if(Prover9).

assign(order, kbo).

assign(eq_defs,fold).

end_if.

formulas(assumptions).

% identity element

0 * x = x.

x * 0 = x.

% quasigroup

x * (x \ y) = y.

x \ (x * y) = y.

(x * y) / y = x.

(x / y) * y = x.

% left Bol (universally LIP, universally LAP)

(x * (y * x)) * z = x * (y * (x * z)).

% C an A1y-element

z * ((x * C) * z) = (z * x) * (C * z).

% C in commutant

C * x = x * C.

% a defined

(x * y) * z = (x * (y * z)) * a(x,y,z).

end_of_list.

formulas(goals).

a(C,x,y) * (a(C,x,y) * a(C,x,y)) = 0.

a(x,C,y) * (a(x,C,y) * a(x,C,y)) = 0.

a(x,y,C) * (a(x,y,C) * a(x,y,C)) = 0.

end_of_list.

if(Prover9).

assign(order, kbo).

assign(eq_defs,fold).

end_if.

formulas(assumptions).

% identity element

0 * x = x.

x * 0 = x.

% quasigroup

x * (x \ y) = y.

x \ (x * y) = y.

(x * y) / y = x.

(x / y) * y = x.

% left Bol (universally LIP, universally LAP)

(x * (y * x)) * z = x * (y * (x * z)).

% A an A1y-element

z * ((x * A) * z) = (z * x) * (A * z).

% C in commutant

C * x = x * C.

% a defined

(x * y) * z = (x * (y * z)) * a(x,y,z).

end_of_list.

formulas(goals).

a(C,A,x) * (a(C,A,x) * a(C,A,y)) = 0.

a(C,x,A) * (a(C,x,A) * a(C,x,A)) = 0.

a(x,C,A) * (a(x,C,A) * a(x,C,A)) = 0.

a(A,C,x) * (a(A,C,x) * a(A,C,x)) = 0.

a(x,A,C) * (a(x,A,C) * a(x,A,C)) = 0.

a(A,x,C) * (a(A,x,C) * a(A,x,C)) = 0.

end_of_list.