Stumped on a Kaldewaij derivation

[elih...@gmail.com]

Hi all,

On p197 of Kaldewaij's book, I'm having trouble understating the proof that begins (for k=0). The hint "case analysis,Q" is not terribly helpful to me. Can anyone help me undertand how he arrives at the values s.0 and n+1?

Eric m.

[Jeremy Weissmann]

I’d be happy to help — do you think you give a simple firewall interface to let me know the relevant properties of the symbols?

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[Jeremy Weissmann]

Okay, I think I understand:  He’s exploring  Q  under  n := n+1  in the case  k = 0 ,  to derive the update to  s.0 .

By the invariant  Q ,  he is entitled to use  s.0 = (min p : 0 ≤ p ≤ n /\ N.p.n ≤ 0 : p) ,  where  N.p.n  gives the number of zeros in the range  [p..n) .

So to calculate  (min p : 0 ≤ p ≤ n /\ N.p.(n+1) ≤ 0 : p) ,  we need to check if  X.n = 0  or not.  If  X.n ≠ 0 ,  then the minimum is still  s.0  (there are no zeros in the range  [s.0..n+1) ,  so no update to  s.0  is needed.  Otherwise  X.n = 0 ,  so the smallest  p  such that  N.p.(n+1) ≤ 0  is  n+1  (that is, an empty range) ,  so we let  s.0 := n+1 .

+j

On Oct 16, 2025, at 05:31, elih...@gmail.com <elih...@gmail.com> wrote:

[elih...@gmail.com]

Thanks, that helps!