phantom types for dependent types; sized list; less than

type Z = struct end
type S<'n> = struct end

type List<'n,'b> = L of 'b list
let empty : List<Z, 'b> = L []
let cons (b:'b) ((L bs) : List<'n, 'b>) : List<S<'n>,'b> = L (b::bs)
let head ((L bs):List<S<'n>,'b>) = List.head bs
let tail ((L bs):List<S<'n>,'b>) : List<'n,'b> = L(List.tail bs)

type LT<'n> = I of int
let z : LT<'n> = I 0
let s ((I x):LT<'n>) : LT<S<'n>> = I (x+1)

let rec lookup (i:LT<S<'n>>) (xs:List<S<'n>,'b>) : 'b =
    match i, xs with
    | I 0, L (x::_xs) -> x
    | I lt, L (x::xs) -> lookup (I (lt-1)) (L xs)

[<EntryPoint>]
let main args =
  let l3 : List<S<S<S<Z>>>,int> = cons 1 (cons 2 (cons 3 empty))
  let l4 = cons 0 (cons 1 (cons 2 (cons 3 empty)))
  
  let snd = lookup (s (s z)) l3
  
  printfn "%A; %A; %A" l3 l4 snd
  
  // Compiler error and ide warning
  // head empty
  // tail empty
  // lookup z empty
  // lookup (s z) empty
  // lookup (s(s(s z))) (cons 1 (cons 2 empty))
  0