if(Prover9).

assign(order, kbo).

assign(eq_defs,fold).

end_if.

formulas(assumptions).

% identity element

0 * x = x.

x * 0 = x.

% inverses

x * x' = 0.

x' * x = 0.

% 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)).

% LIP

x' * (x * y) = y.

% LAP

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

% L(u,x,y) = u L(x) L(y) L(yx)^{-1}

L(u,x,y) = (y * x)' * (y * (x * u)).

% A is A1y

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

% L(A,B,C) = D.

L(A,B,C) = D.

end_of_list.

formulas(goals).

% D is A1y

z * ((x * D) * z) = (z * x) * (D * z).

end_of_list.

if(Prover9).

assign(order, kbo).

assign(eq_defs,fold).

end_if.

formulas(assumptions).

% identity element

0 * x = x.

x * 0 = x.

% inverses

x * x' = 0.

x' * x = 0.

% 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)).

% LIP

x' * (x * y) = y.

% LAP

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

% L(u,x,y) = u L(x) L(y) L(yx)^{-1}

%L(u,x,y) = (y * x)' * (y * (x * u)).

% R(u,x,y) = u R(x) R(y) R(xy)^{-1}

R(u,x,y) = ((u * x) * y) / (x * y).

% A is A1y

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

% R(A,B,C) = D.

R(A,B,C) = D.

end_of_list.

formulas(goals).

% D is A1y

z * ((x * D) * z) = (z * x) * (D * z).

end_of_list.

if(Prover9).

assign(order, kbo).

assign(eq_defs,fold).

end_if.

formulas(assumptions).

% identity element

0 * x = x.

x * 0 = x.

% inverses

x * x' = 0.

x' * x = 0.

% 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)).

% LIP

x' * (x * y) = y.

% LAP

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

% T(u,x) = u R(x) L(x)^{-1}

T(u,x) = x' * (u * x).

% A is A1y

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

% T(A,B) = D.

T(A,B) = D.

end_of_list.

formulas(goals).

% D is A1y

z * ((x * D) * z) = (z * x) * (D * z).

end_of_list.