(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
0(0(2(5(3(2(2(5(3(5(3(2(x)))))))))))) → 2(0(0(0(3(0(4(0(0(3(0(2(5(0(5(2(0(2(x))))))))))))))))))
0(1(5(5(4(4(2(5(4(2(5(1(x)))))))))))) → 0(0(0(5(0(5(3(4(1(0(4(2(2(4(x))))))))))))))
0(2(2(4(5(0(1(3(3(2(5(5(x)))))))))))) → 1(0(4(0(0(4(3(4(2(1(1(4(4(0(2(3(x))))))))))))))))
0(2(5(3(5(1(5(1(5(5(5(5(x)))))))))))) → 2(0(4(0(4(1(5(1(0(4(5(1(1(1(1(0(3(2(x))))))))))))))))))
0(3(0(2(5(2(5(3(3(3(5(4(x)))))))))))) → 2(4(4(2(0(4(4(2(4(4(2(3(1(2(1(1(1(5(x))))))))))))))))))
0(3(1(2(1(2(3(1(5(3(3(5(x)))))))))))) → 0(1(4(4(2(2(0(0(5(4(2(3(1(0(2(1(0(2(x))))))))))))))))))
0(5(3(2(2(1(2(3(3(2(5(0(x)))))))))))) → 3(0(5(5(4(3(1(1(5(4(0(1(4(0(x))))))))))))))
0(5(5(3(2(4(0(5(4(4(0(4(x)))))))))))) → 2(1(0(4(4(3(0(1(0(4(0(2(2(4(4(4(x))))))))))))))))
1(0(3(3(5(5(0(5(4(5(1(0(x)))))))))))) → 2(3(4(0(3(0(1(4(0(3(4(1(2(4(2(x)))))))))))))))
1(1(2(5(0(1(2(3(5(5(3(4(x)))))))))))) → 2(4(2(4(4(3(3(2(2(3(2(1(4(0(x))))))))))))))
1(2(2(5(3(1(4(5(1(3(4(1(x)))))))))))) → 2(3(4(0(4(0(4(5(1(2(0(0(4(5(0(5(x))))))))))))))))
1(2(4(1(5(3(1(1(0(5(4(2(x)))))))))))) → 2(0(1(2(0(4(2(2(1(1(4(4(1(0(0(4(4(0(x))))))))))))))))))
1(3(0(5(5(0(1(2(5(1(5(4(x)))))))))))) → 4(0(4(3(0(3(0(4(0(1(0(4(5(4(x))))))))))))))
1(3(3(1(5(2(3(3(5(5(5(5(x)))))))))))) → 0(2(4(0(1(4(0(4(4(4(5(2(4(3(1(5(2(3(x))))))))))))))))))
1(3(5(5(1(3(2(4(2(3(2(5(x)))))))))))) → 1(2(0(3(0(4(4(1(4(0(1(5(1(1(x))))))))))))))
1(5(0(5(1(4(5(4(3(5(0(2(x)))))))))))) → 0(1(5(4(0(1(5(0(4(4(4(4(2(0(0(2(x))))))))))))))))
1(5(0(5(5(1(5(5(3(5(3(0(x)))))))))))) → 1(0(5(5(2(0(4(0(5(4(1(1(0(4(0(5(0(4(x))))))))))))))))))
1(5(1(3(3(2(2(1(5(1(5(5(x)))))))))))) → 0(2(1(1(1(4(1(2(4(5(2(2(0(1(2(4(x))))))))))))))))
1(5(2(0(3(3(5(3(1(5(2(3(x)))))))))))) → 5(2(4(0(5(4(5(0(5(3(4(2(0(3(x))))))))))))))
1(5(3(1(1(2(3(3(2(5(1(1(x)))))))))))) → 3(1(1(3(1(2(4(0(0(2(5(1(2(0(0(4(x))))))))))))))))
1(5(3(5(3(3(3(2(2(0(1(1(x)))))))))))) → 2(4(0(2(2(0(0(4(1(1(5(3(5(1(3(4(1(x)))))))))))))))))
1(5(5(0(5(1(5(3(5(0(3(4(x)))))))))))) → 4(1(2(0(0(1(3(1(1(0(5(4(2(3(2(x)))))))))))))))
2(0(0(1(5(1(2(5(3(2(1(4(x)))))))))))) → 2(5(4(0(0(2(2(1(4(1(0(4(4(1(3(0(x))))))))))))))))
2(0(5(4(2(2(3(3(2(5(2(5(x)))))))))))) → 4(5(3(2(0(4(4(0(1(2(2(1(0(2(0(0(4(x)))))))))))))))))
2(1(1(1(3(1(1(3(2(0(3(2(x)))))))))))) → 2(2(1(0(0(0(2(1(2(2(2(1(4(4(x))))))))))))))
2(2(0(1(2(5(0(3(2(5(1(2(x)))))))))))) → 3(2(1(4(4(4(2(1(0(1(2(0(0(0(4(0(4(5(x))))))))))))))))))
2(2(0(1(2(5(5(2(1(1(1(3(x)))))))))))) → 2(4(4(0(4(1(3(0(0(5(2(0(4(0(4(4(4(0(x))))))))))))))))))
2(2(0(3(2(1(0(3(5(5(5(5(x)))))))))))) → 4(1(2(2(3(4(0(2(4(1(0(4(2(4(0(0(4(3(x))))))))))))))))))
2(2(1(5(5(3(4(3(2(5(3(2(x)))))))))))) → 2(2(2(1(3(3(4(0(0(4(3(3(1(3(x))))))))))))))
2(2(4(1(5(5(3(3(5(2(5(5(x)))))))))))) → 5(0(5(0(4(4(1(2(1(3(3(4(2(4(4(x)))))))))))))))
2(3(3(0(2(5(5(2(2(4(5(4(x)))))))))))) → 4(0(0(0(1(0(0(5(4(1(4(1(4(2(1(x)))))))))))))))
2(3(4(3(2(5(3(5(1(0(1(4(x)))))))))))) → 2(0(0(4(4(5(0(2(3(5(1(2(2(2(4(x)))))))))))))))
2(4(3(5(2(1(5(5(2(1(5(2(x)))))))))))) → 4(2(1(4(1(0(4(5(0(4(2(1(2(3(5(x)))))))))))))))
2(5(0(5(5(2(4(3(2(5(1(5(x)))))))))))) → 3(4(1(4(0(1(1(0(4(1(1(0(0(2(4(4(5(2(x))))))))))))))))))
2(5(1(5(5(1(1(0(5(5(5(5(x)))))))))))) → 3(3(2(4(4(4(0(2(5(1(0(2(5(1(3(x)))))))))))))))
2(5(1(5(5(5(5(3(4(0(2(4(x)))))))))))) → 4(2(4(4(2(2(4(5(0(4(2(2(1(0(x))))))))))))))
2(5(2(0(5(3(4(1(5(2(5(5(x)))))))))))) → 1(2(2(0(1(4(1(2(1(2(5(1(3(0(0(4(4(1(x))))))))))))))))))
2(5(3(3(3(3(5(2(1(4(2(0(x)))))))))))) → 0(1(3(4(0(5(5(3(0(2(2(4(3(2(3(x)))))))))))))))
2(5(4(5(1(5(3(1(5(2(4(1(x)))))))))))) → 2(4(1(0(4(1(5(2(1(1(1(4(0(5(2(3(4(0(x))))))))))))))))))
3(0(5(5(5(2(0(3(1(2(5(0(x)))))))))))) → 0(4(0(4(2(2(2(0(1(3(4(4(1(0(5(x)))))))))))))))
3(1(1(5(0(3(2(3(3(3(3(4(x)))))))))))) → 2(0(2(4(4(5(0(5(4(3(1(0(0(3(x))))))))))))))
3(1(3(5(3(5(5(2(5(3(0(1(x)))))))))))) → 2(3(1(2(0(3(2(3(5(4(0(0(0(5(0(0(x))))))))))))))))
3(2(1(1(5(4(1(5(0(2(3(5(x)))))))))))) → 2(5(4(2(3(0(0(4(5(4(4(2(4(0(0(0(2(x)))))))))))))))))
3(2(3(1(5(2(0(3(3(4(0(5(x)))))))))))) → 3(0(0(0(4(0(3(0(2(3(3(2(1(4(1(x)))))))))))))))
3(2(5(3(2(5(4(0(3(1(2(2(x)))))))))))) → 0(2(3(0(4(5(0(5(4(4(4(4(1(4(2(3(0(1(x))))))))))))))))))
3(2(5(4(3(5(2(5(1(5(2(2(x)))))))))))) → 4(2(3(3(0(5(3(5(1(0(0(2(1(5(0(4(x))))))))))))))))
3(2(5(5(5(4(4(3(2(3(1(0(x)))))))))))) → 0(0(2(2(4(5(4(0(4(5(5(4(0(5(x))))))))))))))
3(3(1(1(4(5(3(4(3(5(1(5(x)))))))))))) → 2(4(1(2(5(0(1(3(0(5(0(4(2(0(x))))))))))))))
3(3(3(1(0(2(4(3(3(4(4(5(x)))))))))))) → 0(2(0(4(4(4(0(5(5(0(1(3(3(5(x))))))))))))))
3(4(0(5(3(4(0(0(4(5(2(4(x)))))))))))) → 0(0(0(5(4(5(4(4(0(4(2(4(4(0(0(x)))))))))))))))
3(4(1(5(3(1(2(3(1(0(1(3(x)))))))))))) → 1(3(4(3(3(5(4(0(4(0(4(4(5(0(1(x)))))))))))))))
3(4(3(3(4(5(2(5(4(2(4(5(x)))))))))))) → 4(0(0(3(5(1(0(0(4(2(4(4(3(5(0(2(2(x)))))))))))))))))
3(4(5(3(5(3(1(5(3(1(1(1(x)))))))))))) → 1(2(3(0(4(4(4(4(4(1(4(4(4(0(5(x)))))))))))))))
3(5(1(5(0(1(4(4(3(5(5(1(x)))))))))))) → 4(4(2(5(0(4(0(2(2(4(0(0(4(2(0(1(2(4(x))))))))))))))))))
3(5(1(5(3(3(1(3(3(2(4(5(x)))))))))))) → 2(3(4(5(1(0(0(2(4(2(3(1(3(1(4(4(x))))))))))))))))
3(5(3(2(1(3(2(1(5(5(3(5(x)))))))))))) → 3(3(2(1(4(2(1(2(1(5(0(1(0(4(x))))))))))))))
3(5(3(2(3(3(5(3(2(5(3(4(x)))))))))))) → 4(1(1(3(0(4(2(1(2(1(0(4(4(1(1(1(2(x)))))))))))))))))
4(1(1(5(5(5(3(5(5(2(3(5(x)))))))))))) → 4(0(1(5(1(0(2(4(1(2(0(4(2(2(3(4(0(0(x))))))))))))))))))
4(1(2(4(5(4(1(5(5(5(2(1(x)))))))))))) → 4(4(3(0(4(1(1(2(1(1(0(4(1(3(0(4(0(0(x))))))))))))))))))
4(2(5(3(3(3(3(2(1(5(4(5(x)))))))))))) → 3(0(3(4(3(0(1(1(4(3(4(2(0(4(1(3(1(2(x))))))))))))))))))
4(2(5(5(3(4(5(4(2(5(2(0(x)))))))))))) → 0(2(2(2(5(1(4(0(4(0(2(1(5(0(x))))))))))))))
4(3(0(0(1(5(5(3(0(3(3(2(x)))))))))))) → 3(1(3(5(0(0(0(3(1(4(5(0(4(0(x))))))))))))))
4(3(5(2(5(3(5(1(3(0(5(2(x)))))))))))) → 2(5(0(4(4(1(4(4(0(4(0(0(4(2(4(1(3(x)))))))))))))))))
4(5(0(5(0(1(1(2(1(2(4(0(x)))))))))))) → 4(2(0(0(0(5(4(2(3(0(4(4(0(1(x))))))))))))))
4(5(5(2(3(2(5(0(2(0(3(1(x)))))))))))) → 1(0(0(3(4(1(5(1(0(4(4(0(3(4(0(0(x))))))))))))))))
4(5(5(3(2(1(0(1(1(0(5(5(x)))))))))))) → 0(5(0(4(0(3(0(1(2(2(3(2(3(4(x))))))))))))))
5(0(5(5(3(1(5(5(3(3(5(0(x)))))))))))) → 0(4(4(5(4(0(3(5(3(4(3(5(0(4(1(3(1(2(x))))))))))))))))))
5(1(0(3(1(5(5(3(0(3(5(1(x)))))))))))) → 5(4(1(1(0(4(0(3(3(1(3(0(4(0(1(4(4(4(x))))))))))))))))))
5(1(1(0(3(1(5(3(1(1(0(5(x)))))))))))) → 4(1(0(5(4(0(4(5(2(1(5(1(4(0(3(x)))))))))))))))
5(1(1(4(3(1(2(5(5(4(2(2(x)))))))))))) → 0(1(0(0(2(4(0(2(5(0(0(5(0(0(x))))))))))))))
5(1(2(2(5(2(4(1(3(2(3(1(x)))))))))))) → 2(1(4(4(2(4(4(0(0(4(4(4(3(1(4(0(3(1(x))))))))))))))))))
5(1(2(3(3(1(1(2(3(1(5(1(x)))))))))))) → 5(2(0(3(3(5(2(4(4(1(4(4(3(4(1(x)))))))))))))))
5(1(5(1(3(2(4(5(3(5(4(0(x)))))))))))) → 0(1(4(0(0(4(0(5(2(1(4(4(4(3(3(3(x))))))))))))))))
5(2(1(4(3(5(3(1(5(3(5(1(x)))))))))))) → 0(1(1(1(1(2(0(5(0(0(0(2(3(0(0(1(x))))))))))))))))
5(2(1(5(5(0(3(5(5(5(3(0(x)))))))))))) → 2(3(4(1(1(0(2(3(0(4(1(0(2(0(0(0(3(3(x))))))))))))))))))
5(2(3(5(3(0(3(3(1(4(0(2(x)))))))))))) → 4(5(4(0(0(2(1(1(0(3(4(4(4(3(x))))))))))))))
5(2(4(3(0(2(5(2(1(1(5(0(x)))))))))))) → 0(1(1(0(0(0(0(0(4(0(0(5(4(5(x))))))))))))))
5(2(5(4(4(2(3(5(0(5(2(5(x)))))))))))) → 4(1(4(2(3(4(3(4(4(4(2(1(4(0(0(0(1(x)))))))))))))))))
5(3(2(0(2(0(4(2(5(1(5(2(x)))))))))))) → 4(3(0(4(5(0(3(4(0(3(5(1(1(1(4(x)))))))))))))))
5(3(3(2(1(1(3(3(2(5(5(1(x)))))))))))) → 3(1(4(0(1(4(2(0(3(0(0(3(1(2(4(x)))))))))))))))
5(3(3(3(1(5(2(0(1(5(4(0(x)))))))))))) → 3(5(0(3(5(0(0(4(2(3(4(0(1(2(2(3(0(0(x))))))))))))))))))
5(3(5(2(5(5(2(4(2(1(2(2(x)))))))))))) → 0(1(3(4(4(4(0(1(5(0(1(1(2(1(x))))))))))))))
5(3(5(4(4(5(1(3(4(5(5(5(x)))))))))))) → 5(3(4(4(0(4(4(0(1(5(0(5(2(2(3(4(0(4(x))))))))))))))))))
5(4(3(3(0(3(2(5(4(0(5(5(x)))))))))))) → 1(1(4(1(3(2(2(0(1(2(2(0(0(1(1(4(3(x)))))))))))))))))
5(4(3(3(3(2(5(2(4(1(4(2(x)))))))))))) → 3(0(4(4(4(0(4(4(1(4(4(1(1(2(2(4(5(4(x))))))))))))))))))
5(5(0(0(3(4(3(2(3(4(5(5(x)))))))))))) → 0(2(1(0(1(2(4(0(0(5(3(2(2(2(0(x)))))))))))))))
5(5(1(2(5(2(5(3(3(3(4(5(x)))))))))))) → 3(5(1(3(1(5(2(4(0(2(3(3(4(4(2(x)))))))))))))))
5(5(1(3(3(2(0(5(5(1(0(0(x)))))))))))) → 5(0(4(0(3(4(5(1(2(4(4(4(3(0(0(0(1(x)))))))))))))))))
5(5(1(5(1(5(3(4(2(5(4(5(x)))))))))))) → 0(5(4(0(4(0(4(1(2(2(1(1(3(4(4(0(3(3(x))))))))))))))))))
5(5(1(5(2(5(1(5(1(1(1(4(x)))))))))))) → 1(4(1(3(4(4(5(4(0(5(4(5(0(4(3(5(2(x)))))))))))))))))
5(5(2(5(3(3(1(0(1(4(5(5(x)))))))))))) → 2(4(3(0(5(2(3(0(5(4(4(0(3(2(3(2(5(x)))))))))))))))))
5(5(3(1(0(3(3(0(2(3(5(3(x)))))))))))) → 2(4(5(2(2(2(0(0(4(4(3(0(4(0(0(2(4(4(x))))))))))))))))))
5(5(3(3(5(3(3(0(4(5(3(0(x)))))))))))) → 4(4(0(2(4(0(3(4(1(4(1(3(3(4(4(4(4(3(x))))))))))))))))))
5(5(3(4(0(2(5(5(3(2(4(1(x)))))))))))) → 2(4(3(0(4(0(1(3(2(4(1(1(0(1(1(1(0(1(x))))))))))))))))))
5(5(4(5(5(2(5(3(1(2(1(1(x)))))))))))) → 0(3(0(2(2(4(4(3(1(4(4(2(5(0(0(0(x))))))))))))))))
5(5(5(5(5(3(2(5(2(0(5(5(x)))))))))))) → 5(3(5(3(3(1(5(5(4(1(3(4(0(1(x))))))))))))))
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(0(x1)) = x1
POL(1(x1)) = 11 + x1
POL(2(x1)) = 8 + x1
POL(3(x1)) = 15 + x1
POL(4(x1)) = x1
POL(5(x1)) = 21 + x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
0(0(2(5(3(2(2(5(3(5(3(2(x)))))))))))) → 2(0(0(0(3(0(4(0(0(3(0(2(5(0(5(2(0(2(x))))))))))))))))))
0(1(5(5(4(4(2(5(4(2(5(1(x)))))))))))) → 0(0(0(5(0(5(3(4(1(0(4(2(2(4(x))))))))))))))
0(2(2(4(5(0(1(3(3(2(5(5(x)))))))))))) → 1(0(4(0(0(4(3(4(2(1(1(4(4(0(2(3(x))))))))))))))))
0(2(5(3(5(1(5(1(5(5(5(5(x)))))))))))) → 2(0(4(0(4(1(5(1(0(4(5(1(1(1(1(0(3(2(x))))))))))))))))))
0(3(0(2(5(2(5(3(3(3(5(4(x)))))))))))) → 2(4(4(2(0(4(4(2(4(4(2(3(1(2(1(1(1(5(x))))))))))))))))))
0(3(1(2(1(2(3(1(5(3(3(5(x)))))))))))) → 0(1(4(4(2(2(0(0(5(4(2(3(1(0(2(1(0(2(x))))))))))))))))))
0(5(3(2(2(1(2(3(3(2(5(0(x)))))))))))) → 3(0(5(5(4(3(1(1(5(4(0(1(4(0(x))))))))))))))
0(5(5(3(2(4(0(5(4(4(0(4(x)))))))))))) → 2(1(0(4(4(3(0(1(0(4(0(2(2(4(4(4(x))))))))))))))))
1(0(3(3(5(5(0(5(4(5(1(0(x)))))))))))) → 2(3(4(0(3(0(1(4(0(3(4(1(2(4(2(x)))))))))))))))
1(1(2(5(0(1(2(3(5(5(3(4(x)))))))))))) → 2(4(2(4(4(3(3(2(2(3(2(1(4(0(x))))))))))))))
1(2(2(5(3(1(4(5(1(3(4(1(x)))))))))))) → 2(3(4(0(4(0(4(5(1(2(0(0(4(5(0(5(x))))))))))))))))
1(2(4(1(5(3(1(1(0(5(4(2(x)))))))))))) → 2(0(1(2(0(4(2(2(1(1(4(4(1(0(0(4(4(0(x))))))))))))))))))
1(3(0(5(5(0(1(2(5(1(5(4(x)))))))))))) → 4(0(4(3(0(3(0(4(0(1(0(4(5(4(x))))))))))))))
1(3(3(1(5(2(3(3(5(5(5(5(x)))))))))))) → 0(2(4(0(1(4(0(4(4(4(5(2(4(3(1(5(2(3(x))))))))))))))))))
1(3(5(5(1(3(2(4(2(3(2(5(x)))))))))))) → 1(2(0(3(0(4(4(1(4(0(1(5(1(1(x))))))))))))))
1(5(0(5(1(4(5(4(3(5(0(2(x)))))))))))) → 0(1(5(4(0(1(5(0(4(4(4(4(2(0(0(2(x))))))))))))))))
1(5(0(5(5(1(5(5(3(5(3(0(x)))))))))))) → 1(0(5(5(2(0(4(0(5(4(1(1(0(4(0(5(0(4(x))))))))))))))))))
1(5(1(3(3(2(2(1(5(1(5(5(x)))))))))))) → 0(2(1(1(1(4(1(2(4(5(2(2(0(1(2(4(x))))))))))))))))
1(5(2(0(3(3(5(3(1(5(2(3(x)))))))))))) → 5(2(4(0(5(4(5(0(5(3(4(2(0(3(x))))))))))))))
1(5(3(1(1(2(3(3(2(5(1(1(x)))))))))))) → 3(1(1(3(1(2(4(0(0(2(5(1(2(0(0(4(x))))))))))))))))
1(5(3(5(3(3(3(2(2(0(1(1(x)))))))))))) → 2(4(0(2(2(0(0(4(1(1(5(3(5(1(3(4(1(x)))))))))))))))))
1(5(5(0(5(1(5(3(5(0(3(4(x)))))))))))) → 4(1(2(0(0(1(3(1(1(0(5(4(2(3(2(x)))))))))))))))
2(0(0(1(5(1(2(5(3(2(1(4(x)))))))))))) → 2(5(4(0(0(2(2(1(4(1(0(4(4(1(3(0(x))))))))))))))))
2(0(5(4(2(2(3(3(2(5(2(5(x)))))))))))) → 4(5(3(2(0(4(4(0(1(2(2(1(0(2(0(0(4(x)))))))))))))))))
2(1(1(1(3(1(1(3(2(0(3(2(x)))))))))))) → 2(2(1(0(0(0(2(1(2(2(2(1(4(4(x))))))))))))))
2(2(0(1(2(5(0(3(2(5(1(2(x)))))))))))) → 3(2(1(4(4(4(2(1(0(1(2(0(0(0(4(0(4(5(x))))))))))))))))))
2(2(0(1(2(5(5(2(1(1(1(3(x)))))))))))) → 2(4(4(0(4(1(3(0(0(5(2(0(4(0(4(4(4(0(x))))))))))))))))))
2(2(0(3(2(1(0(3(5(5(5(5(x)))))))))))) → 4(1(2(2(3(4(0(2(4(1(0(4(2(4(0(0(4(3(x))))))))))))))))))
2(2(1(5(5(3(4(3(2(5(3(2(x)))))))))))) → 2(2(2(1(3(3(4(0(0(4(3(3(1(3(x))))))))))))))
2(2(4(1(5(5(3(3(5(2(5(5(x)))))))))))) → 5(0(5(0(4(4(1(2(1(3(3(4(2(4(4(x)))))))))))))))
2(3(3(0(2(5(5(2(2(4(5(4(x)))))))))))) → 4(0(0(0(1(0(0(5(4(1(4(1(4(2(1(x)))))))))))))))
2(3(4(3(2(5(3(5(1(0(1(4(x)))))))))))) → 2(0(0(4(4(5(0(2(3(5(1(2(2(2(4(x)))))))))))))))
2(4(3(5(2(1(5(5(2(1(5(2(x)))))))))))) → 4(2(1(4(1(0(4(5(0(4(2(1(2(3(5(x)))))))))))))))
2(5(0(5(5(2(4(3(2(5(1(5(x)))))))))))) → 3(4(1(4(0(1(1(0(4(1(1(0(0(2(4(4(5(2(x))))))))))))))))))
2(5(1(5(5(1(1(0(5(5(5(5(x)))))))))))) → 3(3(2(4(4(4(0(2(5(1(0(2(5(1(3(x)))))))))))))))
2(5(1(5(5(5(5(3(4(0(2(4(x)))))))))))) → 4(2(4(4(2(2(4(5(0(4(2(2(1(0(x))))))))))))))
2(5(2(0(5(3(4(1(5(2(5(5(x)))))))))))) → 1(2(2(0(1(4(1(2(1(2(5(1(3(0(0(4(4(1(x))))))))))))))))))
2(5(4(5(1(5(3(1(5(2(4(1(x)))))))))))) → 2(4(1(0(4(1(5(2(1(1(1(4(0(5(2(3(4(0(x))))))))))))))))))
3(0(5(5(5(2(0(3(1(2(5(0(x)))))))))))) → 0(4(0(4(2(2(2(0(1(3(4(4(1(0(5(x)))))))))))))))
3(1(1(5(0(3(2(3(3(3(3(4(x)))))))))))) → 2(0(2(4(4(5(0(5(4(3(1(0(0(3(x))))))))))))))
3(1(3(5(3(5(5(2(5(3(0(1(x)))))))))))) → 2(3(1(2(0(3(2(3(5(4(0(0(0(5(0(0(x))))))))))))))))
3(2(1(1(5(4(1(5(0(2(3(5(x)))))))))))) → 2(5(4(2(3(0(0(4(5(4(4(2(4(0(0(0(2(x)))))))))))))))))
3(2(3(1(5(2(0(3(3(4(0(5(x)))))))))))) → 3(0(0(0(4(0(3(0(2(3(3(2(1(4(1(x)))))))))))))))
3(2(5(3(2(5(4(0(3(1(2(2(x)))))))))))) → 0(2(3(0(4(5(0(5(4(4(4(4(1(4(2(3(0(1(x))))))))))))))))))
3(2(5(4(3(5(2(5(1(5(2(2(x)))))))))))) → 4(2(3(3(0(5(3(5(1(0(0(2(1(5(0(4(x))))))))))))))))
3(2(5(5(5(4(4(3(2(3(1(0(x)))))))))))) → 0(0(2(2(4(5(4(0(4(5(5(4(0(5(x))))))))))))))
3(3(1(1(4(5(3(4(3(5(1(5(x)))))))))))) → 2(4(1(2(5(0(1(3(0(5(0(4(2(0(x))))))))))))))
3(3(3(1(0(2(4(3(3(4(4(5(x)))))))))))) → 0(2(0(4(4(4(0(5(5(0(1(3(3(5(x))))))))))))))
3(4(0(5(3(4(0(0(4(5(2(4(x)))))))))))) → 0(0(0(5(4(5(4(4(0(4(2(4(4(0(0(x)))))))))))))))
3(4(1(5(3(1(2(3(1(0(1(3(x)))))))))))) → 1(3(4(3(3(5(4(0(4(0(4(4(5(0(1(x)))))))))))))))
3(4(3(3(4(5(2(5(4(2(4(5(x)))))))))))) → 4(0(0(3(5(1(0(0(4(2(4(4(3(5(0(2(2(x)))))))))))))))))
3(4(5(3(5(3(1(5(3(1(1(1(x)))))))))))) → 1(2(3(0(4(4(4(4(4(1(4(4(4(0(5(x)))))))))))))))
3(5(1(5(0(1(4(4(3(5(5(1(x)))))))))))) → 4(4(2(5(0(4(0(2(2(4(0(0(4(2(0(1(2(4(x))))))))))))))))))
3(5(1(5(3(3(1(3(3(2(4(5(x)))))))))))) → 2(3(4(5(1(0(0(2(4(2(3(1(3(1(4(4(x))))))))))))))))
3(5(3(2(1(3(2(1(5(5(3(5(x)))))))))))) → 3(3(2(1(4(2(1(2(1(5(0(1(0(4(x))))))))))))))
3(5(3(2(3(3(5(3(2(5(3(4(x)))))))))))) → 4(1(1(3(0(4(2(1(2(1(0(4(4(1(1(1(2(x)))))))))))))))))
4(1(1(5(5(5(3(5(5(2(3(5(x)))))))))))) → 4(0(1(5(1(0(2(4(1(2(0(4(2(2(3(4(0(0(x))))))))))))))))))
4(1(2(4(5(4(1(5(5(5(2(1(x)))))))))))) → 4(4(3(0(4(1(1(2(1(1(0(4(1(3(0(4(0(0(x))))))))))))))))))
4(2(5(3(3(3(3(2(1(5(4(5(x)))))))))))) → 3(0(3(4(3(0(1(1(4(3(4(2(0(4(1(3(1(2(x))))))))))))))))))
4(2(5(5(3(4(5(4(2(5(2(0(x)))))))))))) → 0(2(2(2(5(1(4(0(4(0(2(1(5(0(x))))))))))))))
4(3(0(0(1(5(5(3(0(3(3(2(x)))))))))))) → 3(1(3(5(0(0(0(3(1(4(5(0(4(0(x))))))))))))))
4(3(5(2(5(3(5(1(3(0(5(2(x)))))))))))) → 2(5(0(4(4(1(4(4(0(4(0(0(4(2(4(1(3(x)))))))))))))))))
4(5(0(5(0(1(1(2(1(2(4(0(x)))))))))))) → 4(2(0(0(0(5(4(2(3(0(4(4(0(1(x))))))))))))))
4(5(5(2(3(2(5(0(2(0(3(1(x)))))))))))) → 1(0(0(3(4(1(5(1(0(4(4(0(3(4(0(0(x))))))))))))))))
4(5(5(3(2(1(0(1(1(0(5(5(x)))))))))))) → 0(5(0(4(0(3(0(1(2(2(3(2(3(4(x))))))))))))))
5(0(5(5(3(1(5(5(3(3(5(0(x)))))))))))) → 0(4(4(5(4(0(3(5(3(4(3(5(0(4(1(3(1(2(x))))))))))))))))))
5(1(0(3(1(5(5(3(0(3(5(1(x)))))))))))) → 5(4(1(1(0(4(0(3(3(1(3(0(4(0(1(4(4(4(x))))))))))))))))))
5(1(1(0(3(1(5(3(1(1(0(5(x)))))))))))) → 4(1(0(5(4(0(4(5(2(1(5(1(4(0(3(x)))))))))))))))
5(1(1(4(3(1(2(5(5(4(2(2(x)))))))))))) → 0(1(0(0(2(4(0(2(5(0(0(5(0(0(x))))))))))))))
5(1(2(2(5(2(4(1(3(2(3(1(x)))))))))))) → 2(1(4(4(2(4(4(0(0(4(4(4(3(1(4(0(3(1(x))))))))))))))))))
5(1(2(3(3(1(1(2(3(1(5(1(x)))))))))))) → 5(2(0(3(3(5(2(4(4(1(4(4(3(4(1(x)))))))))))))))
5(1(5(1(3(2(4(5(3(5(4(0(x)))))))))))) → 0(1(4(0(0(4(0(5(2(1(4(4(4(3(3(3(x))))))))))))))))
5(2(1(4(3(5(3(1(5(3(5(1(x)))))))))))) → 0(1(1(1(1(2(0(5(0(0(0(2(3(0(0(1(x))))))))))))))))
5(2(1(5(5(0(3(5(5(5(3(0(x)))))))))))) → 2(3(4(1(1(0(2(3(0(4(1(0(2(0(0(0(3(3(x))))))))))))))))))
5(2(3(5(3(0(3(3(1(4(0(2(x)))))))))))) → 4(5(4(0(0(2(1(1(0(3(4(4(4(3(x))))))))))))))
5(2(4(3(0(2(5(2(1(1(5(0(x)))))))))))) → 0(1(1(0(0(0(0(0(4(0(0(5(4(5(x))))))))))))))
5(2(5(4(4(2(3(5(0(5(2(5(x)))))))))))) → 4(1(4(2(3(4(3(4(4(4(2(1(4(0(0(0(1(x)))))))))))))))))
5(3(2(0(2(0(4(2(5(1(5(2(x)))))))))))) → 4(3(0(4(5(0(3(4(0(3(5(1(1(1(4(x)))))))))))))))
5(3(3(2(1(1(3(3(2(5(5(1(x)))))))))))) → 3(1(4(0(1(4(2(0(3(0(0(3(1(2(4(x)))))))))))))))
5(3(3(3(1(5(2(0(1(5(4(0(x)))))))))))) → 3(5(0(3(5(0(0(4(2(3(4(0(1(2(2(3(0(0(x))))))))))))))))))
5(3(5(2(5(5(2(4(2(1(2(2(x)))))))))))) → 0(1(3(4(4(4(0(1(5(0(1(1(2(1(x))))))))))))))
5(3(5(4(4(5(1(3(4(5(5(5(x)))))))))))) → 5(3(4(4(0(4(4(0(1(5(0(5(2(2(3(4(0(4(x))))))))))))))))))
5(4(3(3(0(3(2(5(4(0(5(5(x)))))))))))) → 1(1(4(1(3(2(2(0(1(2(2(0(0(1(1(4(3(x)))))))))))))))))
5(4(3(3(3(2(5(2(4(1(4(2(x)))))))))))) → 3(0(4(4(4(0(4(4(1(4(4(1(1(2(2(4(5(4(x))))))))))))))))))
5(5(0(0(3(4(3(2(3(4(5(5(x)))))))))))) → 0(2(1(0(1(2(4(0(0(5(3(2(2(2(0(x)))))))))))))))
5(5(1(2(5(2(5(3(3(3(4(5(x)))))))))))) → 3(5(1(3(1(5(2(4(0(2(3(3(4(4(2(x)))))))))))))))
5(5(1(3(3(2(0(5(5(1(0(0(x)))))))))))) → 5(0(4(0(3(4(5(1(2(4(4(4(3(0(0(0(1(x)))))))))))))))))
5(5(1(5(1(5(3(4(2(5(4(5(x)))))))))))) → 0(5(4(0(4(0(4(1(2(2(1(1(3(4(4(0(3(3(x))))))))))))))))))
5(5(1(5(2(5(1(5(1(1(1(4(x)))))))))))) → 1(4(1(3(4(4(5(4(0(5(4(5(0(4(3(5(2(x)))))))))))))))))
5(5(2(5(3(3(1(0(1(4(5(5(x)))))))))))) → 2(4(3(0(5(2(3(0(5(4(4(0(3(2(3(2(5(x)))))))))))))))))
5(5(3(1(0(3(3(0(2(3(5(3(x)))))))))))) → 2(4(5(2(2(2(0(0(4(4(3(0(4(0(0(2(4(4(x))))))))))))))))))
5(5(3(3(5(3(3(0(4(5(3(0(x)))))))))))) → 4(4(0(2(4(0(3(4(1(4(1(3(3(4(4(4(4(3(x))))))))))))))))))
5(5(3(4(0(2(5(5(3(2(4(1(x)))))))))))) → 2(4(3(0(4(0(1(3(2(4(1(1(0(1(1(1(0(1(x))))))))))))))))))
5(5(4(5(5(2(5(3(1(2(1(1(x)))))))))))) → 0(3(0(2(2(4(4(3(1(4(4(2(5(0(0(0(x))))))))))))))))
5(5(5(5(5(3(2(5(2(0(5(5(x)))))))))))) → 5(3(5(3(3(1(5(5(4(1(3(4(0(1(x))))))))))))))
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
2(5(3(3(3(3(5(2(1(4(2(0(x)))))))))))) → 0(1(3(4(0(5(5(3(0(2(2(4(3(2(3(x)))))))))))))))
Q is empty.
(3) Overlay + Local Confluence (EQUIVALENT transformation)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
2(5(3(3(3(3(5(2(1(4(2(0(x)))))))))))) → 0(1(3(4(0(5(5(3(0(2(2(4(3(2(3(x)))))))))))))))
The set Q consists of the following terms:
2(5(3(3(3(3(5(2(1(4(2(0(x0))))))))))))
(5) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
21(5(3(3(3(3(5(2(1(4(2(0(x)))))))))))) → 21(2(4(3(2(3(x))))))
21(5(3(3(3(3(5(2(1(4(2(0(x)))))))))))) → 21(4(3(2(3(x)))))
21(5(3(3(3(3(5(2(1(4(2(0(x)))))))))))) → 21(3(x))
The TRS R consists of the following rules:
2(5(3(3(3(3(5(2(1(4(2(0(x)))))))))))) → 0(1(3(4(0(5(5(3(0(2(2(4(3(2(3(x)))))))))))))))
The set Q consists of the following terms:
2(5(3(3(3(3(5(2(1(4(2(0(x0))))))))))))
We have to consider all minimal (P,Q,R)-chains.
(7) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.
(8) TRUE