Hurray, we can leave behind us the lexical order discussion. There
is a more fundamental flaw in the compare/3 implementation.

/* Scryer Prolog 0.9.1-207 and SWI-Prolog 9.1.7 */
?- X = X-0-9-7-6-5-4-3-2-1, Y = Y-7-5-8-2-4-1, X @< Y.
true.

?- H = H-9-7-6-5-4-3-2-1-0, Z = H-9-7-6-5-4-3-2-1,
Y = Y-7-5-8-2-4-1, Z @< Y.
false.

But X and Z are the same ground terms:

?- X = X-0-9-7-6-5-4-3-2-1, H = H-9-7-6-5-4-3-2-1-0,
Z = H-9-7-6-5-4-3-2-1, X == Z.
true.

So there is a violation of substitution of equals for equals,
in that X == Z and X @< Y did not imply Z @< Y.

An alternative name for substitution of equals for equals
is the part indiscernibility of identicals from Leibniz’s law:

A(s) & s = t => A(t)
https://en.wikipedia.org/wiki/Identity_of_indiscernibles

Currently violated by SWI-Prolog and Scryer Prolog

Mostowski Collapse schrieb am Dienstag, 28. März 2023 um 15:07:25 UTC+2:
> Hurray, we can leave behind us the lexical order discussion. There
> is a more fundamental flaw in the compare/3 implementation.
>
> /* Scryer Prolog 0.9.1-207 and SWI-Prolog 9.1.7 */
> ?- X = X-0-9-7-6-5-4-3-2-1, Y = Y-7-5-8-2-4-1, X @< Y.
> true.
>
> ?- H = H-9-7-6-5-4-3-2-1-0, Z = H-9-7-6-5-4-3-2-1,
> Y = Y-7-5-8-2-4-1, Z @< Y.
> false.
>
> But X and Z are the same ground terms:
>
> ?- X = X-0-9-7-6-5-4-3-2-1, H = H-9-7-6-5-4-3-2-1-0,
> Z = H-9-7-6-5-4-3-2-1, X == Z.
> true.
>
> So there is a violation of substitution of equals for equals,
> in that X == Z and X @< Y did not imply Z @< Y.

I also don't find anything like a compare in neo4j. Its even the case
that the database stumbles over cycles in general, an article from 2019
it reports a performance penalty.

So if a Prolog system could do better and also
offer a compare, that would be really great news!

> **Avoid cycles in Cypher queries**
[https://graphaware.com/neo4j/2019/04/26/avoid-cycles-in-cypher-queries.html](https://graphaware.com/neo4j/2019/04/26/avoid-cycles-in-cypher-queries.html)

I think Kuniaki Mukai mentioned that already, we can order regular
expressions? So we should be also able to order graphs, potentially
with cycles, since a graph can be represented by its Kleene form as

a regular expression? The bug here of SWI-Prolog and Scryer Prolog, is
related to two different regular expressions for the same thing. The
period (_) of a rational number is the star operator _* in a Kleene algebra..

To construct the test case, where SWI-Prolog and Scryer Prolog stumbled
I used two different regular expressions for the same rational number.
So I guess this little bug can be cheaply fixed? Or can it not?

10/81 = 0.(123456790) = 0.12345679(012345679)

Kleene form:
https://en.wikipedia.org/wiki/Kleene%27s_algorithm

Mostowski Collapse schrieb am Dienstag, 28. März 2023 um 19:04:15 UTC+2:
> I also don't find anything like a compare in neo4j. Its even the case
> that the database stumbles over cycles in general, an article from 2019
> it reports a performance penalty.
>
> So if a Prolog system could do better and also
> offer a compare, that would be really great news!
>
> > **Avoid cycles in Cypher queries**
> [https://graphaware.com/neo4j/2019/04/26/avoid-cycles-in-cypher-queries.html](https://graphaware.com/neo4j/2019/04/26/avoid-cycles-in-cypher-queries..html)

Discussion was transitivity violation. Now I found also a counter
example for Scryer Prolog. Interestingly the triple is ok in
SWI-Prolog, its only nok in Scryer Prolog.

/* Scryer Prolog 0.9.1-209 */
?- A = s(s(A, _), A),
B = s(B, 0),
C = s(_S1, _), % where
_S1 = s(_S1, 1), A @< B, B @< C, \+ A @< C.
A = s(s(A,_A),A), B = s(B,0), C = s(s(_S1,1),_B), _S1 = s(_S1,1)..

Mostowski Collapse schrieb am Dienstag, 28. März 2023 um 19:05:05 UTC+2:
> I think Kuniaki Mukai mentioned that already, we can order regular
> expressions? So we should be also able to order graphs, potentially
> with cycles, since a graph can be represented by its Kleene form as
>
> a regular expression? The bug here of SWI-Prolog and Scryer Prolog, is
> related to two different regular expressions for the same thing. The
> period (_) of a rational number is the star operator _* in a Kleene algebra.
>
> To construct the test case, where SWI-Prolog and Scryer Prolog stumbled
> I used two different regular expressions for the same rational number.
> So I guess this little bug can be cheaply fixed? Or can it not?
>
> 10/81 = 0.(123456790) = 0.12345679(012345679)
>
> Kleene form:
> https://en.wikipedia.org/wiki/Kleene%27s_algorithm
> Mostowski Collapse schrieb am Dienstag, 28. März 2023 um 19:04:15 UTC+2:
> > I also don't find anything like a compare in neo4j. Its even the case
> > that the database stumbles over cycles in general, an article from 2019
> > it reports a performance penalty.
> >
> > So if a Prolog system could do better and also
> > offer a compare, that would be really great news!
> >
> > > **Avoid cycles in Cypher queries**
> > [https://graphaware.com/neo4j/2019/04/26/avoid-cycles-in-cypher-queries..html](https://graphaware.com/neo4j/2019/04/26/avoid-cycles-in-cypher-queries.html)