case inl
m✝ n k✝ l m k : ℕ
⊢ m % (0 * k) / 0 = m / 0 % k

case inr
m✝ n✝ k✝ l m n k : ℕ
hn : n > 0
⊢ m % (n * k) / n = m / n % k
m✝ n✝ k✝ l m n k : ℕ
⊢ m % (n * k) / n = m / n % k
rcases Nat.eq_zero_or_pos n with (rfl | hn)
case inr
m✝ n✝ k✝ l m n k : ℕ
hn : n > 0
⊢ m % (n * k) / n = m / n % k
case inl
m✝ n k✝ l m k : ℕ
⊢ m % (0 * k) / 0 = m / 0 % k

case inr
m✝ n✝ k✝ l m n k : ℕ
hn : n > 0
⊢ m % (n * k) / n = m / n % k
simp [mod_zero]
case inr.inl
m✝ n✝ k l m n : ℕ
hn : n > 0
⊢ m % (n * 0) / n = m / n % 0

case inr.inr
m✝ n✝ k✝ l m n k : ℕ
hn : n > 0
hk : k > 0
⊢ m % (n * k) / n = m / n % k
case inr
m✝ n✝ k✝ l m n k : ℕ
hn : n > 0
⊢ m % (n * k) / n = m / n % k
rcases Nat.eq_zero_or_pos k with (rfl | hk)
case inr.inr
m✝ n✝ k✝ l m n k : ℕ
hn : n > 0
hk : k > 0
⊢ m % (n * k) / n = m / n % k
case inr.inl
m✝ n✝ k l m n : ℕ
hn : n > 0
⊢ m % (n * 0) / n = m / n % 0

case inr.inr
m✝ n✝ k✝ l m n k : ℕ
hn : n > 0
hk : k > 0
⊢ m % (n * k) / n = m / n % k
simp [mod_zero]
case inr.inr
m✝ n✝ k✝ l m n k : ℕ
hn : n > 0
hk : k > 0
⊢ m % (n * k) / n = (m % (n * k) + n * k * (m / (n * k))) / n % k
case inr.inr
m✝ n✝ k✝ l m n k : ℕ
hn : n > 0
hk : k > 0
⊢ m % (n * k) / n = m / n % k
conv_rhs => rw [← mod_add_div m (n * k)]
no goals
case inr.inr
m✝ n✝ k✝ l m n k : ℕ
hn : n > 0
hk : k > 0
⊢ m % (n * k) / n = (m % (n * k) + n * k * (m / (n * k))) / n % k
rw [mul_assoc, add_mul_div_left _ _ hn, add_mul_mod_self_left,
  mod_eq_of_lt (Nat.div_lt_of_lt_mul (mod_lt _ (mul_pos hn hk)))]