Rules:                              |- Add N1 N2 N3
       -------------- AddZ      ----------------------- AddS
        |- Add Z N Z             |- Add S(N1) N2 S(N3)


Lemma: |- Add N Z N
Proof: By induction on the structure of the natural number N in the 
judgement |- Add N Z N.

Case 1:	

-----
Z Nat

#   Judgement            |    Justification

1)  |- Add Z Z Z         |    AddZ.  This is what we needed to show for this case.

Case 2:    

N Nat
--------
S(N) Nat

#   Judgement            |    Justification

1)  |- Add N Z N         |    By induction hypothesis
2)  |- Add S(N) Z S(N)   |    AddS and (1).  This is what we needed to show for this case.

We have covered all cases. //