denis_berthier wrote:And I wonder also which kind of more general proof he was expecting.

I didn't post anything about this, but on further reflection, it turned out that there is a very easy proof for the following: if the pencil marks for a puzzle, contain an embeded minimal BUG pattern, and the BUG pattern has solutions (and so, being minimal, has exactly two solutions) ... then every "rank-0" pattern that justifies the elimination of a candidate from either solution, justifies the elimination of at least one candidate from each solution.

For the proof, I considered a "rank-0" pattern, to be defined by a "base/cover" problem -- not necessarily a "single digit" problem -- that satisfies the "construction rule" from the opening post of

Obi-Wahn's thread,

About the arithmetic of Ultimate Fish, and has "fin sector count" equal to zero (i.e. an equal number of base and cover sectors). The eliminations that it justifies, are those specified in his "exclusion rule".

In other words: N base and N cover sectors are given. Base sectors can overlap. The the list of base sectors is allowed to contain duplicates, and the list of cover sectors is allowed to contains duplicates, and (although it's pointless), a base sector can also be a cover sector. The construction rule requires that each candidate occurs in at least as many cover as base sectors. The exclusion rule (in this case), states that any candidate that occurs in more cover than base sectors, can be eliminated.

Standard patterns of that type, include: hidden and naked N-tuples, unfinned N-fish (including "simple fish"), line/box eliminations, "continuous nice loops", grouped or ungrouped "continuous loop" AICs, SK-loops and variants, Su-de-Coq, and (?) more. "Line of sight" eliminations that follow as singles are placed, are also included, since they can be done as "one base, one cover sector" eliminations before the single is placed.

Note:

Obi-Wahn's thread was short on proofs, and it's an amazing thing (to me), but I only just now realized that (at least in the opening post), his patterns were restricted to being "single digit" patterns. His assertions remain valid, even for

"non-single-digit" base/cover problems. If anyone is interested in proofs, we can open a thread on it, or continue in

Obi-Wahn's thread.

For Denis and

logel:

Obi-Wahn's principle, extended to non-single-digit problems, as far as I know, is the most general case, where a single base\cover proof, can be given, and the valildity of the resulting eliminations, can be "easily verified" -- i.e. it's quick and easy, to verify that the "construction rule" is satisfied, and that the target candidate(s) satisfy the "exclusion rule".

"Hats off" to

Obi-Wahn -- well done

Blue.