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).

> Comments?

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)