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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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gnasher729

Posts: 417
Registered: 10/7/06
Re: Double Induction -- A brief note that may help
Posted: Jul 16, 2013 7:08 PM
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On Wednesday, July 17, 2013 12:02:52 AM UTC+1, christian.bau wrote:

Apologies, I hope I get it right this time:

That would indeed prove P (x, y) for all pairs x, y. However, it may be impossible to prove one of these. For example, I might be able to prove P (x, y) implies P (x+1, y), and I might only be able to prove that if P (x, y) is true for all integers x, then P (0, y + 1) is true.

In general you need to show: There is a way to order all pairs (x, y) in a list in such a way that

1. P (x, y) is true for the first element in the list.
2. For every x, y other than the first element in the list, there is an earlier element (x', y') such that P (x', y') implies P (x, y).



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