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