Induction on Dependent Pairs

Stream: lean4

[Calvin Lee]

I'm trying to write a function that will generate all elements of Fin n for any n. To do that I wrote a helper inductive function on Fin n that would return all the elements of Fin n below a certain value. However, even though the first element of the pair decreases, lean cannot prove termination. Is there anything I can do to remedy this? This is the failing function

theorem Nat.ltSucc : (n : Nat) -> Less n (Nat.succ n)
  | 0               => rfl
  | n'@(Nat.succ n) => ltSucc n

def allBelow {n : Nat} : (x : Fin n) -> List (Fin n)
  | x@⟨0, _⟩ => [x]
  | x@⟨Nat.succ m, lt⟩ => x :: allBelow ⟨m, Nat.ltTrans (Nat.ltSucc m) lt⟩

I'm using lean 4 nightly and the specific error is

error: fail to show termination for
  allBelow
with errors
structural recursion cannot be used

well founded recursion has not been implemented yet

[Mario Carneiro]

You have to write this as a function which is structurally recursive, by first untupling

[Mario Carneiro]

theorem Nat.ltSucc : (n : Nat) -> Less n (Nat.succ n)
  | 0               => rfl
  | n'@(Nat.succ n) => ltSucc n

def allBelow {n : Nat} (x : Fin n) : List (Fin n) := aux x.1 x.2
  where aux : (x : Nat) -> x < n -> List (Fin n)
  | 0, h => [⟨0, h⟩]
  | Nat.succ m, lt => ⟨Nat.succ m, lt⟩ :: aux m (Nat.ltTrans (Nat.ltSucc m) lt)