Question

Can somebody explain step by step type inference in following F# program:

let rec sumList lst =
    match lst with
    | [] -> 0
    | hd :: tl -> hd + sumList tl

I specifically want to see step by step how process of unification in Hindley Milner works.

Answer 1

Fun stuff!

First we invent a generic type for sumList:
x -> y

And get the simple equations:
t(lst) = x;
t(match ...) = y

Now you add the equation:
t(lst) = [a] because of (match lst with [] ...)

Then the equation:
b = t(0) = Int; y = b

Since 0 is a possible result of the match:
c = t(match lst with ...) = b

From the second pattern:
t(lst) = [d];
t(hd) = e;
t(tl) = f;
f = [e];
t(lst) = t(tl);
t(lst) = [t(hd)]

Guess a type (a generic type) for hd:
g = t(hd); e = g

Then we need a type for sumList, so we’ll just get a meaningless function type for now:
h -> i = t(sumList)

So now we know:
h = f;
t(sumList tl) = i

Then from the addition we get:
Addable g;
Addable i;
g = i;
t(hd + sumList tl) = g

Now we can start unification:

t(lst) = t(tl) => [a] = f = [e] => a = e

t(lst) = x = [a] = f = [e]; h = t(tl) = x

t(hd) = g = i /\ i = y => y = t(hd)

x = t(lst) = [t(hd)] /\ t(hd) = y => x = [y]

y = b = Int /\ x = [y] => x = [Int] => t(sumList) = [Int] -> Int

I skipped some trivial steps, but I think you can get how it works.