F# insert/remove item from list

If you need random access in a list, consider using System.Collections.Generic.List<T> or System.Collections.Generic.LinkedList<T> instead of a F# list.

Seems the most idiomatic (not tail recursive):

let rec insert v i l =
    match i, l with
    | 0, xs -> v::xs
    | i, x::xs -> x::insert v (i - 1) xs
    | i, [] -> failwith "index out of range"

let rec remove i l =
    match i, l with
    | 0, x::xs -> xs
    | i, x::xs -> x::remove (i - 1) xs
    | i, [] -> failwith "index out of range"

it seems rather inefficient to traverse the start of the list when I already know the index.

F# lists are singly-linked lists, so you don't have indexed access to them. But most of the time, you don't need it. The majority of indexed operations on arrays are iteration from front to end, which is exactly the most common operation on immutable lists. Its also pretty common to add items to the end of an array, which isn't really the most efficient operation on singly linked lists, but most of the time you can use the "cons and reverse" idiom or use an immutable queue to get the same result.

Arrays and ResizeArrays are really the best choice if you need indexed access, but they aren't immutable. A handful of immutable data structures like VLists allow you to create list-like data structures supporting O(1) cons and O(log n) indexed random access if you really need it.

Removing element at the specified index isn't a typical operation in functional programming - that's why it seems difficult to find the right implementation of these operations. In functional programming, you'll usually process the list element-by-element using recursion, or implement the processing in terms of higher-level declarative operations. Perhaps if you could clarfiy what is your motivation, we can give a better answer.

Anyway, to implement the two operations you wanted, you can use existing higher-order functions (that traverse the entire list a few times, because there is really no good way of doing this without traversing the list):

let removeAt index input =
  input 
  // Associate each element with a boolean flag specifying whether 
  // we want to keep the element in the resulting list
  |> List.mapi (fun i el -> (i <> index, el)) 
  // Remove elements for which the flag is 'false' and drop the flags
  |> List.filter fst |> List.map snd

To insert element to the specified index, you could write:

let insertAt index newEl input =
  // For each element, we generate a list of elements that should
  // replace the original one - either singleton list or two elements
  // for the specified index
  input |> List.mapi (fun i el -> if i = index then [newEl; el] else [el])
        |> List.concat

However, as noted earlier - unless you have a very good reasons for using these functions, you should probably consider describing your goals more broadly and use an alternative (more functional) solution.