MAYBE Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

begin(end(x0)) → rewrite(end(x0))
begin(0(x0)) → rotate(cut(C0(guess(x0))))
begin(1(x0)) → rotate(cut(C1(guess(x0))))
begin(2(x0)) → rotate(cut(C2(guess(x0))))
guess(0(x0)) → C0(guess(x0))
guess(1(x0)) → C1(guess(x0))
guess(2(x0)) → C2(guess(x0))
guess(0(x0)) → moveleft(B0(wait(x0)))
guess(1(x0)) → moveleft(B1(wait(x0)))
guess(2(x0)) → moveleft(B2(wait(x0)))
guess(end(x0)) → finish(end(x0))
C0(moveleft(B0(x0))) → moveleft(B0(A0(x0)))
C1(moveleft(B0(x0))) → moveleft(B0(A1(x0)))
C2(moveleft(B0(x0))) → moveleft(B0(A2(x0)))
C0(moveleft(B1(x0))) → moveleft(B1(A0(x0)))
C1(moveleft(B1(x0))) → moveleft(B1(A1(x0)))
C2(moveleft(B1(x0))) → moveleft(B1(A2(x0)))
C0(moveleft(B2(x0))) → moveleft(B2(A0(x0)))
C1(moveleft(B2(x0))) → moveleft(B2(A1(x0)))
C2(moveleft(B2(x0))) → moveleft(B2(A2(x0)))
cut(moveleft(B0(x0))) → D0(cut(goright(x0)))
cut(moveleft(B1(x0))) → D1(cut(goright(x0)))
cut(moveleft(B2(x0))) → D2(cut(goright(x0)))
goright(A0(x0)) → C0(goright(x0))
goright(A1(x0)) → C1(goright(x0))
goright(A2(x0)) → C2(goright(x0))
goright(wait(0(x0))) → moveleft(B0(wait(x0)))
goright(wait(1(x0))) → moveleft(B1(wait(x0)))
goright(wait(2(x0))) → moveleft(B2(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
C0(finish(x0)) → finish(0(x0))
C1(finish(x0)) → finish(1(x0))
C2(finish(x0)) → finish(2(x0))
cut(finish(x0)) → finish2(x0)
D0(finish2(x0)) → finish2(0(x0))
D1(finish2(x0)) → finish2(1(x0))
D2(finish2(x0)) → finish2(2(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(0(0(0(x0)))) → begin(0(1(2(0(x0)))))
rewrite(0(1(1(x0)))) → begin(0(1(0(1(x0)))))
rewrite(2(0(2(1(x0))))) → begin(2(0(0(1(2(x0))))))
rewrite(0(2(0(2(1(x0)))))) → begin(0(1(0(1(1(2(2(x0))))))))
rewrite(0(0(0(2(1(2(2(x0)))))))) → begin(0(0(1(2(2(2(0(x0))))))))
rewrite(1(2(1(2(0(2(1(x0)))))))) → begin(1(0(1(0(1(2(2(0(x0)))))))))
rewrite(2(0(0(0(1(1(0(x0)))))))) → begin(2(0(1(0(1(0(0(x0))))))))
rewrite(2(0(2(1(1(2(2(x0)))))))) → begin(2(2(2(0(1(2(1(x0))))))))
rewrite(2(2(1(1(0(0(2(1(2(1(x0))))))))))) → begin(2(0(1(2(1(0(1(2(1(2(x0)))))))))))
rewrite(0(0(0(0(0(0(1(2(2(2(0(x0)))))))))))) → begin(0(1(2(0(1(2(0(0(1(2(1(0(1(2(x0)))))))))))))))
rewrite(0(0(1(0(0(1(1(2(1(0(1(x0)))))))))))) → begin(0(0(1(0(1(2(0(1(1(0(1(x0))))))))))))
rewrite(1(2(2(0(2(1(0(0(2(0(2(2(1(x0)))))))))))))) → begin(1(2(0(2(0(1(2(1(2(2(0(2(0(x0))))))))))))))
rewrite(0(2(1(0(0(2(0(0(2(2(2(1(0(1(1(x0)))))))))))))))) → begin(2(1(0(1(2(1(2(0(2(1(1(1(0(1(0(2(x0)))))))))))))))))
rewrite(2(0(2(2(2(0(0(2(2(1(2(0(0(0(2(2(x0))))))))))))))))) → begin(0(1(1(1(2(0(1(2(2(2(1(2(1(2(0(0(2(x0))))))))))))))))))
rewrite(1(0(2(1(1(1(0(1(1(2(0(1(1(2(2(1(1(x0)))))))))))))))))) → begin(1(1(0(1(2(2(0(2(1(0(1(1(1(2(1(1(1(x0))))))))))))))))))
rewrite(2(0(1(0(1(1(2(1(2(0(0(0(0(0(0(2(0(1(x0))))))))))))))))))) → begin(0(1(2(0(1(2(2(0(0(1(2(0(1(2(0(2(1(0(1(2(1(2(1(x0))))))))))))))))))))))))
rewrite(0(0(2(2(1(0(1(1(0(2(1(1(2(1(1(2(1(0(1(x0)))))))))))))))))))) → begin(1(2(1(1(1(2(0(2(0(2(1(0(2(0(1(1(1(0(1(x0))))))))))))))))))))
rewrite(0(0(0(0(1(0(0(2(2(1(2(0(1(1(1(1(2(2(2(2(x0))))))))))))))))))))) → begin(1(2(1(1(2(0(1(0(1(2(1(0(1(1(0(1(2(2(1(1(0(0(x0)))))))))))))))))))))))
rewrite(1(0(1(0(2(1(0(0(1(0(2(0(2(0(2(2(1(2(1(1(x0))))))))))))))))))))) → begin(1(0(0(2(2(0(2(1(0(0(1(2(1(2(1(1(1(0(0(2(x0)))))))))))))))))))))
rewrite(1(2(2(0(2(2(2(1(0(2(1(0(2(2(0(2(1(1(2(2(x0))))))))))))))))))))) → begin(1(2(2(2(2(2(1(1(2(1(0(0(0(0(1(2(2(2(2(2(x0)))))))))))))))))))))
rewrite(2(1(2(0(0(1(0(0(0(0(1(1(1(2(1(1(2(1(1(1(x0))))))))))))))))))))) → begin(0(1(1(1(0(1(0(1(1(2(0(2(0(1(2(1(2(1(1(0(x0)))))))))))))))))))))
rewrite(0(1(1(0(2(1(2(2(1(2(2(1(0(2(0(1(1(0(0(2(1(x0)))))))))))))))))))))) → begin(0(1(0(2(0(1(0(0(0(1(0(1(1(2(1(2(2(1(0(2(0(2(x0)))))))))))))))))))))))
rewrite(0(2(1(2(1(1(2(0(1(2(2(0(1(0(2(0(2(0(0(2(1(2(0(2(x0))))))))))))))))))))))))) → begin(2(2(1(1(0(0(2(0(0(1(1(0(1(2(1(0(1(2(2(2(2(1(2(1(2(0(x0)))))))))))))))))))))))))))
rewrite(0(0(2(0(0(1(1(0(2(1(0(1(1(0(1(0(2(0(0(1(2(2(1(2(2(x0)))))))))))))))))))))))))) → begin(0(0(0(1(0(0(1(1(2(0(1(2(1(2(1(2(0(0(1(0(1(2(1(1(0(2(x0)))))))))))))))))))))))))))
rewrite(0(0(2(0(2(2(2(2(2(0(0(0(2(2(0(1(2(1(2(0(1(1(1(1(2(x0)))))))))))))))))))))))))) → begin(0(1(2(1(0(1(2(2(0(1(0(2(2(1(1(1(0(2(2(0(1(0(2(0(2(1(0(x0))))))))))))))))))))))))))))
rewrite(0(2(0(2(0(2(1(2(1(1(2(2(2(1(1(2(0(1(0(0(2(2(2(1(1(x0)))))))))))))))))))))))))) → begin(2(1(2(0(0(1(2(1(2(0(2(0(0(0(2(1(0(0(2(2(1(2(2(2(1(2(0(x0))))))))))))))))))))))))))))
rewrite(2(2(2(2(1(2(2(1(2(2(2(1(2(2(1(2(0(1(0(0(2(1(0(2(0(x0)))))))))))))))))))))))))) → begin(2(0(2(0(1(2(2(1(2(2(2(2(0(1(2(2(1(1(0(2(2(1(2(2(0(x0))))))))))))))))))))))))))
rewrite(0(0(1(2(0(2(2(1(1(0(1(1(2(0(0(0(0(1(0(2(2(1(1(1(2(1(x0))))))))))))))))))))))))))) → begin(1(1(0(1(2(0(1(0(1(2(1(0(1(1(0(0(1(0(1(0(0(1(1(1(0(2(0(2(x0)))))))))))))))))))))))))))))
rewrite(0(1(0(0(1(1(2(1(0(1(0(2(2(2(1(0(0(2(2(2(2(0(1(1(0(1(2(0(x0))))))))))))))))))))))))))))) → begin(2(1(0(2(2(1(0(1(0(1(2(0(1(2(1(0(1(1(2(1(2(1(0(2(2(0(2(0(0(x0))))))))))))))))))))))))))))))
rewrite(0(0(0(2(2(1(0(2(1(1(0(1(1(0(1(1(1(1(1(1(1(1(2(0(0(0(2(1(1(0(0(1(2(2(x0))))))))))))))))))))))))))))))))))) → begin(2(0(0(1(2(0(2(1(1(1(0(2(0(0(1(0(1(2(2(0(0(1(2(2(2(1(0(0(0(0(1(2(1(0(1(0(1(0(2(0(0(x0))))))))))))))))))))))))))))))))))))))))))
rewrite(2(2(1(1(1(1(1(2(1(0(2(1(1(0(1(0(0(0(0(1(1(2(1(1(0(1(0(2(1(2(1(0(0(0(x0))))))))))))))))))))))))))))))))))) → begin(2(1(0(1(2(0(1(2(1(0(0(0(2(1(2(2(0(2(0(1(1(1(2(1(1(0(0(2(0(1(1(0(1(0(1(2(x0)))))))))))))))))))))))))))))))))))))
rewrite(0(0(0(0(2(0(0(0(2(1(1(0(1(1(0(1(1(1(2(2(2(0(1(1(1(2(2(2(1(1(0(1(0(2(0(x0)))))))))))))))))))))))))))))))))))) → begin(2(0(0(1(2(1(1(1(2(2(0(2(2(1(0(0(1(1(0(1(2(2(1(0(0(2(0(1(0(0(1(1(0(1(0(x0))))))))))))))))))))))))))))))))))))
rewrite(2(1(1(0(2(2(2(1(2(1(1(0(0(2(0(1(0(2(0(0(2(0(2(2(0(0(0(0(0(2(2(1(0(2(0(x0)))))))))))))))))))))))))))))))))))) → begin(2(1(0(0(0(0(1(1(2(2(0(1(0(2(0(1(2(2(0(1(0(0(2(2(2(0(2(2(0(0(2(1(0(2(0(x0))))))))))))))))))))))))))))))))))))
rewrite(1(0(2(1(0(0(1(1(2(0(0(0(2(0(1(1(1(1(0(1(2(2(0(0(0(2(0(0(2(2(2(0(1(1(2(2(2(x0)))))))))))))))))))))))))))))))))))))) → begin(1(0(2(1(1(0(1(1(1(0(1(0(1(2(2(1(0(0(0(1(2(2(0(2(1(2(0(2(2(2(2(0(0(0(2(0(0(x0))))))))))))))))))))))))))))))))))))))
rewrite(0(1(0(2(1(1(1(1(1(0(0(0(2(1(0(1(0(0(0(1(0(1(0(0(0(0(2(1(0(0(0(0(1(1(2(1(0(1(1(x0)))))))))))))))))))))))))))))))))))))))) → begin(2(2(2(1(1(1(0(2(0(1(2(1(2(2(0(1(1(2(1(0(1(0(2(1(1(2(2(2(0(2(2(1(1(1(0(2(2(2(1(1(2(x0))))))))))))))))))))))))))))))))))))))))))
rewrite(1(2(0(2(2(2(0(2(1(1(2(0(1(1(2(0(1(1(1(1(1(2(2(2(1(0(0(1(1(1(1(0(0(2(0(0(2(2(1(x0)))))))))))))))))))))))))))))))))))))))) → begin(1(1(1(2(1(2(1(2(2(2(2(2(0(0(2(1(1(1(0(0(2(1(2(2(0(0(1(0(1(0(1(1(1(2(2(0(0(1(1(x0))))))))))))))))))))))))))))))))))))))))
rewrite(0(2(1(0(1(0(0(0(2(0(1(2(0(2(1(2(1(1(0(2(1(1(0(0(0(0(0(2(2(0(0(1(1(0(1(0(2(2(2(1(1(x0)))))))))))))))))))))))))))))))))))))))))) → begin(1(0(1(2(2(2(1(0(2(1(1(1(1(2(0(1(2(2(0(1(0(2(0(1(2(2(1(0(0(2(0(2(0(1(0(2(0(1(2(0(1(0(1(2(0(1(x0)))))))))))))))))))))))))))))))))))))))))))))))
rewrite(1(0(2(1(0(0(2(2(0(0(2(2(2(0(0(2(1(1(1(2(1(2(1(0(1(1(0(0(2(0(0(2(1(0(1(2(1(1(1(1(1(x0)))))))))))))))))))))))))))))))))))))))))) → begin(1(1(2(1(2(0(2(1(1(2(1(2(2(2(1(2(1(0(1(0(0(0(2(0(1(1(1(2(0(0(0(0(1(0(1(2(1(2(0(1(0(x0))))))))))))))))))))))))))))))))))))))))))
rewrite(1(2(1(1(2(0(1(0(1(0(2(0(2(1(1(2(0(0(0(0(1(1(0(2(1(2(0(0(0(2(1(2(0(0(0(2(2(2(2(1(0(x0)))))))))))))))))))))))))))))))))))))))))) → begin(1(0(1(2(1(0(0(1(0(1(1(0(2(0(2(1(2(0(1(2(0(0(2(0(0(2(0(0(0(1(2(2(0(0(1(0(2(1(2(2(0(0(x0)))))))))))))))))))))))))))))))))))))))))))
rewrite(0(0(1(2(2(2(0(1(1(2(1(2(0(1(2(2(0(2(2(2(0(1(0(0(2(2(2(2(2(1(1(2(0(1(1(2(0(0(2(2(1(0(x0))))))))))))))))))))))))))))))))))))))))))) → begin(1(0(1(2(0(2(1(2(1(2(1(2(0(0(2(2(2(2(2(1(0(0(2(0(1(2(0(1(2(2(1(1(2(0(1(2(0(2(2(2(0(0(x0)))))))))))))))))))))))))))))))))))))))))))
rewrite(1(0(2(2(0(2(2(1(0(2(1(0(0(2(1(1(2(2(1(0(2(2(2(2(0(2(1(2(0(2(1(2(1(1(2(0(1(2(1(1(2(2(x0))))))))))))))))))))))))))))))))))))))))))) → begin(1(0(1(0(0(1(2(0(1(1(1(2(2(1(2(0(0(2(0(0(1(2(0(1(2(2(1(0(0(1(2(2(2(1(1(2(0(0(2(2(1(2(1(2(x0)))))))))))))))))))))))))))))))))))))))))))))
rewrite(2(2(0(0(0(0(0(0(1(0(2(2(2(1(1(2(2(0(0(2(2(1(1(0(2(2(1(1(0(2(0(2(1(1(0(1(1(2(2(1(1(2(x0))))))))))))))))))))))))))))))))))))))))))) → begin(2(2(1(0(0(1(0(0(0(1(2(2(2(1(2(1(2(0(1(1(1(2(1(0(0(2(2(2(1(2(0(0(1(0(1(0(0(1(2(1(2(1(2(0(0(0(2(x0))))))))))))))))))))))))))))))))))))))))))))))))
rewrite(0(2(0(1(0(2(2(2(2(0(0(0(1(2(1(0(0(0(2(1(1(2(2(1(2(2(2(0(2(0(0(1(2(0(2(0(2(0(1(0(2(1(1(1(x0))))))))))))))))))))))))))))))))))))))))))))) → begin(1(2(2(0(2(1(2(0(0(1(1(2(0(0(1(2(1(0(1(0(1(2(1(0(1(0(2(0(1(1(0(2(2(2(1(2(1(0(0(1(2(0(2(0(2(2(1(2(1(x0))))))))))))))))))))))))))))))))))))))))))))))))))
rewrite(1(0(0(2(1(2(1(0(2(0(1(1(1(0(1(1(1(0(0(0(1(2(0(1(2(2(0(0(1(0(0(1(1(2(0(0(2(0(1(1(0(2(1(1(x0))))))))))))))))))))))))))))))))))))))))))))) → begin(1(0(1(2(2(1(1(2(0(1(1(1(1(0(1(2(2(0(0(0(1(2(0(0(2(0(0(0(1(1(0(1(1(0(1(0(0(2(1(1(0(2(0(1(x0)))))))))))))))))))))))))))))))))))))))))))))
rewrite(2(0(0(2(0(2(0(0(2(1(0(1(2(2(0(2(1(2(2(0(2(2(0(2(1(1(0(1(0(2(1(1(0(1(2(1(2(2(0(0(0(1(1(2(x0))))))))))))))))))))))))))))))))))))))))))))) → begin(2(2(0(0(2(2(0(1(2(0(0(1(2(2(0(0(1(2(2(1(2(2(1(0(0(1(1(2(2(2(2(2(0(1(0(1(1(1(0(0(0(1(0(2(x0)))))))))))))))))))))))))))))))))))))))))))))
rewrite(2(2(1(2(0(0(1(1(2(2(2(1(0(2(1(0(1(1(2(1(2(1(1(0(0(0(0(1(1(0(2(2(0(1(2(2(1(2(1(0(0(0(1(1(x0))))))))))))))))))))))))))))))))))))))))))))) → begin(2(2(2(2(0(0(1(2(1(2(0(1(1(2(1(1(0(1(2(1(1(1(0(1(1(2(0(0(1(0(0(2(2(0(2(2(0(2(1(1(0(0(1(1(x0)))))))))))))))))))))))))))))))))))))))))))))
rewrite(0(0(0(2(2(1(1(2(1(2(1(0(2(1(0(2(2(0(2(2(0(1(1(1(2(2(2(0(2(1(2(2(0(0(0(2(1(1(2(1(0(0(0(0(0(2(0(x0)))))))))))))))))))))))))))))))))))))))))))))))) → begin(0(0(1(2(2(1(1(2(0(1(0(0(2(1(0(2(2(1(2(2(0(1(1(2(2(2(0(2(2(2(0(0(1(0(2(0(0(0(2(1(1(2(0(1(0(2(0(x0))))))))))))))))))))))))))))))))))))))))))))))))
rewrite(0(0(2(0(0(0(0(2(2(2(1(0(2(2(2(1(2(1(0(2(1(0(2(2(1(1(1(1(2(1(2(0(0(1(1(1(0(1(2(2(1(1(1(2(1(0(0(0(x0))))))))))))))))))))))))))))))))))))))))))))))))) → begin(0(1(1(0(2(0(0(2(1(2(1(2(2(0(2(1(2(0(2(0(1(0(2(2(1(1(1(0(0(1(1(0(2(0(1(2(0(1(2(2(2(1(1(2(1(0(1(0(x0)))))))))))))))))))))))))))))))))))))))))))))))))
rewrite(0(0(2(0(2(1(0(2(1(1(0(0(0(2(2(0(2(1(2(1(0(1(1(0(2(2(0(1(0(0(1(0(2(2(2(1(0(2(1(0(0(2(1(2(0(2(1(2(x0))))))))))))))))))))))))))))))))))))))))))))))))) → begin(2(2(2(2(0(2(2(1(1(1(1(1(2(0(0(1(1(0(1(2(1(1(0(0(2(0(0(0(2(1(1(0(2(0(0(2(2(2(0(1(2(0(2(2(0(1(0(0(0(x0))))))))))))))))))))))))))))))))))))))))))))))))))
rewrite(1(2(0(1(0(0(0(2(0(2(2(2(2(1(1(0(1(1(2(0(2(1(1(0(2(0(2(2(0(1(1(2(0(0(2(0(0(1(2(2(2(0(0(2(1(1(1(2(x0))))))))))))))))))))))))))))))))))))))))))))))))) → begin(1(2(0(1(2(2(0(2(2(2(2(0(0(0(1(2(0(1(2(0(2(0(1(2(0(1(0(2(0(0(1(1(2(0(2(0(0(1(2(2(2(2(1(0(1(1(1(1(x0)))))))))))))))))))))))))))))))))))))))))))))))))

Proof

1 Termination Assumption

We assume termination of the following TRS
begin(end(x0)) → rewrite(end(x0))
begin(0(x0)) → rotate(cut(C0(guess(x0))))
begin(1(x0)) → rotate(cut(C1(guess(x0))))
begin(2(x0)) → rotate(cut(C2(guess(x0))))
guess(0(x0)) → C0(guess(x0))
guess(1(x0)) → C1(guess(x0))
guess(2(x0)) → C2(guess(x0))
guess(0(x0)) → moveleft(B0(wait(x0)))
guess(1(x0)) → moveleft(B1(wait(x0)))
guess(2(x0)) → moveleft(B2(wait(x0)))
guess(end(x0)) → finish(end(x0))
C0(moveleft(B0(x0))) → moveleft(B0(A0(x0)))
C1(moveleft(B0(x0))) → moveleft(B0(A1(x0)))
C2(moveleft(B0(x0))) → moveleft(B0(A2(x0)))
C0(moveleft(B1(x0))) → moveleft(B1(A0(x0)))
C1(moveleft(B1(x0))) → moveleft(B1(A1(x0)))
C2(moveleft(B1(x0))) → moveleft(B1(A2(x0)))
C0(moveleft(B2(x0))) → moveleft(B2(A0(x0)))
C1(moveleft(B2(x0))) → moveleft(B2(A1(x0)))
C2(moveleft(B2(x0))) → moveleft(B2(A2(x0)))
cut(moveleft(B0(x0))) → D0(cut(goright(x0)))
cut(moveleft(B1(x0))) → D1(cut(goright(x0)))
cut(moveleft(B2(x0))) → D2(cut(goright(x0)))
goright(A0(x0)) → C0(goright(x0))
goright(A1(x0)) → C1(goright(x0))
goright(A2(x0)) → C2(goright(x0))
goright(wait(0(x0))) → moveleft(B0(wait(x0)))
goright(wait(1(x0))) → moveleft(B1(wait(x0)))
goright(wait(2(x0))) → moveleft(B2(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
C0(finish(x0)) → finish(0(x0))
C1(finish(x0)) → finish(1(x0))
C2(finish(x0)) → finish(2(x0))
cut(finish(x0)) → finish2(x0)
D0(finish2(x0)) → finish2(0(x0))
D1(finish2(x0)) → finish2(1(x0))
D2(finish2(x0)) → finish2(2(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(0(0(0(x0)))) → begin(0(1(2(0(x0)))))
rewrite(0(1(1(x0)))) → begin(0(1(0(1(x0)))))
rewrite(2(0(2(1(x0))))) → begin(2(0(0(1(2(x0))))))
rewrite(0(2(0(2(1(x0)))))) → begin(0(1(0(1(1(2(2(x0))))))))
rewrite(0(0(0(2(1(2(2(x0)))))))) → begin(0(0(1(2(2(2(0(x0))))))))
rewrite(1(2(1(2(0(2(1(x0)))))))) → begin(1(0(1(0(1(2(2(0(x0)))))))))
rewrite(2(0(0(0(1(1(0(x0)))))))) → begin(2(0(1(0(1(0(0(x0))))))))
rewrite(2(0(2(1(1(2(2(x0)))))))) → begin(2(2(2(0(1(2(1(x0))))))))
rewrite(2(2(1(1(0(0(2(1(2(1(x0))))))))))) → begin(2(0(1(2(1(0(1(2(1(2(x0)))))))))))
rewrite(0(0(0(0(0(0(1(2(2(2(0(x0)))))))))))) → begin(0(1(2(0(1(2(0(0(1(2(1(0(1(2(x0)))))))))))))))
rewrite(0(0(1(0(0(1(1(2(1(0(1(x0)))))))))))) → begin(0(0(1(0(1(2(0(1(1(0(1(x0))))))))))))
rewrite(1(2(2(0(2(1(0(0(2(0(2(2(1(x0)))))))))))))) → begin(1(2(0(2(0(1(2(1(2(2(0(2(0(x0))))))))))))))
rewrite(0(2(1(0(0(2(0(0(2(2(2(1(0(1(1(x0)))))))))))))))) → begin(2(1(0(1(2(1(2(0(2(1(1(1(0(1(0(2(x0)))))))))))))))))
rewrite(2(0(2(2(2(0(0(2(2(1(2(0(0(0(2(2(x0))))))))))))))))) → begin(0(1(1(1(2(0(1(2(2(2(1(2(1(2(0(0(2(x0))))))))))))))))))
rewrite(1(0(2(1(1(1(0(1(1(2(0(1(1(2(2(1(1(x0)))))))))))))))))) → begin(1(1(0(1(2(2(0(2(1(0(1(1(1(2(1(1(1(x0))))))))))))))))))
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