YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/ICFP_2010/138142.srs

Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS
↳1 QTRS Reverse (⇔, 0 ms)
↳2 QTRS
↳3 QTRSRRRProof (⇔, 280 ms)
↳4 QTRS
↳5 DependencyPairsProof (⇔, 29 ms)
↳6 QDP
↳7 DependencyGraphProof (⇔, 0 ms)
↳8 QDP
↳9 UsableRulesProof (⇔, 0 ms)
↳10 QDP
↳11 QDPSizeChangeProof (⇔, 0 ms)
↳12 YES

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

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

Q is empty.

(1) QTRS Reverse (EQUIVALENT transformation)

We applied the QTRS Reverse Processor [REVERSE].

(2) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

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

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0(x1)) = 4 + x1   
POL(1(x1)) = 1 + x1   
POL(2(x1)) = x1   
POL(3(x1)) = 2 + x1   
POL(4(x1)) = 3 + x1   
POL(5(x1)) = x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

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


(4) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

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

Q is empty.

(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:

51(3(0(1(3(1(3(5(1(0(1(2(x)))))))))))) → 51(3(0(4(4(2(5(3(5(x)))))))))
51(3(0(1(3(1(3(5(1(0(1(2(x)))))))))))) → 41(4(2(5(3(5(x))))))
51(3(0(1(3(1(3(5(1(0(1(2(x)))))))))))) → 41(2(5(3(5(x)))))
51(3(0(1(3(1(3(5(1(0(1(2(x)))))))))))) → 51(3(5(x)))
51(3(0(1(3(1(3(5(1(0(1(2(x)))))))))))) → 51(x)
51(3(4(3(4(5(0(5(4(5(3(4(x)))))))))))) → 51(5(1(0(4(3(4(5(5(2(2(0(2(2(4(3(5(x)))))))))))))))))
51(3(4(3(4(5(0(5(4(5(3(4(x)))))))))))) → 51(1(0(4(3(4(5(5(2(2(0(2(2(4(3(5(x))))))))))))))))
51(3(4(3(4(5(0(5(4(5(3(4(x)))))))))))) → 41(3(4(5(5(2(2(0(2(2(4(3(5(x)))))))))))))
51(3(4(3(4(5(0(5(4(5(3(4(x)))))))))))) → 41(5(5(2(2(0(2(2(4(3(5(x)))))))))))
51(3(4(3(4(5(0(5(4(5(3(4(x)))))))))))) → 51(5(2(2(0(2(2(4(3(5(x))))))))))
51(3(4(3(4(5(0(5(4(5(3(4(x)))))))))))) → 51(2(2(0(2(2(4(3(5(x)))))))))
51(3(4(3(4(5(0(5(4(5(3(4(x)))))))))))) → 41(3(5(x)))
51(3(4(3(4(5(0(5(4(5(3(4(x)))))))))))) → 51(x)
41(1(2(4(2(5(4(1(1(5(1(5(x)))))))))))) → 51(2(2(2(2(2(4(1(4(1(1(3(2(1(2(2(1(5(x))))))))))))))))))
41(1(2(4(2(5(4(1(1(5(1(5(x)))))))))))) → 41(1(4(1(1(3(2(1(2(2(1(5(x))))))))))))
41(1(2(4(2(5(4(1(1(5(1(5(x)))))))))))) → 41(1(1(3(2(1(2(2(1(5(x))))))))))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 14 less nodes.

(8) Obligation:

Q DP problem:
The TRS P consists of the following rules:

51(3(4(3(4(5(0(5(4(5(3(4(x)))))))))))) → 51(x)
51(3(0(1(3(1(3(5(1(0(1(2(x)))))))))))) → 51(x)

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(9) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(10) Obligation:

Q DP problem:
The TRS P consists of the following rules:

51(3(4(3(4(5(0(5(4(5(3(4(x)))))))))))) → 51(x)
51(3(0(1(3(1(3(5(1(0(1(2(x)))))))))))) → 51(x)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(11) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • 51(3(4(3(4(5(0(5(4(5(3(4(x)))))))))))) → 51(x)
    The graph contains the following edges 1 > 1

  • 51(3(0(1(3(1(3(5(1(0(1(2(x)))))))))))) → 51(x)
    The graph contains the following edges 1 > 1

(12) YES