   • You can think of the let/in/end syntax as a way to declare a temporary variable which could be used in the expression following in. end just ends the expression. (this is in contrast to ocaml where end is omitted)
2. Returns the sum of 3 plus the function h applying the argument y, or 5.

At a high level, the function g takes some function, applies 5 to that function and adds 3 to the result. (g(h) = 3+h(5))

At this point, three variables are bound in the environment: y = 2, f = function and g = function.

let val y=3
in
    g(f)
end;

Now 3 is bound to a temporary variable y and calls function g with the function f as the argument. You need to remember that when a function is defined, it keeps it's environment along with it so the temporary binding of y here has no affect on the functions g and f. Their behavior does not change.

g (g(h) = 3+h(5)), is called with argument f (f(x) = x*2). Performing the substitutions for parameter h, g becomes 3+((5)*2) which evaluates to 13.

I hope this is clear to you.