Inductive natural : Set :=
| Zero : natural
| Succ : natural -> natural.

Fixpoint add (n m : natural ): natural :=
  match n with
  |  Zero => m
  | Succ n' => Succ (add n' m)
  end.

(** The Associative Law of Addition **)
Theorem add_assoc : forall n m o,
    add ( add n m) o = add n (add m o).
Proof.
  induction n.
  intros.
  reflexivity.
  intros.
  simpl.  
  rewrite IHn.
  reflexivity.
Qed.

Lemma add_Zero : forall n, add n Zero = n.
  induction n; simpl.
  reflexivity.
  rewrite IHn.  
  reflexivity.
Qed.

Lemma add_Succ: forall n m,
    add n (Succ m ) = Succ (add n m).
  induction n; simpl; intros.
  reflexivity.
  rewrite IHn.  
  reflexivity.
Qed.

Theorem add_comm : forall n m,
    add n m = add m n.
Proof.
  induction n;intros;simpl.
  rewrite add_Zero.
  reflexivity.
  rewrite IHn.
  rewrite add_Succ.
  reflexivity.
Qed.