natural-number-game

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lemma zero_add (n : ℕ) : 0 + n = n :=
begin
  induction n,
  rw add_zero,
  refl,
  rw add_succ,
  rw n_ih,
  refl,
end

lemma add_assoc (a b c : ℕ) : (a + b) + c = a + (b + c) :=
begin
  induction c,
  repeat {rw add_zero},
  repeat {rw add_succ},
  rw c_ih,
  refl,
end

lemma succ_add (a b : ℕ) : succ a + b = succ (a + b) :=
begin
  induction b with c,
  repeat {rw add_zero},
  repeat {rw add_succ},
  rw b_ih,
  refl,
end

lemma add_comm (a b : ℕ) : a + b = b + a :=
begin
  induction b,
  rw add_zero,
  rw zero_add,
  refl,
  rw add_succ,
  rw succ_add,
  rw b_ih,
  refl,
end

theorem succ_eq_add_one (n : ℕ) : succ n = n + 1 :=
begin
  rw one_eq_succ_zero,
  rw add_succ,
  rw add_zero,
  refl,
end

lemma add_right_comm (a b c : ℕ) : a + b + c = a + c + b :=
begin
  induction c with d,
  repeat {rw add_zero},
  repeat {rw add_succ},
  rw c_ih,
  rw succ_add,
  refl,
end