Pack Exp lt

theorem packExp_lt (α κ : ℕ) (hk : 2 * 2 ^ κ ∣ 2 ^ α) (j : Fin (2 ^ α / 2 ^ κ)) :    packExp α (2 ^ κ) (j : ℕ) < 2 ^ α := by  have hκ : κ + 1 ≤ α := succ_le_of_two_mul_two_pow_dvd hk  have hMle : 2 ^ (α - κ - 1) ≤ 2 ^ (α - 1) := Nat.pow_le_pow_right (by norm_num) (by omega)  have hd2 : (2 : ℕ) ^ α = 2 * 2 ^ (α - 1) := by rw [← pow_succ']; congr 1; omega  have hN : 2 ^ α / 2 ^ κ = 2 * 2 ^ (α - κ - 1) := by    have hfac : (2 : ℕ) ^ κ * (2 * 2 ^ (α - κ - 1)) = 2 ^ α := by      rw [← Nat.mul_assoc, show (2 : ℕ) ^ κ * 2 = 2 ^ (κ + 1) from by rw [pow_succ], ← pow_add]      congr 1; omega    rw [← hfac, Nat.mul_div_cancel_left _ (by positivity : 0 < 2 ^ κ)]  have hMM : 2 ^ α / (2 * 2 ^ κ) = 2 ^ (α - κ - 1) := by    rw [show 2 * 2 ^ κ = 2 ^ (κ + 1) from by rw [pow_succ]; ring,      Nat.pow_div (by omega) (by norm_num), show α - (κ + 1) = α - κ - 1 from by omega]  have hj : (j : ℕ) < 2 * 2 ^ (α - κ - 1) := hN ▸ j.isLt  unfold packExp; rw [hMM]; split_ifs with h <;> omega