Stream: lean4

Topic: Induction on Dependent Pairs

Calvin Lee (Jan 23 2021 at 00:14):

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 (Jan 23 2021 at 02:34):

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

Mario Carneiro (Jan 23 2021 at 02:38):

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)

Last updated: May 07 2021 at 13:21 UTC