% --------------------------------------------------------------------------
% Gagola:  R(C,x,y) * (z * (z * z)) = (z * (z * z)) * R(C,x,y).
% --------------------------------------------------------------------------

assign(order, kbo).
set(lex_order_vars).

set(para_from_small).

clear(back_demod).
set(restrict_denials).

assign(max_weight, 5).

formulas(assumptions).

% two-sided identity
1 * x = x.
x * 1 = x.

% inverses (and IP and AAIP)
x * x' = 1.
x' * x = 1.
x' * (x * y) = y.
(y * x) * x' = y.
(x * y)' = y' * x'.

% alternative
(x * x) * y = x * (x * y).
(y * x) * x = y * (x * x).

% flexible
x * (y * x) = (x * y) * x.

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

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

% C in commutant
C * x = x * C.

% ***********
% Definitions
% ***********

% commutator (for Normality Conditions 3 and 4)

   x * y = (y * x) * K(x,y) # label("commutator").

% associator (for Normality Condition 4)

   (x * y) * z = (x * (y * z)) * a(x,y,z) # label("associator").

end_of_list.

formulas(goals).

   R(C,x,y) * (z * (z * z)) = (z * (z * z)) * R(C,x,y).

end_of_list.

%
% Hints extracted from previously found proofs.
%

formulas(hints).

R(C,x,y) * (z * (z * z)) = (z * (z * z)) * R(C,x,y).
1 * x = x.
x * 1 = x.
x * x' = 1.
x' * x = 1.
x' * (x * y) = y.
(x * y) * y' = x.
(x * y)' = y' * x'.
(x * x) * y = x * (x * y).
(x * y) * y = x * (y * y).
x * (y * x) = (x * y) * x.
(x * y) * x = x * (y * x).
((x * y) * x) * z = x * (y * (x * z)).
((x * y) * z) * y = x * (y * (z * y)).
(x * y) * (z * x) = x * ((y * z) * x).
R(x,y,z) = ((x * y) * z) * (y * z)'.
C * x = x * C.
x * y = (y * x) * K(x,y).
(x * y) * K(y,x) = y * x.
(x * y) * z = (x * (y * z)) * a(x,y,z).
(x * (y * z)) * a(x,y,z) = (x * y) * z.
1 = 1'.
1' = 1.
x'' * 1 = x.
(x * y)' * x = y'.
x * (y * x)' = y'.
(x' * y') * (y * x) = 1.
(x' * y') * ((y * x) * z) = z.
x' * ((x * y) * x) = y * x.
x * y' = y' * ((y * x) * y').
x' * ((x * y) * x') = y * x'.
(x * (y * x)) * z = x * (y * (x * z)).
(x * y) * (z * x) = z' * ((z * x) * (y * (z * x))).
x' * ((x * y) * (z * (x * y))) = (y * z) * (x * y).
(x * (y * (z * y))) * y' = (x * y) * z.
(x * (y * y)) * y = x * (y * (y * y)).
x' * ((x * y) * (z * x)) = (y * z) * x.
((x * y) * z)' * R(x,y,z) = (y * z)'.
R(x',x * y,z) = (y * z) * ((x * y) * z)'.
R(x,y,z) = ((x * y) * z) * (z' * y').
C' * (x * C) = x.
x' * (C * x) = C.
(x * C) * x' = C.
(C * x) * C' = x.
(C * (C * x)) * y = C * (x * (C * y)).
C * ((x * C) * y) = x * (C * (y * C)).
(x * C) * (y * C) = C * ((x * y) * C).
(x * y) * (x * C) = x * ((y * C) * x).
(C * x) * (C * y) = C * ((x * y) * C).
(C * x) * (y * C) = C * (C * (x * y)).
R(x,C,y) = ((x * C) * y) * (y * C)'.
((x * C) * y) * (y * C)' = R(x,C,y).
R(x,y,C) = (C * (x * y)) * (y * C)'.
(C * (x * y)) * (y * C)' = R(x,y,C).
(x * y)' * (y * x) = K(y,x).
(x * (y * z))' * ((x * y) * z) = a(x,y,z).
((x * y) * z) * a(x,y,z)' = x * (y * z).
x = x''.
x'' = x.
x * (x' * y) = y.
(x * y') * y = x.
(x * y')' = y * x'.
(x' * y)' = y' * x.
x * (C * x') = C.
(x * (C * y)) * (C' * x) = x * (y * x).
x * (y' * x)' = y.
(x * y) * ((y' * x') * z) = z.
x * y = y * ((y' * x) * y).
x * ((x' * y) * x) = y * x.
(x * y) * z = y * ((y' * x) * (y * z)).
x * ((x' * y) * (x * z)) = (y * x) * z.
((x * y) * x) * (x' * z) = x * (y * z).
x * (y * z) = z * (((z' * x) * y) * z).
x * (((x' * y) * z) * x) = y * (z * x).
x * C' = C' * x.
C' * x = x * C'.
(x * y')' * x = y.
(x * (y' * z')) * (z * y) = x.
(x * (x * y')) * y = x * x.
(x * y) * z = (x * z') * (z * (y * z)).
(x * y') * (y * (z * y)) = (x * z) * y.
(x * y) * z = x * ((y * (z * x')) * x).
x * ((y * (z * x')) * x) = (x * y) * z.
R(x,y,z) * (y * z) = (x * y) * z.
C * (x * C') = x.
(x' * y) * z = (z * x)' * (z * (y * z)).
(x * y)' * (x * (z * x)) = (y' * z) * x.
(x * y)' * (y * x) = K(x,y)'.
((x * y) * z)' * (x * (y * z)) = a(x,y,z)'.
x * (x * (y * x))' = (x * y)'.
x * (y * (x * (z * x)))' = ((y * x) * z)'.
(x * (y * x)')' = y.
((x * y) * C') * y = x * (y * (y * C')).
R(x,y,(z * y)') = ((x * y) * (z * y)') * z.
((x * y) * (z * y)') * z = R(x,y,(z * y)').
R(x,y,z)' = (y * z) * ((x * y) * z)'.
(x * y) * ((z * x) * y)' = R(z,x,y)'.
(x' * y) * ((y' * x) * z) = z.
(x * (y' * z))' = (z' * y) * x'.
((x' * y') * z') * (z * (y * x)) = 1.
x' * (y * x) = (x' * y) * x.
(x' * y) * x = x' * (y * x).
(x' * y') * (x * y) = K(x,y).
x * ((x * y)' * y) = K((x * y)',y).
(x * y') * (x' * y) = K(x',y).
(x' * y) * (x * y') = K(x,y').
((x' * y') * z') * ((z * (y * x)) * u) = u.
(C' * x') * ((C * x) * y) = y.
K(x,y)' = K(y,x).
(x * K(y,z))' * x = K(z,y).
(C' * x')' * y = (C * x) * y.
x * (y * x') = (x * y) * x'.
(x * y) * x' = x * (y * x').
((x * y') * z') * (z * y) = R(x,y',z').
(x * (x * y)') * y = K(x',x * y).
((x * y)' * y) * x = K(x * y,y').
x' * ((x * (y * x)) * z) = y * (x * z).
C * (x * (y * x)) = x * (y * (x * C)).
(x * (y * x)) * (x' * z) = x * (y * z).
C * ((x * C) * y) = x * (C * (C * y)).
(x * C) * (y * C) = C * (C * (x * y)).
x * (y * C) = y * (((y' * x) * C) * y).
x * (((x' * y) * C) * x) = y * (x * C).
(C * x) * (C * y) = C * (C * (x * y)).
x * (y * C) = C * (C * ((x * C') * y)).
C * (C * ((x * C') * y)) = x * (y * C).
(C * (x * y)) * (C' * y') = R(x,y,C).
C * (x * y) = (x * C') * (C * (C * y)).
(x * C') * (C * (C * y)) = C * (x * y).
(x * C) * (y * y) = C * (C * (x * (y * (y * C')))).
C * (C * (x * (y * (y * C')))) = (x * C) * (y * y).
(C * x) * y = C * (C * (x * (C' * y))).
C * (C * (x * (C' * y))) = (C * x) * y.
(x * (y * z)) * z' = (x * z) * (z' * y).
(x * (y * (x * C))) * x' = (C * x) * y.
x' * ((x * y) * z) = (y * (z * x')) * x.
(x * (y * z')) * z = z' * ((z * x) * y).
x' * ((x * y) * z') = (y * (x * z)') * x.
(x * (y * z)') * y = y' * ((y * x) * z').
x * K(x,y) = (y' * x) * y.
(x' * y) * x = y * K(y,x).
x * ((y' * x') * y) = K(x',y).
x * (y * K(y,x)) = y * x.
x * K(x,y) = y' * (x * y).
x' * (y * x) = y * K(y,x).
(x * y)' * y = (x * K(x,y))'.
(x * K(x,y))' = (x * y)' * y.
x' * (y' * (x * y)) = K(x,y).
((x * y)' * y) * x = K(y,x).
K(x,y) = K(y * x,x').
K(x * y,y') = K(y,x).
K(x * y',y) = K(y',x).
K(x,y) = K(y',x * y).
K(x',y * x) = K(y,x).
x * ((x * y)' * y) = K(x',y).
K(x',y) = K((x * y)',y).
K((x * y)',y) = K(x',y).
K((x' * y)',y) = K(x,y).
K(x' * y,x) = K(y,x).
K(x,y)' = K(y,y' * x).
K(x,x' * y) = K(y,x)'.
K(x,x' * y) = K(x,y).
K(x',x * y) = K(x',y).
(x * (x * y)') * y = K(x',y).
x * (y * (z * x))' = (x * ((x' * y) * z))'.
(x * ((x' * y) * z))' = x * (y * (z * x))'.
x * ((y * (z' * x))' * x) = z * (y' * x).
(x * y') * (y * z) = (x * (z * y')) * y.
(x * (y * z')) * z = (x * z') * (z * y).
(x * y') * (z * y) = (x * (y' * z)) * y.
(x * (y' * z)) * y = (x * y') * (z * y).
C * (x * y) = x * ((y * (C * x')) * x).
x * ((y * (C * x')) * x) = C * (x * y).
x' * ((x * y) * z) = (y * x') * (x * z).
(x * y') * (y * z) = y' * ((y * x) * z).
x' * ((x * y') * z) = (x * y)' * (x * z).
(x * y)' * (x * z) = x' * ((x * y') * z).
x' * (((x * y) * z) * u) = (y * (z * x')) * (x * u).
(x * (y * z')) * (z * u) = z' * (((z * x) * y) * u).
(x * y) * (x' * z) = x * ((y * x') * z).
x * ((C * (x' * y)) * x) = y * (x * C).
(x * C') * (C * y) = C * (x * (C' * y)).
(C * x) * y = x * ((y * (x * C)) * x').
x * ((y * (x * C)) * x') = (C * x) * y.
x * ((y * (x * C)) * x') = (x * C) * y.
x * (y * x') = (x * C) * (y * (C' * x')).
(x * C) * (y * (C' * x')) = x * (y * x').
(x * y)' * ((x * (y * z)) * z') = a(x,y * z,z').
((x * y) * (y' * z))' * (x * z) = a(x * y,y',z).
(x * y)' * ((x * z) * (z' * y)) = a(x,z,z' * y).
(x * ((x' * y) * z))' * (y * z) = a(x,x' * y,z).
((x * y') * (y * z))' * (x * z) = a(x * y',y,z).
(x' * (y * z)') * (y * (z * x)) = a(y,z,x)'.
x * (y * (z * x))' = ((y * x) * (x' * z))'.
((x * y) * (y' * z))' = y * (x * (z * y))'.
C * (C * (x * (y * (y * C')))) = (x * y) * (y * C).
R(x,y,z) = ((x * y) * (y' * z)) * z'.
R(x,y,z) * z = (x * y) * (y' * z).
R(x * y,x,z) = (x * (y * z)) * z'.
R(x * y,x,z) * z = x * (y * z).
R(x * C,C,y) * y = C * (x * y).
R(x,y,z) * z = y * ((y' * x) * z).
(C * (x * y)) * y' = R(x * C,C,y).
R(x * y,x,z) = (x * z) * (z' * y).
R(x,y',z) = (y' * z) * (z' * (y * x)).
(x' * y) * (y' * (x * z)) = R(z,x',y).
R(x * (y * z),x,y) = (x * y) * z.
R(x * y',x,z) = (x * z) * (y * z)'.
R(x * (y * C),x,C) = (x * C) * y.
R(x,y,z) * z = R(x * z,x,y).
R(x * y,x,z) = R(x,z,y) * y.
C * R(x,y,C) = R(x * C,x,y).
R(x * C,x,y) = C * R(x,y,C).
R(x * C,C,y) = R(C,y,x) * x.
(R(x,y,z) * z) * y = x * (z * y).
R(x * y,x,z) = z * ((z' * x) * y).
R(x * C,C,y) = y * ((y' * C) * x).
x * ((x' * C) * y) = R(y * C,C,x).
((R(x,y,z) * z) * y) * (z * y)' = x.
R(x,y',z) = z * ((z' * y') * (y * x)).
x * ((x' * y') * (y * z)) = R(z,y',x).
R(x,y,z) = z * ((z' * y) * (y' * x)).
R(x * y',x,z) = z * (y * (x' * z))'.
x * ((C * x') * y) = R(y * C,C,x).
R(R(x,y,z),z,y) = x.
R((x * y) * z,y,x) = x * (y * z).
R(x,y,z) = z * (y * ((y' * z') * x)).
R(x,y,z') = z' * (y * ((y' * z) * x)).
x' * (y * ((y' * x) * z)) = R(z,y,x').
R(x,y,z) = (y * z) * ((x' * y) * z)'.
R(x,y,z) = z * (R(z',y,x) * x).
x * (R(x',y,z) * z) = R(z,y,x).
R(x'' * y,z,x) = x * R(y,x',z).
R(x,y,z) = R(x',y,z)'.
R(x',y,z)' = R(x,y,z).
R(x,y,z)' = R(x',y,z).
R(x' * y,z,u)' = R(y' * x,z,u).
x * (R(x,y,z)' * z) = R(z,y,x).
x * R(y,z,x') = R(x',z,y)' * y.
R(x',y,z)' * z = x * R(z,y,x').
R(x * y,z,x) = x * R(y,x',z).
R(x,y,z) * z = x * R(z,y,x').
R(x * y,x,z) = x * R(y,z,x').
x' * R(x * y,x,z) = R(y,z,x').
x' * ((x * (y * z)) * z') = R(y,z,x').
x * ((C * x') * (y * x)) = y * (x * C).
x * (y * C) = R((x * y) * C,C,y).
R((x * y) * C,C,y) = x * (y * C).
R(C * (x * y),C,y) = x * (y * C).
R(x * C,C,y) = (x * y') * (y * C).
(x * y') * (y * C) = R(x * C,C,y).
R(C * x',C,y) = (y * x)' * (y * C).
(x * y)' * (x * C) = R(C * y',C,x).
(x * (C * y')) * y = R(x * C,C,y).
x * R(y * C,C,x) = C * (x * y).
R(x * C,C,y) = y' * ((y * x) * C).
x' * ((x * y) * C) = R(y * C,C,x).
x * ((C * (y * x)) * x') = C * (x * y).
x * ((x' * y) * C) = R(y * C,C,x').
(x * (C * (y * x))) * x' = C * (x * y).
(C * (x * y)) * x = x * (C * (y * x)).
(C * x) * y = y * (C * ((y' * x) * y)).
x * (C * ((x' * y) * x)) = (C * y) * x.
x * (C * (x' * (y * x))) = (C * y) * x.
(x * y') * (C * ((y * x') * (z * (x * y')))) = (C * z) * (x * y').
(x * C) * (x' * (y * x)) = y * (x * C).
(x * C)' * (y * (x * C)) = x' * (y * x).
(x * C') * (y * (z * (y * C))) = C * (x * (C' * (y * (z * y)))).
x * ((y * x)' * (y * z)) = R(z,y',y * x).
x * ((y * x)' * (y * z)) = R(z,y',x).
x' * ((x * y') * (y * z)) = R(z,y',x').
x * R(C * x',C,y) = R(C,y',x).
x * (y * (z' * x))' = (z * x) * (y * x)'.
(x * y) * (z * y)' = y * (z * (x' * y))'.
(x * y)' * (x * (y * z)) = R(z,x,(x * y)').
x' * (y' * (x * y)) = K(x,y * C).
K(x,y * C) = K(x,y).
K(x,C * y) = K(x,y).
R(x',x * y,z) = R(x,y,z)'.
R(x,x' * y,z) = R(x',y,z)'.
R(x,(y * x)',z) = R(x',y',z)'.
R(x,x' * y,z) = R(x,y,z).
R(x,y,z) = R(x,x * y,z).
R(x,x * y,z) = R(x,y,z).
R(x,(y * x)',z) = R(x,y',z).
R(x * C,C,y) = R(x * C,x',y).
R(x * C,x',y) = R(x * C,C,y).
((x * y') * (y * z))' * (x * z) = a(x,y',y * z)'.
((x' * y') * z') * ((z * y) * x) = a(z,y,x).
((x * y) * (z * y)') * z = R(x,y,z').
C * ((x * y) * (C * y)') = R(x,y,C').
R(x,y,z') = R(x,y,(z * y)').
R(x,y,(z * y)') = R(x,y,z').
C * ((x * y) * (y * C)') = R(x,y,C').
(x * (y' * z)) * (z' * u) = z * ((y * (x' * z))' * u).
R(x,y',y * z) = R(x,y',z).
R(x,y'',z) = R(x,y'',y * z).
R(x,y'',y * z) = R(x,y'',z).
R(x,y'',y * z) = R(x,y,z).
R(x,y,y * z) = R(x,y,z).
R(x,y,C) = R(x,y,C * y').
R(x,y,C * y') = R(x,y,C).
R(x,C,y) = R(x,C,y * C').
R(x,C,y * C') = R(x,C,y).
(x * y)' * (x * (y * z)) = R(z,y',x').
R(x,y,(y * z)') = R(x,z',y').
R(x,y,z * y') = R(x,z'',y').
R(x,y,z * y') = R(x,z,y').
R(x,y,z) = R(x,z * y,y').
R(x,y * z,z') = R(x,z,y).
R(x,C,y') = R(x,y,C).
R(x,y,C') = R(x,C,y).
R(x,C,y) = R(x,y,C').
R(x,C,y) = R(x,y',C).
R(x,y',C) = R(x,C,y).
R(x * y,x,C) = x * R(y,x,C).
x * ((x' * y) * C) = R(y * C,x,C).
C * ((x * y) * (y * C)') = R(x,C,y).
R(x,(y * x)',C) = R(x',C,y)'.
R(x,y * z',z) = R(x,z',y).
x * R(y * C,x,C) = (x * C) * y.
R(x,y,z) = R(x,z',y * z).
R(x,y',z * y) = R(x,z,y).
R(x * y,z * x,x) = x * R(y,z,x).
R(x * C,y,x) = R(x * C,C,y * x).
R(x * C,C,y * x) = R(x * C,y,x).
R(x',C,y)' = R(x,C,y * x).
R(x,C,y * x) = R(x',C,y)'.
R(x,C,y * x) = R(x,C,y).
((x * C) * y)' * x = R(y * C,x,C)'.
R(x * C,x,y) = R(x * C,y,C).
R(x * C,y,C) = R(x * C,x,y).
R(x * C,y,C) = C * R(x,y,C).
C * R(x,C,y) = R(x * C,C,y).
R(x * C,C,y) = C * R(x,C,y).
R(x * C,C,y * x) = C * R(x,C,y).
R(x * C,C,y) = R(x * C,C,y * x).
R(x * C,C,y * x) = R(x * C,C,y).
R(x * C,C,y) = R(x * C,y,x).
R(x * C,y * x,x) = R(x * C,C,y).
R(x * C,C,y) = x * R(C,y,x).
R(C * x,C,y) = x * R(C,y,x).
x * R(C,y,x) = R(C,y,x) * x.
R(C,x,y) * y = y * R(C,x,y).
x * R(C * x',C,y) = R(C,y,x').
R(C,x,y') = R(C,x',y).
R(C,x',y) = R(C,x,y').
R(C,x,y) = R(C,x',y').
R(C,x',y') = R(C,x,y).
R(C,x,(y * x)') = R(C,y,x).
R(C,x,y) = R(C,y,x').
R(C,x,y') = R(C,y,x).
R(C,x,y) = R(C,y',x).
R(C,x',y) = R(C,y,x).
(x' * (y' * z')) * (z * (y * x)) = a(z,y,x)'.
x * ((y * z) * (x * y)) = (x * y) * (z * (x * y)).
(x * y) * (z * (x * y)) = x * ((y * z) * (x * y)).
(C' * x') * R(y,x,C) = C' * ((x' * (C * (y * x))) * (C' * x')).
C' * ((x' * (C * (y * x))) * (C' * x')) = (C' * x') * R(y,x,C).
(x * y) * (y * C) = (x * C) * (y * y).
(x * C) * (y * y) = (x * y) * (y * C).
((x * C) * (y * y)) * y = x * (y * ((y * C) * y)).
x * (y * ((y * C) * y)) = (x * C) * (y * (y * y)).
(x * C) * (y * (y * y)) = x * (y * ((y * C) * y)).
(x * C) * (y * (y * y)) = x * (y * (y * (C * y))).
(x * C) * (y * (y * y)) = x * (y * (y * (y * C))).
x * (y * (y * y)) = (x * C') * (y * (y * (y * C))).
(x * C') * (y * (y * (y * C))) = x * (y * (y * y)).
x' * (((x * y') * z') * (z * y)) = a(x,z,y)'.
x' * R(x,y',z') = a(x,z,y)'.
x * a(x,y,z)' = R(x,z',y').
x * a(x,y,z')' = R(x,z,y').
R(x,y',z')' * x = a(x,z,y).
R(x,y,z')' * x = a(x,z,y').
R(x,y',z')' * x = a(x,z,y * x).
a(x,y,z * x) = R(x,z',y')' * x.
R(x,y,z)' * x = a(x,z',y').
C * R(C,x',y')' = a(C,y,x * C).
a(C,x,y * C) = C * R(C,y',x')'.
a(x,y,z * x) = a(x,y,z).
a(x * y,z,y) = a(x * y,z,x').
a(x * y,z,x') = a(x * y,z,y).
a(x',y,z) = a(x',y,z * x).
a(x',y,z * x) = a(x',y,z).
a(C,x,y * C) = C * R(C,y,x)'.
(x * y)' * ((x * z) * (z' * y)) = a(x,y * z,z').
(x * (y * (z * x))') * (y * z) = a(y * x,x',z).
(x * (y * (z * x))') * (y * z) = a(x,x' * y,z).
a(x,x' * y,z) = a(y * x,x',z).
a(x * y,y',z) = a(y,y' * x,z).
a(x,y',z) = a(y,y' * (x * y'),z).
a(x,x' * (y * x'),z) = a(y,x',z).
a(x,y * x',z) = a(x * y,x',z).
a(x * y,x',z) = a(x,y * x',z).
a(x,y'',z) = a(y',(y * x) * y'',z).
a(x',(x * y) * x'',z) = a(y,x'',z).
a(x',(x * y) * x'',z) = a(y,x,z).
a(x',(x * y) * x,z) = a(y,x,z).
a(x',x * (y * x),z) = a(y,x,z).
a(x',x * y,z) = a(y * x',x,z).
a(x * y',y,z) = a(y',y * x,z).
a(x,y * z,z') = a(x,z,z' * y).
a(x,y,z'') = a(x,z',z'' * (y * z)).
a(x,y',y'' * (z * y)) = a(x,z,y'').
a(x,y,z') = a(x,z,z' * (y * z')).
a(x,y,y' * (z * y')) = a(x,z,y').
a(x,y,z * y') = a(x,y * z,y').
a(x,y * z,y') = a(x,y,z * y').
a(x,y',y'' * (z * y)) = a(x,z,y).
a(x,y',y * (z * y)) = a(x,z,y).
a(x,y',(y * z) * y) = a(x,z,y).
a(x,y',y * z)' = a(x * y',y,z).
a(x,y,z)' = a(x * z',z,y * z).
a(x * y',y,z * y) = a(x,z,y)'.
a(x,y,z * y) = a(x * y,z,y)'.
a(x * y,z,y)' = a(x,y,z * y).
a(x,y,z)' = a(z',z * x,y * z).
a(x',x * y,z * x) = a(y,z,x)'.
a(x,y,z)' = a(z',z * x,y).
a(x',x * y,z) = a(y,z,x)'.
a(x,x' * y,z) = a(y,z,x')'.
a(x * y,z,y)' = a(x,y,z).
a(x,y,z)' = a(x * y,z,y).
a(x * y,z,y) = a(x,y,z)'.
a(x,y,z) = a(x,y,z * y).
a(x,y,z * y) = a(x,y,z).
a(x,y',z) = a(x,y',z * y).
a(x,y',z * y) = a(x,y',z).
a(x,C,C * y) = a(x,C,y).
a(x,y',C) = a(x,y',y * C).
a(x,y',y * C) = a(x,y',C).
a(x,y,z) = a(x,y,z * y').
a(x,y,z * y') = a(x,y,z).
a(x,C,y) = a(x,C,C' * y).
a(x,C,C' * y) = a(x,C,y).
a(x * y,z,x') = a(x,y,z)'.
a(x,y * z,y') = a(x,y,z).
a(x,y,z') = a(x,z,z' * y).
a(x,y,y' * z) = a(x,z,y').
a(x,y',y * z) = a(x,z,y).
a(x,y * z,z) = a(x,y,z).
a(x,y',C) = a(x,C,y).
a(x,y,C) = a(x,C,y').
a(x,C,y') = a(x,y,C).
a(x,C,y' * z') = a(x,z * y,C).
a(x,y * z,C) = a(x,C,z' * y').
a(x,y,C') = a(x,C,y).
a(x,C,y) = a(x,y,C').
a(x,y',C') = a(x,y,C).
a(C * x,C,y) = a(C,x,y)'.
R(x,C,y)' * x = a(x,y,C).
R(x,C,y) * a(x,y,C) = x.
a(x,y,C)' = x' * R(x,C,y).
x' * R(x,C,y) = a(x,y,C)'.
(R(x',C,y) * a(x',y,C))' = x.
a(x,x' * y,C) = a(y,x,C)'.
(R(x,C,y)' * a(x',y,C))' = x.
a(x,C,y' * x'') = a(y,x,C)'.
a(x,C,y' * x) = a(y,x,C)'.
a(x,y,C)' = a(y,C,x').
a(x,y,C)' = a(y,x,C).
a(x,y * C,C) = a(y,C,x).
x' * R(x,C,y) = a(y,x,C).
a(x,C,y) = a(y,x,C).
a(x,y',C) = a(y,x,C).
a(x,C,y') = a(y,C,x).
a(x,y,C) = a(y,x,C').
a(x,y,C') = a(y,x,C).
a(x,C,y)' = a(x,y,C).
a(x,y,C)' = a(x,C,y).
a(x * C,C,y) = a(x,C,y).
a(x,C,y) = a(x',y,C).
a(x',y,C) = a(x,C,y).
a(x,y,C')' = a(x,y,C).
a(x,y,C) * a(x,C,y) = 1.
a(x,y,C) * (a(x,C,y) * z) = z.
(a(x,C,y) * z)' = z' * a(x,y,C).
(x * a(y,z,C)) * a(y,C,z) = x.
a(x * C,y,C) = a(x,y,C).
a(x,C,y) = a(y,C,x)'.
a(x,C,y)' = a(y,C,x).
x' * R(x,C,y) = a(x,C,y).
(R(x,C,y)' * a(x,C,y))' = x.
a(C * x,C,y) = a(x,C,y).
x * a(x,y,C) = R(x,C,y').
a(x,C,y) = a(C,x,y)'.
a(C,x,y)' = a(x,C,y).
a(x,C,y)' = a(C,x,y).
a(C,x,y) = a(x,y,C).
a(C,x,y) = a(y,C,x).
a(C,x,y * C) = a(x,y,C).
a(x,y',C) = a(C,y,x).
a(C,x,y) = a(y,x',C).
a(C,x',y) = a(x,C,y).
a(x,y,C) = C * R(C,y,x)'.
C * R(C,x,y)' = a(y,x,C).
x * a(y,x,C) = R(x,C,y).
x * a(x,C,y) = R(x,C,y).
x * a(x,y,C) = R(x,y,C).
x' * R(x,y,C) = a(x,y,C).
x * R(x',y,C) = a(x',y,C).
(x * y') * R(y,z,C) = y' * ((y * x) * a(y,z,C)).
x' * ((x * y) * a(x,z,C)) = (y * x') * R(x,z,C).
x * a(y,C,x) = R(x,y,C).
R(C,x,y) * a(y,x,C) = C.
C * a(x,C,y) = R(C,y,x).
K(x,R(C,y,z)) = K(x,a(z,C,y)).
K(x,a(y,C,z)) = K(x,R(C,z,y)).
C * a(x,y,C) = R(C,x,y).
C * a(C,x,y) = R(C,x,y).
K(x,R(C,y,z)) = K(x,a(y,z,C)).
K(x,a(y,z,C)) = K(x,R(C,y,z)).
a(x,C,y) * x = R(x,C,y).
a(x,y,C) * R(x,C,y) = x.
x * R(x,C,y)' = a(x,y,C).
x * (x * a(x,C,y))' = a(x,y,C).
x * R(x,y,C)' = a(x',y,C).
x * R(x,y,C)' = a(x,C,y).
a(x,C,y)' * x = R(x,y,C).
a(x,y,C) * x = R(x,y,C).
R(x,y,C) * x' = a(x,y,C).
(C * R(x,y,C)) * (x * C)' = R(a(x,y,C),x,C).
a(x,y,C) * a(x,C,y) = K(x',a(x,C,y)).
K(x',a(x,C,y)) = 1.
1 = K(x',R(C,y,x)).
K(x',R(C,y,x)) = 1.
K(x',x * R(C,y,x)) = 1.
K(x',R(C,y,x) * x) = 1.
1 = K(R(C,x,y),y).
K(R(C,x,y),y) = 1.
1' = K(x,R(C,y,x)).
K(x,R(C,y,x)) = 1'.
K(x,R(C,y,x)) = 1.
K(x,R(C,x,y)) = 1.
R(x * C,y,C) * (x * C)' = R(a(x,y,C),x,C).
R(a(x,y,C),x,C) = a(x * C,y,C).
R(a(x,y,C),x,C) = a(x,y,C).
R(a(x,y,C),C,x) = a(x,y,C).
C * (x * (C' * (y * (y * y)))) = x * (y * (y * y)).
C * (x * (y * (y * y))) = (C * x) * (y * (y * y)).
(C * x) * (y * (y * y)) = C * (x * (y * (y * y))).
x' * (y * (y * y)) = C * ((x * C)' * (y * (y * y))).
C * ((x * C)' * (y * (y * y))) = x' * (y * (y * y)).
(((x * (x * x))' * y') * C') * ((C * y) * (x * (x * x))) = 1.
R(C,x,y) * (z * (z * z)) = C * (a(x,y,C) * (z * (z * z))).
C * (a(x,y,C) * (z * (z * z))) = R(C,x,y) * (z * (z * z)).
R(x,C,(y * C)' * (z * (z * z))) = ((x * C) * ((y * C)' * (z * (z * z)))) * (y' * (z * (z * z)))'.
((x * C) * ((y * C)' * (z * (z * z)))) * (y' * (z * (z * z)))' = R(x,C,(y * C)' * (z * (z * z))).
R(x,C,y' * (z * (z * z))) = R(x,C,(y * C)' * (z * (z * z))).
R(x,C,(y * C)' * (z * (z * z))) = R(x,C,y' * (z * (z * z))).
1 = a(C,x,y * (y * y)).
a(C,x,y * (y * y)) = 1.
1 = a(x,C,y * (y * y)).
a(x,C,y * (y * y)) = 1.
x * 1 = R(x,C,y * (y * y)).
R(x,C,y * (y * y)) = x * 1.
R(x,C,y * (y * y)) = x.
(x * (x * x)) * (y * C) = C * ((x * (x * x)) * y).
(C * ((x * (x * x)) * y)) * (y * C)' = x * (x * x).
(x * (x * x)) * (C * y) = C * ((x * (x * x)) * y).
(x * (x * x)) * R(C,y,z) = C * ((x * (x * x)) * a(z,C,y)).
C * ((x * (x * x)) * a(y,C,z)) = (x * (x * x)) * R(C,z,y).
x * (x * x) = R(x * (x * x),y,C).
R(x * (x * x),y,C) = x * (x * x).
R(x * (x * x),C,y) = x * (x * x).
(x' * y') * (C * ((y * x) * (z * (x' * y')))) = (C * z) * (x' * y').
C' * (x' * ((y * x) * x')) = (C' * x') * R(y,x,C).
C' * (x' * y) = (C' * x') * R(y,x,C).
(C' * x') * R(y,x,C) = C' * (x' * y).
(C' * x') * ((C * (y * x)) * (C' * x')) = C' * (x' * y).
(C' * x') * (C * (x * (y * x'))) = (C * y) * (C' * x').
(C' * x') * (C * (x * y)) = (C * (y * x)) * (C' * x').
(C * (x * y)) * (C' * y') = (C' * y') * (C * (y * x)).
(C * ((x * y) * z)) * (C' * z') = (C' * z') * (C * ((z * (x * z)) * (z' * y))).
(C' * x') * (C * ((x * (y * x)) * (x' * z))) = (C * ((y * z) * x)) * (C' * x').
(C' * x') * (C * ((x * (y * x)) * (z * (z' * (x' * u))))) = (C * ((y * u) * x)) * (C' * x').
(C' * x') * ((C * (C * (x * (y * x)))) * (C' * (x' * z))) = (C * ((y * z) * x)) * (C' * x').
(C' * x') * ((C * (C * (x * (y * x)))) * ((C' * x') * ((C * (z * x)) * (C' * x')))) = (C * ((y * z) * x)) * (C' * x').
((x * C) * ((y * C)' * (z * (z * z)))) * ((z * (z * z))' * y) = R(x,C,(y * C)' * (z * (z * z))).
R(x,C,(y * C)' * (z * (z * z))) = (z * (z * z)) * (((y * C) * ((x * C)' * (z * (z * z))))' * y).
(x * (x * x)) * (((y * C) * ((z * C)' * (x * (x * x))))' * y) = R(z,C,(y * C)' * (x * (x * x))).
(x * (x * x)) * (((y * C) * ((z * C)' * (x * (x * x))))' * y) = R(z,C,y' * (x * (x * x))).
(x * (x * x)) * R(((y * C)' * (x * (x * x))) * C,z,C)' = R(y,C,z' * (x * (x * x))).
(x * (x * x)) * R(C * ((y * C)' * (x * (x * x))),z,C)' = R(y,C,z' * (x * (x * x))).
(x * (x * x)) * R(y' * (x * (x * x)),z,C)' = R(y,C,z' * (x * (x * x))).
(x * (x * x)) * R((x * (x * x))' * y,z,C) = R(y,C,z' * (x * (x * x))).
(C' * x') * (((C * x) * (C * (y * x))) * ((C' * x') * ((C * (z * x)) * (C' * x')))) = (C * ((y * z) * x)) * (C' * x').
(C' * x') * (((C' * x')' * (C * (y * x))) * ((C' * x') * ((C * (z * x)) * (C' * x')))) = (C * ((y * z) * x)) * (C' * x').
(C * ((x * y) * z)) * (C' * z') = ((C * (x * z)) * (C' * z')) * ((C * (y * z)) * (C' * z')).
((C * (x * y)) * (C' * y')) * ((C * (z * y)) * (C' * y')) = (C * ((x * z) * y)) * (C' * y').
((C * (x * y)) * (C' * y')) * R(z,y,C) = (C * ((x * z) * y)) * (C' * y').
R(x,y,C) * R(z,y,C) = (C * ((x * z) * y)) * (C' * y').
(C * ((x * y) * z)) * (C' * z') = R(x,z,C) * R(y,z,C).
R(x,y,C) * R(z,y,C) = R(x * z,y,C).
R(x,C,y) * R(z,y',C) = R(x * z,y',C).
(x * (x * x)) * R(y,z,C) = R((x * (x * x)) * y,z,C).
R((x * (x * x)) * y,z,C) = (x * (x * x)) * R(y,z,C).
R(x,y,C) * (z * (z * z)) = R(x * (z * (z * z)),y,C).
R(x * (y * (y * y)),z,C) = R(x,z,C) * (y * (y * y)).
R(x,C,y) * R(z,y',C) = R(x * z,C,y).
R(x,C,y) * R(z,C,y) = R(x * z,C,y).
R(x,C,y) * R(z,C,y)' = R(x * z',C,y).
a(x,y,C) * R(z,C,x) = R(a(x,y,C) * z,C,x).
R(a(x,y,C) * z,C,x) = a(x,y,C) * R(z,C,x).
R(x,C,y) * (z * (z * z))' = R(x * (z * (z * z))',C,y).
R(x * (y * (y * y))',C,z) = R(x,C,z) * (y * (y * y))'.
R(a(x,y',C) * z,C,x) = a(y,x,C) * R(z,C,x).
R(a(x',y',C) * z,C,x') = a(y,C,x) * R(z,C,x').
R((x * (x * x)) * ((x * (x * x))' * y),z,C) = R(y,C,z' * (x * (x * x))).
((x * (x * x)) * y)' * ((x * (x * x)) * R(z * (x * (x * x)),u,C)) = (y' * R(z,u,C)) * (x * (x * x)).
R(a(x',y',C) * z,C,x') = a(y,C,x) * R(z,x,C).
R(a(x,C,y') * z,C,x') = a(y,C,x) * R(z,x,C).
R(a(x,C,y) * z,C,y') = a(x,C,y) * R(z,y,C).
a(x,C,y) * R(z,y,C) = R(a(x,C,y) * z,y,C).
R(a(x,C,y) * z,y,C) = a(x,C,y) * R(z,y,C).
R(x,y,C) = R(x,C,y' * (z * (z * z))).
R(x,C,y' * (z * (z * z))) = R(x,y,C).
R(x,C,y * (z * (z * z))) = R(x,y',C).
R(x,C,y * (z * (z * z))) = R(x,C,y).
x' * R(x,C,y) = a(x,C,y * (z * (z * z))).
a(x,C,y * (z * (z * z))) = x' * R(x,C,y).
x * R(x,C,y)' = a(x,y * (z * (z * z)),C).
a(x,y * (z * (z * z)),C) = x * R(x,C,y)'.
a(x,C,y * (z * (z * z))) = a(y,x,C).
(x * (y * (y * y))) * a(x,z,C) = R(x * (y * (y * y)),z,C).
a(x,y * (z * (z * z)),C) = a(x,y,C).
a(x,y',C) = a(x,((z * (z * z)) * y)',C).
a(x,((y * (y * y)) * z)',C) = a(x,z',C).
a(x,((y * (y * y)) * z)',C) = a(x,C,z).
a(C,(x * (x * x)) * y,z) = a(z,C,y).
C * a(C,(x * (x * x)) * y,z) = R(C,y,z).
R(C,x,y) = R(C,(z * (z * z)) * x,y).
R(C,(x * (x * x)) * y,z) = R(C,y,z).
R(C,x',y) = R(C,(x * (z * (z * z)))',y).
R(C,(x * (y * (y * y)))',z) = R(C,x',z).
R(C,(x * (y * (y * y)))',z') = R(C,x,z).
R(C,x,y) = R(C,x * (z * (z * z)),y).
R(C,x * (y * (y * y)),z) = R(C,x,z).
K(x * (y * (y * y)),R(C,x,z)) = 1.
R(x,y * (y * y),a(x,z,C)) = R(x * (y * (y * y)),z,C) * ((y * (y * y)) * a(x,z,C))'.
R(x * (y * (y * y)),z,C) * ((y * (y * y)) * a(x,z,C))' = R(x,y * (y * y),a(x,z,C)).
R(x * (y * (y * y)),z,C)' * R(x,y * (y * y),a(x,z,C)) = ((y * (y * y)) * a(x,z,C))'.
a(x,y,C)' * R(x * (z * (z * z)),y,C) = (x * (z * (z * z))) * K(x * (z * (z * z)),a(x,y,C)).
R(x * (y * (y * y)),z,C) * (u * a(x,z,C))' = a(x,z,C) * (u * ((x * (y * (y * y)))' * a(x,z,C)))'.
a(x,y,C) * (z * ((x * (u * (u * u)))' * a(x,y,C)))' = R(x * (u * (u * u)),y,C) * (z * a(x,y,C))'.
x' * R(x * (y * (y * y)),z,C) = ((y * (y * y)) * x') * R(x,z,C).
((x * (x * x)) * y') * R(y,z,C) = y' * R(y * (x * (x * x)),z,C).
(R(x * (y * (y * y)),z,C)' * x)' = ((y * (y * y)) * x') * R(x,z,C).
a(x,C,y) * R(x * (z * (z * z)),y,C) = (x * (z * (z * z))) * K(x * (z * (z * z)),a(x,y,C)).
R(a(x,C,y) * (x * (z * (z * z))),y,C) = (x * (z * (z * z))) * K(x * (z * (z * z)),a(x,y,C)).
R(a(x,C,y) * (x * (z * (z * z))),y,C) = (x * (z * (z * z))) * K(x * (z * (z * z)),R(C,x,y)).
R(a(x,C,y) * (x * (z * (z * z))),y,C) = (x * (z * (z * z))) * 1.
R(a(x,C,y) * (x * (z * (z * z))),y,C) = x * (z * (z * z)).
R(x * (y * (y * y)),C,z) = a(x,C,z) * (x * (y * (y * y))).
a(x,C,y) * (x * (z * (z * z))) = R(x * (z * (z * z)),C,y).
R(x * (y * (y * y)),C,z)' = (x * (y * (y * y)))' * a(x,z,C).
(x * (y * (y * y)))' * a(x,z,C) = R(x * (y * (y * y)),C,z)'.
((x * (x * x)) * y)' * R((x * (x * x)) * (z * (x * (x * x))),u,C) = (y' * R(z,u,C)) * (x * (x * x)).
a(x,y,C) * (z * R(x * (u * (u * u)),C,y)')' = R(x * (u * (u * u)),y,C) * (z * a(x,y,C))'.
R(x * (y * (y * y)),z,C) * (u * a(x,z,C))' = a(x,z,C) * (u * R(x * (y * (y * y)),C,z)')'.
R(x * (y * (y * y)),z,C) * (u * a(x,z,C))' = a(x,z,C) * (R(x * (y * (y * y)),C,z) * u').
a(x,y,C) * (R(x * (z * (z * z)),C,y) * (z * (z * z))') = R(x,z * (z * z),a(x,y,C)).
a(x,y,C) * R((x * (z * (z * z))) * (z * (z * z))',C,y) = R(x,z * (z * z),a(x,y,C)).
a(x,y,C) * R(x,C,y) = R(x,z * (z * z),a(x,y,C)).
R(x,y * (y * y),a(x,z,C)) = a(x,z,C) * R(x,C,z).
R(x,y * (y * y),a(x,z,C)) = x.
R(x * (y * (y * y)),z,C)' * x = ((y * (y * y)) * a(x,z,C))'.
((x * (x * x)) * a(y,z,C))'' = ((x * (x * x)) * y') * R(y,z,C).
((x * (x * x)) * y') * R(y,z,C) = ((x * (x * x)) * a(y,z,C))''.
((x * (x * x)) * y') * R(y,z,C) = (x * (x * x)) * a(y,z,C).
(x * (x * x)) * a(y,z,C) = y' * R(y * (x * (x * x)),z,C).
x' * R(x * (y * (y * y)),z,C) = (y * (y * y)) * a(x,z,C).
((x * (x * x)) * y)' * R((x * (x * x)) * (y * (x * (x * x))),z,C) = (x * (x * x)) * a((x * (x * x)) * y,z,C).
(x * (x * x)) * a((x * (x * x)) * y,z,C) = (y' * R(y,z,C)) * (x * (x * x)).
(x' * R(x,y,C)) * (z * (z * z)) = (z * (z * z)) * a((z * (z * z)) * x,y,C).
a(x,y,C) * (z * (z * z)) = (z * (z * z)) * a((z * (z * z)) * x,y,C).
(x * (x * x)) * a((x * (x * x)) * y,z,C) = a(y,z,C) * (x * (x * x)).
(x * (x * x)) * a(C,(x * (x * x)) * y,z) = a(y,z,C) * (x * (x * x)).
(x * (x * x)) * a(y,C,z) = a(z,y,C) * (x * (x * x)).
a(x,y,C) * (z * (z * z)) = (z * (z * z)) * a(y,C,x).
C * (a(x,y,C) * (z * (z * z))) = (z * (z * z)) * R(C,x,y).
(x * (x * x)) * R(C,y,z) = R(C,y,z) * (x * (x * x)).

end_of_list.
