natural-number-game

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lemma zero_mul (m : ℕ) : 0 * m = 0 :=
begin
  induction m,
  rw mul_zero,
  refl,
  rw mul_succ,
  rw add_zero,
  rw m_ih,
  refl,
end

lemma mul_one (m : ℕ) : m * 1 = m :=
begin
  rw one_eq_succ_zero,
  rw mul_succ,
  rw mul_zero,
  rw zero_add,
  refl,
end

lemma one_mul (m : ℕ) : 1 * m = m :=
begin
  induction m,
  rw mul_zero,
  refl,
  rw mul_succ,
  rw m_ih,
  rw succ_eq_add_one,
  refl,
end

lemma mul_add (t a b : ℕ) : t * (a + b) = t * a + t * b :=
begin
  induction b,
  rw mul_zero,
  repeat {rw add_zero},
  rw add_succ,
  repeat {rw mul_succ},
  rw ← add_assoc,
  rw b_ih,
  refl,
end

lemma mul_assoc (a b c : ℕ) : (a * b) * c = a * (b * c) :=
begin
  induction c,
  repeat {rw mul_zero},
  repeat {rw mul_succ},
  rw mul_add,
  rw c_ih,
  refl,
end

lemma succ_mul (a b : ℕ) : succ a * b = a * b + b :=
begin
  induction b,
  repeat {rw mul_zero},
  rw add_zero,
  refl,
  repeat {rw mul_succ},
  rw b_ih,
  repeat {rw add_succ},
  rw add_right_comm,
  refl,
end

lemma add_mul (a b t : ℕ) : (a + b) * t = a * t + b * t :=
begin
  induction b,
  rw zero_mul,
  repeat {rw add_zero},
  rw add_succ,
  repeat {rw succ_mul},
  rw b_ih,
  rw add_assoc,
  refl,
end

lemma mul_comm (a b : ℕ) : a * b = b * a :=
begin
  induction b,
  rw mul_zero,
  rw zero_mul,
  refl,
  rw mul_succ,
  rw succ_mul,
  rw b_ih,
  refl,
end

lemma mul_left_comm (a b c : ℕ) : a * (b * c) = b * (a * c) :=
begin
  rw ← mul_assoc,
  rw mul_comm a b,
  rw mul_assoc,
  refl,
end