Advanced Induction

There are other forms of inductive proofs, which may be required based on what you need to prove. These listed types of inductive proofs should be pretty much all you would ever need, even in graduate courses in Formal Languages.

Complete Induction
Form I:

  1. Prove P(0) = True
  2. Assume that for some arbitray n, P(0)=True, P(1)=True,...P(n)=True, and Prove that P(n+1)=True.
It follows from (1) and (2) that P(n) is True for every n.

Complete Induction
Form II:

In order to prove P(n)=True for every n in N, prove the following assertion:
(*)For every n in N, if for every k<n, P(k)=True, then P(N)=True. From (*) it follows that P(n) is true for every n.
To use (*), assume n is such that whenever k<n then P(k)=True in order to prove that P(n)=True

Nested Induction

Must prove property P(n,m) on two natural numbers n, m in N
To prove P(n,m)=True for every n,m in N, try to use induction on n.
What is the role of m? We may assume m is fixed but arbitrary.
For every m: (For every n: p(n,m) is true).

Example: n+m = m+n.
Proof by induction on n: n+m = m+n.
BASIS: 0+m=m+0
IH: Assume true for n: n+m = m+n
IS: (n+1) + m = (n+m) + 1 = (m+n) + 1 = m + (n+1)
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