MAYBE Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

0(0(1(x0))) → 1(2(0(2(0(x0)))))
0(0(1(x0))) → 2(0(2(1(0(x0)))))
0(0(1(x0))) → 2(1(0(2(0(x0)))))
0(0(1(x0))) → 1(1(2(0(2(0(x0))))))
0(0(1(x0))) → 1(2(0(2(0(2(x0))))))
0(0(1(x0))) → 1(2(0(2(0(3(x0))))))
0(0(1(x0))) → 1(2(0(2(0(4(x0))))))
0(0(1(x0))) → 1(2(0(2(2(0(x0))))))
0(0(1(x0))) → 1(2(0(2(3(0(x0))))))
0(0(1(x0))) → 1(2(0(2(5(0(x0))))))
0(0(1(x0))) → 1(2(0(3(2(0(x0))))))
0(0(1(x0))) → 1(2(1(0(2(0(x0))))))
0(0(1(x0))) → 1(2(2(0(2(0(x0))))))
0(0(1(x0))) → 1(2(2(0(3(0(x0))))))
0(0(1(x0))) → 1(2(2(0(4(0(x0))))))
0(0(1(x0))) → 1(2(3(0(2(0(x0))))))
0(0(1(x0))) → 1(2(5(0(2(0(x0))))))
0(0(1(x0))) → 1(3(2(0(2(0(x0))))))
0(0(1(x0))) → 2(0(1(2(0(2(x0))))))
0(0(1(x0))) → 2(0(1(2(2(0(x0))))))
0(0(1(x0))) → 2(0(1(3(0(2(x0))))))
0(0(1(x0))) → 2(0(2(0(2(1(x0))))))
0(0(1(x0))) → 2(0(2(1(0(2(x0))))))
0(0(1(x0))) → 2(0(2(1(0(4(x0))))))
0(0(1(x0))) → 2(0(2(1(2(0(x0))))))
0(0(1(x0))) → 2(0(2(3(1(0(x0))))))
0(0(1(x0))) → 2(0(2(5(1(0(x0))))))
0(0(1(x0))) → 2(1(0(2(0(2(x0))))))
0(0(1(x0))) → 2(1(0(2(2(0(x0))))))
0(0(1(x0))) → 2(1(0(2(5(0(x0))))))
0(0(1(x0))) → 2(1(2(0(2(0(x0))))))
0(0(1(x0))) → 2(1(2(0(4(0(x0))))))
0(0(1(x0))) → 2(1(4(0(2(0(x0))))))
0(0(1(x0))) → 2(1(4(0(4(0(x0))))))
0(0(1(x0))) → 2(2(0(2(1(0(x0))))))
0(0(1(x0))) → 2(2(0(4(1(0(x0))))))
0(0(1(x0))) → 2(2(1(0(2(0(x0))))))
0(0(1(x0))) → 3(1(2(0(2(0(x0))))))
0(0(0(1(x0)))) → 0(0(1(2(0(2(x0))))))
0(0(0(1(x0)))) → 0(2(1(0(2(0(x0))))))
0(0(0(1(x0)))) → 2(0(2(1(0(0(x0))))))
0(0(1(0(x0)))) → 0(0(2(0(2(1(x0))))))
0(0(1(1(x0)))) → 2(0(1(2(1(0(x0))))))
0(0(1(1(x0)))) → 2(0(2(0(1(1(x0))))))
0(0(1(1(x0)))) → 2(1(0(2(0(1(x0))))))
0(0(3(1(x0)))) → 2(0(1(3(0(x0)))))
0(0(3(1(x0)))) → 2(0(1(3(0(2(x0))))))
0(0(3(1(x0)))) → 2(0(2(3(1(0(x0))))))
0(0(3(1(x0)))) → 2(1(0(2(0(3(x0))))))
0(0(3(1(x0)))) → 3(2(0(2(1(0(x0))))))
0(0(5(1(x0)))) → 0(2(5(1(0(x0)))))
0(0(5(1(x0)))) → 0(5(0(2(1(x0)))))
0(0(5(1(x0)))) → 2(5(0(0(1(x0)))))
0(0(5(1(x0)))) → 2(5(1(0(0(x0)))))
0(0(5(1(x0)))) → 0(1(2(5(2(0(x0))))))
0(0(5(1(x0)))) → 0(2(1(2(5(0(x0))))))
0(0(5(1(x0)))) → 0(2(1(4(5(0(x0))))))
0(0(5(1(x0)))) → 1(2(0(0(2(5(x0))))))
0(0(5(1(x0)))) → 2(0(2(1(0(5(x0))))))
0(0(5(1(x0)))) → 2(1(0(4(5(0(x0))))))
0(0(5(1(x0)))) → 2(5(0(2(1(0(x0))))))
0(1(0(1(x0)))) → 0(0(1(2(1(x0)))))
0(1(0(1(x0)))) → 1(2(1(0(0(x0)))))
0(1(0(1(x0)))) → 2(1(0(0(1(x0)))))
0(1(0(1(x0)))) → 0(0(1(1(2(2(x0))))))
0(1(0(1(x0)))) → 0(1(1(3(2(0(x0))))))
0(1(0(1(x0)))) → 1(1(3(2(0(0(x0))))))
0(1(0(1(x0)))) → 1(2(0(2(0(1(x0))))))
0(1(0(1(x0)))) → 1(2(2(0(0(1(x0))))))
0(1(0(1(x0)))) → 1(2(5(1(0(0(x0))))))
0(1(0(1(x0)))) → 2(1(2(1(0(0(x0))))))
0(1(0(1(x0)))) → 3(1(2(1(0(0(x0))))))
0(3(0(1(x0)))) → 1(2(3(0(2(0(x0))))))
0(3(0(1(x0)))) → 3(1(2(0(2(0(x0))))))
0(5(0(1(x0)))) → 2(5(0(0(1(x0)))))
0(5(0(1(x0)))) → 0(0(2(1(2(5(x0))))))
0(5(0(1(x0)))) → 0(0(2(5(2(1(x0))))))
0(5(0(1(x0)))) → 0(1(2(0(2(5(x0))))))
0(5(0(1(x0)))) → 1(2(0(2(5(0(x0))))))
0(5(0(1(x0)))) → 2(0(0(1(2(5(x0))))))
0(5(0(1(x0)))) → 2(0(1(5(2(0(x0))))))
0(5(0(1(x0)))) → 2(0(2(1(5(0(x0))))))
0(5(0(1(x0)))) → 2(5(0(0(1(2(x0))))))
0(5(0(1(x0)))) → 2(5(0(2(0(1(x0))))))
0(5(1(1(x0)))) → 1(2(5(1(0(x0)))))
0(5(1(1(x0)))) → 2(5(0(1(1(x0)))))
0(5(1(1(x0)))) → 1(2(1(0(4(5(x0))))))
0(5(1(1(x0)))) → 1(2(1(4(5(0(x0))))))
0(5(1(1(x0)))) → 2(0(1(2(5(1(x0))))))
0(5(1(1(x0)))) → 2(5(0(2(1(1(x0))))))
0(5(1(1(x0)))) → 2(5(1(2(0(1(x0))))))
0(5(1(1(x0)))) → 2(5(1(2(1(0(x0))))))
0(5(1(1(x0)))) → 2(5(2(1(0(1(x0))))))
0(5(5(1(x0)))) → 2(5(5(1(0(x0)))))
0(5(5(1(x0)))) → 0(1(2(5(2(5(x0))))))
0(5(5(1(x0)))) → 1(2(0(2(5(5(x0))))))
0(5(5(1(x0)))) → 2(0(2(5(1(5(x0))))))
0(5(5(1(x0)))) → 2(2(5(0(5(1(x0))))))
0(5(5(1(x0)))) → 2(2(5(5(1(0(x0))))))
0(5(5(1(x0)))) → 2(5(0(2(5(1(x0))))))
0(5(5(1(x0)))) → 2(5(2(5(0(1(x0))))))
5(0(0(1(x0)))) → 1(2(5(2(0(0(x0))))))
5(0(0(1(x0)))) → 1(4(5(0(2(0(x0))))))
5(0(0(1(x0)))) → 2(0(2(5(1(0(x0))))))
5(0(0(1(x0)))) → 2(5(1(0(2(0(x0))))))
0(0(0(3(1(x0))))) → 2(0(3(0(0(1(x0))))))
0(0(0(5(1(x0))))) → 2(5(1(0(0(0(x0))))))
0(0(1(4(0(x0))))) → 0(2(1(0(4(0(x0))))))
0(0(1(5(0(x0))))) → 0(2(5(1(0(0(x0))))))
0(0(1(5(0(x0))))) → 0(5(0(2(1(0(x0))))))
0(0(5(1(1(x0))))) → 0(0(2(5(1(1(x0))))))
0(0(5(1(1(x0))))) → 2(1(0(5(0(1(x0))))))
0(0(5(3(1(x0))))) → 3(0(0(2(5(1(x0))))))
0(0(5(3(1(x0))))) → 3(2(5(1(0(0(x0))))))
0(1(0(1(1(x0))))) → 0(0(1(1(1(3(x0))))))
0(1(0(3(1(x0))))) → 2(0(1(3(1(0(x0))))))
0(1(1(0(1(x0))))) → 0(0(1(1(1(3(x0))))))
0(1(4(1(1(x0))))) → 1(2(1(4(0(1(x0))))))
0(1(5(0(1(x0))))) → 1(2(5(1(0(0(x0))))))
0(3(0(5(1(x0))))) → 2(5(0(0(3(1(x0))))))
0(3(1(0(1(x0))))) → 2(3(1(0(0(1(x0))))))
0(3(5(0(1(x0))))) → 0(0(3(2(5(1(x0))))))
0(3(5(1(1(x0))))) → 1(2(5(1(0(3(x0))))))
0(4(4(1(1(x0))))) → 1(2(1(4(0(4(x0))))))
0(4(5(5(1(x0))))) → 5(0(2(1(4(5(x0))))))
0(5(0(0(1(x0))))) → 2(5(0(0(0(1(x0))))))
0(5(0(1(0(x0))))) → 0(0(4(1(0(5(x0))))))
0(5(1(0(1(x0))))) → 2(1(0(1(5(0(x0))))))
0(5(1(3(1(x0))))) → 2(5(0(1(3(1(x0))))))
0(5(1(3(1(x0))))) → 2(5(1(0(1(3(x0))))))
0(5(2(5(1(x0))))) → 2(0(2(5(5(1(x0))))))
0(5(2(5(1(x0))))) → 2(5(5(0(2(1(x0))))))
0(5(3(0(1(x0))))) → 0(0(3(2(5(1(x0))))))
0(5(3(0(1(x0))))) → 2(0(3(0(5(1(x0))))))
0(5(3(1(1(x0))))) → 1(1(3(0(2(5(x0))))))
0(5(3(1(1(x0))))) → 1(2(5(1(3(0(x0))))))
0(5(4(1(1(x0))))) → 2(5(0(4(1(1(x0))))))
5(0(1(0(1(x0))))) → 5(1(2(1(0(0(x0))))))
5(5(4(1(1(x0))))) → 1(4(5(2(5(1(x0))))))
5(5(4(1(1(x0))))) → 2(5(1(4(5(1(x0))))))

Proof

1 Termination Assumption

We assume termination of the following TRS
0(0(1(x0))) → 1(2(0(2(0(x0)))))
0(0(1(x0))) → 2(0(2(1(0(x0)))))
0(0(1(x0))) → 2(1(0(2(0(x0)))))
0(0(1(x0))) → 1(1(2(0(2(0(x0))))))
0(0(1(x0))) → 1(2(0(2(0(2(x0))))))
0(0(1(x0))) → 1(2(0(2(0(3(x0))))))
0(0(1(x0))) → 1(2(0(2(0(4(x0))))))
0(0(1(x0))) → 1(2(0(2(2(0(x0))))))
0(0(1(x0))) → 1(2(0(2(3(0(x0))))))
0(0(1(x0))) → 1(2(0(2(5(0(x0))))))
0(0(1(x0))) → 1(2(0(3(2(0(x0))))))
0(0(1(x0))) → 1(2(1(0(2(0(x0))))))
0(0(1(x0))) → 1(2(2(0(2(0(x0))))))
0(0(1(x0))) → 1(2(2(0(3(0(x0))))))
0(0(1(x0))) → 1(2(2(0(4(0(x0))))))
0(0(1(x0))) → 1(2(3(0(2(0(x0))))))
0(0(1(x0))) → 1(2(5(0(2(0(x0))))))
0(0(1(x0))) → 1(3(2(0(2(0(x0))))))
0(0(1(x0))) → 2(0(1(2(0(2(x0))))))
0(0(1(x0))) → 2(0(1(2(2(0(x0))))))
0(0(1(x0))) → 2(0(1(3(0(2(x0))))))
0(0(1(x0))) → 2(0(2(0(2(1(x0))))))
0(0(1(x0))) → 2(0(2(1(0(2(x0))))))
0(0(1(x0))) → 2(0(2(1(0(4(x0))))))
0(0(1(x0))) → 2(0(2(1(2(0(x0))))))
0(0(1(x0))) → 2(0(2(3(1(0(x0))))))
0(0(1(x0))) → 2(0(2(5(1(0(x0))))))
0(0(1(x0))) → 2(1(0(2(0(2(x0))))))
0(0(1(x0))) → 2(1(0(2(2(0(x0))))))
0(0(1(x0))) → 2(1(0(2(5(0(x0))))))
0(0(1(x0))) → 2(1(2(0(2(0(x0))))))
0(0(1(x0))) → 2(1(2(0(4(0(x0))))))
0(0(1(x0))) → 2(1(4(0(2(0(x0))))))
0(0(1(x0))) → 2(1(4(0(4(0(x0))))))
0(0(1(x0))) → 2(2(0(2(1(0(x0))))))
0(0(1(x0))) → 2(2(0(4(1(0(x0))))))
0(0(1(x0))) → 2(2(1(0(2(0(x0))))))
0(0(1(x0))) → 3(1(2(0(2(0(x0))))))
0(0(0(1(x0)))) → 0(0(1(2(0(2(x0))))))
0(0(0(1(x0)))) → 0(2(1(0(2(0(x0))))))
0(0(0(1(x0)))) → 2(0(2(1(0(0(x0))))))
0(0(1(0(x0)))) → 0(0(2(0(2(1(x0))))))
0(0(1(1(x0)))) → 2(0(1(2(1(0(x0))))))
0(0(1(1(x0)))) → 2(0(2(0(1(1(x0))))))
0(0(1(1(x0)))) → 2(1(0(2(0(1(x0))))))
0(0(3(1(x0)))) → 2(0(1(3(0(x0)))))
0(0(3(1(x0)))) → 2(0(1(3(0(2(x0))))))
0(0(3(1(x0)))) → 2(0(2(3(1(0(x0))))))
0(0(3(1(x0)))) → 2(1(0(2(0(3(x0))))))
0(0(3(1(x0)))) → 3(2(0(2(1(0(x0))))))
0(0(5(1(x0)))) → 0(2(5(1(0(x0)))))
0(0(5(1(x0)))) → 0(5(0(2(1(x0)))))
0(0(5(1(x0)))) → 2(5(0(0(1(x0)))))
0(0(5(1(x0)))) → 2(5(1(0(0(x0)))))
0(0(5(1(x0)))) → 0(1(2(5(2(0(x0))))))
0(0(5(1(x0)))) → 0(2(1(2(5(0(x0))))))
0(0(5(1(x0)))) → 0(2(1(4(5(0(x0))))))
0(0(5(1(x0)))) → 1(2(0(0(2(5(x0))))))
0(0(5(1(x0)))) → 2(0(2(1(0(5(x0))))))
0(0(5(1(x0)))) → 2(1(0(4(5(0(x0))))))
0(0(5(1(x0)))) → 2(5(0(2(1(0(x0))))))
0(1(0(1(x0)))) → 0(0(1(2(1(x0)))))
0(1(0(1(x0)))) → 1(2(1(0(0(x0)))))
0(1(0(1(x0)))) → 2(1(0(0(1(x0)))))
0(1(0(1(x0)))) → 0(0(1(1(2(2(x0))))))
0(1(0(1(x0)))) → 0(1(1(3(2(0(x0))))))
0(1(0(1(x0)))) → 1(1(3(2(0(0(x0))))))
0(1(0(1(x0)))) → 1(2(0(2(0(1(x0))))))
0(1(0(1(x0)))) → 1(2(2(0(0(1(x0))))))
0(1(0(1(x0)))) → 1(2(5(1(0(0(x0))))))
0(1(0(1(x0)))) → 2(1(2(1(0(0(x0))))))
0(1(0(1(x0)))) → 3(1(2(1(0(0(x0))))))
0(3(0(1(x0)))) → 1(2(3(0(2(0(x0))))))
0(3(0(1(x0)))) → 3(1(2(0(2(0(x0))))))
0(5(0(1(x0)))) → 2(5(0(0(1(x0)))))
0(5(0(1(x0)))) → 0(0(2(1(2(5(x0))))))
0(5(0(1(x0)))) → 0(0(2(5(2(1(x0))))))
0(5(0(1(x0)))) → 0(1(2(0(2(5(x0))))))
0(5(0(1(x0)))) → 1(2(0(2(5(0(x0))))))
0(5(0(1(x0)))) → 2(0(0(1(2(5(x0))))))
0(5(0(1(x0)))) → 2(0(1(5(2(0(x0))))))
0(5(0(1(x0)))) → 2(0(2(1(5(0(x0))))))
0(5(0(1(x0)))) → 2(5(0(0(1(2(x0))))))
0(5(0(1(x0)))) → 2(5(0(2(0(1(x0))))))
0(5(1(1(x0)))) → 1(2(5(1(0(x0)))))
0(5(1(1(x0)))) → 2(5(0(1(1(x0)))))
0(5(1(1(x0)))) → 1(2(1(0(4(5(x0))))))
0(5(1(1(x0)))) → 1(2(1(4(5(0(x0))))))
0(5(1(1(x0)))) → 2(0(1(2(5(1(x0))))))
0(5(1(1(x0)))) → 2(5(0(2(1(1(x0))))))
0(5(1(1(x0)))) → 2(5(1(2(0(1(x0))))))
0(5(1(1(x0)))) → 2(5(1(2(1(0(x0))))))
0(5(1(1(x0)))) → 2(5(2(1(0(1(x0))))))
0(5(5(1(x0)))) → 2(5(5(1(0(x0)))))
0(5(5(1(x0)))) → 0(1(2(5(2(5(x0))))))
0(5(5(1(x0)))) → 1(2(0(2(5(5(x0))))))
0(5(5(1(x0)))) → 2(0(2(5(1(5(x0))))))
0(5(5(1(x0)))) → 2(2(5(0(5(1(x0))))))
0(5(5(1(x0)))) → 2(2(5(5(1(0(x0))))))
0(5(5(1(x0)))) → 2(5(0(2(5(1(x0))))))
0(5(5(1(x0)))) → 2(5(2(5(0(1(x0))))))
5(0(0(1(x0)))) → 1(2(5(2(0(0(x0))))))
5(0(0(1(x0)))) → 1(4(5(0(2(0(x0))))))
5(0(0(1(x0)))) → 2(0(2(5(1(0(x0))))))
5(0(0(1(x0)))) → 2(5(1(0(2(0(x0))))))
0(0(0(3(1(x0))))) → 2(0(3(0(0(1(x0))))))
0(0(0(5(1(x0))))) → 2(5(1(0(0(0(x0))))))
0(0(1(4(0(x0))))) → 0(2(1(0(4(0(x0))))))
0(0(1(5(0(x0))))) → 0(2(5(1(0(0(x0))))))
0(0(1(5(0(x0))))) → 0(5(0(2(1(0(x0))))))
0(0(5(1(1(x0))))) → 0(0(2(5(1(1(x0))))))
0(0(5(1(1(x0))))) → 2(1(0(5(0(1(x0))))))
0(0(5(3(1(x0))))) → 3(0(0(2(5(1(x0))))))
0(0(5(3(1(x0))))) → 3(2(5(1(0(0(x0))))))
0(1(0(1(1(x0))))) → 0(0(1(1(1(3(x0))))))
0(1(0(3(1(x0))))) → 2(0(1(3(1(0(x0))))))
0(1(1(0(1(x0))))) → 0(0(1(1(1(3(x0))))))
0(1(4(1(1(x0))))) → 1(2(1(4(0(1(x0))))))
0(1(5(0(1(x0))))) → 1(2(5(1(0(0(x0))))))
0(3(0(5(1(x0))))) → 2(5(0(0(3(1(x0))))))
0(3(1(0(1(x0))))) → 2(3(1(0(0(1(x0))))))
0(3(5(0(1(x0))))) → 0(0(3(2(5(1(x0))))))
0(3(5(1(1(x0))))) → 1(2(5(1(0(3(x0))))))
0(4(4(1(1(x0))))) → 1(2(1(4(0(4(x0))))))
0(4(5(5(1(x0))))) → 5(0(2(1(4(5(x0))))))
0(5(0(0(1(x0))))) → 2(5(0(0(0(1(x0))))))
0(5(0(1(0(x0))))) → 0(0(4(1(0(5(x0))))))
0(5(1(0(1(x0))))) → 2(1(0(1(5(0(x0))))))
0(5(1(3(1(x0))))) → 2(5(0(1(3(1(x0))))))
0(5(1(3(1(x0))))) → 2(5(1(0(1(3(x0))))))
0(5(2(5(1(x0))))) → 2(0(2(5(5(1(x0))))))
0(5(2(5(1(x0))))) → 2(5(5(0(2(1(x0))))))
0(5(3(0(1(x0))))) → 0(0(3(2(5(1(x0))))))
0(5(3(0(1(x0))))) → 2(0(3(0(5(1(x0))))))
0(5(3(1(1(x0))))) → 1(1(3(0(2(5(x0))))))
0(5(3(1(1(x0))))) → 1(2(5(1(3(0(x0))))))
0(5(4(1(1(x0))))) → 2(5(0(4(1(1(x0))))))
5(0(1(0(1(x0))))) → 5(1(2(1(0(0(x0))))))
5(5(4(1(1(x0))))) → 1(4(5(2(5(1(x0))))))
5(5(4(1(1(x0))))) → 2(5(1(4(5(1(x0))))))