Date: Jul 15, 2013 10:12 PM
Author: David Petry
Subject: Re: Double Induction -- A brief note that may help
On Monday, July 15, 2013 1:50:39 PM UTC-7, Dan Christensen wrote:
> The explanations of double induction online can be quite confusing. No doubt I am re-inventing wheel here, but you may find the following analogy to ordinary induction to be useful.
> With ordinary induction, we want to prove that for all x in N, we have P(x) where P is a unary predicate.
> With double induction, we want to prove that for all x, y in N, we have P(x,y) where P is a binary predicate.
> 1. Base case:
> Ordinary induction: Prove P(1)
> Double induction: Prove P(1,1)
> 2. Inductive step:
> Ordinary induction: For x in N, assume P(x) and prove P(x+1)
> Double induction: For x, y in N, assume P(x,y) and prove P(x+1,y) and P(x,y+1).
I don't think that's right.
Base case: prove for all n, P(1,n) and P(n,1)
Inductive step: Assume P(x,n) for all n <= y, and assume P(n,m) for all m and for all n < x, prove P(x, y+1)