induction' n with n ih <;> simp[ball, Nat.lt_succ_iff] at*
p : ℕ → ℕ
n : ℕ
⊢ 0 < n.ball p ↔ ∀ m < n, 0 < p m
case succ
p : ℕ → ℕ
n : ℕ
ih : 0 < rec 1 (fun n ih => (p n).pos.and ih) n ↔ ∀ m < n, 0 < p m
⊢ 0 < p n ∧ 0 < rec 1 (fun n ih => (p n).pos.and ih) n ↔ ∀ m ≤ n, 0 < p m