YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/ICFP_2010/214169.srs

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

0 QTRS
↳1 QTRS Reverse (⇔, 0 ms)
↳2 QTRS
↳3 DependencyPairsProof (⇔, 184 ms)
↳4 QDP
↳5 DependencyGraphProof (⇔, 0 ms)
↳6 AND
    ↳7 QDP
        ↳8 UsableRulesProof (⇔, 0 ms)
        ↳9 QDP
        ↳10 QDPOrderProof (⇔, 15 ms)
        ↳11 QDP
        ↳12 PisEmptyProof (⇔, 0 ms)
        ↳13 YES
    ↳14 QDP
        ↳15 QDPOrderProof (⇔, 154 ms)
        ↳16 QDP
        ↳17 QDPOrderProof (⇔, 102 ms)
        ↳18 QDP
        ↳19 PisEmptyProof (⇔, 0 ms)
        ↳20 YES
    ↳21 QDP
        ↳22 QDPSizeChangeProof (⇔, 0 ms)
        ↳23 YES
    ↳24 QDP
        ↳25 QDPOrderProof (⇔, 154 ms)
        ↳26 QDP
        ↳27 QDPOrderProof (⇔, 97 ms)
        ↳28 QDP
        ↳29 PisEmptyProof (⇔, 0 ms)
        ↳30 YES

(0) Obligation:

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

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

1(0(0(x))) → 3(1(2(0(0(x)))))
1(0(0(x))) → 1(3(2(2(0(0(x))))))
1(4(0(x))) → 1(2(2(4(0(x)))))
1(4(0(x))) → 2(4(1(3(2(0(x))))))
1(0(1(x))) → 1(1(2(0(x))))
1(0(1(x))) → 2(1(1(2(0(x)))))
1(0(1(x))) → 1(2(3(0(1(x)))))
0(4(1(x))) → 2(4(2(3(0(1(x))))))
1(4(1(x))) → 1(1(2(4(x))))
1(4(1(x))) → 1(2(1(1(2(4(x))))))
1(0(4(x))) → 1(2(0(4(x))))
1(0(5(x))) → 1(2(0(5(x))))
1(0(5(x))) → 3(5(1(2(0(x)))))
1(0(5(x))) → 5(5(1(2(3(0(x))))))
1(0(5(x))) → 1(5(3(2(0(3(x))))))
5(0(5(x))) → 5(3(5(2(3(0(x))))))
1(4(5(x))) → 3(1(5(2(4(2(x))))))
1(4(5(x))) → 3(3(1(2(5(4(x))))))
5(4(0(0(x)))) → 2(4(2(0(5(0(x))))))
0(2(1(0(x)))) → 0(1(2(0(2(x)))))
1(0(2(0(x)))) → 3(1(2(0(0(x)))))
0(4(3(0(x)))) → 3(0(0(4(2(3(x))))))
4(0(4(0(x)))) → 4(4(2(3(0(0(x))))))
4(1(0(1(x)))) → 0(4(1(1(3(2(x))))))
1(3(0(1(x)))) → 0(1(1(3(2(4(x))))))
4(0(2(1(x)))) → 2(3(0(2(4(1(x))))))
4(0(3(1(x)))) → 3(1(2(3(0(4(x))))))
5(1(4(1(x)))) → 4(1(5(2(1(2(x))))))
1(5(0(4(x)))) → 5(1(2(4(2(0(x))))))
0(0(1(4(x)))) → 0(0(4(1(2(3(x))))))
4(0(1(4(x)))) → 1(2(0(4(4(x)))))
1(0(0(5(x)))) → 0(1(5(2(3(0(x))))))
1(3(0(5(x)))) → 1(2(3(5(0(x)))))
1(3(0(5(x)))) → 1(2(2(5(3(0(x))))))
1(5(0(5(x)))) → 0(1(3(5(2(5(x))))))
1(4(2(5(x)))) → 5(2(4(1(2(x)))))
5(1(4(5(x)))) → 1(2(5(5(4(x)))))
5(1(4(5(x)))) → 5(2(4(5(1(2(x))))))
1(3(4(5(x)))) → 3(1(2(4(5(x)))))
1(3(4(5(x)))) → 1(2(4(2(5(3(x))))))
1(3(4(5(x)))) → 1(2(3(4(2(5(x))))))
4(0(5(5(x)))) → 2(4(5(2(0(5(x))))))
1(3(3(0(0(x))))) → 3(2(1(0(3(0(x))))))
1(3(4(2(0(x))))) → 2(4(1(3(2(0(x))))))
1(1(4(5(0(x))))) → 1(1(5(2(0(4(x))))))
1(3(4(2(1(x))))) → 4(1(3(3(2(1(x))))))
1(3(0(0(4(x))))) → 0(1(2(3(0(4(x))))))
1(3(0(0(4(x))))) → 1(0(0(2(3(4(x))))))
5(0(0(1(5(x))))) → 5(1(2(0(0(5(x))))))
1(0(2(4(5(x))))) → 1(2(0(4(5(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(0(x))) → 11(2(0(0(x))))
11(0(0(x))) → 11(3(2(2(0(0(x))))))
11(4(0(x))) → 11(2(2(4(0(x)))))
11(4(0(x))) → 41(1(3(2(0(x)))))
11(4(0(x))) → 11(3(2(0(x))))
11(0(1(x))) → 11(1(2(0(x))))
11(0(1(x))) → 11(2(0(x)))
11(0(1(x))) → 01(x)
11(0(1(x))) → 11(2(3(0(1(x)))))
01(4(1(x))) → 41(2(3(0(1(x)))))
01(4(1(x))) → 01(1(x))
11(4(1(x))) → 11(1(2(4(x))))
11(4(1(x))) → 11(2(4(x)))
11(4(1(x))) → 41(x)
11(4(1(x))) → 11(2(1(1(2(4(x))))))
11(0(4(x))) → 11(2(0(4(x))))
11(0(5(x))) → 11(2(0(5(x))))
11(0(5(x))) → 51(1(2(0(x))))
11(0(5(x))) → 11(2(0(x)))
11(0(5(x))) → 01(x)
11(0(5(x))) → 51(5(1(2(3(0(x))))))
11(0(5(x))) → 51(1(2(3(0(x)))))
11(0(5(x))) → 11(2(3(0(x))))
11(0(5(x))) → 11(5(3(2(0(3(x))))))
11(0(5(x))) → 51(3(2(0(3(x)))))
11(0(5(x))) → 01(3(x))
51(0(5(x))) → 51(3(5(2(3(0(x))))))
51(0(5(x))) → 51(2(3(0(x))))
51(0(5(x))) → 01(x)
11(4(5(x))) → 11(5(2(4(2(x)))))
11(4(5(x))) → 51(2(4(2(x))))
11(4(5(x))) → 41(2(x))
11(4(5(x))) → 11(2(5(4(x))))
11(4(5(x))) → 51(4(x))
11(4(5(x))) → 41(x)
51(4(0(0(x)))) → 41(2(0(5(0(x)))))
51(4(0(0(x)))) → 01(5(0(x)))
51(4(0(0(x)))) → 51(0(x))
01(2(1(0(x)))) → 01(1(2(0(2(x)))))
01(2(1(0(x)))) → 11(2(0(2(x))))
01(2(1(0(x)))) → 01(2(x))
11(0(2(0(x)))) → 11(2(0(0(x))))
11(0(2(0(x)))) → 01(0(x))
01(4(3(0(x)))) → 01(0(4(2(3(x)))))
01(4(3(0(x)))) → 01(4(2(3(x))))
01(4(3(0(x)))) → 41(2(3(x)))
41(0(4(0(x)))) → 41(4(2(3(0(0(x))))))
41(0(4(0(x)))) → 41(2(3(0(0(x)))))
41(0(4(0(x)))) → 01(0(x))
41(1(0(1(x)))) → 01(4(1(1(3(2(x))))))
41(1(0(1(x)))) → 41(1(1(3(2(x)))))
41(1(0(1(x)))) → 11(1(3(2(x))))
41(1(0(1(x)))) → 11(3(2(x)))
11(3(0(1(x)))) → 01(1(1(3(2(4(x))))))
11(3(0(1(x)))) → 11(1(3(2(4(x)))))
11(3(0(1(x)))) → 11(3(2(4(x))))
11(3(0(1(x)))) → 41(x)
41(0(2(1(x)))) → 01(2(4(1(x))))
41(0(2(1(x)))) → 41(1(x))
41(0(3(1(x)))) → 11(2(3(0(4(x)))))
41(0(3(1(x)))) → 01(4(x))
41(0(3(1(x)))) → 41(x)
51(1(4(1(x)))) → 41(1(5(2(1(2(x))))))
51(1(4(1(x)))) → 11(5(2(1(2(x)))))
51(1(4(1(x)))) → 51(2(1(2(x))))
51(1(4(1(x)))) → 11(2(x))
11(5(0(4(x)))) → 51(1(2(4(2(0(x))))))
11(5(0(4(x)))) → 11(2(4(2(0(x)))))
11(5(0(4(x)))) → 41(2(0(x)))
11(5(0(4(x)))) → 01(x)
01(0(1(4(x)))) → 01(0(4(1(2(3(x))))))
01(0(1(4(x)))) → 01(4(1(2(3(x)))))
01(0(1(4(x)))) → 41(1(2(3(x))))
01(0(1(4(x)))) → 11(2(3(x)))
41(0(1(4(x)))) → 11(2(0(4(4(x)))))
41(0(1(4(x)))) → 01(4(4(x)))
41(0(1(4(x)))) → 41(4(x))
11(0(0(5(x)))) → 01(1(5(2(3(0(x))))))
11(0(0(5(x)))) → 11(5(2(3(0(x)))))
11(0(0(5(x)))) → 51(2(3(0(x))))
11(0(0(5(x)))) → 01(x)
11(3(0(5(x)))) → 11(2(3(5(0(x)))))
11(3(0(5(x)))) → 51(0(x))
11(3(0(5(x)))) → 01(x)
11(3(0(5(x)))) → 11(2(2(5(3(0(x))))))
11(3(0(5(x)))) → 51(3(0(x)))
11(5(0(5(x)))) → 01(1(3(5(2(5(x))))))
11(5(0(5(x)))) → 11(3(5(2(5(x)))))
11(5(0(5(x)))) → 51(2(5(x)))
11(4(2(5(x)))) → 51(2(4(1(2(x)))))
11(4(2(5(x)))) → 41(1(2(x)))
11(4(2(5(x)))) → 11(2(x))
51(1(4(5(x)))) → 11(2(5(5(4(x)))))
51(1(4(5(x)))) → 51(5(4(x)))
51(1(4(5(x)))) → 51(4(x))
51(1(4(5(x)))) → 41(x)
51(1(4(5(x)))) → 51(2(4(5(1(2(x))))))
51(1(4(5(x)))) → 41(5(1(2(x))))
51(1(4(5(x)))) → 51(1(2(x)))
51(1(4(5(x)))) → 11(2(x))
11(3(4(5(x)))) → 11(2(4(5(x))))
11(3(4(5(x)))) → 11(2(4(2(5(3(x))))))
11(3(4(5(x)))) → 41(2(5(3(x))))
11(3(4(5(x)))) → 51(3(x))
11(3(4(5(x)))) → 11(2(3(4(2(5(x))))))
11(3(4(5(x)))) → 41(2(5(x)))
41(0(5(5(x)))) → 41(5(2(0(5(x)))))
41(0(5(5(x)))) → 51(2(0(5(x))))
41(0(5(5(x)))) → 01(5(x))
11(3(3(0(0(x))))) → 11(0(3(0(x))))
11(3(3(0(0(x))))) → 01(3(0(x)))
11(3(4(2(0(x))))) → 41(1(3(2(0(x)))))
11(3(4(2(0(x))))) → 11(3(2(0(x))))
11(1(4(5(0(x))))) → 11(1(5(2(0(4(x))))))
11(1(4(5(0(x))))) → 11(5(2(0(4(x)))))
11(1(4(5(0(x))))) → 51(2(0(4(x))))
11(1(4(5(0(x))))) → 01(4(x))
11(1(4(5(0(x))))) → 41(x)
11(3(4(2(1(x))))) → 41(1(3(3(2(1(x))))))
11(3(4(2(1(x))))) → 11(3(3(2(1(x)))))
11(3(0(0(4(x))))) → 01(1(2(3(0(4(x))))))
11(3(0(0(4(x))))) → 11(2(3(0(4(x)))))
11(3(0(0(4(x))))) → 11(0(0(2(3(4(x))))))
11(3(0(0(4(x))))) → 01(0(2(3(4(x)))))
11(3(0(0(4(x))))) → 01(2(3(4(x))))
51(0(0(1(5(x))))) → 51(1(2(0(0(5(x))))))
51(0(0(1(5(x))))) → 11(2(0(0(5(x)))))
51(0(0(1(5(x))))) → 01(0(5(x)))
51(0(0(1(5(x))))) → 01(5(x))
11(0(2(4(5(x))))) → 11(2(0(4(5(2(x))))))
11(0(2(4(5(x))))) → 01(4(5(2(x))))
11(0(2(4(5(x))))) → 41(5(2(x)))
11(0(2(4(5(x))))) → 51(2(x))

The TRS R consists of the following rules:

1(0(0(x))) → 3(1(2(0(0(x)))))
1(0(0(x))) → 1(3(2(2(0(0(x))))))
1(4(0(x))) → 1(2(2(4(0(x)))))
1(4(0(x))) → 2(4(1(3(2(0(x))))))
1(0(1(x))) → 1(1(2(0(x))))
1(0(1(x))) → 2(1(1(2(0(x)))))
1(0(1(x))) → 1(2(3(0(1(x)))))
0(4(1(x))) → 2(4(2(3(0(1(x))))))
1(4(1(x))) → 1(1(2(4(x))))
1(4(1(x))) → 1(2(1(1(2(4(x))))))
1(0(4(x))) → 1(2(0(4(x))))
1(0(5(x))) → 1(2(0(5(x))))
1(0(5(x))) → 3(5(1(2(0(x)))))
1(0(5(x))) → 5(5(1(2(3(0(x))))))
1(0(5(x))) → 1(5(3(2(0(3(x))))))
5(0(5(x))) → 5(3(5(2(3(0(x))))))
1(4(5(x))) → 3(1(5(2(4(2(x))))))
1(4(5(x))) → 3(3(1(2(5(4(x))))))
5(4(0(0(x)))) → 2(4(2(0(5(0(x))))))
0(2(1(0(x)))) → 0(1(2(0(2(x)))))
1(0(2(0(x)))) → 3(1(2(0(0(x)))))
0(4(3(0(x)))) → 3(0(0(4(2(3(x))))))
4(0(4(0(x)))) → 4(4(2(3(0(0(x))))))
4(1(0(1(x)))) → 0(4(1(1(3(2(x))))))
1(3(0(1(x)))) → 0(1(1(3(2(4(x))))))
4(0(2(1(x)))) → 2(3(0(2(4(1(x))))))
4(0(3(1(x)))) → 3(1(2(3(0(4(x))))))
5(1(4(1(x)))) → 4(1(5(2(1(2(x))))))
1(5(0(4(x)))) → 5(1(2(4(2(0(x))))))
0(0(1(4(x)))) → 0(0(4(1(2(3(x))))))
4(0(1(4(x)))) → 1(2(0(4(4(x)))))
1(0(0(5(x)))) → 0(1(5(2(3(0(x))))))
1(3(0(5(x)))) → 1(2(3(5(0(x)))))
1(3(0(5(x)))) → 1(2(2(5(3(0(x))))))
1(5(0(5(x)))) → 0(1(3(5(2(5(x))))))
1(4(2(5(x)))) → 5(2(4(1(2(x)))))
5(1(4(5(x)))) → 1(2(5(5(4(x)))))
5(1(4(5(x)))) → 5(2(4(5(1(2(x))))))
1(3(4(5(x)))) → 3(1(2(4(5(x)))))
1(3(4(5(x)))) → 1(2(4(2(5(3(x))))))
1(3(4(5(x)))) → 1(2(3(4(2(5(x))))))
4(0(5(5(x)))) → 2(4(5(2(0(5(x))))))
1(3(3(0(0(x))))) → 3(2(1(0(3(0(x))))))
1(3(4(2(0(x))))) → 2(4(1(3(2(0(x))))))
1(1(4(5(0(x))))) → 1(1(5(2(0(4(x))))))
1(3(4(2(1(x))))) → 4(1(3(3(2(1(x))))))
1(3(0(0(4(x))))) → 0(1(2(3(0(4(x))))))
1(3(0(0(4(x))))) → 1(0(0(2(3(4(x))))))
5(0(0(1(5(x))))) → 5(1(2(0(0(5(x))))))
1(0(2(4(5(x))))) → 1(2(0(4(5(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 4 SCCs with 123 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

01(2(1(0(x)))) → 01(2(x))

The TRS R consists of the following rules:

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

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

(8) 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.

(9) Obligation:

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

01(2(1(0(x)))) → 01(2(x))

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

(10) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.


01(2(1(0(x)))) → 01(2(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0(x1)) = 1 + x1   
POL(01(x1)) = x1   
POL(1(x1)) = 1 + x1   
POL(2(x1)) = x1   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none

(11) Obligation:

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

(12) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(13) YES

(14) Obligation:

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

01(0(1(4(x)))) → 01(0(4(1(2(3(x))))))
01(4(1(x))) → 01(1(x))
01(0(1(4(x)))) → 01(4(1(2(3(x)))))

The TRS R consists of the following rules:

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

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

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.


01(4(1(x))) → 01(1(x))
01(0(1(4(x)))) → 01(4(1(2(3(x)))))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0(x1)) = 1 + x1   
POL(01(x1)) = x1   
POL(1(x1)) = x1   
POL(2(x1)) = 0   
POL(3(x1)) = x1   
POL(4(x1)) = 1 + x1   
POL(5(x1)) = x1   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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

(16) Obligation:

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

01(0(1(4(x)))) → 01(0(4(1(2(3(x))))))

The TRS R consists of the following rules:

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

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

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.


01(0(1(4(x)))) → 01(0(4(1(2(3(x))))))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0(x1)) = 1 + x1   
POL(01(x1)) = x1   
POL(1(x1)) = 1   
POL(2(x1)) = 1   
POL(3(x1)) = 0   
POL(4(x1)) = 0   
POL(5(x1)) = 0   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

0(4(1(x))) → 2(4(2(3(0(1(x))))))

(18) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

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

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

(19) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(20) YES

(21) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(22) 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:

  • 41(0(3(1(x)))) → 41(x)
    The graph contains the following edges 1 > 1

  • 41(0(2(1(x)))) → 41(1(x))
    The graph contains the following edges 1 > 1

  • 41(0(1(4(x)))) → 41(4(x))
    The graph contains the following edges 1 > 1

(23) YES

(24) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(25) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.


51(4(0(0(x)))) → 51(0(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0(x1)) = 1 + x1   
POL(1(x1)) = x1   
POL(2(x1)) = 0   
POL(3(x1)) = 0   
POL(4(x1)) = x1   
POL(5(x1)) = x1   
POL(51(x1)) = x1   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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

(26) Obligation:

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

51(1(4(5(x)))) → 51(5(4(x)))
51(1(4(5(x)))) → 51(4(x))

The TRS R consists of the following rules:

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

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

(27) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.


51(1(4(5(x)))) → 51(5(4(x)))
51(1(4(5(x)))) → 51(4(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0(x1)) = 1 + x1   
POL(1(x1)) = 1 + x1   
POL(2(x1)) = 0   
POL(3(x1)) = 0   
POL(4(x1)) = 1 + x1   
POL(5(x1)) = x1   
POL(51(x1)) = x1   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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

(28) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

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

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

(29) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(30) YES