 158    show (a + b) + succ c = a + (b + succ c)
 159    rw [add_succ, add_succ, add_succ, ih]
 160
 161theorem add_comm (a b : LogicNat) : a + b = b + a := by
 162  induction b with
 163  | identity =>
 164    show a + zero = zero + a
 165    rw [add_zero, zero_add]
 166  | step b ih =>
 167    show a + succ b = succ b + a
 168    rw [add_succ, succ_add, ih]
 169
 170/-- Multiplication by recursion on the second argument. -/
 171def mul : LogicNat → LogicNat → LogicNat
 172  | _, .identity => zero
 173  | n, .step m   => mul n m + n
 174
 175instance : Mul LogicNat := ⟨mul⟩
 176
 177@[simp] theorem mul_def (n m : LogicNat) : n * m = mul n m := rfl
 178
 179@[simp] theorem mul_zero (n : LogicNat) : n * zero = zero := rfl
 180
 181@[simp] theorem mul_succ (n m : LogicNat) : n * succ m = n * m + n := rfl
 182
 183theorem zero_mul (n : LogicNat) : zero * n = zero := by
 184  induction n with
 185  | identity => rfl
 186  | step n ih =>
 187    show zero * succ n = zero
 188    rw [mul_succ, ih, zero_add]
 189
 190theorem mul_one (n : LogicNat) : n * succ zero = n := by
 191  show n * succ zero = n

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