YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/ICFP_2010/132611.srs

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

0 QTRS
↳1 QTRSRRRProof (⇔, 271 ms)
↳2 QTRS
↳3 DependencyPairsProof (⇔, 15 ms)
↳4 QDP
↳5 DependencyGraphProof (⇔, 0 ms)
↳6 QDP
↳7 UsableRulesProof (⇔, 0 ms)
↳8 QDP
↳9 QDPSizeChangeProof (⇔, 0 ms)
↳10 YES

(0) Obligation:

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

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

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0(x1)) = 1 + x1   
POL(1(x1)) = 1 + x1   
POL(2(x1)) = 1 + x1   
POL(3(x1)) = 1 + x1   
POL(4(x1)) = 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(0(2(0(1(1(1(1(3(3(2(3(x)))))))))))) → 3(2(4(0(0(5(4(4(0(4(4(4(1(3(1(4(5(x)))))))))))))))))
0(0(2(2(1(1(2(3(1(5(1(2(x)))))))))))) → 0(5(4(5(4(4(5(2(0(0(5(5(4(0(2(4(5(1(x))))))))))))))))))
0(1(0(1(3(5(0(2(0(5(0(4(x)))))))))))) → 0(1(4(4(5(4(1(4(3(0(0(1(5(2(5(4(4(4(x))))))))))))))))))
0(1(2(3(2(3(5(1(3(3(5(3(x)))))))))))) → 0(4(5(0(4(2(4(1(4(4(4(5(5(1(5(x)))))))))))))))
0(1(3(1(3(3(1(1(0(5(1(0(x)))))))))))) → 4(4(3(5(2(1(5(2(5(4(3(4(1(3(5(4(x))))))))))))))))
0(1(4(0(3(2(2(2(4(2(2(3(x)))))))))))) → 4(4(4(4(4(1(3(1(5(5(4(4(1(3(4(4(x))))))))))))))))
0(1(5(1(5(1(2(4(5(2(0(0(x)))))))))))) → 3(4(4(5(5(5(4(1(0(3(4(0(4(5(4(5(3(x)))))))))))))))))
0(2(0(1(1(1(1(2(5(1(2(2(x)))))))))))) → 5(3(3(0(3(2(4(4(2(3(4(1(5(4(4(4(0(4(x))))))))))))))))))
0(2(2(0(1(1(0(1(3(0(3(5(x)))))))))))) → 4(2(1(3(3(4(5(2(5(4(5(5(4(4(2(3(4(1(x))))))))))))))))))
0(2(2(2(2(0(4(0(2(3(5(1(x)))))))))))) → 3(2(1(4(5(2(5(5(4(4(5(5(0(0(4(1(x))))))))))))))))
0(2(3(3(1(0(5(2(2(0(0(0(x)))))))))))) → 1(1(4(5(3(5(2(2(3(1(4(5(4(4(3(x)))))))))))))))
0(2(5(1(5(0(2(0(2(3(2(0(x)))))))))))) → 5(2(3(4(4(1(4(5(4(5(0(0(1(4(5(5(1(1(x))))))))))))))))))
0(2(5(5(0(3(0(3(2(2(0(1(x)))))))))))) → 0(4(5(2(4(2(5(4(1(2(2(2(0(5(4(1(x))))))))))))))))
0(3(3(1(3(0(4(2(2(3(1(0(x)))))))))))) → 5(5(4(5(3(3(4(5(4(5(0(1(0(3(0(1(x))))))))))))))))
0(3(5(1(0(3(1(1(1(0(2(2(x)))))))))))) → 4(3(4(4(0(3(2(1(2(4(2(5(5(5(4(5(4(4(x))))))))))))))))))
0(3(5(2(1(3(0(3(2(3(1(0(x)))))))))))) → 5(2(5(3(1(3(4(0(4(0(0(4(4(3(2(x)))))))))))))))
0(4(3(3(2(2(3(4(5(3(2(1(x)))))))))))) → 0(5(4(1(4(4(4(3(5(4(5(2(4(4(2(2(1(x)))))))))))))))))
0(5(3(1(3(2(5(2(0(0(2(1(x)))))))))))) → 0(4(4(5(4(3(5(2(5(5(2(5(4(5(5(4(5(x)))))))))))))))))
1(0(0(1(3(3(3(2(0(3(0(4(x)))))))))))) → 4(4(3(0(4(5(0(5(4(4(0(4(0(0(x))))))))))))))
1(0(0(2(4(2(2(2(2(4(2(0(x)))))))))))) → 5(1(5(3(5(4(5(0(3(3(4(1(4(4(x))))))))))))))
1(0(2(0(3(3(3(3(3(4(3(3(x)))))))))))) → 5(1(5(3(1(4(4(5(3(1(5(4(5(4(4(1(2(x)))))))))))))))))
1(0(3(2(0(2(2(2(1(2(3(2(x)))))))))))) → 3(3(2(4(4(0(3(4(0(1(4(5(2(2(4(x)))))))))))))))
1(0(3(3(2(2(1(1(0(2(3(1(x)))))))))))) → 2(4(3(4(4(1(4(5(1(5(1(0(5(4(3(x)))))))))))))))
1(1(0(0(5(2(3(2(1(1(0(5(x)))))))))))) → 5(4(5(4(0(1(2(1(2(3(3(0(4(1(x))))))))))))))
1(1(0(2(3(2(2(2(0(5(1(4(x)))))))))))) → 4(4(4(1(4(4(4(4(5(5(2(1(4(3(0(5(4(2(x))))))))))))))))))
1(1(1(3(1(2(2(4(3(0(0(2(x)))))))))))) → 5(2(4(3(4(4(1(1(3(1(4(5(2(4(0(2(x))))))))))))))))
1(2(1(1(3(0(2(2(4(1(1(3(x)))))))))))) → 1(4(5(4(1(1(5(1(1(2(5(4(1(1(3(4(x))))))))))))))))
1(2(2(2(3(2(0(2(5(3(5(3(x)))))))))))) → 1(4(0(5(3(3(5(3(3(4(3(4(2(1(x))))))))))))))
1(2(5(0(2(3(0(0(2(0(5(2(x)))))))))))) → 5(2(2(1(4(0(4(5(4(1(0(1(5(4(1(3(5(4(x))))))))))))))))))
1(3(0(0(2(3(0(4(2(0(2(5(x)))))))))))) → 0(2(1(4(2(1(4(5(0(5(5(4(5(5(4(1(4(5(x))))))))))))))))))
1(3(1(1(3(0(3(3(3(5(1(5(x)))))))))))) → 5(3(3(3(4(4(2(5(0(0(4(4(4(1(1(x)))))))))))))))
1(3(1(2(0(2(0(3(3(1(1(1(x)))))))))))) → 1(5(4(4(5(4(4(4(3(0(1(5(4(1(x))))))))))))))
1(3(3(1(0(0(0(2(5(1(1(1(x)))))))))))) → 2(1(2(4(4(4(4(1(2(1(3(5(4(5(0(5(2(x)))))))))))))))))
1(5(2(2(1(1(5(0(3(3(0(4(x)))))))))))) → 4(2(4(5(4(5(4(3(5(5(4(4(4(1(1(2(x))))))))))))))))
1(5(3(3(4(0(3(1(1(2(1(0(x)))))))))))) → 4(4(4(2(4(2(2(4(4(5(4(5(4(0(0(5(2(x)))))))))))))))))
2(0(3(2(2(0(2(2(3(0(0(1(x)))))))))))) → 5(4(3(4(4(1(5(4(2(4(1(5(4(5(3(0(0(x)))))))))))))))))
2(0(3(3(2(2(3(1(2(3(0(3(x)))))))))))) → 5(4(5(5(1(5(4(4(4(5(4(0(2(1(2(0(2(x)))))))))))))))))
2(1(0(0(0(3(3(3(3(0(1(1(x)))))))))))) → 5(3(2(2(4(2(2(4(0(4(5(3(4(2(5(x)))))))))))))))
2(1(0(2(1(3(1(2(5(1(3(5(x)))))))))))) → 2(5(1(3(5(1(4(4(3(1(0(1(4(5(x))))))))))))))
2(1(0(2(2(1(1(3(1(1(1(3(x)))))))))))) → 5(5(4(5(5(2(2(0(1(0(4(4(2(5(2(0(x))))))))))))))))
2(1(1(0(0(5(5(0(0(2(5(5(x)))))))))))) → 2(2(5(2(3(2(5(4(1(1(4(4(5(5(x))))))))))))))
2(1(3(0(0(1(0(3(3(1(0(4(x)))))))))))) → 4(4(5(3(3(4(2(3(5(4(2(4(4(5(3(3(x))))))))))))))))
2(1(5(1(1(5(0(2(5(1(3(0(x)))))))))))) → 3(5(4(2(3(4(4(3(3(4(0(5(2(0(5(4(x))))))))))))))))
2(2(0(3(2(0(1(0(2(3(3(2(x)))))))))))) → 5(5(5(0(2(2(4(1(4(4(3(3(0(0(2(x)))))))))))))))
2(2(1(1(0(2(0(0(2(3(0(1(x)))))))))))) → 5(4(1(1(4(3(5(5(0(1(4(3(5(4(1(1(4(x)))))))))))))))))
2(2(2(0(3(1(0(2(5(0(0(1(x)))))))))))) → 5(3(0(5(1(4(4(5(2(4(4(2(4(4(0(5(x))))))))))))))))
2(2(2(0(4(2(0(1(3(1(2(2(x)))))))))))) → 4(4(5(4(0(4(2(5(4(3(0(4(4(4(4(5(3(2(x))))))))))))))))))
2(2(2(4(1(1(0(1(1(1(4(0(x)))))))))))) → 3(3(5(4(1(4(4(3(5(4(4(4(1(4(4(1(4(4(x))))))))))))))))))
2(2(3(0(4(2(3(1(1(0(4(5(x)))))))))))) → 4(1(3(2(5(5(5(3(3(5(2(4(1(5(x))))))))))))))
2(2(3(4(3(3(5(1(2(4(2(2(x)))))))))))) → 3(4(3(5(3(4(1(4(0(4(5(5(0(2(x))))))))))))))
2(2(3(5(0(2(2(2(3(3(2(2(x)))))))))))) → 3(4(0(5(3(2(0(4(5(4(0(1(5(4(1(2(4(x)))))))))))))))))
2(3(0(2(0(3(0(0(3(2(2(3(x)))))))))))) → 4(5(4(0(2(3(4(5(5(4(1(3(3(1(5(1(1(x)))))))))))))))))
2(3(1(2(1(1(2(0(4(5(1(3(x)))))))))))) → 4(0(4(1(4(1(4(5(3(3(3(4(1(5(0(4(1(x)))))))))))))))))
2(3(1(3(5(1(4(5(1(1(5(1(x)))))))))))) → 5(0(4(0(4(1(0(4(1(5(5(3(4(4(3(x)))))))))))))))
2(3(2(1(2(3(1(5(5(2(3(0(x)))))))))))) → 2(4(5(2(4(1(4(0(1(4(3(2(3(4(1(x)))))))))))))))
2(3(2(2(3(2(0(3(4(5(2(1(x)))))))))))) → 3(4(1(5(0(0(2(5(2(0(5(3(4(2(x))))))))))))))
2(3(3(1(1(0(0(2(2(1(2(5(x)))))))))))) → 5(3(4(4(1(5(2(1(1(5(0(4(5(3(x))))))))))))))
2(3(3(2(4(3(0(2(3(3(5(1(x)))))))))))) → 0(0(4(3(4(4(3(0(4(1(0(1(4(4(x))))))))))))))
2(4(0(0(0(3(1(1(2(3(2(3(x)))))))))))) → 0(0(3(5(4(2(5(4(3(3(4(4(1(0(3(x)))))))))))))))
2(5(1(5(1(4(3(3(0(4(1(1(x)))))))))))) → 4(3(5(5(5(3(1(3(4(0(3(5(4(2(x))))))))))))))
3(0(0(0(2(2(4(1(2(1(1(1(x)))))))))))) → 4(0(5(2(5(4(3(2(5(4(3(5(4(2(5(4(3(x)))))))))))))))))
3(0(0(3(1(1(3(3(1(2(0(3(x)))))))))))) → 5(0(1(4(5(3(0(5(0(0(4(4(2(4(4(2(5(3(x))))))))))))))))))
3(0(1(4(2(0(3(0(5(1(4(2(x)))))))))))) → 0(5(4(5(2(1(5(5(4(3(5(4(5(5(4(3(2(x)))))))))))))))))
3(0(2(3(3(0(5(3(4(5(0(0(x)))))))))))) → 5(4(4(0(4(5(3(3(5(3(5(0(4(4(4(x)))))))))))))))
3(0(3(0(0(3(0(3(5(1(3(2(x)))))))))))) → 3(4(0(5(4(4(4(5(4(1(0(2(1(4(1(5(3(4(x))))))))))))))))))
3(0(3(3(4(0(5(1(3(2(3(1(x)))))))))))) → 3(4(5(2(1(1(3(4(4(4(3(5(5(0(4(4(5(x)))))))))))))))))
3(1(0(0(5(4(3(2(2(3(3(2(x)))))))))))) → 5(2(1(4(1(5(3(3(4(4(0(5(2(5(1(x)))))))))))))))
3(1(0(5(1(1(3(0(5(1(2(1(x)))))))))))) → 4(0(4(4(5(4(4(3(4(5(4(4(2(1(0(1(x))))))))))))))))
3(1(1(1(2(0(1(0(0(5(0(4(x)))))))))))) → 5(0(1(5(5(5(0(1(5(4(4(3(2(0(4(x)))))))))))))))
3(1(1(1(3(0(3(1(0(3(4(0(x)))))))))))) → 0(5(3(1(4(4(1(5(5(5(5(4(1(5(x))))))))))))))
3(1(1(2(0(2(5(5(3(3(1(3(x)))))))))))) → 4(2(1(5(1(5(4(0(3(3(0(4(2(4(x))))))))))))))
3(1(5(0(0(1(0(4(2(1(5(3(x)))))))))))) → 1(5(0(0(4(1(4(5(4(5(4(1(4(4(0(4(x))))))))))))))))
3(2(0(0(3(0(2(3(1(3(3(2(x)))))))))))) → 1(4(5(4(4(0(3(1(5(4(3(3(0(4(4(5(4(x)))))))))))))))))
3(2(2(5(1(5(0(3(4(2(4(3(x)))))))))))) → 1(5(4(5(2(4(1(5(4(4(4(2(4(4(5(4(0(4(x))))))))))))))))))
3(3(0(2(4(3(0(1(5(5(3(0(x)))))))))))) → 5(1(4(3(4(5(3(4(4(0(1(3(1(4(0(x)))))))))))))))
3(3(1(1(2(3(5(5(2(4(0(0(x)))))))))))) → 0(1(5(5(4(5(5(2(4(4(4(1(2(5(4(5(4(x)))))))))))))))))
3(3(1(2(2(1(3(3(0(2(0(4(x)))))))))))) → 2(5(4(3(1(4(4(4(0(4(2(3(0(2(x))))))))))))))
3(3(1(2(3(0(2(0(3(0(1(0(x)))))))))))) → 3(0(1(3(0(4(5(5(1(0(5(2(4(1(2(4(x))))))))))))))))
3(3(2(3(0(1(3(3(5(1(1(2(x)))))))))))) → 2(5(5(5(4(4(2(5(4(1(5(5(4(1(4(5(2(x)))))))))))))))))
3(5(1(2(1(0(3(2(2(2(0(1(x)))))))))))) → 1(5(5(4(1(2(5(5(4(4(4(4(2(1(2(5(4(5(x))))))))))))))))))
4(0(0(0(2(5(0(2(0(5(5(0(x)))))))))))) → 4(5(3(4(3(4(2(4(4(3(0(4(1(2(5(x)))))))))))))))
4(0(0(3(1(3(5(0(3(3(2(4(x)))))))))))) → 4(0(2(4(0(1(2(4(4(0(4(1(4(2(4(4(x))))))))))))))))
4(0(1(3(1(2(3(3(1(1(2(2(x)))))))))))) → 1(4(5(3(0(1(4(3(2(4(1(5(3(4(0(x)))))))))))))))
4(1(3(1(1(1(1(2(1(0(4(3(x)))))))))))) → 0(5(4(2(4(4(5(4(2(4(0(0(4(4(4(4(5(x)))))))))))))))))
4(1(3(3(0(0(3(3(3(3(3(0(x)))))))))))) → 4(0(2(1(4(0(5(1(0(4(2(4(5(2(x))))))))))))))
4(1(5(0(2(2(1(3(0(0(3(1(x)))))))))))) → 4(5(4(1(4(1(4(4(5(5(4(4(3(4(4(3(2(4(x))))))))))))))))))
4(3(3(2(0(1(1(2(0(1(5(5(x)))))))))))) → 3(0(2(4(1(4(4(4(5(4(4(4(5(0(4(x)))))))))))))))
4(4(3(0(0(2(2(5(5(3(1(1(x)))))))))))) → 5(5(4(4(0(4(2(5(4(2(4(1(4(1(1(x)))))))))))))))


(2) Obligation:

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

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

Q is empty.

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(4) Obligation:

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

11(0(2(2(5(1(3(1(3(3(2(5(x)))))))))))) → 31(5(3(5(0(5(2(3(0(2(4(0(2(3(4(x)))))))))))))))
11(0(2(2(5(1(3(1(3(3(2(5(x)))))))))))) → 31(5(0(5(2(3(0(2(4(0(2(3(4(x)))))))))))))
11(0(2(2(5(1(3(1(3(3(2(5(x)))))))))))) → 21(3(0(2(4(0(2(3(4(x)))))))))
11(0(2(2(5(1(3(1(3(3(2(5(x)))))))))))) → 31(0(2(4(0(2(3(4(x))))))))
11(0(2(2(5(1(3(1(3(3(2(5(x)))))))))))) → 21(4(0(2(3(4(x))))))
11(0(2(2(5(1(3(1(3(3(2(5(x)))))))))))) → 21(3(4(x)))
11(0(2(2(5(1(3(1(3(3(2(5(x)))))))))))) → 31(4(x))
11(5(1(1(1(5(1(3(2(3(1(5(x)))))))))))) → 11(3(4(4(4(5(3(3(3(5(2(1(5(4(1(5(2(5(x))))))))))))))))))
11(5(1(1(1(5(1(3(2(3(1(5(x)))))))))))) → 31(4(4(4(5(3(3(3(5(2(1(5(4(1(5(2(5(x)))))))))))))))))
11(5(1(1(1(5(1(3(2(3(1(5(x)))))))))))) → 31(3(3(5(2(1(5(4(1(5(2(5(x))))))))))))
11(5(1(1(1(5(1(3(2(3(1(5(x)))))))))))) → 31(3(5(2(1(5(4(1(5(2(5(x)))))))))))
11(5(1(1(1(5(1(3(2(3(1(5(x)))))))))))) → 31(5(2(1(5(4(1(5(2(5(x))))))))))
11(5(1(1(1(5(1(3(2(3(1(5(x)))))))))))) → 21(1(5(4(1(5(2(5(x))))))))
11(5(1(1(1(5(1(3(2(3(1(5(x)))))))))))) → 11(5(4(1(5(2(5(x)))))))
11(5(1(1(1(5(1(3(2(3(1(5(x)))))))))))) → 11(5(2(5(x))))
11(5(1(1(1(5(1(3(2(3(1(5(x)))))))))))) → 21(5(x))
21(0(2(4(5(1(2(2(5(3(5(0(x)))))))))))) → 11(5(4(1(4(3(4(5(3(4(5(2(x))))))))))))
21(0(2(4(5(1(2(2(5(3(5(0(x)))))))))))) → 11(4(3(4(5(3(4(5(2(x)))))))))
21(0(2(4(5(1(2(2(5(3(5(0(x)))))))))))) → 31(4(5(3(4(5(2(x)))))))
21(0(2(4(5(1(2(2(5(3(5(0(x)))))))))))) → 31(4(5(2(x))))
21(0(2(4(5(1(2(2(5(3(5(0(x)))))))))))) → 21(x)
21(2(1(0(2(5(1(2(1(5(4(5(x)))))))))))) → 31(5(4(5(4(0(2(2(0(1(3(4(4(3(5(x)))))))))))))))
21(2(1(0(2(5(1(2(1(5(4(5(x)))))))))))) → 21(2(0(1(3(4(4(3(5(x)))))))))
21(2(1(0(2(5(1(2(1(5(4(5(x)))))))))))) → 21(0(1(3(4(4(3(5(x))))))))
21(2(1(0(2(5(1(2(1(5(4(5(x)))))))))))) → 11(3(4(4(3(5(x))))))
21(2(1(0(2(5(1(2(1(5(4(5(x)))))))))))) → 31(4(4(3(5(x)))))
21(2(1(0(2(5(1(2(1(5(4(5(x)))))))))))) → 31(5(x))
21(3(1(1(3(0(3(0(4(0(5(3(x)))))))))))) → 21(0(1(2(4(3(4(1(4(4(3(4(1(4(4(0(3(x)))))))))))))))))
21(3(1(1(3(0(3(0(4(0(5(3(x)))))))))))) → 11(2(4(3(4(1(4(4(3(4(1(4(4(0(3(x)))))))))))))))
21(3(1(1(3(0(3(0(4(0(5(3(x)))))))))))) → 21(4(3(4(1(4(4(3(4(1(4(4(0(3(x))))))))))))))
21(3(1(1(3(0(3(0(4(0(5(3(x)))))))))))) → 31(4(1(4(4(3(4(1(4(4(0(3(x))))))))))))
21(3(1(1(3(0(3(0(4(0(5(3(x)))))))))))) → 11(4(4(3(4(1(4(4(0(3(x))))))))))
21(3(1(1(3(0(3(0(4(0(5(3(x)))))))))))) → 31(4(1(4(4(0(3(x)))))))
21(3(1(1(3(0(3(0(4(0(5(3(x)))))))))))) → 11(4(4(0(3(x)))))
21(5(1(3(0(0(4(5(3(1(3(4(x)))))))))))) → 11(5(4(0(5(4(0(2(4(0(1(2(x))))))))))))
21(5(1(3(0(0(4(5(3(1(3(4(x)))))))))))) → 21(4(0(1(2(x)))))
21(5(1(3(0(0(4(5(3(1(3(4(x)))))))))))) → 11(2(x))
21(5(1(3(0(0(4(5(3(1(3(4(x)))))))))))) → 21(x)
31(3(1(2(5(1(2(0(5(5(0(0(x)))))))))))) → 31(4(4(5(4(1(3(1(4(4(0(5(4(3(2(0(2(x)))))))))))))))))
31(3(1(2(5(1(2(0(5(5(0(0(x)))))))))))) → 11(3(1(4(4(0(5(4(3(2(0(2(x))))))))))))
31(3(1(2(5(1(2(0(5(5(0(0(x)))))))))))) → 31(1(4(4(0(5(4(3(2(0(2(x)))))))))))
31(3(1(2(5(1(2(0(5(5(0(0(x)))))))))))) → 11(4(4(0(5(4(3(2(0(2(x))))))))))
31(3(1(2(5(1(2(0(5(5(0(0(x)))))))))))) → 31(2(0(2(x))))
31(3(1(2(5(1(2(0(5(5(0(0(x)))))))))))) → 21(0(2(x)))
31(3(1(2(5(1(2(0(5(5(0(0(x)))))))))))) → 21(x)
31(5(5(0(2(3(5(0(5(0(4(0(x)))))))))))) → 21(1(3(0(5(4(4(2(x))))))))
31(5(5(0(2(3(5(0(5(0(4(0(x)))))))))))) → 11(3(0(5(4(4(2(x)))))))
31(5(5(0(2(3(5(0(5(0(4(0(x)))))))))))) → 31(0(5(4(4(2(x))))))
31(5(5(0(2(3(5(0(5(0(4(0(x)))))))))))) → 21(x)

The TRS R consists of the following rules:

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

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 45 less nodes.

(6) Obligation:

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

21(2(1(0(2(5(1(2(1(5(4(5(x)))))))))))) → 31(5(x))
31(5(5(0(2(3(5(0(5(0(4(0(x)))))))))))) → 21(x)
21(0(2(4(5(1(2(2(5(3(5(0(x)))))))))))) → 21(x)
21(5(1(3(0(0(4(5(3(1(3(4(x)))))))))))) → 21(x)

The TRS R consists of the following rules:

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

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

(7) 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.

(8) Obligation:

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

21(2(1(0(2(5(1(2(1(5(4(5(x)))))))))))) → 31(5(x))
31(5(5(0(2(3(5(0(5(0(4(0(x)))))))))))) → 21(x)
21(0(2(4(5(1(2(2(5(3(5(0(x)))))))))))) → 21(x)
21(5(1(3(0(0(4(5(3(1(3(4(x)))))))))))) → 21(x)

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

(9) 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:

  • 31(5(5(0(2(3(5(0(5(0(4(0(x)))))))))))) → 21(x)
    The graph contains the following edges 1 > 1

  • 21(2(1(0(2(5(1(2(1(5(4(5(x)))))))))))) → 31(5(x))
    The graph contains the following edges 1 > 1

  • 21(0(2(4(5(1(2(2(5(3(5(0(x)))))))))))) → 21(x)
    The graph contains the following edges 1 > 1

  • 21(5(1(3(0(0(4(5(3(1(3(4(x)))))))))))) → 21(x)
    The graph contains the following edges 1 > 1

(10) YES