can somebody write down a quick explanation
for each of the following operators:

'=', '==', '=:=', '\='. '\==', '=\='

thanks, you help is much appreciated!

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Such explanations ahve been written out too often already.
Go to http://www.clip.dia.fi.upm.es/Software/Ciao/ciao_html/ciao_toc.html.
It has an online manual for CIAO and it contains Prolog. At the bottom of that
page there is a Global Index link that makes it easy to find any builtin
you want explanation about. If you find it not helpful, nag the CIAO team :-)

Bart Demoen

From GNU Prolog documentation

'=' prolog unification
'\=' not unifiable (X \= Y :- not(X=Y). )
'==' terms are equals (but doesnot the unification)
'==' terms are not equals
'=:=' arithmetic equal
'=\=' not arithmetic equal

Jean Michel LECONTE