How to figure out the type of this function?

I try to figure out the type of the OCaml function below:

let subst = fun x -> fun y -> fun z -> x z (y z)(* Or more concisely *)let subst x y z = x z (y z)

Here's my derivation.

Step 1, assume the type of x, y, and z, and the output value of subst as below:

x: 'ay: 'bz: 'csubst: 'a -> 'b -> 'c -> 'd

Step 2, notice that both x and y can be applied to z, so we get: (two more types are introduced, 'e and 'f)

'a = 'c -> 'e'b = 'c -> 'f(y z):'f

Step 3, notice that (x z) can be applied to (y z):'f, so we get:

(x z) : 'f -> 'd  (* note that 'd is the type of the final output value *)

Step 4, recall that we have x : 'c -> 'e, and the type of x's output is unique, so with step 3 we get:

'e = 'f -> 'd

Step 5, expand all and we get:

'a = 'c -> 'e = 'c -> 'f -> 'd   (* type of x, actually I am a bit uncertain about the expansion of 'e to 'f -> 'd *)'b = 'c -> 'f  (* type of y *)'c    (* type of z *)

Finally, the type of the function subst is:

subst: ('c -> 'f -> 'd) -> ('c -> 'f) -> 'c ->   'd       ----------------    ----------    ---   -------              x                y          z     output

Replace c, f, d with a, b, c, we get:

subst: ('a -> 'b -> 'c) -> ('a -> 'b) -> 'a -> 'c

I am new to type theory and functional programming. I'm not sure if my derivation is optimal or even correct. Could anyone provide some insight?

Is there any better solution?