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Topic: Double Induction -- A brief note that may help
Replies: 13   Last Post: Jul 18, 2013 12:42 AM

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William Elliot

Posts: 2,637
Registered: 1/8/12
Re: Double Induction -- A brief note that may help
Posted: Jul 16, 2013 4:35 AM
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On Mon, 15 Jul 2013, david petry wrote:
> 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)

It correct. From P(1,1) comes P(1,2), P(1,3),.. so by simple induction
for all n P(1,n). What he's omitted is that proving doubled induction
requires simple induction.

> 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)

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