induction l with
| nil => exact sign_one
| cons j l IH =>
  dsimp
  rewrite [List.mem_cons, not_or] at h
  rewrite [sign_swap _ i j h.1, IH h.2, Bool.eq_iff_iff]
  conv =>
    lhs; rewrite [Nat.not_beq_eq_true_eq]
  conv =>
    rhs; rewrite [beq_iff_eq, Nat.succ_eq_add_one, Nat.add_mod]
    change (l.length % 2 + 1) % 2 = 1
  cases l.length.mod_two_eq_zero_or_one
  all_goals (rename_i h; simp only [h])
m n✝ n : Nat
i : Fin n
l : List (Fin n)
h : ¬i ∈ l
⊢ sign (List.foldr (fun j x => swap x i j) 1 l) = (List.length l % 2 == 1)
no goals