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(1(x0)) → rotate(cut(C1(guess(x0))))
begin(3(x0)) → rotate(cut(C3(guess(x0))))
begin(5(x0)) → rotate(cut(C5(guess(x0))))
begin(2(x0)) → rotate(cut(C2(guess(x0))))
begin(0(x0)) → rotate(cut(C0(guess(x0))))
begin(4(x0)) → rotate(cut(C4(guess(x0))))
guess(1(x0)) → C1(guess(x0))
guess(3(x0)) → C3(guess(x0))
guess(5(x0)) → C5(guess(x0))
guess(2(x0)) → C2(guess(x0))
guess(0(x0)) → C0(guess(x0))
guess(4(x0)) → C4(guess(x0))
guess(1(x0)) → moveleft(B1(wait(x0)))
guess(3(x0)) → moveleft(B3(wait(x0)))
guess(5(x0)) → moveleft(B5(wait(x0)))
guess(2(x0)) → moveleft(B2(wait(x0)))
guess(0(x0)) → moveleft(B0(wait(x0)))
guess(4(x0)) → moveleft(B4(wait(x0)))
guess(end(x0)) → finish(end(x0))
C1(moveleft(B1(x0))) → moveleft(B1(A1(x0)))
C3(moveleft(B1(x0))) → moveleft(B1(A3(x0)))
C5(moveleft(B1(x0))) → moveleft(B1(A5(x0)))
C2(moveleft(B1(x0))) → moveleft(B1(A2(x0)))
C0(moveleft(B1(x0))) → moveleft(B1(A0(x0)))
C4(moveleft(B1(x0))) → moveleft(B1(A4(x0)))
C1(moveleft(B3(x0))) → moveleft(B3(A1(x0)))
C3(moveleft(B3(x0))) → moveleft(B3(A3(x0)))
C5(moveleft(B3(x0))) → moveleft(B3(A5(x0)))
C2(moveleft(B3(x0))) → moveleft(B3(A2(x0)))
C0(moveleft(B3(x0))) → moveleft(B3(A0(x0)))
C4(moveleft(B3(x0))) → moveleft(B3(A4(x0)))
C1(moveleft(B5(x0))) → moveleft(B5(A1(x0)))
C3(moveleft(B5(x0))) → moveleft(B5(A3(x0)))
C5(moveleft(B5(x0))) → moveleft(B5(A5(x0)))
C2(moveleft(B5(x0))) → moveleft(B5(A2(x0)))
C0(moveleft(B5(x0))) → moveleft(B5(A0(x0)))
C4(moveleft(B5(x0))) → moveleft(B5(A4(x0)))
C1(moveleft(B2(x0))) → moveleft(B2(A1(x0)))
C3(moveleft(B2(x0))) → moveleft(B2(A3(x0)))
C5(moveleft(B2(x0))) → moveleft(B2(A5(x0)))
C2(moveleft(B2(x0))) → moveleft(B2(A2(x0)))
C0(moveleft(B2(x0))) → moveleft(B2(A0(x0)))
C4(moveleft(B2(x0))) → moveleft(B2(A4(x0)))
C1(moveleft(B0(x0))) → moveleft(B0(A1(x0)))
C3(moveleft(B0(x0))) → moveleft(B0(A3(x0)))
C5(moveleft(B0(x0))) → moveleft(B0(A5(x0)))
C2(moveleft(B0(x0))) → moveleft(B0(A2(x0)))
C0(moveleft(B0(x0))) → moveleft(B0(A0(x0)))
C4(moveleft(B0(x0))) → moveleft(B0(A4(x0)))
C1(moveleft(B4(x0))) → moveleft(B4(A1(x0)))
C3(moveleft(B4(x0))) → moveleft(B4(A3(x0)))
C5(moveleft(B4(x0))) → moveleft(B4(A5(x0)))
C2(moveleft(B4(x0))) → moveleft(B4(A2(x0)))
C0(moveleft(B4(x0))) → moveleft(B4(A0(x0)))
C4(moveleft(B4(x0))) → moveleft(B4(A4(x0)))
cut(moveleft(B1(x0))) → D1(cut(goright(x0)))
cut(moveleft(B3(x0))) → D3(cut(goright(x0)))
cut(moveleft(B5(x0))) → D5(cut(goright(x0)))
cut(moveleft(B2(x0))) → D2(cut(goright(x0)))
cut(moveleft(B0(x0))) → D0(cut(goright(x0)))
cut(moveleft(B4(x0))) → D4(cut(goright(x0)))
goright(A1(x0)) → C1(goright(x0))
goright(A3(x0)) → C3(goright(x0))
goright(A5(x0)) → C5(goright(x0))
goright(A2(x0)) → C2(goright(x0))
goright(A0(x0)) → C0(goright(x0))
goright(A4(x0)) → C4(goright(x0))
goright(wait(1(x0))) → moveleft(B1(wait(x0)))
goright(wait(3(x0))) → moveleft(B3(wait(x0)))
goright(wait(5(x0))) → moveleft(B5(wait(x0)))
goright(wait(2(x0))) → moveleft(B2(wait(x0)))
goright(wait(0(x0))) → moveleft(B0(wait(x0)))
goright(wait(4(x0))) → moveleft(B4(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
C1(finish(x0)) → finish(1(x0))
C3(finish(x0)) → finish(3(x0))
C5(finish(x0)) → finish(5(x0))
C2(finish(x0)) → finish(2(x0))
C0(finish(x0)) → finish(0(x0))
C4(finish(x0)) → finish(4(x0))
cut(finish(x0)) → finish2(x0)
D1(finish2(x0)) → finish2(1(x0))
D3(finish2(x0)) → finish2(3(x0))
D5(finish2(x0)) → finish2(5(x0))
D2(finish2(x0)) → finish2(2(x0))
D0(finish2(x0)) → finish2(0(x0))
D4(finish2(x0)) → finish2(4(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(1(3(3(x0)))) → begin(3(5(3(2(5(0(2(4(5(4(x0)))))))))))
rewrite(0(3(3(3(x0))))) → begin(5(4(3(5(3(0(5(4(4(0(x0)))))))))))
rewrite(0(3(3(3(1(x0)))))) → begin(5(4(4(0(3(1(0(5(1(0(x0)))))))))))
rewrite(1(2(3(3(3(x0)))))) → begin(4(1(1(2(3(5(0(4(0(5(x0)))))))))))
rewrite(1(4(4(2(2(x0)))))) → begin(1(1(2(0(1(1(1(0(2(2(x0)))))))))))
rewrite(0(3(3(1(4(3(x0))))))) → begin(4(4(3(0(2(3(0(3(0(0(x0)))))))))))
rewrite(3(3(3(3(4(0(x0))))))) → begin(3(0(0(2(1(0(5(3(5(4(x0)))))))))))
rewrite(4(0(1(3(4(0(x0))))))) → begin(2(2(3(0(0(0(5(0(0(0(x0)))))))))))
rewrite(4(1(4(4(4(1(x0))))))) → begin(4(1(0(3(3(5(5(5(4(1(x0)))))))))))
rewrite(0(1(3(5(2(2(3(x0)))))))) → begin(0(3(0(0(5(0(0(4(4(3(x0)))))))))))
rewrite(0(2(3(1(3(2(5(x0)))))))) → begin(0(4(3(1(2(3(2(3(2(0(x0)))))))))))
rewrite(1(1(3(3(5(3(1(x0)))))))) → begin(3(5(0(5(3(2(5(0(0(1(x0)))))))))))
rewrite(3(5(2(0(1(3(3(x0)))))))) → begin(3(4(3(2(3(2(4(4(5(5(x0)))))))))))
rewrite(4(1(4(2(4(0(1(x0)))))))) → begin(5(2(2(1(0(5(5(4(5(1(x0)))))))))))
rewrite(4(5(1(2(4(4(4(x0)))))))) → begin(4(1(1(4(5(3(0(1(0(4(x0)))))))))))
rewrite(5(1(4(5(3(3(3(x0)))))))) → begin(5(1(4(5(3(4(4(2(3(2(x0)))))))))))

Proof

1 Termination Assumption

We assume termination of the following TRS
begin(end(x0)) → rewrite(end(x0))
begin(1(x0)) → rotate(cut(C1(guess(x0))))
begin(3(x0)) → rotate(cut(C3(guess(x0))))
begin(5(x0)) → rotate(cut(C5(guess(x0))))
begin(2(x0)) → rotate(cut(C2(guess(x0))))
begin(0(x0)) → rotate(cut(C0(guess(x0))))
begin(4(x0)) → rotate(cut(C4(guess(x0))))
guess(1(x0)) → C1(guess(x0))
guess(3(x0)) → C3(guess(x0))
guess(5(x0)) → C5(guess(x0))
guess(2(x0)) → C2(guess(x0))
guess(0(x0)) → C0(guess(x0))
guess(4(x0)) → C4(guess(x0))
guess(1(x0)) → moveleft(B1(wait(x0)))
guess(3(x0)) → moveleft(B3(wait(x0)))
guess(5(x0)) → moveleft(B5(wait(x0)))
guess(2(x0)) → moveleft(B2(wait(x0)))
guess(0(x0)) → moveleft(B0(wait(x0)))
guess(4(x0)) → moveleft(B4(wait(x0)))
guess(end(x0)) → finish(end(x0))
C1(moveleft(B1(x0))) → moveleft(B1(A1(x0)))
C3(moveleft(B1(x0))) → moveleft(B1(A3(x0)))
C5(moveleft(B1(x0))) → moveleft(B1(A5(x0)))
C2(moveleft(B1(x0))) → moveleft(B1(A2(x0)))
C0(moveleft(B1(x0))) → moveleft(B1(A0(x0)))
C4(moveleft(B1(x0))) → moveleft(B1(A4(x0)))
C1(moveleft(B3(x0))) → moveleft(B3(A1(x0)))
C3(moveleft(B3(x0))) → moveleft(B3(A3(x0)))
C5(moveleft(B3(x0))) → moveleft(B3(A5(x0)))
C2(moveleft(B3(x0))) → moveleft(B3(A2(x0)))
C0(moveleft(B3(x0))) → moveleft(B3(A0(x0)))
C4(moveleft(B3(x0))) → moveleft(B3(A4(x0)))
C1(moveleft(B5(x0))) → moveleft(B5(A1(x0)))
C3(moveleft(B5(x0))) → moveleft(B5(A3(x0)))
C5(moveleft(B5(x0))) → moveleft(B5(A5(x0)))
C2(moveleft(B5(x0))) → moveleft(B5(A2(x0)))
C0(moveleft(B5(x0))) → moveleft(B5(A0(x0)))
C4(moveleft(B5(x0))) → moveleft(B5(A4(x0)))
C1(moveleft(B2(x0))) → moveleft(B2(A1(x0)))
C3(moveleft(B2(x0))) → moveleft(B2(A3(x0)))
C5(moveleft(B2(x0))) → moveleft(B2(A5(x0)))
C2(moveleft(B2(x0))) → moveleft(B2(A2(x0)))
C0(moveleft(B2(x0))) → moveleft(B2(A0(x0)))
C4(moveleft(B2(x0))) → moveleft(B2(A4(x0)))
C1(moveleft(B0(x0))) → moveleft(B0(A1(x0)))
C3(moveleft(B0(x0))) → moveleft(B0(A3(x0)))
C5(moveleft(B0(x0))) → moveleft(B0(A5(x0)))
C2(moveleft(B0(x0))) → moveleft(B0(A2(x0)))
C0(moveleft(B0(x0))) → moveleft(B0(A0(x0)))
C4(moveleft(B0(x0))) → moveleft(B0(A4(x0)))
C1(moveleft(B4(x0))) → moveleft(B4(A1(x0)))
C3(moveleft(B4(x0))) → moveleft(B4(A3(x0)))
C5(moveleft(B4(x0))) → moveleft(B4(A5(x0)))
C2(moveleft(B4(x0))) → moveleft(B4(A2(x0)))
C0(moveleft(B4(x0))) → moveleft(B4(A0(x0)))
C4(moveleft(B4(x0))) → moveleft(B4(A4(x0)))
cut(moveleft(B1(x0))) → D1(cut(goright(x0)))
cut(moveleft(B3(x0))) → D3(cut(goright(x0)))
cut(moveleft(B5(x0))) → D5(cut(goright(x0)))
cut(moveleft(B2(x0))) → D2(cut(goright(x0)))
cut(moveleft(B0(x0))) → D0(cut(goright(x0)))
cut(moveleft(B4(x0))) → D4(cut(goright(x0)))
goright(A1(x0)) → C1(goright(x0))
goright(A3(x0)) → C3(goright(x0))
goright(A5(x0)) → C5(goright(x0))
goright(A2(x0)) → C2(goright(x0))
goright(A0(x0)) → C0(goright(x0))
goright(A4(x0)) → C4(goright(x0))
goright(wait(1(x0))) → moveleft(B1(wait(x0)))
goright(wait(3(x0))) → moveleft(B3(wait(x0)))
goright(wait(5(x0))) → moveleft(B5(wait(x0)))
goright(wait(2(x0))) → moveleft(B2(wait(x0)))
goright(wait(0(x0))) → moveleft(B0(wait(x0)))
goright(wait(4(x0))) → moveleft(B4(wait(x0)))
goright(wait(end(x0))) → finish(end(x0))
C1(finish(x0)) → finish(1(x0))
C3(finish(x0)) → finish(3(x0))
C5(finish(x0)) → finish(5(x0))
C2(finish(x0)) → finish(2(x0))
C0(finish(x0)) → finish(0(x0))
C4(finish(x0)) → finish(4(x0))
cut(finish(x0)) → finish2(x0)
D1(finish2(x0)) → finish2(1(x0))
D3(finish2(x0)) → finish2(3(x0))
D5(finish2(x0)) → finish2(5(x0))
D2(finish2(x0)) → finish2(2(x0))
D0(finish2(x0)) → finish2(0(x0))
D4(finish2(x0)) → finish2(4(x0))
rotate(finish2(x0)) → rewrite(x0)
rewrite(1(3(3(x0)))) → begin(3(5(3(2(5(0(2(4(5(4(x0)))))))))))
rewrite(0(3(3(3(x0))))) → begin(5(4(3(5(3(0(5(4(4(0(x0)))))))))))
rewrite(0(3(3(3(1(x0)))))) → begin(5(4(4(0(3(1(0(5(1(0(x0)))))))))))
rewrite(1(2(3(3(3(x0)))))) → begin(4(1(1(2(3(5(0(4(0(5(x0)))))))))))
rewrite(1(4(4(2(2(x0)))))) → begin(1(1(2(0(1(1(1(0(2(2(x0)))))))))))
rewrite(0(3(3(1(4(3(x0))))))) → begin(4(4(3(0(2(3(0(3(0(0(x0)))))))))))
rewrite(3(3(3(3(4(0(x0))))))) → begin(3(0(0(2(1(0(5(3(5(4(x0)))))))))))
rewrite(4(0(1(3(4(0(x0))))))) → begin(2(2(3(0(0(0(5(0(0(0(x0)))))))))))
rewrite(4(1(4(4(4(1(x0))))))) → begin(4(1(0(3(3(5(5(5(4(1(x0)))))))))))
rewrite(0(1(3(5(2(2(3(x0)))))))) → begin(0(3(0(0(5(0(0(4(4(3(x0)))))))))))
rewrite(0(2(3(1(3(2(5(x0)))))))) → begin(0(4(3(1(2(3(2(3(2(0(x0)))))))))))
rewrite(1(1(3(3(5(3(1(x0)))))))) → begin(3(5(0(5(3(2(5(0(0(1(x0)))))))))))
rewrite(3(5(2(0(1(3(3(x0)))))))) → begin(3(4(3(2(3(2(4(4(5(5(x0)))))))))))
rewrite(4(1(4(2(4(0(1(x0)))))))) → begin(5(2(2(1(0(5(5(4(5(1(x0)))))))))))
rewrite(4(5(1(2(4(4(4(x0)))))))) → begin(4(1(1(4(5(3(0(1(0(4(x0)))))))))))
rewrite(5(1(4(5(3(3(3(x0)))))))) → begin(5(1(4(5(3(4(4(2(3(2(x0)))))))))))