Functionally find the index of the minimum value in an array
I think there's no need to use the match keyword here because you're not actually doing any pattern matching, only shadowing the name index with i. I've change it to use if/then/else.
Removing all of the type annotations makes the function more generic and more concise. I've done this below and used the function Array.length to allow the compiler to infer that array is an array. The input type has changed from int [] to 'a [] where 'a has comparison, making the function reusable for other array types. However, this also makes the function a lot slower for some reason so I've marked it as inline to get the re-usability benefits without the performance hit.
I think the findNext function isn't really necessary for readability, but that is probably down to personal preference. I noticed that it makes the function much much slower for a large array so I have removed it. Making it inline has a similar effect.


That leaves us with this:

let inline findMinimumValueIndex array =
    let rec find i minimumValueIndex = 
        if i &gt;= Array.length array then minimumValueIndex // base case
        elif array.[i] &lt; array.[minimumValueIndex] then find (i + 1) i
        else find (i + 1) minimumValueIndex
    find 0 0




If I actually needed to do this in code, I would probably do this instead:

array |&gt; Seq.indexed |&gt; Seq.minBy snd |&gt; fst

This is simple enough that it might not even be worth defining a function for, though it is quite slow compared to my refactoring of your function above.

However, for an empty input array this throws an exception, which isn't very functional. And your function returns 0 for an empty array, which can be ambiguous as it's also the result if the first item is smallest. So you could handle that case and return int option instead of int:

let inline minIndexSafe array =
    if array = [||] then None
    else array |&gt; Seq.indexed |&gt; Seq.minBy snd |&gt; fst |&gt; Some