## Stream: new members

### 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


#### Notification Bot (Jan 23 2021 at 02:32):

This topic was moved by Mario Carneiro to #lean4 > Induction on Dependent Pairs

Last updated: May 10 2021 at 00:31 UTC