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I am learning semantics of Haskell and there I came across this question:

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I have tried it but still unable to conclude the answer. It will be great if someone explains me how to prove this one. Thank you.

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Waqar Ahmed
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    I'm voting to close this question as off-topic because it is not a programming question. – Renzo Aug 14 '16 at 07:35
  • @Renzo can you please tell me then where to ask this kind of question? Thank you – Waqar Ahmed Aug 14 '16 at 07:39
  • You can try http://cs.stackexchange.com or maybe http://math.stackexchange.com – Renzo Aug 14 '16 at 07:43
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    This kind of questions are bettre suited to ComputerScience StackExchange. In any case: you should start from the definition `<=` is transitive if given `s <= s'` and `s' <= s''` it holds that `s <= s''`. Then you take the definition of `<=` and check which cases are possible. `s <= s'` may hold because `s == s'`, or because there is an `n` such that ... same for `s' <= s''`, you just prove case by case. – Bakuriu Aug 14 '16 at 09:03
  • @Bakuriu.. i have posted question there as well, and thank you again for helping me out. I try to solve it now, and can you explain the 2nd question. How to find the chain of this series. – Waqar Ahmed Aug 14 '16 at 09:13

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  1. Just a sketch -> Since pn(s) for fixed n is morphism Ninf -> N , that is set of Integers into Integer, this proof can be simplified using this relation into proof of transitivity over integers

  2. [1,0,0 .. ] -> [2,0,0 ..] -> [3,0,0 ..] -> ...
    I am sure you can find even more interesting one

Luka Rahne
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  • Can you please explain a bit, how you find the chain. – Waqar Ahmed Aug 14 '16 at 08:05
  • pn([x,0,0 .. ]) == x for any n. If chain x-es satisfy transitivity, chain of sets X-es also. – Luka Rahne Aug 14 '16 at 08:35
  • thank you for this, but i didn't get it, how pn(x,0,0,0..) ? , how do you derive this. can you explain this, as Im still struggling with this question. – Waqar Ahmed Aug 14 '16 at 08:41
  • can this be the chain : [0,0,1,1,2,2...] ? – Waqar Ahmed Aug 14 '16 at 10:16
  • Chan is chan of elements. One element is sequence of integers, so to build chain it is needed chain of sequences of integers. This chan has to ordered (transitivity rule). Comparison of two elements (two sequences of integers) is defined as mapping trough function pn to get integer from sequence of integers and then comparing. So [0,0,1,1,2,2, ..] is just single element, what you need is chain of such elements. – Luka Rahne Aug 14 '16 at 14:20