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(1(2(2(1(x0)))))))) → begin(1(1(0(1(0(1(1(0(1(2(1(x0))))))))))))
rewrite(0(2(0(2(0(0(0(2(x0))))))))) → begin(2(1(1(0(1(2(1(2(2(x0))))))))))
rewrite(2(2(0(2(1(2(2(2(2(x0)))))))))) → begin(1(1(0(1(1(2(1(1(2(0(x0)))))))))))
rewrite(1(0(0(2(2(2(2(0(2(2(0(x0)))))))))))) → begin(1(1(1(0(1(2(0(2(1(1(2(1(x0)))))))))))))
rewrite(2(2(1(1(1(2(0(0(0(0(2(x0)))))))))))) → begin(0(0(1(0(1(2(1(0(0(2(1(2(1(x0))))))))))))))
rewrite(0(2(2(1(1(2(0(2(0(0(0(2(x0))))))))))))) → begin(0(0(0(2(1(1(1(0(1(2(1(1(0(2(x0)))))))))))))))
rewrite(1(0(0(1(1(1(0(2(0(2(2(2(2(x0)))))))))))))) → begin(1(0(2(1(2(0(2(2(2(1(1(0(0(x0))))))))))))))
rewrite(2(2(1(2(0(2(2(2(2(0(1(2(0(x0)))))))))))))) → begin(1(2(0(0(1(0(1(2(1(2(1(0(0(1(1(x0))))))))))))))))
rewrite(0(0(2(0(2(0(2(1(2(0(0(0(0(2(x0))))))))))))))) → begin(2(2(0(1(1(0(1(1(2(1(2(0(0(0(0(x0))))))))))))))))
rewrite(0(2(1(2(1(2(2(0(0(0(0(2(0(2(0(x0)))))))))))))))) → begin(0(1(1(2(0(1(2(1(0(1(0(1(1(1(2(1(1(2(x0)))))))))))))))))))
rewrite(0(0(1(2(2(0(0(2(2(2(2(2(2(1(1(2(x0))))))))))))))))) → begin(1(0(1(2(1(1(2(0(0(0(2(0(2(1(2(0(0(x0))))))))))))))))))
rewrite(0(2(1(0(0(2(1(1(1(1(2(0(1(1(2(0(x0))))))))))))))))) → begin(2(0(1(0(1(1(2(1(1(2(0(1(1(0(0(2(1(x0))))))))))))))))))
rewrite(2(2(0(0(1(2(1(2(2(2(2(0(2(1(0(2(x0))))))))))))))))) → begin(1(0(1(2(1(0(1(1(1(0(2(0(1(1(0(1(2(1(0(x0))))))))))))))))))))
rewrite(2(2(2(0(2(2(1(0(1(1(0(2(1(0(0(1(x0))))))))))))))))) → begin(0(1(1(0(1(1(2(1(0(0(0(2(1(0(1(1(1(1(x0)))))))))))))))))))
rewrite(0(0(1(1(0(1(2(2(2(0(2(2(0(2(0(0(2(0(x0))))))))))))))))))) → begin(1(0(0(0(1(0(1(1(2(1(0(1(2(2(0(1(0(1(0(x0))))))))))))))))))))
rewrite(1(0(0(1(1(0(1(1(0(1(1(0(0(2(1(0(2(2(x0))))))))))))))))))) → begin(1(0(1(1(1(2(1(2(0(1(1(1(2(0(1(2(0(2(1(x0))))))))))))))))))))
rewrite(2(0(2(0(0(1(1(2(2(2(0(1(1(1(2(2(0(0(0(x0)))))))))))))))))))) → begin(2(1(2(2(0(1(0(1(2(2(1(1(0(2(1(0(1(0(1(2(1(x0))))))))))))))))))))))
rewrite(2(0(2(1(1(0(0(0(0(2(1(2(2(2(0(2(2(0(2(x0)))))))))))))))))))) → begin(0(2(0(1(2(2(0(1(0(1(2(1(1(2(2(1(2(0(0(0(x0)))))))))))))))))))))
rewrite(1(2(2(1(2(2(2(2(0(2(1(0(1(1(1(2(0(2(1(2(x0))))))))))))))))))))) → begin(1(2(1(1(2(1(1(1(2(0(1(2(1(2(1(2(1(0(1(2(2(x0))))))))))))))))))))))
rewrite(2(0(1(0(1(0(2(1(0(1(2(1(1(0(0(1(0(2(2(0(x0))))))))))))))))))))) → begin(2(2(2(2(1(1(2(0(1(0(2(2(1(0(1(0(1(2(1(2(1(x0))))))))))))))))))))))
rewrite(2(2(0(2(2(0(2(0(2(1(1(0(0(0(2(2(0(1(1(2(0(x0)))))))))))))))))))))) → begin(0(0(0(1(0(1(2(2(2(1(1(0(2(0(1(0(1(2(1(1(1(1(x0)))))))))))))))))))))))
rewrite(2(0(2(2(2(0(2(0(0(2(1(2(2(2(0(0(1(0(2(2(2(2(x0))))))))))))))))))))))) → begin(1(0(0(0(1(1(1(2(1(1(2(1(1(0(1(2(0(1(1(2(0(0(1(1(x0)))))))))))))))))))))))))
rewrite(0(0(1(1(0(2(2(2(2(0(0(0(1(0(0(2(0(2(1(0(2(0(0(x0)))))))))))))))))))))))) → begin(1(2(1(1(0(1(2(1(0(2(1(1(0(2(1(0(1(2(2(2(0(1(1(2(1(x0))))))))))))))))))))))))))
rewrite(0(2(0(2(2(0(1(2(2(1(2(2(2(2(2(0(1(0(2(0(2(2(2(0(x0))))))))))))))))))))))))) → begin(1(1(0(0(2(0(1(1(0(2(2(1(0(0(2(2(2(1(1(0(1(2(1(0(0(x0))))))))))))))))))))))))))
rewrite(0(2(1(0(1(2(2(2(2(2(0(2(1(1(1(1(1(2(2(0(1(2(2(2(x0))))))))))))))))))))))))) → begin(0(1(1(0(1(0(0(1(1(0(1(0(2(1(2(1(0(0(1(0(1(2(1(1(0(x0))))))))))))))))))))))))))
rewrite(1(0(0(2(2(1(0(2(2(2(2(1(0(1(2(2(0(2(0(1(1(1(2(0(x0))))))))))))))))))))))))) → begin(1(1(1(2(1(1(0(1(0(1(1(1(0(1(2(1(0(2(2(1(2(1(1(0(1(2(2(0(x0)))))))))))))))))))))))))))))
rewrite(0(0(0(2(2(1(2(1(0(0(0(1(2(2(0(1(0(0(2(0(2(2(0(2(0(2(0(x0)))))))))))))))))))))))))))) → begin(1(0(1(2(2(0(2(1(1(0(1(2(1(1(0(1(1(1(0(2(2(2(0(0(2(1(0(2(x0)))))))))))))))))))))))))))))
rewrite(0(2(1(2(0(2(1(2(0(2(1(2(0(1(1(2(2(2(0(1(0(1(1(0(0(2(0(x0)))))))))))))))))))))))))))) → begin(1(0(0(1(0(1(0(1(1(2(1(1(2(0(0(2(1(1(2(2(1(1(0(0(1(1(2(2(2(x0))))))))))))))))))))))))))))))
rewrite(0(0(0(2(1(1(2(2(1(1(0(2(1(0(1(0(0(1(1(0(1(1(2(2(2(2(2(2(x0))))))))))))))))))))))))))))) → begin(0(1(0(1(2(1(0(1(0(1(1(0(2(2(0(1(2(1(0(1(1(2(1(1(1(1(2(0(1(2(1(x0))))))))))))))))))))))))))))))))
rewrite(2(2(2(0(2(0(2(0(2(2(2(0(2(0(2(0(0(1(0(0(0(0(0(2(2(1(2(2(x0))))))))))))))))))))))))))))) → begin(0(0(0(1(1(1(2(1(2(1(1(0(1(0(1(1(2(1(0(2(2(2(1(1(1(0(2(2(2(1(2(1(0(x0))))))))))))))))))))))))))))))))))
rewrite(0(2(0(0(1(1(1(2(2(0(0(2(2(0(1(2(1(0(1(0(2(2(1(1(2(2(2(0(2(x0)))))))))))))))))))))))))))))) → begin(0(2(2(1(0(0(1(0(0(1(0(0(1(1(1(1(1(1(2(1(0(0(2(2(2(2(0(1(2(2(x0)))))))))))))))))))))))))))))))
rewrite(0(2(0(0(2(0(2(0(0(1(1(1(1(0(0(0(0(2(1(1(2(0(0(1(1(0(0(2(1(x0)))))))))))))))))))))))))))))) → begin(1(2(2(1(1(2(1(1(2(1(0(0(0(1(0(1(1(1(2(1(2(2(0(2(0(0(2(0(1(2(1(x0))))))))))))))))))))))))))))))))
rewrite(2(0(0(2(2(0(1(0(0(0(0(1(1(0(0(0(2(1(1(2(0(0(2(2(2(0(1(1(0(x0)))))))))))))))))))))))))))))) → begin(0(1(1(1(2(1(0(1(0(1(1(1(0(1(1(1(0(0(1(1(1(1(2(1(0(2(1(2(1(2(x0)))))))))))))))))))))))))))))))
rewrite(0(0(0(2(2(0(0(0(1(0(2(0(1(0(2(2(1(1(0(0(0(2(1(0(2(2(1(2(2(2(x0))))))))))))))))))))))))))))))) → begin(0(0(2(2(1(2(0(2(1(2(1(2(2(0(1(1(2(1(2(1(0(1(1(0(1(2(1(1(2(0(2(1(0(1(x0)))))))))))))))))))))))))))))))))))
rewrite(0(0(0(2(2(2(2(0(0(2(0(1(0(2(2(0(2(1(2(1(1(1(2(2(2(2(2(0(0(1(2(x0)))))))))))))))))))))))))))))))) → begin(1(0(2(0(0(2(2(0(0(1(0(1(1(0(0(1(0(1(0(2(1(0(1(1(1(2(1(2(2(2(0(0(x0)))))))))))))))))))))))))))))))))
rewrite(0(1(2(2(1(1(1(2(2(2(1(2(0(0(0(2(2(0(0(0(2(0(2(0(0(1(1(0(0(1(1(0(x0))))))))))))))))))))))))))))))))) → begin(0(1(1(2(1(1(1(2(0(2(2(2(1(0(0(0(2(0(0(0(2(2(2(0(0(1(2(0(0(1(1(0(x0)))))))))))))))))))))))))))))))))
rewrite(2(2(2(2(1(0(1(2(0(0(1(1(2(2(1(2(2(1(2(2(2(2(2(0(2(0(1(0(1(0(2(1(x0))))))))))))))))))))))))))))))))) → begin(0(0(1(2(1(0(0(2(0(0(2(0(1(0(1(0(2(1(0(1(0(2(1(1(0(1(1(1(2(1(2(1(1(x0))))))))))))))))))))))))))))))))))
rewrite(0(2(0(1(0(0(2(2(0(2(2(2(0(0(1(2(0(1(2(0(2(0(0(2(1(0(2(0(1(1(2(2(0(x0)))))))))))))))))))))))))))))))))) → begin(1(0(1(2(2(0(1(2(1(2(1(2(0(1(1(1(0(1(0(1(2(1(2(1(2(0(1(2(0(0(2(2(2(2(0(2(x0)))))))))))))))))))))))))))))))))))))
rewrite(0(2(1(0(0(2(0(2(2(0(2(0(2(0(0(2(1(2(2(2(2(1(1(0(2(2(1(1(2(2(2(2(0(x0)))))))))))))))))))))))))))))))))) → begin(0(2(1(0(1(1(0(2(0(1(2(0(1(1(1(0(1(1(0(0(2(2(0(1(2(1(2(1(1(1(0(1(1(1(x0)))))))))))))))))))))))))))))))))))
rewrite(0(2(2(2(1(1(0(2(1(0(1(2(2(0(0(2(1(1(2(0(1(1(1(2(1(2(0(1(0(2(2(0(1(x0)))))))))))))))))))))))))))))))))) → begin(2(0(0(1(1(2(1(0(1(1(1(0(1(2(2(0(1(2(1(2(1(0(0(0(1(0(1(1(0(1(2(0(0(1(x0)))))))))))))))))))))))))))))))))))
rewrite(2(2(2(1(0(1(1(2(2(2(0(0(1(2(2(2(0(2(0(1(0(1(0(1(0(0(2(1(0(1(0(1(2(x0)))))))))))))))))))))))))))))))))) → begin(2(0(0(1(2(2(1(0(1(2(2(0(1(1(0(1(2(1(1(0(1(0(1(2(1(1(1(2(1(1(2(0(0(0(1(1(2(x0))))))))))))))))))))))))))))))))))))))
rewrite(0(0(2(1(1(1(2(2(0(2(1(1(1(1(0(1(0(0(2(2(2(0(2(0(2(2(2(2(2(2(2(1(0(1(x0))))))))))))))))))))))))))))))))))) → begin(2(0(2(2(2(0(1(0(0(1(1(2(1(2(0(2(1(0(0(2(2(1(2(1(2(2(2(0(2(2(1(1(0(1(x0)))))))))))))))))))))))))))))))))))
rewrite(1(1(2(0(0(0(2(0(1(1(0(0(0(0(1(1(1(0(0(0(0(1(1(1(0(0(2(2(0(0(1(1(2(0(x0))))))))))))))))))))))))))))))))))) → begin(1(1(1(0(0(2(1(2(2(0(0(1(0(1(2(1(0(1(1(2(0(2(0(0(1(2(0(2(0(0(2(0(0(2(2(x0))))))))))))))))))))))))))))))))))))
rewrite(2(0(1(1(2(2(2(0(1(0(1(1(0(2(0(1(0(0(2(1(2(0(0(1(2(0(0(2(0(1(1(0(2(0(x0))))))))))))))))))))))))))))))))))) → begin(2(0(0(2(1(1(0(1(0(1(1(0(2(2(0(0(0(2(2(0(1(0(2(2(0(0(1(1(2(1(1(0(0(2(x0)))))))))))))))))))))))))))))))))))
rewrite(0(0(2(0(2(2(2(0(0(0(2(1(1(2(0(1(2(1(0(0(1(2(0(2(2(0(2(1(1(2(1(0(0(0(2(x0)))))))))))))))))))))))))))))))))))) → begin(2(1(2(1(1(2(0(1(0(1(1(1(1(2(1(0(2(0(1(1(1(2(1(2(1(0(2(0(0(1(2(1(2(1(1(1(0(x0))))))))))))))))))))))))))))))))))))))
rewrite(2(0(2(2(1(1(2(0(0(2(1(2(0(0(1(0(2(2(1(2(1(0(0(1(2(1(1(1(1(1(1(1(0(0(2(x0)))))))))))))))))))))))))))))))))))) → begin(1(1(1(2(0(0(1(0(1(0(2(0(1(0(1(1(0(1(0(1(2(1(2(1(1(1(1(2(1(0(0(0(2(2(2(0(x0)))))))))))))))))))))))))))))))))))))
rewrite(2(1(2(0(1(1(2(1(1(0(1(1(1(1(2(2(1(2(1(2(2(2(1(2(2(1(1(2(0(1(1(0(0(0(0(x0)))))))))))))))))))))))))))))))))))) → begin(2(1(0(1(2(1(0(0(1(2(2(2(0(0(2(1(0(0(0(0(1(1(2(2(2(1(1(2(1(1(0(2(2(1(1(2(x0)))))))))))))))))))))))))))))))))))))
rewrite(0(2(2(1(1(1(0(0(0(2(2(2(1(2(2(2(2(2(0(1(1(2(2(0(1(2(2(0(2(1(0(2(2(0(1(2(x0))))))))))))))))))))))))))))))))))))) → begin(2(0(2(2(1(1(1(2(2(0(2(2(0(0(2(1(0(1(1(2(1(1(1(2(2(1(2(1(0(1(2(1(1(2(1(2(1(1(0(x0))))))))))))))))))))))))))))))))))))))))
rewrite(0(2(1(0(0(2(0(0(2(0(2(0(2(1(1(1(0(2(2(2(2(2(2(0(2(2(2(1(0(1(1(1(2(0(2(2(0(x0)))))))))))))))))))))))))))))))))))))) → begin(1(0(0(1(0(1(2(1(0(1(2(0(0(1(0(0(1(0(1(0(2(1(0(1(2(1(2(0(0(0(2(0(0(1(2(2(2(0(0(2(x0)))))))))))))))))))))))))))))))))))))))))
rewrite(1(2(2(2(0(0(0(2(1(1(1(2(1(2(0(2(2(1(1(1(2(0(1(2(0(0(2(0(1(1(1(1(1(1(0(0(1(0(2(x0)))))))))))))))))))))))))))))))))))))))) → begin(1(0(1(1(1(2(1(0(1(0(0(2(1(0(0(0(1(2(1(1(2(1(0(2(2(1(2(1(2(0(1(0(1(0(0(0(1(1(1(2(x0)))))))))))))))))))))))))))))))))))))))))
rewrite(0(0(2(1(1(1(0(1(0(2(0(1(2(2(1(1(1(0(0(2(0(0(1(2(2(2(0(0(1(2(0(2(2(2(2(0(0(1(1(0(x0))))))))))))))))))))))))))))))))))))))))) → begin(1(1(1(2(2(0(1(1(0(1(2(1(2(1(2(0(0(0(2(2(0(1(1(0(2(2(2(1(0(1(1(2(1(0(0(2(0(1(0(1(1(0(x0)))))))))))))))))))))))))))))))))))))))))))
rewrite(2(0(0(2(2(2(0(0(0(2(0(2(0(0(1(1(0(0(1(1(0(2(1(0(2(2(0(0(1(0(1(0(1(2(0(2(2(2(2(2(2(x0)))))))))))))))))))))))))))))))))))))))))) → begin(2(2(2(1(1(2(0(2(1(2(1(1(1(2(1(1(2(0(1(0(1(0(1(2(2(1(1(1(2(1(0(0(2(0(2(0(0(0(0(0(1(2(x0)))))))))))))))))))))))))))))))))))))))))))
rewrite(2(0(1(1(2(1(0(0(2(2(1(1(0(0(0(0(2(1(0(1(1(1(1(2(0(2(1(1(2(2(1(2(2(1(2(1(0(1(1(0(0(0(x0))))))))))))))))))))))))))))))))))))))))))) → begin(0(2(1(2(1(0(1(2(1(1(1(2(0(0(2(2(0(2(2(1(0(2(2(0(2(1(1(1(2(1(2(1(1(2(1(2(1(2(2(1(1(0(0(x0))))))))))))))))))))))))))))))))))))))))))))
rewrite(2(2(1(1(2(0(2(1(0(0(2(1(0(1(0(0(2(2(0(0(2(1(2(0(2(2(2(0(0(1(1(0(0(0(1(0(0(0(1(0(2(2(x0))))))))))))))))))))))))))))))))))))))))))) → begin(1(2(0(0(2(1(2(0(0(0(1(2(1(1(0(0(2(1(1(1(2(0(0(1(1(2(2(2(2(1(2(0(1(1(0(1(1(1(2(1(1(1(2(0(x0)))))))))))))))))))))))))))))))))))))))))))))
rewrite(0(1(1(1(2(0(2(2(2(1(1(0(0(1(1(0(1(0(0(2(1(1(2(0(1(1(1(2(0(0(2(0(1(1(0(0(1(2(1(1(2(0(0(x0)))))))))))))))))))))))))))))))))))))))))))) → begin(1(1(2(2(1(2(0(2(2(0(2(0(2(2(2(1(1(1(1(2(1(1(1(1(2(0(1(1(0(1(0(1(1(2(1(1(2(2(2(1(1(0(2(0(2(x0))))))))))))))))))))))))))))))))))))))))))))))
rewrite(0(1(2(0(1(0(0(0(2(2(1(2(2(2(1(2(2(1(1(2(1(0(0(0(0(0(1(1(2(2(0(2(0(0(0(0(0(0(2(0(2(2(0(0(x0))))))))))))))))))))))))))))))))))))))))))))) → begin(0(0(2(1(1(2(1(1(0(1(0(2(0(2(0(2(2(0(0(1(0(1(1(0(1(1(1(1(2(1(2(1(2(0(1(2(0(1(1(0(2(0(0(2(0(0(x0)))))))))))))))))))))))))))))))))))))))))))))))
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rewrite(0(2(0(2(1(2(0(0(0(2(0(0(0(1(0(2(2(1(2(0(2(2(0(1(2(2(1(1(2(1(1(2(1(2(0(1(0(1(2(2(2(2(2(2(x0))))))))))))))))))))))))))))))))))))))))))))) → begin(1(0(0(2(0(1(2(2(1(1(1(1(1(1(2(1(1(0(2(1(1(1(1(0(1(2(1(0(1(0(1(1(1(0(0(2(2(1(1(0(0(0(1(2(1(x0))))))))))))))))))))))))))))))))))))))))))))))
rewrite(1(2(2(2(0(1(1(0(0(0(0(0(2(0(2(2(0(1(2(2(0(2(1(1(1(1(2(2(0(1(1(0(2(0(0(1(1(0(1(2(0(0(0(0(x0))))))))))))))))))))))))))))))))))))))))))))) → begin(1(0(1(2(1(1(1(1(0(2(2(1(0(0(0(1(2(1(1(1(2(1(1(2(1(1(0(2(0(2(0(0(2(2(1(2(0(2(1(0(2(2(1(1(1(x0))))))))))))))))))))))))))))))))))))))))))))))
rewrite(2(2(0(0(0(2(1(0(1(2(0(2(0(1(2(0(1(0(1(0(0(2(2(2(0(2(1(1(0(1(0(1(0(0(2(1(0(1(0(0(1(2(1(0(x0))))))))))))))))))))))))))))))))))))))))))))) → begin(1(0(2(1(0(1(2(1(2(0(0(0(0(1(1(2(1(2(1(2(0(1(1(1(0(1(1(1(1(2(1(1(1(0(2(2(1(0(1(0(1(2(2(2(0(1(x0)))))))))))))))))))))))))))))))))))))))))))))))
rewrite(2(2(0(0(2(1(1(1(2(1(2(0(2(0(1(2(0(2(0(2(0(2(0(0(2(1(0(1(0(1(1(0(1(1(0(1(1(1(2(2(1(0(1(1(x0))))))))))))))))))))))))))))))))))))))))))))) → begin(1(2(1(0(1(1(2(1(0(0(1(2(2(2(2(1(1(1(2(1(2(0(2(0(2(1(1(2(1(2(2(1(2(0(2(1(1(2(0(1(0(2(1(1(0(1(x0)))))))))))))))))))))))))))))))))))))))))))))))
rewrite(1(0(1(2(0(2(1(0(1(2(0(1(0(1(1(2(2(2(1(1(1(0(2(2(2(1(0(2(1(2(2(0(2(1(2(2(2(0(2(1(1(2(2(0(2(x0)))))))))))))))))))))))))))))))))))))))))))))) → begin(1(0(2(2(0(2(2(1(2(0(0(1(0(1(0(0(1(2(2(1(1(2(2(1(1(1(1(0(1(2(1(0(0(1(1(2(0(1(1(0(0(1(0(2(1(0(x0)))))))))))))))))))))))))))))))))))))))))))))))
rewrite(0(0(2(0(0(0(2(0(0(0(2(0(2(1(1(1(2(2(2(0(1(1(2(2(1(2(1(0(1(1(0(0(1(1(2(2(1(2(1(2(0(0(0(1(1(2(x0))))))))))))))))))))))))))))))))))))))))))))))) → begin(2(1(1(2(2(1(1(1(0(1(2(2(0(1(2(1(1(0(0(1(1(2(1(1(1(1(1(0(0(2(1(2(2(2(1(2(1(2(2(2(2(2(0(2(1(2(1(x0))))))))))))))))))))))))))))))))))))))))))))))))
rewrite(0(0(2(1(1(2(2(0(1(0(2(0(0(1(0(2(1(2(2(0(0(0(0(2(2(2(1(0(1(2(2(1(0(0(1(2(2(0(1(2(0(1(2(1(0(2(x0))))))))))))))))))))))))))))))))))))))))))))))) → begin(1(2(0(2(2(0(1(2(1(1(1(0(1(1(2(2(1(2(0(0(0(0(1(2(1(2(0(1(2(0(0(2(2(1(2(0(1(1(0(1(1(2(1(2(1(0(0(x0))))))))))))))))))))))))))))))))))))))))))))))))
rewrite(0(2(1(0(1(1(1(0(1(0(2(1(0(0(2(0(0(1(2(0(1(0(2(0(2(1(0(1(1(0(1(2(0(1(2(2(2(0(0(2(2(2(0(2(2(2(x0))))))))))))))))))))))))))))))))))))))))))))))) → begin(2(0(1(0(0(1(0(1(1(2(1(2(2(0(1(2(0(1(2(0(1(1(1(0(1(0(0(1(0(2(1(0(2(2(2(1(2(1(1(1(2(2(1(2(0(2(1(x0))))))))))))))))))))))))))))))))))))))))))))))))
rewrite(1(2(0(0(1(2(1(1(2(2(1(0(0(0(1(2(1(2(1(0(1(1(2(2(0(0(2(2(1(1(2(0(2(2(2(0(2(0(0(0(0(0(1(1(2(1(x0))))))))))))))))))))))))))))))))))))))))))))))) → begin(1(2(0(1(1(1(1(0(0(0(1(0(2(1(0(1(1(0(1(1(1(0(1(0(1(2(1(1(2(1(0(0(2(2(0(2(2(1(0(1(1(0(0(1(0(1(0(0(1(x0))))))))))))))))))))))))))))))))))))))))))))))))))
rewrite(2(0(1(1(0(1(2(2(2(0(1(1(2(2(2(1(0(2(2(1(1(0(2(1(2(2(0(2(0(0(2(1(2(0(0(1(1(1(2(1(1(2(2(1(2(2(x0))))))))))))))))))))))))))))))))))))))))))))))) → begin(2(0(0(1(1(0(2(1(0(1(1(0(1(2(1(1(0(0(2(0(0(0(1(2(2(2(0(1(1(2(1(1(2(2(1(1(1(1(0(0(1(0(2(1(0(0(2(0(x0)))))))))))))))))))))))))))))))))))))))))))))))))
rewrite(2(2(0(0(2(0(2(0(2(0(1(0(2(1(1(1(0(2(1(0(2(1(0(0(0(1(1(1(1(1(2(0(2(0(1(0(2(2(0(0(1(0(2(1(2(0(x0))))))))))))))))))))))))))))))))))))))))))))))) → begin(1(0(2(1(0(2(2(2(1(0(2(0(1(2(1(1(2(2(0(1(2(1(2(2(1(1(2(2(0(2(2(0(2(1(1(0(1(2(1(2(2(0(1(1(1(2(1(2(x0)))))))))))))))))))))))))))))))))))))))))))))))))
rewrite(1(1(1(1(0(0(2(1(1(0(1(2(0(2(0(1(1(1(0(1(0(2(1(1(0(0(0(2(2(1(2(1(2(2(0(2(1(1(0(2(1(1(1(1(2(2(2(x0)))))))))))))))))))))))))))))))))))))))))))))))) → begin(1(1(1(1(1(1(1(0(1(0(0(1(2(2(0(1(1(0(0(2(2(2(1(1(1(0(0(2(2(1(2(1(0(0(2(2(2(1(2(2(1(0(1(1(0(1(2(x0))))))))))))))))))))))))))))))))))))))))))))))))
rewrite(2(2(1(0(1(2(1(1(0(0(1(2(0(1(2(0(1(2(0(2(0(0(0(2(0(0(2(2(1(1(2(2(0(0(2(2(1(0(0(2(2(1(0(0(0(2(2(x0)))))))))))))))))))))))))))))))))))))))))))))))) → begin(2(0(2(1(1(0(1(0(2(2(1(0(2(0(2(2(1(1(2(1(0(1(2(1(2(1(2(2(0(2(1(1(0(1(2(0(2(1(2(1(0(1(0(1(1(1(2(2(x0)))))))))))))))))))))))))))))))))))))))))))))))))
rewrite(0(2(1(1(1(2(2(2(2(2(2(1(1(0(0(0(2(2(2(2(2(2(2(2(2(0(2(2(0(0(1(1(0(0(2(0(2(2(1(0(2(0(0(1(0(1(1(2(x0))))))))))))))))))))))))))))))))))))))))))))))))) → begin(0(1(2(2(1(2(1(0(1(0(2(2(2(0(0(0(1(1(2(1(1(0(0(1(1(2(2(1(0(1(0(1(1(2(1(2(0(2(2(2(0(1(0(0(0(2(2(2(2(1(x0)))))))))))))))))))))))))))))))))))))))))))))))))))
rewrite(2(0(1(0(0(0(0(1(0(0(1(1(2(0(1(1(2(1(0(2(2(0(0(0(1(1(1(0(2(0(1(2(1(1(2(0(0(1(1(0(0(1(1(1(2(2(2(1(x0))))))))))))))))))))))))))))))))))))))))))))))))) → begin(2(1(1(1(0(2(0(2(1(1(2(1(2(1(0(1(1(0(2(1(1(2(1(2(0(0(2(1(2(1(0(0(1(1(0(1(2(1(0(2(2(1(2(2(2(1(2(1(1(x0))))))))))))))))))))))))))))))))))))))))))))))))))
rewrite(2(0(2(2(1(0(2(1(0(2(0(1(0(0(2(1(0(2(2(1(0(0(1(1(0(0(2(1(2(2(1(1(1(2(0(0(0(0(2(1(2(1(2(0(1(0(2(0(x0))))))))))))))))))))))))))))))))))))))))))))))))) → begin(1(2(2(2(2(0(1(0(2(1(1(2(1(2(1(0(2(1(0(2(1(1(0(2(2(0(2(0(2(1(1(2(0(2(0(2(0(0(0(2(0(0(1(2(1(2(1(2(1(x0))))))))))))))))))))))))))))))))))))))))))))))))))
rewrite(0(2(0(1(1(2(0(2(1(2(0(0(1(2(2(2(2(0(2(1(2(0(1(2(1(1(1(0(1(2(0(0(1(0(1(0(1(1(2(2(2(1(2(0(1(2(2(0(1(x0)))))))))))))))))))))))))))))))))))))))))))))))))) → begin(1(0(1(1(1(1(1(0(1(2(1(0(2(0(0(1(2(0(1(2(1(0(0(2(0(0(1(2(2(2(1(0(1(1(1(1(0(2(0(2(1(1(1(1(0(1(1(0(1(1(x0)))))))))))))))))))))))))))))))))))))))))))))))))))
rewrite(1(1(0(2(2(0(2(2(1(2(1(0(1(1(0(0(1(1(0(2(1(2(0(2(1(0(0(1(2(2(1(0(2(2(1(1(0(0(2(0(1(1(2(0(2(2(1(1(1(0(x0))))))))))))))))))))))))))))))))))))))))))))))))))) → begin(1(1(0(2(2(0(1(2(0(2(1(0(2(2(0(0(1(1(1(2(0(1(0(2(2(0(0(1(1(2(1(0(2(2(1(1(0(0(2(0(1(1(2(2(0(1(1(2(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(1(2(2(1(x0)))))))) → begin(1(1(0(1(0(1(1(0(1(2(1(x0))))))))))))
rewrite(0(2(0(2(0(0(0(2(x0))))))))) → begin(2(1(1(0(1(2(1(2(2(x0))))))))))
rewrite(2(2(0(2(1(2(2(2(2(x0)))))))))) → begin(1(1(0(1(1(2(1(1(2(0(x0)))))))))))
rewrite(1(0(0(2(2(2(2(0(2(2(0(x0)))))))))))) → begin(1(1(1(0(1(2(0(2(1(1(2(1(x0)))))))))))))
rewrite(2(2(1(1(1(2(0(0(0(0(2(x0)))))))))))) → begin(0(0(1(0(1(2(1(0(0(2(1(2(1(x0))))))))))))))
rewrite(0(2(2(1(1(2(0(2(0(0(0(2(x0))))))))))))) → begin(0(0(0(2(1(1(1(0(1(2(1(1(0(2(x0)))))))))))))))
rewrite(1(0(0(1(1(1(0(2(0(2(2(2(2(x0)))))))))))))) → begin(1(0(2(1(2(0(2(2(2(1(1(0(0(x0))))))))))))))
rewrite(2(2(1(2(0(2(2(2(2(0(1(2(0(x0)))))))))))))) → begin(1(2(0(0(1(0(1(2(1(2(1(0(0(1(1(x0))))))))))))))))
rewrite(0(0(2(0(2(0(2(1(2(0(0(0(0(2(x0))))))))))))))) → begin(2(2(0(1(1(0(1(1(2(1(2(0(0(0(0(x0))))))))))))))))
rewrite(0(2(1(2(1(2(2(0(0(0(0(2(0(2(0(x0)))))))))))))))) → begin(0(1(1(2(0(1(2(1(0(1(0(1(1(1(2(1(1(2(x0)))))))))))))))))))
rewrite(0(0(1(2(2(0(0(2(2(2(2(2(2(1(1(2(x0))))))))))))))))) → begin(1(0(1(2(1(1(2(0(0(0(2(0(2(1(2(0(0(x0))))))))))))))))))
rewrite(0(2(1(0(0(2(1(1(1(1(2(0(1(1(2(0(x0))))))))))))))))) → begin(2(0(1(0(1(1(2(1(1(2(0(1(1(0(0(2(1(x0))))))))))))))))))
rewrite(2(2(0(0(1(2(1(2(2(2(2(0(2(1(0(2(x0))))))))))))))))) → begin(1(0(1(2(1(0(1(1(1(0(2(0(1(1(0(1(2(1(0(x0))))))))))))))))))))
rewrite(2(2(2(0(2(2(1(0(1(1(0(2(1(0(0(1(x0))))))))))))))))) → begin(0(1(1(0(1(1(2(1(0(0(0(2(1(0(1(1(1(1(x0)))))))))))))))))))
rewrite(0(0(1(1(0(1(2(2(2(0(2(2(0(2(0(0(2(0(x0))))))))))))))))))) → begin(1(0(0(0(1(0(1(1(2(1(0(1(2(2(0(1(0(1(0(x0))))))))))))))))))))
rewrite(1(0(0(1(1(0(1(1(0(1(1(0(0(2(1(0(2(2(x0))))))))))))))))))) → begin(1(0(1(1(1(2(1(2(0(1(1(1(2(0(1(2(0(2(1(x0))))))))))))))))))))
rewrite(2(0(2(0(0(1(1(2(2(2(0(1(1(1(2(2(0(0(0(x0)))))))))))))))))))) → begin(2(1(2(2(0(1(0(1(2(2(1(1(0(2(1(0(1(0(1(2(1(x0))))))))))))))))))))))
rewrite(2(0(2(1(1(0(0(0(0(2(1(2(2(2(0(2(2(0(2(x0)))))))))))))))))))) → begin(0(2(0(1(2(2(0(1(0(1(2(1(1(2(2(1(2(0(0(0(x0)))))))))))))))))))))
rewrite(1(2(2(1(2(2(2(2(0(2(1(0(1(1(1(2(0(2(1(2(x0))))))))))))))))))))) → begin(1(2(1(1(2(1(1(1(2(0(1(2(1(2(1(2(1(0(1(2(2(x0))))))))))))))))))))))
rewrite(2(0(1(0(1(0(2(1(0(1(2(1(1(0(0(1(0(2(2(0(x0))))))))))))))))))))) → begin(2(2(2(2(1(1(2(0(1(0(2(2(1(0(1(0(1(2(1(2(1(x0))))))))))))))))))))))
rewrite(2(2(0(2(2(0(2(0(2(1(1(0(0(0(2(2(0(1(1(2(0(x0)))))))))))))))))))))) → begin(0(0(0(1(0(1(2(2(2(1(1(0(2(0(1(0(1(2(1(1(1(1(x0)))))))))))))))))))))))
rewrite(2(0(2(2(2(0(2(0(0(2(1(2(2(2(0(0(1(0(2(2(2(2(x0))))))))))))))))))))))) → begin(1(0(0(0(1(1(1(2(1(1(2(1(1(0(1(2(0(1(1(2(0(0(1(1(x0)))))))))))))))))))))))))
rewrite(0(0(1(1(0(2(2(2(2(0(0(0(1(0(0(2(0(2(1(0(2(0(0(x0)))))))))))))))))))))))) → begin(1(2(1(1(0(1(2(1(0(2(1(1(0(2(1(0(1(2(2(2(0(1(1(2(1(x0))))))))))))))))))))))))))
rewrite(0(2(0(2(2(0(1(2(2(1(2(2(2(2(2(0(1(0(2(0(2(2(2(0(x0))))))))))))))))))))))))) → begin(1(1(0(0(2(0(1(1(0(2(2(1(0(0(2(2(2(1(1(0(1(2(1(0(0(x0))))))))))))))))))))))))))
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