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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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Dan Christensen

Posts: 2,248
Registered: 7/9/08
Re: Double Induction -- A brief note that may help
Posted: Jul 15, 2013 10:22 PM
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On Monday, July 15, 2013 8:22:09 PM UTC-4, David Melik wrote:
> On Mon, 15 Jul 2013 13:50:39 -0700 (PDT)
>
> Dan Christensen <Dan_Christensen@sympatico.ca> 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?
>
> >
>
>
> Sounds like an interesting idea... is there an example proof somewhere?


I have a formal proof of the concept starting from Peano-like axioms (163 lines) using my DC Proof system. I am working on some sample applications. I hope to post them here over the next week or so.

Dan
Download my DC Proof 2.0 software at http://www.dcproof.com



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