i (0) => 0.
i (i (X)) => f (0,X).
i (f (X,Y)) => f (i (Y),i (X)).
f (X,0) => X.
f (i (X),X) => 0.
f (f (X,Y),Z) => f (X,f (Y,Z)).
f (0,i (X)) => i (X).
f (X,f (0,Y)) => f (X,Y).
f (i (X),f (X,Y)) => f (0,Y).
f (X,f (Y,i (Y))) => X.
f (0,f (i (X),Y)) => f (i (X),Y).
f (X,f (Y,f (i (Y),Z))) => f (X,Z).
.
f(i(i(X)), i(X)) = f(i(i(i(X))), i(i(X))).
f(i(i(X)), i(X)) = f(i(X), i(i(X))).
f(i(i(X)), i(X)) = X.

i(i(Z))=i(f(Y,f(f(f(Y,Z),f(Y,X)),f(i(Y),i(X))))).
i(f(f(i(i(Y)),0),i(f(i(Z),i(Y)))))=i(i(f(0,i(i(Z))))).
i(i(i(f(f(Y,X),Z))))=f(i(i(f(i(X),f(X,0)))),i(i(i(f(X,Z))))).
0=X.
f(f(f(i(Y),i(0)),f(f(f(0,Z),i(Y)),f(i(Y),f(0,Z)))),f(i(f(X,f(0,Z))),i(i(f(0,X)))))=f(f(f(i(f(Z,0)),f(f(X,X),f(0,X))),f(0,f(f(X,Y),i(Y)))),i(i(f(0,Y)))).

i(f(f(f(f(Y,0),i(Y)),i(i(0))),i(Y)))=i(0).

0=f(i(i(f(i(0),f(0,Z)))),i(f(f(i(X),Y),f(f(0,X),X)))).

f(f(0,i(f(f(Z,Y),f(Z,Z)))),i(f(0,i(i(0)))))=f(f(X,f(0,0)),0).
f(Z,i(X))=f(Z,i(f(f(f(X,Z),i(0)),Y))).
f(i(i(i(i(Z)))),0)=f(0,f(f(i(f(Y,Y)),Z),X)).
i(X)=i(0).
f(X,i(f(i(i(0)),X)))=i(i(i(f(i(Z),f(Z,Y))))).
i(i(f(f(f(Y,0),Y),i(i(Y)))))=f(f(Y,0),Z).
f(f(f(i(i(0)),f(f(Z,Y),i(X))),i(i(f(X,Z)))),0)=0.
f(i(i(i(0))),f(X,Y))=f(f(f(f(f(Z,X),Y),f(0,i(0))),f(f(f(Y,0),f(Y,0)),i(Z))),f(f(f(i(Y),Y),X),f(i(i(0)),Y))).
f(i(i(f(i(Z),i(Y)))),Z)=f(i(f(i(i(Y)),f(f(0,0),0))),0).
X=f(f(i(i(f(Z,0))),f(f(i(Y),i(0)),i(f(X,0)))),f(f(i(f(Y,0)),i(X)),i(Y))).
i(i(i(i(f(0,Y)))))=i(f(i(0),f(i(f(0,X)),Y))).
0=i(i(i(i(f(X,Z))))).
f(f(0,i(f(Z,i(X)))),f(0,f(f(i(0),f(0,0)),f(i(0),f(Z,X)))))=Z.
X=i(i(i(Z))).
i(i(Y))=f(i(f(i(Y),f(i(0),Z))),i(X)).
f(Z,f(i(i(i(Y))),f(Z,i(i(0)))))=X.
X=f(f(f(f(i(X),f(0,X)),Y),i(f(f(0,0),f(X,Y)))),f(f(f(i(Z),i(Z)),i(f(0,Y))),f(i(i(X)),f(f(X,Z),f(Z,X))))).
Y=f(f(i(f(i(Y),f(Y,X))),f(X,i(i(X)))),f(i(i(Z)),X)).
Y=i(i(i(f(i(0),i(0))))).
f(f(i(f(0,f(Y,0))),f(i(0),f(i(X),i(0)))),f(Y,i(f(f(Y,Y),f(Y,Z)))))=f(f(Z,i(f(f(Y,X),i(Z)))),f(f(f(i(Y),f(Z,Z)),f(f(0,Y),i(0))),i(i(f(0,0))))).
f(i(i(X)),i(Z))=Z.
Z=Y.
i(i(i(i(Y))))=f(f(i(Z),f(i(f(X,Z)),i(i(0)))),i(f(i(f(Y,Y)),f(i(X),f(0,Y))))).
i(i(Z))=i(i(f(i(f(Z,Y)),f(i(Y),i(0))))).
f(f(f(X,i(f(Z,0))),i(f(f(Z,Y),Y))),f(f(i(i(X)),f(f(0,X),i(Y))),i(i(i(Z)))))=i(f(i(f(i(X),0)),i(f(f(X,Z),i(Y))))).
f(i(f(f(i(X),i(Y)),i(i(Z)))),i(f(i(i(Z)),f(i(X),Y))))=Z.
i(i(i(i(i(Z)))))=i(f(i(i(X)),f(i(i(X)),i(i(Y))))).
i(i(Y))=i(i(Z)).
f(f(f(Y,Y),Z),i(i(i(i(X)))))=f(f(i(i(i(X))),i(0)),Z).
Y=Y.
X=i(f(i(X),Z)).
f(f(f(X,f(i(0),f(0,X))),X),i(i(f(f(Z,X),f(Y,Z)))))=f(f(i(Z),i(f(f(X,X),Y))),f(f(f(f(Y,X),f(0,Z)),f(i(Y),f(Y,0))),i(f(i(0),Z)))).
i(0)=0.
i(i(X))=i(0).
f(i(f(i(f(Y,Z)),i(0))),f(i(i(f(Y,0))),f(f(i(Y),0),i(X))))=i(f(f(f(f(X,X),i(Y)),i(i(X))),i(0))).
i(i(Y))=Z.
X=0.
f(0,f(f(Y,i(i(Z))),i(i(i(X)))))=X.
f(i(Z),f(i(Y),X))=f(i(Y),i(f(0,f(Z,i(Y))))).
i(f(i(f(X,i(X))),i(f(f(X,0),i(Z)))))=i(X).
f(i(Z),i(Y))=i(i(Z)).
i(i(Z))=0.
f(0,i(0))=i(i(f(i(i(0)),f(i(Y),i(X))))).
i(i(i(i(i(Z)))))=Z.
X=i(i(f(i(f(Y,Z)),Y))).
Z=f(i(i(f(Z,f(Z,Z)))),0).
f(f(i(X),f(f(i(0),f(Y,0)),Z)),f(X,0))=i(f(0,f(i(f(X,0)),i(f(Y,0))))).
0=0.
Z=Y.
X=f(i(f(f(i(X),f(0,Y)),f(i(Z),i(Z)))),X).
f(i(f(f(f(X,X),i(0)),i(Z))),i(i(i(f(Y,Z)))))=i(i(i(0))).
f(f(i(f(i(Y),Z)),X),Y)=i(i(i(f(i(X),f(0,Z))))).
f(f(Z,Z),0)=i(f(i(Y),i(i(i(X))))).
f(f(f(f(X,i(Y)),f(Z,f(Z,0))),i(f(Y,Z))),i(f(f(i(Y),i(0)),i(X))))=f(f(f(f(f(Z,Y),f(Z,X)),i(i(0))),Y),i(i(i(0)))).
f(0,Z)=i(i(i(i(i(Z))))).
i(f(f(i(f(Y,Z)),0),i(X)))=f(f(f(i(f(X,Y)),Y),i(i(i(X)))),f(i(i(f(Y,Z))),f(i(i(Z)),X))).
f(f(Y,f(f(Z,i(X)),X)),f(f(i(f(X,Z)),f(f(0,Y),X)),i(i(f(Z,0)))))=i(i(X)).
i(f(f(Z,i(f(Z,X))),i(i(f(0,Y)))))=i(i(f(f(f(X,0),f(0,X)),i(i(Z))))).
X=f(i(f(Z,f(X,f(X,Z)))),0).
X=i(f(f(i(i(Z)),f(Z,f(Z,Z))),i(i(i(Y))))).
i(i(X))=X.
i(f(i(i(Z)),f(Z,f(f(0,Z),Z))))=f(i(i(i(f(Y,Y)))),i(f(i(f(X,0)),f(Y,i(0))))).
f(Z,f(i(X),f(i(Y),f(f(X,X),i(X)))))=f(i(i(f(i(Z),f(Y,Z)))),f(i(f(f(0,Y),f(0,0))),f(f(f(0,Y),f(Z,0)),i(f(X,0))))).
i(f(f(i(f(0,0)),f(f(X,Z),f(Z,Z))),f(i(f(X,Z)),X)))=X.
Z=f(0,Y).
i(f(Y,i(i(i(X)))))=f(f(f(f(f(Z,Z),f(0,Z)),i(f(Y,0))),X),0).
X=i(i(f(Y,i(i(X))))).
0=f(i(i(f(f(Z,Y),f(Y,Y)))),f(i(f(X,f(Y,0))),f(i(f(Y,0)),f(f(0,Y),i(X))))).
f(X,i(i(f(f(0,0),f(Z,Z)))))=f(i(i(f(f(Y,0),Z))),i(f(i(f(Z,Y)),f(i(Z),0)))).
0=f(f(f(Y,Y),i(i(f(Y,Y)))),i(f(i(Z),i(f(X,Y))))).
i(i(Y))=i(X).
i(f(X,Y))=i(i(i(i(i(0))))).
X=f(f(f(f(i(Z),i(X)),f(i(Z),i(X))),f(i(0),i(i(Y)))),f(X,i(Z))).
i(i(f(f(f(0,0),f(X,0)),f(i(Z),i(Z)))))=f(f(Y,f(i(i(Z)),f(f(Z,Y),i(X)))),Z).
f(f(f(f(f(X,Z),f(Z,0)),f(f(0,X),i(X))),f(f(i(Z),i(0)),i(f(X,Y)))),f(i(f(i(Y),f(Z,Y))),i(i(f(0,Y)))))=i(i(f(i(X),i(i(Y))))).
f(f(i(i(X)),i(f(f(Y,Z),f(X,0)))),Y)=f(i(Z),i(0)).
f(i(i(f(f(X,Z),f(Z,Y)))),Y)=f(i(f(f(0,i(0)),i(Y))),i(i(X))).
i(i(i(f(i(0),f(Z,X)))))=i(i(f(i(i(0)),f(i(X),i(Z))))).
f(Z,i(Z))=0.
f(0,i(f(i(i(Z)),X)))=f(i(f(0,i(i(0)))),i(i(i(i(X))))).
i(i(i(f(Z,Z))))=f(i(f(f(i(Z),i(0)),f(i(Y),X))),i(i(f(0,Y)))).
0=f(Z,X).
i(i(f(f(f(0,0),f(Z,Z)),i(f(Z,Z)))))=i(X).
f(X,f(f(X,f(f(X,X),Z)),Y))=i(f(i(Z),f(f(i(Y),i(Y)),0))).
i(i(f(i(f(0,Z)),f(f(Y,X),X))))=f(i(f(f(i(0),i(Z)),f(0,f(Z,Z)))),X).
f(X,0)=i(Y).
0=f(f(i(i(0)),i(f(i(Z),f(0,Y)))),i(i(f(i(0),f(0,Y))))).
f(i(i(f(f(X,Z),i(X)))),i(f(f(f(0,0),i(Y)),Z)))=i(i(X)).
f(i(i(f(0,i(Z)))),i(i(i(X))))=Y.
i(f(i(Z),i(i(f(Y,X)))))=i(0).
Y=i(f(f(i(f(Y,Y)),i(Y)),i(i(Z)))).
Y=X.
i(f(i(i(f(X,0))),X))=i(f(i(X),i(f(i(Y),f(Z,Y))))).
f(f(Z,Z),i(Z))=i(X).
X=i(i(i(0))).
i(f(f(f(f(Y,Y),X),i(i(Y))),i(f(0,i(Z)))))=f(0,f(f(i(i(Y)),Y),i(i(f(Y,0))))).
f(f(f(i(X),i(f(X,Z))),i(i(f(Y,X)))),f(f(i(f(0,Y)),i(f(Z,Y))),f(f(i(Z),f(X,Z)),i(0))))=i(i(f(i(f(Z,Y)),f(f(0,Z),f(Z,0))))).
i(i(X))=i(Y).
0=i(i(Y)).
i(f(X,i(f(0,0))))=i(i(i(i(Z)))).
f(i(f(f(f(0,0),X),i(f(X,0)))),i(f(f(f(X,Y),f(Y,Y)),i(f(Y,0)))))=i(i(i(i(f(Z,0))))).
X=f(f(i(Y),0),i(f(i(f(Y,0)),f(f(X,X),i(0))))).
X=f(f(f(0,f(Z,0)),i(f(0,Z))),Z).
Y=f(i(f(Z,i(f(Y,X)))),f(f(i(f(X,Z)),i(f(Z,Z))),i(f(f(Y,X),f(Z,X))))).
i(Z)=Z.
Z=i(i(f(f(i(X),Y),f(i(X),f(Y,0))))).
Y=0.
f(Y,f(f(f(0,X),f(i(Y),i(0))),i(i(i(Y)))))=i(f(Z,i(f(Z,i(0))))).
0=i(f(i(i(i(0))),i(f(f(0,Z),i(X))))).
Z=0.
f(i(0),f(f(0,i(X)),f(i(i(Y)),i(i(X)))))=i(f(X,Z)).
i(i(f(f(i(Y),f(X,Z)),f(i(0),i(Z)))))=i(i(i(Z))).
0=i(i(f(i(f(Z,Y)),f(X,X)))).
i(i(i(f(X,f(Z,X)))))=i(Y).
0=i(f(f(0,i(f(Z,Y))),f(i(i(Z)),i(Z)))).
f(f(0,f(i(f(0,X)),f(i(0),f(Z,0)))),f(Z,f(f(i(Y),i(Z)),i(Y))))=i(i(X)).
X=f(f(f(f(i(Z),f(0,0)),f(f(Z,Y),Z)),f(i(i(Y)),i(f(0,Z)))),i(i(i(i(Y))))).
i(f(f(i(i(X)),f(f(Y,0),0)),Y))=f(Y,i(0)).
Y=f(Z,f(0,i(X))).
f(i(f(0,f(Y,f(0,0)))),i(i(i(i(X)))))=f(f(Y,i(i(i(X)))),i(0)).
i(i(Z))=Z.
X=f(i(f(f(i(X),i(Y)),i(Y))),f(f(f(i(X),Y),i(i(Z))),f(f(f(0,Z),i(X)),f(X,i(X))))).
i(f(f(i(f(Z,0)),f(i(X),Z)),f(i(Y),f(i(Z),f(Z,Y)))))=i(i(i(i(f(Y,Y))))).
X=i(i(i(f(i(Y),f(Y,Z))))).
f(X,i(f(f(i(Z),i(0)),0)))=f(i(Z),i(f(Y,i(f(X,X))))).
i(Z)=i(f(f(i(0),f(i(Z),f(X,0))),f(f(i(Z),i(0)),i(i(Z))))).
f(f(Y,i(i(i(X)))),i(i(i(i(0)))))=f(0,i(i(f(i(0),f(X,0))))).
f(f(i(i(i(X))),f(0,f(i(X),i(X)))),f(f(f(i(Z),f(X,X)),f(Z,X)),f(f(f(0,Y),f(0,X)),f(i(X),f(0,0)))))=Z.
i(i(i(i(f(0,0)))))=i(f(X,i(i(i(Z))))).
f(i(Y),f(f(i(i(Z)),i(f(0,Z))),f(f(i(Z),Z),f(i(Y),f(0,0)))))=i(i(f(i(f(Z,Y)),i(f(Z,0))))).
X=f(f(f(i(f(0,Z)),Y),i(i(f(0,Z)))),i(Z)).
f(f(i(i(f(0,Z))),i(i(f(Y,0)))),i(f(i(i(Z)),f(i(Z),i(Z)))))=f(f(i(f(i(Y),Z)),i(f(f(0,0),i(Z)))),i(f(i(i(0)),Y))).
f(0,i(Y))=0.
i(X)=X.
f(f(Z,f(i(f(Z,Z)),f(f(0,X),X))),f(i(i(f(0,Y))),f(X,i(i(Z)))))=i(i(i(f(X,i(X))))).
f(i(Y),i(i(i(i(Z)))))=f(i(X),i(i(f(i(X),f(0,X))))).
i(f(f(i(i(Y)),f(i(X),Y)),i(i(i(X)))))=i(X).
i(i(f(f(i(X),f(X,0)),f(i(Z),f(Z,0)))))=i(f(i(i(i(Y))),i(i(f(0,Z))))).
f(f(i(i(f(Y,X))),i(f(f(Y,Y),i(0)))),i(f(0,Z)))=f(Z,X).
f(i(i(f(X,0))),f(f(0,f(0,0)),i(f(X,i(X)))))=Y.
i(f(i(Z),i(f(i(Z),Z))))=Z.
i(i(Z))=f(f(0,i(i(i(Z)))),f(Y,Z)).
Z=Z.
Z=i(i(Z)).
f(f(i(0),X),Z)=f(f(Z,f(f(i(Z),i(X)),i(f(Y,Y)))),i(f(i(i(0)),f(f(0,Z),i(Z))))).
i(i(i(f(f(0,X),f(0,Z)))))=Z.
Y=f(0,Z).
f(i(i(X)),f(i(i(Z)),i(f(f(Z,Y),f(0,X)))))=f(0,f(f(i(i(Z)),i(f(0,Y))),i(i(Y)))).
f(f(f(i(f(X,0)),i(i(X))),i(i(f(Y,Z)))),f(i(i(Z)),X))=f(f(i(0),i(f(Z,Y))),i(i(i(f(0,Y))))).
f(i(f(i(f(Z,Z)),f(f(Y,Y),i(Z)))),f(f(Z,f(i(X),i(Y))),Z))=i(f(Z,f(X,i(0)))).
X=f(f(Z,0),f(f(i(f(0,Y)),i(f(Y,X))),i(0))).
Z=i(f(Y,i(i(Y)))).
i(i(i(i(i(0)))))=f(f(f(i(i(Z)),Y),f(i(i(X)),f(i(Z),X))),i(f(Z,Y))).
i(i(f(f(X,Y),i(i(Y)))))=i(f(X,i(i(0)))).
f(i(f(i(i(Y)),0)),f(f(f(f(Z,0),f(Y,X)),Y),X))=f(i(f(i(X),f(f(0,X),i(0)))),f(i(f(f(0,0),f(Z,X))),i(0))).
f(i(i(f(i(Z),i(X)))),Y)=f(i(Z),i(i(i(i(X))))).
i(f(f(f(f(0,Z),X),f(i(Z),i(0))),i(f(i(Z),Y))))=f(i(i(i(i(X)))),f(i(Y),X)).
i(f(i(i(i(0))),i(f(f(0,Z),f(Y,0)))))=X.
i(f(i(f(f(X,Y),f(X,Y))),Y))=i(f(f(Z,f(Y,Y)),f(Y,X))).
f(i(Z),i(Y))=f(f(f(i(i(0)),i(i(Y))),f(f(i(0),f(Y,0)),f(X,f(Y,0)))),Y).
i(i(f(f(f(0,Y),i(X)),f(f(Y,X),f(X,0)))))=f(X,i(i(i(f(Y,Y))))).
i(i(f(i(f(Z,0)),f(f(Y,X),i(Z)))))=f(f(0,f(f(i(Z),f(X,0)),f(f(X,Z),i(Y)))),i(X)).
0=f(Y,i(f(f(0,f(Z,X)),i(f(Y,X))))).
i(f(f(i(f(0,X)),f(i(X),i(X))),f(f(Z,f(0,X)),f(f(Z,Z),i(Z)))))=i(Y).
i(f(f(f(i(Z),i(0)),f(Z,i(Y))),X))=f(0,i(f(Y,f(i(Y),f(0,Z))))).
f(f(f(i(f(Y,Y)),f(Y,i(Y))),i(Y)),f(f(i(i(Y)),i(f(X,Z))),X))=i(Z).
f(f(f(f(f(X,0),f(0,Y)),f(f(X,X),i(0))),i(i(i(X)))),Y)=i(f(f(f(Y,i(Y)),i(i(X))),f(f(i(0),0),i(f(Y,0))))).
