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  • $\begingroup$ Your ind term looks a bit odd to me. I would have expected the "exists" term to be something like (∀ b, f b < f a -> P b). Your term says that there exists some b such that P b. $\endgroup$
    – Rupert Swarbrick
    Commented Mar 14, 2022 at 16:30
  • $\begingroup$ Aha sorry, I actually wrote it incorrectly, the ind term should actually be something like (ind : ∀ a : α, f a > 0 → ∃ b : α, f b < f a). But I don't apriori know that P b. I fixed it in the question. Perhaps ind is not the best name for this term. $\endgroup$
    – burek
    Commented Mar 14, 2022 at 16:39
  • $\begingroup$ Well, that hypothesis doesn't give you an inductive step: you still need something that says how to get P a out of P b when f b < f a. $\endgroup$
    – Rupert Swarbrick
    Commented Mar 14, 2022 at 17:00
  • $\begingroup$ Add (∀ a : α, f a = 0 + f a > 0) in your definition of foo. Then pattern match on case. You should provide some term of type ∀ n : ℕ, n = 0 + n > 0. $\endgroup$
    – KANG Rongji
    Commented Mar 14, 2022 at 17:44
  • $\begingroup$ P.S. By + I mean the sum type. I am not sure what symbol Lean uses. $\endgroup$
    – KANG Rongji
    Commented Mar 14, 2022 at 17:46