Termination for Wrapped `Fin n` in Lean4

Question

The following example type checks in Lean 4, but I am confused why the termination_by declaration is required.

import Mathlib.Tactic.Linarith

def Idx (n:Nat) := Fin n

def sum(k:Idx n) : Nat :=
  if p:k.val=0 then 0
  else
    have p : k.val-1 < k.val := (Nat.pred_lt p);
    let km1 : Idx n := {val:=k.val-1,isLt:=(by linarith [k.isLt])};
    1+(sum km1)
  termination_by sum k => k.val

Specifically, a very similar example does not require it:

def sum2(k:Fin n) : Nat :=
  if p:k.val=0 then 0
  else
    have p : k.val-1 < k.val := (Nat.pred_lt p);
    let km1 : Fin n := {val:=k.val-1,isLt:=(by linarith [k.isLt])};
    1+(sum2 km1)

Obviously, it has something to do with the fact that I've wrapped the Fin n with Idx n. I tried adding a @[simp] annotation, but that didn't help.

(also am interested in whether the example can be improved)