% At one time the equation below was a candidate single
% axiom for Boolean algebra in terms of the Sheffer stroke.
% This file shows that it is not, by giving a noncommutative
% model.  It's interesting that order 8 succeeds immediately,
% but order 6 takes a long time to fail.  In other words,
% iterate_up_to would be a bad choice for this problem.
%
% See www.mcs.anl.gov/-mccune/papers/basax for more information.

assign(domain_size, 8).

formulas(assumptions).

f(f(f(y,f(z,x)),y),f(x,f(z,y))) = x.  % 8-model

f(A,B) != f(B,A).

% Here are 6 other non-single-axioms (size of counterexample is shown).

% f(f(y,f(f(x,z),y)),f(x,f(z,y))) = x.   % 15  5-model
% f(f(f(y,f(x,z)),y),f(x,f(z,y))) = x.   % 16  5-model
% f(f(f(y,f(x,y)),y),f(x,f(y,z))) = x.   % 5   6-model (b)
% f(f(y,f(y,f(x,y))),f(x,f(y,z))) = x.   % 9   6-model (a)
% f(f(f(y,f(x,x)),y),f(x,f(y,z))) = x.   % 23  6-model (b)
% f(f(y,f(y,f(x,x))),f(x,f(y,z))) = x.   % 25  6-model (a)

% Here is a candidate that turns out to be a single axiom.

% f(f(y,f(f(x,y),y)),f(x,f(z,y))) = x.   % 1   single axiom

end_of_list.
