/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) QTRS
(1) DependencyPairsProof [EQUIVALENT, 544 ms]
(2) QDP
(3) DependencyGraphProof [EQUIVALENT, 0 ms]
(4) QDP
(5) QDPOrderProof [EQUIVALENT, 556 ms]
(6) QDP
(7) DependencyGraphProof [EQUIVALENT, 2 ms]
(8) AND
    (9) QDP
        (10) QDPOrderProof [EQUIVALENT, 180 ms]
        (11) QDP
        (12) PisEmptyProof [EQUIVALENT, 0 ms]
        (13) YES
    (14) QDP
        (15) QDPOrderProof [EQUIVALENT, 132 ms]
        (16) QDP
        (17) PisEmptyProof [EQUIVALENT, 0 ms]
        (18) YES
    (19) QDP
        (20) QDPOrderProof [EQUIVALENT, 151 ms]
        (21) QDP
        (22) DependencyGraphProof [EQUIVALENT, 0 ms]
        (23) QDP
        (24) QDPOrderProof [EQUIVALENT, 161 ms]
        (25) QDP
        (26) TransformationProof [EQUIVALENT, 0 ms]
        (27) QDP
        (28) DependencyGraphProof [EQUIVALENT, 0 ms]
        (29) QDP
        (30) TransformationProof [EQUIVALENT, 0 ms]
        (31) QDP
        (32) DependencyGraphProof [EQUIVALENT, 0 ms]
        (33) QDP
        (34) QDPOrderProof [EQUIVALENT, 2700 ms]
        (35) QDP
        (36) SplitQDPProof [EQUIVALENT, 0 ms]
        (37) AND
            (38) QDP
                (39) SemLabProof [SOUND, 0 ms]
                (40) QDP
                (41) DependencyGraphProof [EQUIVALENT, 0 ms]
                (42) QDP
                (43) PisEmptyProof [SOUND, 0 ms]
                (44) TRUE
            (45) QDP
                (46) QDPOrderProof [EQUIVALENT, 93 ms]
                (47) QDP
                (48) QDPOrderProof [EQUIVALENT, 65 ms]
                (49) QDP
                (50) PisEmptyProof [EQUIVALENT, 0 ms]
                (51) YES


----------------------------------------

(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   0(0(1(2(x1)))) -> 0(3(1(0(2(x1)))))
   0(1(2(2(x1)))) -> 1(2(0(3(2(2(x1))))))
   0(1(2(4(x1)))) -> 0(3(2(3(1(4(x1))))))
   0(5(0(5(x1)))) -> 0(3(0(5(5(x1)))))
   0(5(1(2(x1)))) -> 1(0(1(5(2(x1)))))
   0(5(1(2(x1)))) -> 0(1(0(1(5(2(x1))))))
   0(5(1(2(x1)))) -> 0(3(2(3(1(5(x1))))))
   0(5(4(2(x1)))) -> 0(4(5(3(2(x1)))))
   0(5(5(2(x1)))) -> 5(0(1(5(2(x1)))))
   1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1)))))))
   1(0(1(2(x1)))) -> 1(1(3(0(2(x1)))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1))))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1))))))
   1(0(5(4(x1)))) -> 0(1(1(5(4(x1)))))
   1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1))))))
   1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1))))))
   1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1)))))))
   1(5(1(2(x1)))) -> 0(1(1(5(2(x1)))))
   1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1))))))
   5(0(0(2(x1)))) -> 5(0(3(0(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1))))))
   0(0(0(1(2(x1))))) -> 0(2(0(1(0(3(3(4(x1))))))))
   0(0(2(5(2(x1))))) -> 0(3(2(0(5(2(x1))))))
   0(1(2(5(0(x1))))) -> 3(3(2(2(0(0(1(5(x1))))))))
   0(1(2(5(2(x1))))) -> 0(3(2(1(5(3(2(x1)))))))
   0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1))))))
   0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1)))))))
   0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1)))))))
   0(5(0(2(2(x1))))) -> 0(2(5(0(3(2(x1))))))
   0(5(0(5(1(x1))))) -> 0(1(0(3(5(5(x1))))))
   0(5(1(3(0(x1))))) -> 0(0(1(1(5(3(x1))))))
   0(5(2(2(4(x1))))) -> 0(5(3(2(2(4(x1))))))
   0(5(2(3(1(x1))))) -> 0(1(5(3(2(2(2(x1)))))))
   0(5(2(4(1(x1))))) -> 0(4(3(2(5(1(x1))))))
   0(5(3(5(2(x1))))) -> 0(0(3(5(5(2(x1))))))
   0(5(5(3(1(x1))))) -> 5(0(1(5(3(3(2(x1)))))))
   1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1))))))
   1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1))))))
   1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1))))))
   1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1))))))
   1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1)))))))
   1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1))))))))
   5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1)))))))
   5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1))))))
   5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1))))))
   0(0(0(5(1(2(x1)))))) -> 0(0(1(5(0(2(4(x1)))))))
   0(0(1(2(4(1(x1)))))) -> 1(3(0(2(3(0(4(1(x1))))))))
   0(0(2(1(2(0(x1)))))) -> 0(1(0(3(2(2(2(0(x1))))))))
   0(0(2(3(0(5(x1)))))) -> 0(0(3(0(3(2(5(x1)))))))
   0(0(5(2(3(4(x1)))))) -> 0(4(0(3(1(2(5(x1)))))))
   0(0(5(5(3(4(x1)))))) -> 1(4(1(0(0(3(5(5(x1))))))))
   0(1(2(0(1(2(x1)))))) -> 1(0(3(2(2(1(1(0(x1))))))))
   0(1(2(2(0(5(x1)))))) -> 0(4(1(5(0(3(2(2(x1))))))))
   0(1(2(5(5(5(x1)))))) -> 0(2(5(1(5(3(5(x1)))))))
   0(1(3(1(5(2(x1)))))) -> 1(0(1(5(3(3(2(x1)))))))
   0(1(4(4(0(5(x1)))))) -> 4(3(0(0(1(5(4(x1)))))))
   0(2(5(3(5(1(x1)))))) -> 0(3(3(2(2(1(5(5(x1))))))))
   0(5(0(0(5(4(x1)))))) -> 0(1(0(1(0(4(5(5(x1))))))))
   0(5(1(2(1(4(x1)))))) -> 1(1(5(3(2(2(0(4(x1))))))))
   0(5(5(1(2(5(x1)))))) -> 0(2(1(5(5(4(5(x1)))))))
   1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1)))))))
   1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1))))))))
   1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1)))))))
   1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1))))))))
   1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1)))))))
   1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1)))))))
   1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1)))))))
   0(0(1(2(3(5(5(x1))))))) -> 5(1(5(0(3(0(2(1(x1))))))))
   0(0(2(3(4(2(1(x1))))))) -> 0(0(4(1(3(2(3(2(x1))))))))
   0(1(0(0(5(3(4(x1))))))) -> 0(3(0(5(0(4(3(1(x1))))))))
   0(1(2(4(4(0(5(x1))))))) -> 5(4(0(1(0(3(2(4(x1))))))))
   0(5(2(5(1(3(4(x1))))))) -> 1(4(5(2(0(3(1(5(x1))))))))
   0(5(5(0(2(5(1(x1))))))) -> 5(3(0(0(1(5(2(5(x1))))))))
   0(5(5(2(5(3(4(x1))))))) -> 0(3(2(1(4(5(5(5(x1))))))))
   1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1))))))))
   1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1))))))))
   1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1))))))))
   1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(1(5(4(x1))))))))

Q is empty.

----------------------------------------

(1) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
----------------------------------------

(2)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   0^1(0(1(2(x1)))) -> 0^1(3(1(0(2(x1)))))
   0^1(0(1(2(x1)))) -> 1^1(0(2(x1)))
   0^1(0(1(2(x1)))) -> 0^1(2(x1))
   0^1(1(2(2(x1)))) -> 1^1(2(0(3(2(2(x1))))))
   0^1(1(2(2(x1)))) -> 0^1(3(2(2(x1))))
   0^1(1(2(4(x1)))) -> 0^1(3(2(3(1(4(x1))))))
   0^1(1(2(4(x1)))) -> 1^1(4(x1))
   0^1(5(0(5(x1)))) -> 0^1(3(0(5(5(x1)))))
   0^1(5(0(5(x1)))) -> 0^1(5(5(x1)))
   0^1(5(0(5(x1)))) -> 5^1(5(x1))
   0^1(5(1(2(x1)))) -> 1^1(0(1(5(2(x1)))))
   0^1(5(1(2(x1)))) -> 0^1(1(5(2(x1))))
   0^1(5(1(2(x1)))) -> 1^1(5(2(x1)))
   0^1(5(1(2(x1)))) -> 5^1(2(x1))
   0^1(5(1(2(x1)))) -> 0^1(1(0(1(5(2(x1))))))
   0^1(5(1(2(x1)))) -> 0^1(3(2(3(1(5(x1))))))
   0^1(5(1(2(x1)))) -> 1^1(5(x1))
   0^1(5(1(2(x1)))) -> 5^1(x1)
   0^1(5(4(2(x1)))) -> 0^1(4(5(3(2(x1)))))
   0^1(5(4(2(x1)))) -> 5^1(3(2(x1)))
   0^1(5(5(2(x1)))) -> 5^1(0(1(5(2(x1)))))
   0^1(5(5(2(x1)))) -> 0^1(1(5(2(x1))))
   0^1(5(5(2(x1)))) -> 1^1(5(2(x1)))
   1^1(0(0(5(x1)))) -> 1^1(1(0(0(1(5(4(x1)))))))
   1^1(0(0(5(x1)))) -> 1^1(0(0(1(5(4(x1))))))
   1^1(0(0(5(x1)))) -> 0^1(0(1(5(4(x1)))))
   1^1(0(0(5(x1)))) -> 0^1(1(5(4(x1))))
   1^1(0(0(5(x1)))) -> 1^1(5(4(x1)))
   1^1(0(0(5(x1)))) -> 5^1(4(x1))
   1^1(0(1(2(x1)))) -> 1^1(1(3(0(2(x1)))))
   1^1(0(1(2(x1)))) -> 1^1(3(0(2(x1))))
   1^1(0(1(2(x1)))) -> 0^1(2(x1))
   1^1(0(1(2(x1)))) -> 1^1(1(0(3(2(2(x1))))))
   1^1(0(1(2(x1)))) -> 1^1(0(3(2(2(x1)))))
   1^1(0(1(2(x1)))) -> 0^1(3(2(2(x1))))
   1^1(0(1(2(x1)))) -> 1^1(1(0(3(2(3(x1))))))
   1^1(0(1(2(x1)))) -> 1^1(0(3(2(3(x1)))))
   1^1(0(1(2(x1)))) -> 0^1(3(2(3(x1))))
   1^1(0(5(4(x1)))) -> 0^1(1(1(5(4(x1)))))
   1^1(0(5(4(x1)))) -> 1^1(1(5(4(x1))))
   1^1(0(5(4(x1)))) -> 1^1(5(4(x1)))
   1^1(2(0(5(x1)))) -> 0^1(3(2(3(1(5(x1))))))
   1^1(2(0(5(x1)))) -> 1^1(5(x1))
   1^1(2(0(5(x1)))) -> 5^1(0(3(3(2(1(x1))))))
   1^1(2(0(5(x1)))) -> 0^1(3(3(2(1(x1)))))
   1^1(2(0(5(x1)))) -> 1^1(x1)
   1^1(5(0(2(x1)))) -> 1^1(1(0(1(1(5(2(x1)))))))
   1^1(5(0(2(x1)))) -> 1^1(0(1(1(5(2(x1))))))
   1^1(5(0(2(x1)))) -> 0^1(1(1(5(2(x1)))))
   1^1(5(0(2(x1)))) -> 1^1(1(5(2(x1))))
   1^1(5(0(2(x1)))) -> 1^1(5(2(x1)))
   1^1(5(0(2(x1)))) -> 5^1(2(x1))
   1^1(5(1(2(x1)))) -> 0^1(1(1(5(2(x1)))))
   1^1(5(1(2(x1)))) -> 1^1(1(5(2(x1))))
   1^1(5(1(2(x1)))) -> 1^1(5(2(x1)))
   1^1(5(1(2(x1)))) -> 5^1(2(x1))
   1^1(5(1(2(x1)))) -> 1^1(0(1(5(3(2(x1))))))
   1^1(5(1(2(x1)))) -> 0^1(1(5(3(2(x1)))))
   1^1(5(1(2(x1)))) -> 1^1(5(3(2(x1))))
   1^1(5(1(2(x1)))) -> 5^1(3(2(x1)))
   5^1(0(0(2(x1)))) -> 5^1(0(3(0(2(x1)))))
   5^1(0(0(2(x1)))) -> 0^1(3(0(2(x1))))
   5^1(0(1(2(x1)))) -> 5^1(1(0(3(2(x1)))))
   5^1(0(1(2(x1)))) -> 1^1(0(3(2(x1))))
   5^1(0(1(2(x1)))) -> 0^1(3(2(x1)))
   5^1(0(1(2(x1)))) -> 5^1(1(0(3(2(3(x1))))))
   5^1(0(1(2(x1)))) -> 1^1(0(3(2(3(x1)))))
   5^1(0(1(2(x1)))) -> 0^1(3(2(3(x1))))
   0^1(0(0(1(2(x1))))) -> 0^1(2(0(1(0(3(3(4(x1))))))))
   0^1(0(0(1(2(x1))))) -> 0^1(1(0(3(3(4(x1))))))
   0^1(0(0(1(2(x1))))) -> 1^1(0(3(3(4(x1)))))
   0^1(0(0(1(2(x1))))) -> 0^1(3(3(4(x1))))
   0^1(0(2(5(2(x1))))) -> 0^1(3(2(0(5(2(x1))))))
   0^1(0(2(5(2(x1))))) -> 0^1(5(2(x1)))
   0^1(1(2(5(0(x1))))) -> 0^1(0(1(5(x1))))
   0^1(1(2(5(0(x1))))) -> 0^1(1(5(x1)))
   0^1(1(2(5(0(x1))))) -> 1^1(5(x1))
   0^1(1(2(5(0(x1))))) -> 5^1(x1)
   0^1(1(2(5(2(x1))))) -> 0^1(3(2(1(5(3(2(x1)))))))
   0^1(1(2(5(2(x1))))) -> 1^1(5(3(2(x1))))
   0^1(1(2(5(2(x1))))) -> 5^1(3(2(x1)))
   0^1(3(5(2(2(x1))))) -> 0^1(4(5(3(2(2(x1))))))
   0^1(3(5(2(2(x1))))) -> 5^1(3(2(2(x1))))
   0^1(4(2(0(5(x1))))) -> 0^1(4(0(3(2(1(5(x1)))))))
   0^1(4(2(0(5(x1))))) -> 0^1(3(2(1(5(x1)))))
   0^1(4(2(0(5(x1))))) -> 1^1(5(x1))
   0^1(4(2(5(2(x1))))) -> 0^1(5(4(3(3(2(2(x1)))))))
   0^1(4(2(5(2(x1))))) -> 5^1(4(3(3(2(2(x1))))))
   0^1(5(0(2(2(x1))))) -> 0^1(2(5(0(3(2(x1))))))
   0^1(5(0(2(2(x1))))) -> 5^1(0(3(2(x1))))
   0^1(5(0(2(2(x1))))) -> 0^1(3(2(x1)))
   0^1(5(0(5(1(x1))))) -> 0^1(1(0(3(5(5(x1))))))
   0^1(5(0(5(1(x1))))) -> 1^1(0(3(5(5(x1)))))
   0^1(5(0(5(1(x1))))) -> 0^1(3(5(5(x1))))
   0^1(5(0(5(1(x1))))) -> 5^1(5(x1))
   0^1(5(0(5(1(x1))))) -> 5^1(x1)
   0^1(5(1(3(0(x1))))) -> 0^1(0(1(1(5(3(x1))))))
   0^1(5(1(3(0(x1))))) -> 0^1(1(1(5(3(x1)))))
   0^1(5(1(3(0(x1))))) -> 1^1(1(5(3(x1))))
   0^1(5(1(3(0(x1))))) -> 1^1(5(3(x1)))
   0^1(5(1(3(0(x1))))) -> 5^1(3(x1))
   0^1(5(2(2(4(x1))))) -> 0^1(5(3(2(2(4(x1))))))
   0^1(5(2(2(4(x1))))) -> 5^1(3(2(2(4(x1)))))
   0^1(5(2(3(1(x1))))) -> 0^1(1(5(3(2(2(2(x1)))))))
   0^1(5(2(3(1(x1))))) -> 1^1(5(3(2(2(2(x1))))))
   0^1(5(2(3(1(x1))))) -> 5^1(3(2(2(2(x1)))))
   0^1(5(2(4(1(x1))))) -> 0^1(4(3(2(5(1(x1))))))
   0^1(5(2(4(1(x1))))) -> 5^1(1(x1))
   0^1(5(3(5(2(x1))))) -> 0^1(0(3(5(5(2(x1))))))
   0^1(5(3(5(2(x1))))) -> 0^1(3(5(5(2(x1)))))
   0^1(5(3(5(2(x1))))) -> 5^1(5(2(x1)))
   0^1(5(5(3(1(x1))))) -> 5^1(0(1(5(3(3(2(x1)))))))
   0^1(5(5(3(1(x1))))) -> 0^1(1(5(3(3(2(x1))))))
   0^1(5(5(3(1(x1))))) -> 1^1(5(3(3(2(x1)))))
   0^1(5(5(3(1(x1))))) -> 5^1(3(3(2(x1))))
   1^1(0(5(5(1(x1))))) -> 0^1(4(5(1(5(1(x1))))))
   1^1(0(5(5(1(x1))))) -> 5^1(1(5(1(x1))))
   1^1(0(5(5(1(x1))))) -> 1^1(5(1(x1)))
   1^1(1(2(2(0(x1))))) -> 1^1(1(3(2(2(0(x1))))))
   1^1(1(2(2(0(x1))))) -> 1^1(3(2(2(0(x1)))))
   1^1(1(2(3(4(x1))))) -> 1^1(1(3(2(2(4(x1))))))
   1^1(1(2(3(4(x1))))) -> 1^1(3(2(2(4(x1)))))
   1^1(1(3(5(2(x1))))) -> 1^1(1(5(3(3(2(x1))))))
   1^1(1(3(5(2(x1))))) -> 1^1(5(3(3(2(x1)))))
   1^1(1(3(5(2(x1))))) -> 5^1(3(3(2(x1))))
   1^1(5(0(5(0(x1))))) -> 0^1(1(5(3(5(1(0(x1)))))))
   1^1(5(0(5(0(x1))))) -> 1^1(5(3(5(1(0(x1))))))
   1^1(5(0(5(0(x1))))) -> 5^1(3(5(1(0(x1)))))
   1^1(5(0(5(0(x1))))) -> 5^1(1(0(x1)))
   1^1(5(0(5(0(x1))))) -> 1^1(0(x1))
   1^1(5(5(1(2(x1))))) -> 1^1(5(1(1(5(3(2(2(x1))))))))
   1^1(5(5(1(2(x1))))) -> 5^1(1(1(5(3(2(2(x1)))))))
   1^1(5(5(1(2(x1))))) -> 1^1(1(5(3(2(2(x1))))))
   1^1(5(5(1(2(x1))))) -> 1^1(5(3(2(2(x1)))))
   1^1(5(5(1(2(x1))))) -> 5^1(3(2(2(x1))))
   5^1(0(2(0(5(x1))))) -> 5^1(0(3(3(2(0(5(x1)))))))
   5^1(0(2(0(5(x1))))) -> 0^1(3(3(2(0(5(x1))))))
   5^1(0(2(3(4(x1))))) -> 5^1(0(3(2(3(4(x1))))))
   5^1(0(2(3(4(x1))))) -> 0^1(3(2(3(4(x1)))))
   5^1(5(0(1(2(x1))))) -> 5^1(5(3(0(2(1(x1))))))
   5^1(5(0(1(2(x1))))) -> 5^1(3(0(2(1(x1)))))
   5^1(5(0(1(2(x1))))) -> 0^1(2(1(x1)))
   5^1(5(0(1(2(x1))))) -> 1^1(x1)
   0^1(0(0(5(1(2(x1)))))) -> 0^1(0(1(5(0(2(4(x1)))))))
   0^1(0(0(5(1(2(x1)))))) -> 0^1(1(5(0(2(4(x1))))))
   0^1(0(0(5(1(2(x1)))))) -> 1^1(5(0(2(4(x1)))))
   0^1(0(0(5(1(2(x1)))))) -> 5^1(0(2(4(x1))))
   0^1(0(0(5(1(2(x1)))))) -> 0^1(2(4(x1)))
   0^1(0(1(2(4(1(x1)))))) -> 1^1(3(0(2(3(0(4(1(x1))))))))
   0^1(0(1(2(4(1(x1)))))) -> 0^1(2(3(0(4(1(x1))))))
   0^1(0(1(2(4(1(x1)))))) -> 0^1(4(1(x1)))
   0^1(0(2(1(2(0(x1)))))) -> 0^1(1(0(3(2(2(2(0(x1))))))))
   0^1(0(2(1(2(0(x1)))))) -> 1^1(0(3(2(2(2(0(x1)))))))
   0^1(0(2(1(2(0(x1)))))) -> 0^1(3(2(2(2(0(x1))))))
   0^1(0(2(3(0(5(x1)))))) -> 0^1(0(3(0(3(2(5(x1)))))))
   0^1(0(2(3(0(5(x1)))))) -> 0^1(3(0(3(2(5(x1))))))
   0^1(0(2(3(0(5(x1)))))) -> 0^1(3(2(5(x1))))
   0^1(0(5(2(3(4(x1)))))) -> 0^1(4(0(3(1(2(5(x1)))))))
   0^1(0(5(2(3(4(x1)))))) -> 0^1(3(1(2(5(x1)))))
   0^1(0(5(2(3(4(x1)))))) -> 1^1(2(5(x1)))
   0^1(0(5(2(3(4(x1)))))) -> 5^1(x1)
   0^1(0(5(5(3(4(x1)))))) -> 1^1(4(1(0(0(3(5(5(x1))))))))
   0^1(0(5(5(3(4(x1)))))) -> 1^1(0(0(3(5(5(x1))))))
   0^1(0(5(5(3(4(x1)))))) -> 0^1(0(3(5(5(x1)))))
   0^1(0(5(5(3(4(x1)))))) -> 0^1(3(5(5(x1))))
   0^1(0(5(5(3(4(x1)))))) -> 5^1(5(x1))
   0^1(0(5(5(3(4(x1)))))) -> 5^1(x1)
   0^1(1(2(0(1(2(x1)))))) -> 1^1(0(3(2(2(1(1(0(x1))))))))
   0^1(1(2(0(1(2(x1)))))) -> 0^1(3(2(2(1(1(0(x1)))))))
   0^1(1(2(0(1(2(x1)))))) -> 1^1(1(0(x1)))
   0^1(1(2(0(1(2(x1)))))) -> 1^1(0(x1))
   0^1(1(2(0(1(2(x1)))))) -> 0^1(x1)
   0^1(1(2(2(0(5(x1)))))) -> 0^1(4(1(5(0(3(2(2(x1))))))))
   0^1(1(2(2(0(5(x1)))))) -> 1^1(5(0(3(2(2(x1))))))
   0^1(1(2(2(0(5(x1)))))) -> 5^1(0(3(2(2(x1)))))
   0^1(1(2(2(0(5(x1)))))) -> 0^1(3(2(2(x1))))
   0^1(1(2(5(5(5(x1)))))) -> 0^1(2(5(1(5(3(5(x1)))))))
   0^1(1(2(5(5(5(x1)))))) -> 5^1(1(5(3(5(x1)))))
   0^1(1(2(5(5(5(x1)))))) -> 1^1(5(3(5(x1))))
   0^1(1(2(5(5(5(x1)))))) -> 5^1(3(5(x1)))
   0^1(1(3(1(5(2(x1)))))) -> 1^1(0(1(5(3(3(2(x1)))))))
   0^1(1(3(1(5(2(x1)))))) -> 0^1(1(5(3(3(2(x1))))))
   0^1(1(3(1(5(2(x1)))))) -> 1^1(5(3(3(2(x1)))))
   0^1(1(3(1(5(2(x1)))))) -> 5^1(3(3(2(x1))))
   0^1(1(4(4(0(5(x1)))))) -> 0^1(0(1(5(4(x1)))))
   0^1(1(4(4(0(5(x1)))))) -> 0^1(1(5(4(x1))))
   0^1(1(4(4(0(5(x1)))))) -> 1^1(5(4(x1)))
   0^1(1(4(4(0(5(x1)))))) -> 5^1(4(x1))
   0^1(2(5(3(5(1(x1)))))) -> 0^1(3(3(2(2(1(5(5(x1))))))))
   0^1(2(5(3(5(1(x1)))))) -> 1^1(5(5(x1)))
   0^1(2(5(3(5(1(x1)))))) -> 5^1(5(x1))
   0^1(2(5(3(5(1(x1)))))) -> 5^1(x1)
   0^1(5(0(0(5(4(x1)))))) -> 0^1(1(0(1(0(4(5(5(x1))))))))
   0^1(5(0(0(5(4(x1)))))) -> 1^1(0(1(0(4(5(5(x1)))))))
   0^1(5(0(0(5(4(x1)))))) -> 0^1(1(0(4(5(5(x1))))))
   0^1(5(0(0(5(4(x1)))))) -> 1^1(0(4(5(5(x1)))))
   0^1(5(0(0(5(4(x1)))))) -> 0^1(4(5(5(x1))))
   0^1(5(0(0(5(4(x1)))))) -> 5^1(5(x1))
   0^1(5(0(0(5(4(x1)))))) -> 5^1(x1)
   0^1(5(1(2(1(4(x1)))))) -> 1^1(1(5(3(2(2(0(4(x1))))))))
   0^1(5(1(2(1(4(x1)))))) -> 1^1(5(3(2(2(0(4(x1)))))))
   0^1(5(1(2(1(4(x1)))))) -> 5^1(3(2(2(0(4(x1))))))
   0^1(5(1(2(1(4(x1)))))) -> 0^1(4(x1))
   0^1(5(5(1(2(5(x1)))))) -> 0^1(2(1(5(5(4(5(x1)))))))
   0^1(5(5(1(2(5(x1)))))) -> 1^1(5(5(4(5(x1)))))
   0^1(5(5(1(2(5(x1)))))) -> 5^1(5(4(5(x1))))
   0^1(5(5(1(2(5(x1)))))) -> 5^1(4(5(x1)))
   1^1(0(0(2(3(4(x1)))))) -> 1^1(4(0(0(3(3(2(x1)))))))
   1^1(0(0(2(3(4(x1)))))) -> 0^1(0(3(3(2(x1)))))
   1^1(0(0(2(3(4(x1)))))) -> 0^1(3(3(2(x1))))
   1^1(0(1(3(5(1(x1)))))) -> 0^1(1(1(1(5(3(2(2(x1))))))))
   1^1(0(1(3(5(1(x1)))))) -> 1^1(1(1(5(3(2(2(x1)))))))
   1^1(0(1(3(5(1(x1)))))) -> 1^1(1(5(3(2(2(x1))))))
   1^1(0(1(3(5(1(x1)))))) -> 1^1(5(3(2(2(x1)))))
   1^1(0(1(3(5(1(x1)))))) -> 5^1(3(2(2(x1))))
   1^1(0(5(4(2(1(x1)))))) -> 0^1(1(1(5(3(4(2(x1)))))))
   1^1(0(5(4(2(1(x1)))))) -> 1^1(1(5(3(4(2(x1))))))
   1^1(0(5(4(2(1(x1)))))) -> 1^1(5(3(4(2(x1)))))
   1^1(0(5(4(2(1(x1)))))) -> 5^1(3(4(2(x1))))
   1^1(1(0(1(2(2(x1)))))) -> 1^1(0(1(1(3(1(2(2(x1))))))))
   1^1(1(0(1(2(2(x1)))))) -> 0^1(1(1(3(1(2(2(x1)))))))
   1^1(1(0(1(2(2(x1)))))) -> 1^1(1(3(1(2(2(x1))))))
   1^1(1(0(1(2(2(x1)))))) -> 1^1(3(1(2(2(x1)))))
   1^1(2(1(2(0(0(x1)))))) -> 0^1(3(2(2(1(0(1(x1)))))))
   1^1(2(1(2(0(0(x1)))))) -> 1^1(0(1(x1)))
   1^1(2(1(2(0(0(x1)))))) -> 0^1(1(x1))
   1^1(2(1(2(0(0(x1)))))) -> 1^1(x1)
   1^1(4(1(0(0(5(x1)))))) -> 0^1(0(1(5(4(2(1(x1)))))))
   1^1(4(1(0(0(5(x1)))))) -> 0^1(1(5(4(2(1(x1))))))
   1^1(4(1(0(0(5(x1)))))) -> 1^1(5(4(2(1(x1)))))
   1^1(4(1(0(0(5(x1)))))) -> 5^1(4(2(1(x1))))
   1^1(4(1(0(0(5(x1)))))) -> 1^1(x1)
   1^1(4(3(5(0(2(x1)))))) -> 0^1(3(3(2(1(5(4(x1)))))))
   1^1(4(3(5(0(2(x1)))))) -> 1^1(5(4(x1)))
   1^1(4(3(5(0(2(x1)))))) -> 5^1(4(x1))
   0^1(0(1(2(3(5(5(x1))))))) -> 5^1(1(5(0(3(0(2(1(x1))))))))
   0^1(0(1(2(3(5(5(x1))))))) -> 1^1(5(0(3(0(2(1(x1)))))))
   0^1(0(1(2(3(5(5(x1))))))) -> 5^1(0(3(0(2(1(x1))))))
   0^1(0(1(2(3(5(5(x1))))))) -> 0^1(3(0(2(1(x1)))))
   0^1(0(1(2(3(5(5(x1))))))) -> 0^1(2(1(x1)))
   0^1(0(1(2(3(5(5(x1))))))) -> 1^1(x1)
   0^1(0(2(3(4(2(1(x1))))))) -> 0^1(0(4(1(3(2(3(2(x1))))))))
   0^1(0(2(3(4(2(1(x1))))))) -> 0^1(4(1(3(2(3(2(x1)))))))
   0^1(0(2(3(4(2(1(x1))))))) -> 1^1(3(2(3(2(x1)))))
   0^1(1(0(0(5(3(4(x1))))))) -> 0^1(3(0(5(0(4(3(1(x1))))))))
   0^1(1(0(0(5(3(4(x1))))))) -> 0^1(5(0(4(3(1(x1))))))
   0^1(1(0(0(5(3(4(x1))))))) -> 5^1(0(4(3(1(x1)))))
   0^1(1(0(0(5(3(4(x1))))))) -> 0^1(4(3(1(x1))))
   0^1(1(0(0(5(3(4(x1))))))) -> 1^1(x1)
   0^1(1(2(4(4(0(5(x1))))))) -> 5^1(4(0(1(0(3(2(4(x1))))))))
   0^1(1(2(4(4(0(5(x1))))))) -> 0^1(1(0(3(2(4(x1))))))
   0^1(1(2(4(4(0(5(x1))))))) -> 1^1(0(3(2(4(x1)))))
   0^1(1(2(4(4(0(5(x1))))))) -> 0^1(3(2(4(x1))))
   0^1(5(2(5(1(3(4(x1))))))) -> 1^1(4(5(2(0(3(1(5(x1))))))))
   0^1(5(2(5(1(3(4(x1))))))) -> 5^1(2(0(3(1(5(x1))))))
   0^1(5(2(5(1(3(4(x1))))))) -> 0^1(3(1(5(x1))))
   0^1(5(2(5(1(3(4(x1))))))) -> 1^1(5(x1))
   0^1(5(2(5(1(3(4(x1))))))) -> 5^1(x1)
   0^1(5(5(0(2(5(1(x1))))))) -> 5^1(3(0(0(1(5(2(5(x1))))))))
   0^1(5(5(0(2(5(1(x1))))))) -> 0^1(0(1(5(2(5(x1))))))
   0^1(5(5(0(2(5(1(x1))))))) -> 0^1(1(5(2(5(x1)))))
   0^1(5(5(0(2(5(1(x1))))))) -> 1^1(5(2(5(x1))))
   0^1(5(5(0(2(5(1(x1))))))) -> 5^1(2(5(x1)))
   0^1(5(5(0(2(5(1(x1))))))) -> 5^1(x1)
   0^1(5(5(2(5(3(4(x1))))))) -> 0^1(3(2(1(4(5(5(5(x1))))))))
   0^1(5(5(2(5(3(4(x1))))))) -> 1^1(4(5(5(5(x1)))))
   0^1(5(5(2(5(3(4(x1))))))) -> 5^1(5(5(x1)))
   0^1(5(5(2(5(3(4(x1))))))) -> 5^1(5(x1))
   0^1(5(5(2(5(3(4(x1))))))) -> 5^1(x1)
   1^1(0(1(2(3(4(5(x1))))))) -> 0^1(2(1(5(1(3(4(x1)))))))
   1^1(0(1(2(3(4(5(x1))))))) -> 1^1(5(1(3(4(x1)))))
   1^1(0(1(2(3(4(5(x1))))))) -> 5^1(1(3(4(x1))))
   1^1(0(1(2(3(4(5(x1))))))) -> 1^1(3(4(x1)))
   1^1(1(0(2(0(2(2(x1))))))) -> 1^1(1(0(2(0(3(2(2(x1))))))))
   1^1(1(0(2(0(2(2(x1))))))) -> 1^1(0(2(0(3(2(2(x1)))))))
   1^1(1(0(2(0(2(2(x1))))))) -> 0^1(2(0(3(2(2(x1))))))
   1^1(1(0(2(0(2(2(x1))))))) -> 0^1(3(2(2(x1))))
   1^1(2(4(3(5(3(5(x1))))))) -> 5^1(1(3(2(2(4(3(5(x1))))))))
   1^1(2(4(3(5(3(5(x1))))))) -> 1^1(3(2(2(4(3(5(x1)))))))
   1^1(4(3(5(2(5(2(x1))))))) -> 5^1(1(3(2(2(1(5(4(x1))))))))
   1^1(4(3(5(2(5(2(x1))))))) -> 1^1(3(2(2(1(5(4(x1)))))))
   1^1(4(3(5(2(5(2(x1))))))) -> 1^1(5(4(x1)))
   1^1(4(3(5(2(5(2(x1))))))) -> 5^1(4(x1))

The TRS R consists of the following rules:

   0(0(1(2(x1)))) -> 0(3(1(0(2(x1)))))
   0(1(2(2(x1)))) -> 1(2(0(3(2(2(x1))))))
   0(1(2(4(x1)))) -> 0(3(2(3(1(4(x1))))))
   0(5(0(5(x1)))) -> 0(3(0(5(5(x1)))))
   0(5(1(2(x1)))) -> 1(0(1(5(2(x1)))))
   0(5(1(2(x1)))) -> 0(1(0(1(5(2(x1))))))
   0(5(1(2(x1)))) -> 0(3(2(3(1(5(x1))))))
   0(5(4(2(x1)))) -> 0(4(5(3(2(x1)))))
   0(5(5(2(x1)))) -> 5(0(1(5(2(x1)))))
   1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1)))))))
   1(0(1(2(x1)))) -> 1(1(3(0(2(x1)))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1))))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1))))))
   1(0(5(4(x1)))) -> 0(1(1(5(4(x1)))))
   1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1))))))
   1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1))))))
   1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1)))))))
   1(5(1(2(x1)))) -> 0(1(1(5(2(x1)))))
   1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1))))))
   5(0(0(2(x1)))) -> 5(0(3(0(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1))))))
   0(0(0(1(2(x1))))) -> 0(2(0(1(0(3(3(4(x1))))))))
   0(0(2(5(2(x1))))) -> 0(3(2(0(5(2(x1))))))
   0(1(2(5(0(x1))))) -> 3(3(2(2(0(0(1(5(x1))))))))
   0(1(2(5(2(x1))))) -> 0(3(2(1(5(3(2(x1)))))))
   0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1))))))
   0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1)))))))
   0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1)))))))
   0(5(0(2(2(x1))))) -> 0(2(5(0(3(2(x1))))))
   0(5(0(5(1(x1))))) -> 0(1(0(3(5(5(x1))))))
   0(5(1(3(0(x1))))) -> 0(0(1(1(5(3(x1))))))
   0(5(2(2(4(x1))))) -> 0(5(3(2(2(4(x1))))))
   0(5(2(3(1(x1))))) -> 0(1(5(3(2(2(2(x1)))))))
   0(5(2(4(1(x1))))) -> 0(4(3(2(5(1(x1))))))
   0(5(3(5(2(x1))))) -> 0(0(3(5(5(2(x1))))))
   0(5(5(3(1(x1))))) -> 5(0(1(5(3(3(2(x1)))))))
   1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1))))))
   1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1))))))
   1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1))))))
   1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1))))))
   1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1)))))))
   1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1))))))))
   5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1)))))))
   5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1))))))
   5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1))))))
   0(0(0(5(1(2(x1)))))) -> 0(0(1(5(0(2(4(x1)))))))
   0(0(1(2(4(1(x1)))))) -> 1(3(0(2(3(0(4(1(x1))))))))
   0(0(2(1(2(0(x1)))))) -> 0(1(0(3(2(2(2(0(x1))))))))
   0(0(2(3(0(5(x1)))))) -> 0(0(3(0(3(2(5(x1)))))))
   0(0(5(2(3(4(x1)))))) -> 0(4(0(3(1(2(5(x1)))))))
   0(0(5(5(3(4(x1)))))) -> 1(4(1(0(0(3(5(5(x1))))))))
   0(1(2(0(1(2(x1)))))) -> 1(0(3(2(2(1(1(0(x1))))))))
   0(1(2(2(0(5(x1)))))) -> 0(4(1(5(0(3(2(2(x1))))))))
   0(1(2(5(5(5(x1)))))) -> 0(2(5(1(5(3(5(x1)))))))
   0(1(3(1(5(2(x1)))))) -> 1(0(1(5(3(3(2(x1)))))))
   0(1(4(4(0(5(x1)))))) -> 4(3(0(0(1(5(4(x1)))))))
   0(2(5(3(5(1(x1)))))) -> 0(3(3(2(2(1(5(5(x1))))))))
   0(5(0(0(5(4(x1)))))) -> 0(1(0(1(0(4(5(5(x1))))))))
   0(5(1(2(1(4(x1)))))) -> 1(1(5(3(2(2(0(4(x1))))))))
   0(5(5(1(2(5(x1)))))) -> 0(2(1(5(5(4(5(x1)))))))
   1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1)))))))
   1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1))))))))
   1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1)))))))
   1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1))))))))
   1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1)))))))
   1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1)))))))
   1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1)))))))
   0(0(1(2(3(5(5(x1))))))) -> 5(1(5(0(3(0(2(1(x1))))))))
   0(0(2(3(4(2(1(x1))))))) -> 0(0(4(1(3(2(3(2(x1))))))))
   0(1(0(0(5(3(4(x1))))))) -> 0(3(0(5(0(4(3(1(x1))))))))
   0(1(2(4(4(0(5(x1))))))) -> 5(4(0(1(0(3(2(4(x1))))))))
   0(5(2(5(1(3(4(x1))))))) -> 1(4(5(2(0(3(1(5(x1))))))))
   0(5(5(0(2(5(1(x1))))))) -> 5(3(0(0(1(5(2(5(x1))))))))
   0(5(5(2(5(3(4(x1))))))) -> 0(3(2(1(4(5(5(5(x1))))))))
   1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1))))))))
   1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1))))))))
   1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1))))))))
   1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(1(5(4(x1))))))))

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

(3) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 216 less nodes.
----------------------------------------

(4)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   0^1(0(1(2(x1)))) -> 1^1(0(2(x1)))
   1^1(0(1(2(x1)))) -> 0^1(2(x1))
   0^1(2(5(3(5(1(x1)))))) -> 1^1(5(5(x1)))
   1^1(0(5(5(1(x1))))) -> 5^1(1(5(1(x1))))
   5^1(0(0(2(x1)))) -> 5^1(0(3(0(2(x1)))))
   5^1(5(0(1(2(x1))))) -> 0^1(2(1(x1)))
   0^1(2(5(3(5(1(x1)))))) -> 5^1(5(x1))
   5^1(5(0(1(2(x1))))) -> 1^1(x1)
   1^1(2(0(5(x1)))) -> 1^1(5(x1))
   1^1(0(5(5(1(x1))))) -> 1^1(5(1(x1)))
   1^1(5(0(5(0(x1))))) -> 5^1(1(0(x1)))
   1^1(5(0(5(0(x1))))) -> 1^1(0(x1))
   1^1(4(1(0(0(5(x1)))))) -> 1^1(x1)
   1^1(2(0(5(x1)))) -> 1^1(x1)
   1^1(2(1(2(0(0(x1)))))) -> 1^1(0(1(x1)))
   1^1(2(1(2(0(0(x1)))))) -> 0^1(1(x1))
   0^1(0(1(2(x1)))) -> 0^1(2(x1))
   0^1(2(5(3(5(1(x1)))))) -> 5^1(x1)
   0^1(1(2(4(x1)))) -> 1^1(4(x1))
   0^1(5(0(5(x1)))) -> 0^1(5(5(x1)))
   0^1(5(0(5(x1)))) -> 5^1(5(x1))
   0^1(5(1(2(x1)))) -> 1^1(5(x1))
   0^1(5(1(2(x1)))) -> 5^1(x1)
   0^1(1(2(5(0(x1))))) -> 0^1(0(1(5(x1))))
   0^1(1(2(5(0(x1))))) -> 0^1(1(5(x1)))
   0^1(1(2(5(0(x1))))) -> 1^1(5(x1))
   0^1(1(2(5(0(x1))))) -> 5^1(x1)
   0^1(5(0(5(1(x1))))) -> 0^1(1(0(3(5(5(x1))))))
   0^1(5(0(5(1(x1))))) -> 1^1(0(3(5(5(x1)))))
   0^1(5(0(5(1(x1))))) -> 5^1(5(x1))
   0^1(5(0(5(1(x1))))) -> 5^1(x1)
   0^1(0(0(5(1(2(x1)))))) -> 0^1(0(1(5(0(2(4(x1)))))))
   0^1(0(0(5(1(2(x1)))))) -> 0^1(1(5(0(2(4(x1))))))
   0^1(0(5(5(3(4(x1)))))) -> 1^1(4(1(0(0(3(5(5(x1))))))))
   0^1(0(5(5(3(4(x1)))))) -> 1^1(0(0(3(5(5(x1))))))
   0^1(0(5(5(3(4(x1)))))) -> 0^1(0(3(5(5(x1)))))
   0^1(0(5(5(3(4(x1)))))) -> 5^1(5(x1))
   0^1(0(5(5(3(4(x1)))))) -> 5^1(x1)
   0^1(1(2(0(1(2(x1)))))) -> 1^1(1(0(x1)))
   0^1(1(2(0(1(2(x1)))))) -> 1^1(0(x1))
   0^1(1(2(0(1(2(x1)))))) -> 0^1(x1)
   0^1(0(2(5(2(x1))))) -> 0^1(5(2(x1)))
   0^1(5(2(4(1(x1))))) -> 5^1(1(x1))
   0^1(5(2(5(1(3(4(x1))))))) -> 1^1(5(x1))
   0^1(5(2(5(1(3(4(x1))))))) -> 5^1(x1)
   0^1(4(2(0(5(x1))))) -> 1^1(5(x1))
   0^1(0(5(2(3(4(x1)))))) -> 1^1(2(5(x1)))
   1^1(2(1(2(0(0(x1)))))) -> 1^1(x1)
   0^1(0(5(2(3(4(x1)))))) -> 5^1(x1)
   0^1(5(0(0(5(4(x1)))))) -> 0^1(1(0(1(0(4(5(5(x1))))))))
   0^1(5(0(0(5(4(x1)))))) -> 1^1(0(1(0(4(5(5(x1)))))))
   0^1(5(0(0(5(4(x1)))))) -> 0^1(1(0(4(5(5(x1))))))
   0^1(5(0(0(5(4(x1)))))) -> 1^1(0(4(5(5(x1)))))
   0^1(5(0(0(5(4(x1)))))) -> 5^1(5(x1))
   0^1(5(0(0(5(4(x1)))))) -> 5^1(x1)
   0^1(5(1(2(1(4(x1)))))) -> 0^1(4(x1))
   0^1(0(1(2(3(5(5(x1))))))) -> 5^1(1(5(0(3(0(2(1(x1))))))))
   0^1(0(1(2(3(5(5(x1))))))) -> 1^1(5(0(3(0(2(1(x1)))))))
   0^1(0(1(2(3(5(5(x1))))))) -> 5^1(0(3(0(2(1(x1))))))
   0^1(0(1(2(3(5(5(x1))))))) -> 0^1(2(1(x1)))
   0^1(0(1(2(3(5(5(x1))))))) -> 1^1(x1)
   0^1(1(0(0(5(3(4(x1))))))) -> 1^1(x1)
   0^1(5(5(0(2(5(1(x1))))))) -> 5^1(x1)
   0^1(5(5(2(5(3(4(x1))))))) -> 1^1(4(5(5(5(x1)))))
   0^1(5(5(2(5(3(4(x1))))))) -> 5^1(5(5(x1)))
   0^1(5(5(2(5(3(4(x1))))))) -> 5^1(5(x1))
   0^1(5(5(2(5(3(4(x1))))))) -> 5^1(x1)

The TRS R consists of the following rules:

   0(0(1(2(x1)))) -> 0(3(1(0(2(x1)))))
   0(1(2(2(x1)))) -> 1(2(0(3(2(2(x1))))))
   0(1(2(4(x1)))) -> 0(3(2(3(1(4(x1))))))
   0(5(0(5(x1)))) -> 0(3(0(5(5(x1)))))
   0(5(1(2(x1)))) -> 1(0(1(5(2(x1)))))
   0(5(1(2(x1)))) -> 0(1(0(1(5(2(x1))))))
   0(5(1(2(x1)))) -> 0(3(2(3(1(5(x1))))))
   0(5(4(2(x1)))) -> 0(4(5(3(2(x1)))))
   0(5(5(2(x1)))) -> 5(0(1(5(2(x1)))))
   1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1)))))))
   1(0(1(2(x1)))) -> 1(1(3(0(2(x1)))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1))))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1))))))
   1(0(5(4(x1)))) -> 0(1(1(5(4(x1)))))
   1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1))))))
   1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1))))))
   1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1)))))))
   1(5(1(2(x1)))) -> 0(1(1(5(2(x1)))))
   1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1))))))
   5(0(0(2(x1)))) -> 5(0(3(0(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1))))))
   0(0(0(1(2(x1))))) -> 0(2(0(1(0(3(3(4(x1))))))))
   0(0(2(5(2(x1))))) -> 0(3(2(0(5(2(x1))))))
   0(1(2(5(0(x1))))) -> 3(3(2(2(0(0(1(5(x1))))))))
   0(1(2(5(2(x1))))) -> 0(3(2(1(5(3(2(x1)))))))
   0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1))))))
   0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1)))))))
   0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1)))))))
   0(5(0(2(2(x1))))) -> 0(2(5(0(3(2(x1))))))
   0(5(0(5(1(x1))))) -> 0(1(0(3(5(5(x1))))))
   0(5(1(3(0(x1))))) -> 0(0(1(1(5(3(x1))))))
   0(5(2(2(4(x1))))) -> 0(5(3(2(2(4(x1))))))
   0(5(2(3(1(x1))))) -> 0(1(5(3(2(2(2(x1)))))))
   0(5(2(4(1(x1))))) -> 0(4(3(2(5(1(x1))))))
   0(5(3(5(2(x1))))) -> 0(0(3(5(5(2(x1))))))
   0(5(5(3(1(x1))))) -> 5(0(1(5(3(3(2(x1)))))))
   1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1))))))
   1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1))))))
   1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1))))))
   1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1))))))
   1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1)))))))
   1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1))))))))
   5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1)))))))
   5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1))))))
   5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1))))))
   0(0(0(5(1(2(x1)))))) -> 0(0(1(5(0(2(4(x1)))))))
   0(0(1(2(4(1(x1)))))) -> 1(3(0(2(3(0(4(1(x1))))))))
   0(0(2(1(2(0(x1)))))) -> 0(1(0(3(2(2(2(0(x1))))))))
   0(0(2(3(0(5(x1)))))) -> 0(0(3(0(3(2(5(x1)))))))
   0(0(5(2(3(4(x1)))))) -> 0(4(0(3(1(2(5(x1)))))))
   0(0(5(5(3(4(x1)))))) -> 1(4(1(0(0(3(5(5(x1))))))))
   0(1(2(0(1(2(x1)))))) -> 1(0(3(2(2(1(1(0(x1))))))))
   0(1(2(2(0(5(x1)))))) -> 0(4(1(5(0(3(2(2(x1))))))))
   0(1(2(5(5(5(x1)))))) -> 0(2(5(1(5(3(5(x1)))))))
   0(1(3(1(5(2(x1)))))) -> 1(0(1(5(3(3(2(x1)))))))
   0(1(4(4(0(5(x1)))))) -> 4(3(0(0(1(5(4(x1)))))))
   0(2(5(3(5(1(x1)))))) -> 0(3(3(2(2(1(5(5(x1))))))))
   0(5(0(0(5(4(x1)))))) -> 0(1(0(1(0(4(5(5(x1))))))))
   0(5(1(2(1(4(x1)))))) -> 1(1(5(3(2(2(0(4(x1))))))))
   0(5(5(1(2(5(x1)))))) -> 0(2(1(5(5(4(5(x1)))))))
   1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1)))))))
   1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1))))))))
   1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1)))))))
   1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1))))))))
   1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1)))))))
   1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1)))))))
   1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1)))))))
   0(0(1(2(3(5(5(x1))))))) -> 5(1(5(0(3(0(2(1(x1))))))))
   0(0(2(3(4(2(1(x1))))))) -> 0(0(4(1(3(2(3(2(x1))))))))
   0(1(0(0(5(3(4(x1))))))) -> 0(3(0(5(0(4(3(1(x1))))))))
   0(1(2(4(4(0(5(x1))))))) -> 5(4(0(1(0(3(2(4(x1))))))))
   0(5(2(5(1(3(4(x1))))))) -> 1(4(5(2(0(3(1(5(x1))))))))
   0(5(5(0(2(5(1(x1))))))) -> 5(3(0(0(1(5(2(5(x1))))))))
   0(5(5(2(5(3(4(x1))))))) -> 0(3(2(1(4(5(5(5(x1))))))))
   1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1))))))))
   1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1))))))))
   1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1))))))))
   1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(1(5(4(x1))))))))

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

(5) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   1^1(0(5(5(1(x1))))) -> 5^1(1(5(1(x1))))
   5^1(5(0(1(2(x1))))) -> 0^1(2(1(x1)))
   0^1(2(5(3(5(1(x1)))))) -> 5^1(5(x1))
   5^1(5(0(1(2(x1))))) -> 1^1(x1)
   1^1(0(5(5(1(x1))))) -> 1^1(5(1(x1)))
   1^1(5(0(5(0(x1))))) -> 5^1(1(0(x1)))
   1^1(5(0(5(0(x1))))) -> 1^1(0(x1))
   1^1(4(1(0(0(5(x1)))))) -> 1^1(x1)
   1^1(2(0(5(x1)))) -> 1^1(x1)
   0^1(2(5(3(5(1(x1)))))) -> 5^1(x1)
   0^1(5(0(5(x1)))) -> 5^1(5(x1))
   0^1(5(1(2(x1)))) -> 5^1(x1)
   0^1(1(2(5(0(x1))))) -> 5^1(x1)
   0^1(5(0(5(1(x1))))) -> 5^1(5(x1))
   0^1(5(0(5(1(x1))))) -> 5^1(x1)
   0^1(0(5(5(3(4(x1)))))) -> 5^1(5(x1))
   0^1(0(5(5(3(4(x1)))))) -> 5^1(x1)
   0^1(5(2(4(1(x1))))) -> 5^1(1(x1))
   0^1(5(2(5(1(3(4(x1))))))) -> 1^1(5(x1))
   0^1(5(2(5(1(3(4(x1))))))) -> 5^1(x1)
   0^1(0(5(2(3(4(x1)))))) -> 5^1(x1)
   0^1(5(0(0(5(4(x1)))))) -> 5^1(5(x1))
   0^1(5(0(0(5(4(x1)))))) -> 5^1(x1)
   0^1(5(1(2(1(4(x1)))))) -> 0^1(4(x1))
   0^1(0(1(2(3(5(5(x1))))))) -> 5^1(1(5(0(3(0(2(1(x1))))))))
   0^1(0(1(2(3(5(5(x1))))))) -> 1^1(5(0(3(0(2(1(x1)))))))
   0^1(0(1(2(3(5(5(x1))))))) -> 5^1(0(3(0(2(1(x1))))))
   0^1(0(1(2(3(5(5(x1))))))) -> 0^1(2(1(x1)))
   0^1(0(1(2(3(5(5(x1))))))) -> 1^1(x1)
   0^1(1(0(0(5(3(4(x1))))))) -> 1^1(x1)
   0^1(5(5(0(2(5(1(x1))))))) -> 5^1(x1)
   0^1(5(5(2(5(3(4(x1))))))) -> 5^1(5(5(x1)))
   0^1(5(5(2(5(3(4(x1))))))) -> 5^1(5(x1))
   0^1(5(5(2(5(3(4(x1))))))) -> 5^1(x1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(0(x_1)) = x_1
   POL(0^1(x_1)) = x_1
   POL(1(x_1)) = x_1
   POL(1^1(x_1)) = x_1
   POL(2(x_1)) = x_1
   POL(3(x_1)) = x_1
   POL(4(x_1)) = x_1
   POL(5(x_1)) = 1 + x_1
   POL(5^1(x_1)) = x_1

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

   0(2(5(3(5(1(x1)))))) -> 0(3(3(2(2(1(5(5(x1))))))))
   5(0(0(2(x1)))) -> 5(0(3(0(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1))))))
   5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1)))))))
   5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1))))))
   5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1))))))
   1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1)))))))
   1(0(1(2(x1)))) -> 1(1(3(0(2(x1)))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1))))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1))))))
   1(0(5(4(x1)))) -> 0(1(1(5(4(x1)))))
   1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1))))))
   1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1))))))
   1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1)))))))
   1(5(1(2(x1)))) -> 0(1(1(5(2(x1)))))
   1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1))))))
   1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1))))))
   1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1))))))
   1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1))))))
   1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1))))))
   1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1)))))))
   1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1))))))))
   1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1)))))))
   1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1))))))))
   1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1)))))))
   1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1))))))))
   1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1)))))))
   1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1)))))))
   1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1)))))))
   1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1))))))))
   1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1))))))))
   1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1))))))))
   1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(1(5(4(x1))))))))
   0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1))))))
   0(0(1(2(x1)))) -> 0(3(1(0(2(x1)))))
   0(1(2(2(x1)))) -> 1(2(0(3(2(2(x1))))))
   0(1(2(4(x1)))) -> 0(3(2(3(1(4(x1))))))
   0(5(0(5(x1)))) -> 0(3(0(5(5(x1)))))
   0(5(1(2(x1)))) -> 1(0(1(5(2(x1)))))
   0(5(1(2(x1)))) -> 0(1(0(1(5(2(x1))))))
   0(5(1(2(x1)))) -> 0(3(2(3(1(5(x1))))))
   0(5(4(2(x1)))) -> 0(4(5(3(2(x1)))))
   0(5(5(2(x1)))) -> 5(0(1(5(2(x1)))))
   0(0(0(1(2(x1))))) -> 0(2(0(1(0(3(3(4(x1))))))))
   0(0(2(5(2(x1))))) -> 0(3(2(0(5(2(x1))))))
   0(1(2(5(0(x1))))) -> 3(3(2(2(0(0(1(5(x1))))))))
   0(1(2(5(2(x1))))) -> 0(3(2(1(5(3(2(x1)))))))
   0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1)))))))
   0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1)))))))
   0(5(0(2(2(x1))))) -> 0(2(5(0(3(2(x1))))))
   0(0(0(5(1(2(x1)))))) -> 0(0(1(5(0(2(4(x1)))))))
   0(5(0(5(1(x1))))) -> 0(1(0(3(5(5(x1))))))
   0(5(0(0(5(4(x1)))))) -> 0(1(0(1(0(4(5(5(x1))))))))
   0(5(1(3(0(x1))))) -> 0(0(1(1(5(3(x1))))))
   0(5(2(2(4(x1))))) -> 0(5(3(2(2(4(x1))))))
   0(5(2(3(1(x1))))) -> 0(1(5(3(2(2(2(x1)))))))
   0(5(2(4(1(x1))))) -> 0(4(3(2(5(1(x1))))))
   0(5(3(5(2(x1))))) -> 0(0(3(5(5(2(x1))))))
   0(5(5(3(1(x1))))) -> 5(0(1(5(3(3(2(x1)))))))
   0(0(1(2(4(1(x1)))))) -> 1(3(0(2(3(0(4(1(x1))))))))
   0(0(2(1(2(0(x1)))))) -> 0(1(0(3(2(2(2(0(x1))))))))
   0(0(2(3(0(5(x1)))))) -> 0(0(3(0(3(2(5(x1)))))))
   0(0(5(2(3(4(x1)))))) -> 0(4(0(3(1(2(5(x1)))))))
   0(0(5(5(3(4(x1)))))) -> 1(4(1(0(0(3(5(5(x1))))))))
   0(1(2(0(1(2(x1)))))) -> 1(0(3(2(2(1(1(0(x1))))))))
   0(1(2(2(0(5(x1)))))) -> 0(4(1(5(0(3(2(2(x1))))))))
   0(1(2(5(5(5(x1)))))) -> 0(2(5(1(5(3(5(x1)))))))
   0(1(3(1(5(2(x1)))))) -> 1(0(1(5(3(3(2(x1)))))))
   0(1(4(4(0(5(x1)))))) -> 4(3(0(0(1(5(4(x1)))))))
   0(5(1(2(1(4(x1)))))) -> 1(1(5(3(2(2(0(4(x1))))))))
   0(5(5(1(2(5(x1)))))) -> 0(2(1(5(5(4(5(x1)))))))
   0(0(1(2(3(5(5(x1))))))) -> 5(1(5(0(3(0(2(1(x1))))))))
   0(0(2(3(4(2(1(x1))))))) -> 0(0(4(1(3(2(3(2(x1))))))))
   0(1(0(0(5(3(4(x1))))))) -> 0(3(0(5(0(4(3(1(x1))))))))
   0(1(2(4(4(0(5(x1))))))) -> 5(4(0(1(0(3(2(4(x1))))))))
   0(5(2(5(1(3(4(x1))))))) -> 1(4(5(2(0(3(1(5(x1))))))))
   0(5(5(0(2(5(1(x1))))))) -> 5(3(0(0(1(5(2(5(x1))))))))
   0(5(5(2(5(3(4(x1))))))) -> 0(3(2(1(4(5(5(5(x1))))))))


----------------------------------------

(6)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   0^1(0(1(2(x1)))) -> 1^1(0(2(x1)))
   1^1(0(1(2(x1)))) -> 0^1(2(x1))
   0^1(2(5(3(5(1(x1)))))) -> 1^1(5(5(x1)))
   5^1(0(0(2(x1)))) -> 5^1(0(3(0(2(x1)))))
   1^1(2(0(5(x1)))) -> 1^1(5(x1))
   1^1(2(1(2(0(0(x1)))))) -> 1^1(0(1(x1)))
   1^1(2(1(2(0(0(x1)))))) -> 0^1(1(x1))
   0^1(0(1(2(x1)))) -> 0^1(2(x1))
   0^1(1(2(4(x1)))) -> 1^1(4(x1))
   0^1(5(0(5(x1)))) -> 0^1(5(5(x1)))
   0^1(5(1(2(x1)))) -> 1^1(5(x1))
   0^1(1(2(5(0(x1))))) -> 0^1(0(1(5(x1))))
   0^1(1(2(5(0(x1))))) -> 0^1(1(5(x1)))
   0^1(1(2(5(0(x1))))) -> 1^1(5(x1))
   0^1(5(0(5(1(x1))))) -> 0^1(1(0(3(5(5(x1))))))
   0^1(5(0(5(1(x1))))) -> 1^1(0(3(5(5(x1)))))
   0^1(0(0(5(1(2(x1)))))) -> 0^1(0(1(5(0(2(4(x1)))))))
   0^1(0(0(5(1(2(x1)))))) -> 0^1(1(5(0(2(4(x1))))))
   0^1(0(5(5(3(4(x1)))))) -> 1^1(4(1(0(0(3(5(5(x1))))))))
   0^1(0(5(5(3(4(x1)))))) -> 1^1(0(0(3(5(5(x1))))))
   0^1(0(5(5(3(4(x1)))))) -> 0^1(0(3(5(5(x1)))))
   0^1(1(2(0(1(2(x1)))))) -> 1^1(1(0(x1)))
   0^1(1(2(0(1(2(x1)))))) -> 1^1(0(x1))
   0^1(1(2(0(1(2(x1)))))) -> 0^1(x1)
   0^1(0(2(5(2(x1))))) -> 0^1(5(2(x1)))
   0^1(4(2(0(5(x1))))) -> 1^1(5(x1))
   0^1(0(5(2(3(4(x1)))))) -> 1^1(2(5(x1)))
   1^1(2(1(2(0(0(x1)))))) -> 1^1(x1)
   0^1(5(0(0(5(4(x1)))))) -> 0^1(1(0(1(0(4(5(5(x1))))))))
   0^1(5(0(0(5(4(x1)))))) -> 1^1(0(1(0(4(5(5(x1)))))))
   0^1(5(0(0(5(4(x1)))))) -> 0^1(1(0(4(5(5(x1))))))
   0^1(5(0(0(5(4(x1)))))) -> 1^1(0(4(5(5(x1)))))
   0^1(5(5(2(5(3(4(x1))))))) -> 1^1(4(5(5(5(x1)))))

The TRS R consists of the following rules:

   0(0(1(2(x1)))) -> 0(3(1(0(2(x1)))))
   0(1(2(2(x1)))) -> 1(2(0(3(2(2(x1))))))
   0(1(2(4(x1)))) -> 0(3(2(3(1(4(x1))))))
   0(5(0(5(x1)))) -> 0(3(0(5(5(x1)))))
   0(5(1(2(x1)))) -> 1(0(1(5(2(x1)))))
   0(5(1(2(x1)))) -> 0(1(0(1(5(2(x1))))))
   0(5(1(2(x1)))) -> 0(3(2(3(1(5(x1))))))
   0(5(4(2(x1)))) -> 0(4(5(3(2(x1)))))
   0(5(5(2(x1)))) -> 5(0(1(5(2(x1)))))
   1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1)))))))
   1(0(1(2(x1)))) -> 1(1(3(0(2(x1)))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1))))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1))))))
   1(0(5(4(x1)))) -> 0(1(1(5(4(x1)))))
   1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1))))))
   1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1))))))
   1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1)))))))
   1(5(1(2(x1)))) -> 0(1(1(5(2(x1)))))
   1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1))))))
   5(0(0(2(x1)))) -> 5(0(3(0(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1))))))
   0(0(0(1(2(x1))))) -> 0(2(0(1(0(3(3(4(x1))))))))
   0(0(2(5(2(x1))))) -> 0(3(2(0(5(2(x1))))))
   0(1(2(5(0(x1))))) -> 3(3(2(2(0(0(1(5(x1))))))))
   0(1(2(5(2(x1))))) -> 0(3(2(1(5(3(2(x1)))))))
   0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1))))))
   0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1)))))))
   0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1)))))))
   0(5(0(2(2(x1))))) -> 0(2(5(0(3(2(x1))))))
   0(5(0(5(1(x1))))) -> 0(1(0(3(5(5(x1))))))
   0(5(1(3(0(x1))))) -> 0(0(1(1(5(3(x1))))))
   0(5(2(2(4(x1))))) -> 0(5(3(2(2(4(x1))))))
   0(5(2(3(1(x1))))) -> 0(1(5(3(2(2(2(x1)))))))
   0(5(2(4(1(x1))))) -> 0(4(3(2(5(1(x1))))))
   0(5(3(5(2(x1))))) -> 0(0(3(5(5(2(x1))))))
   0(5(5(3(1(x1))))) -> 5(0(1(5(3(3(2(x1)))))))
   1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1))))))
   1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1))))))
   1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1))))))
   1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1))))))
   1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1)))))))
   1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1))))))))
   5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1)))))))
   5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1))))))
   5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1))))))
   0(0(0(5(1(2(x1)))))) -> 0(0(1(5(0(2(4(x1)))))))
   0(0(1(2(4(1(x1)))))) -> 1(3(0(2(3(0(4(1(x1))))))))
   0(0(2(1(2(0(x1)))))) -> 0(1(0(3(2(2(2(0(x1))))))))
   0(0(2(3(0(5(x1)))))) -> 0(0(3(0(3(2(5(x1)))))))
   0(0(5(2(3(4(x1)))))) -> 0(4(0(3(1(2(5(x1)))))))
   0(0(5(5(3(4(x1)))))) -> 1(4(1(0(0(3(5(5(x1))))))))
   0(1(2(0(1(2(x1)))))) -> 1(0(3(2(2(1(1(0(x1))))))))
   0(1(2(2(0(5(x1)))))) -> 0(4(1(5(0(3(2(2(x1))))))))
   0(1(2(5(5(5(x1)))))) -> 0(2(5(1(5(3(5(x1)))))))
   0(1(3(1(5(2(x1)))))) -> 1(0(1(5(3(3(2(x1)))))))
   0(1(4(4(0(5(x1)))))) -> 4(3(0(0(1(5(4(x1)))))))
   0(2(5(3(5(1(x1)))))) -> 0(3(3(2(2(1(5(5(x1))))))))
   0(5(0(0(5(4(x1)))))) -> 0(1(0(1(0(4(5(5(x1))))))))
   0(5(1(2(1(4(x1)))))) -> 1(1(5(3(2(2(0(4(x1))))))))
   0(5(5(1(2(5(x1)))))) -> 0(2(1(5(5(4(5(x1)))))))
   1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1)))))))
   1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1))))))))
   1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1)))))))
   1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1))))))))
   1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1)))))))
   1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1)))))))
   1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1)))))))
   0(0(1(2(3(5(5(x1))))))) -> 5(1(5(0(3(0(2(1(x1))))))))
   0(0(2(3(4(2(1(x1))))))) -> 0(0(4(1(3(2(3(2(x1))))))))
   0(1(0(0(5(3(4(x1))))))) -> 0(3(0(5(0(4(3(1(x1))))))))
   0(1(2(4(4(0(5(x1))))))) -> 5(4(0(1(0(3(2(4(x1))))))))
   0(5(2(5(1(3(4(x1))))))) -> 1(4(5(2(0(3(1(5(x1))))))))
   0(5(5(0(2(5(1(x1))))))) -> 5(3(0(0(1(5(2(5(x1))))))))
   0(5(5(2(5(3(4(x1))))))) -> 0(3(2(1(4(5(5(5(x1))))))))
   1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1))))))))
   1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1))))))))
   1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1))))))))
   1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(1(5(4(x1))))))))

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

(7) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 17 less nodes.
----------------------------------------

(8)
Complex Obligation (AND)

----------------------------------------

(9)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   5^1(0(0(2(x1)))) -> 5^1(0(3(0(2(x1)))))

The TRS R consists of the following rules:

   0(0(1(2(x1)))) -> 0(3(1(0(2(x1)))))
   0(1(2(2(x1)))) -> 1(2(0(3(2(2(x1))))))
   0(1(2(4(x1)))) -> 0(3(2(3(1(4(x1))))))
   0(5(0(5(x1)))) -> 0(3(0(5(5(x1)))))
   0(5(1(2(x1)))) -> 1(0(1(5(2(x1)))))
   0(5(1(2(x1)))) -> 0(1(0(1(5(2(x1))))))
   0(5(1(2(x1)))) -> 0(3(2(3(1(5(x1))))))
   0(5(4(2(x1)))) -> 0(4(5(3(2(x1)))))
   0(5(5(2(x1)))) -> 5(0(1(5(2(x1)))))
   1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1)))))))
   1(0(1(2(x1)))) -> 1(1(3(0(2(x1)))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1))))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1))))))
   1(0(5(4(x1)))) -> 0(1(1(5(4(x1)))))
   1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1))))))
   1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1))))))
   1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1)))))))
   1(5(1(2(x1)))) -> 0(1(1(5(2(x1)))))
   1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1))))))
   5(0(0(2(x1)))) -> 5(0(3(0(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1))))))
   0(0(0(1(2(x1))))) -> 0(2(0(1(0(3(3(4(x1))))))))
   0(0(2(5(2(x1))))) -> 0(3(2(0(5(2(x1))))))
   0(1(2(5(0(x1))))) -> 3(3(2(2(0(0(1(5(x1))))))))
   0(1(2(5(2(x1))))) -> 0(3(2(1(5(3(2(x1)))))))
   0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1))))))
   0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1)))))))
   0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1)))))))
   0(5(0(2(2(x1))))) -> 0(2(5(0(3(2(x1))))))
   0(5(0(5(1(x1))))) -> 0(1(0(3(5(5(x1))))))
   0(5(1(3(0(x1))))) -> 0(0(1(1(5(3(x1))))))
   0(5(2(2(4(x1))))) -> 0(5(3(2(2(4(x1))))))
   0(5(2(3(1(x1))))) -> 0(1(5(3(2(2(2(x1)))))))
   0(5(2(4(1(x1))))) -> 0(4(3(2(5(1(x1))))))
   0(5(3(5(2(x1))))) -> 0(0(3(5(5(2(x1))))))
   0(5(5(3(1(x1))))) -> 5(0(1(5(3(3(2(x1)))))))
   1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1))))))
   1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1))))))
   1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1))))))
   1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1))))))
   1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1)))))))
   1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1))))))))
   5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1)))))))
   5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1))))))
   5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1))))))
   0(0(0(5(1(2(x1)))))) -> 0(0(1(5(0(2(4(x1)))))))
   0(0(1(2(4(1(x1)))))) -> 1(3(0(2(3(0(4(1(x1))))))))
   0(0(2(1(2(0(x1)))))) -> 0(1(0(3(2(2(2(0(x1))))))))
   0(0(2(3(0(5(x1)))))) -> 0(0(3(0(3(2(5(x1)))))))
   0(0(5(2(3(4(x1)))))) -> 0(4(0(3(1(2(5(x1)))))))
   0(0(5(5(3(4(x1)))))) -> 1(4(1(0(0(3(5(5(x1))))))))
   0(1(2(0(1(2(x1)))))) -> 1(0(3(2(2(1(1(0(x1))))))))
   0(1(2(2(0(5(x1)))))) -> 0(4(1(5(0(3(2(2(x1))))))))
   0(1(2(5(5(5(x1)))))) -> 0(2(5(1(5(3(5(x1)))))))
   0(1(3(1(5(2(x1)))))) -> 1(0(1(5(3(3(2(x1)))))))
   0(1(4(4(0(5(x1)))))) -> 4(3(0(0(1(5(4(x1)))))))
   0(2(5(3(5(1(x1)))))) -> 0(3(3(2(2(1(5(5(x1))))))))
   0(5(0(0(5(4(x1)))))) -> 0(1(0(1(0(4(5(5(x1))))))))
   0(5(1(2(1(4(x1)))))) -> 1(1(5(3(2(2(0(4(x1))))))))
   0(5(5(1(2(5(x1)))))) -> 0(2(1(5(5(4(5(x1)))))))
   1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1)))))))
   1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1))))))))
   1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1)))))))
   1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1))))))))
   1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1)))))))
   1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1)))))))
   1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1)))))))
   0(0(1(2(3(5(5(x1))))))) -> 5(1(5(0(3(0(2(1(x1))))))))
   0(0(2(3(4(2(1(x1))))))) -> 0(0(4(1(3(2(3(2(x1))))))))
   0(1(0(0(5(3(4(x1))))))) -> 0(3(0(5(0(4(3(1(x1))))))))
   0(1(2(4(4(0(5(x1))))))) -> 5(4(0(1(0(3(2(4(x1))))))))
   0(5(2(5(1(3(4(x1))))))) -> 1(4(5(2(0(3(1(5(x1))))))))
   0(5(5(0(2(5(1(x1))))))) -> 5(3(0(0(1(5(2(5(x1))))))))
   0(5(5(2(5(3(4(x1))))))) -> 0(3(2(1(4(5(5(5(x1))))))))
   1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1))))))))
   1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1))))))))
   1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1))))))))
   1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(1(5(4(x1))))))))

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

(10) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   5^1(0(0(2(x1)))) -> 5^1(0(3(0(2(x1)))))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(0(x_1)) = 1 + x_1
   POL(1(x_1)) = 0
   POL(2(x_1)) = 0
   POL(3(x_1)) = 0
   POL(4(x_1)) = 0
   POL(5(x_1)) = 0
   POL(5^1(x_1)) = x_1

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

   0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1))))))


----------------------------------------

(11)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

   0(0(1(2(x1)))) -> 0(3(1(0(2(x1)))))
   0(1(2(2(x1)))) -> 1(2(0(3(2(2(x1))))))
   0(1(2(4(x1)))) -> 0(3(2(3(1(4(x1))))))
   0(5(0(5(x1)))) -> 0(3(0(5(5(x1)))))
   0(5(1(2(x1)))) -> 1(0(1(5(2(x1)))))
   0(5(1(2(x1)))) -> 0(1(0(1(5(2(x1))))))
   0(5(1(2(x1)))) -> 0(3(2(3(1(5(x1))))))
   0(5(4(2(x1)))) -> 0(4(5(3(2(x1)))))
   0(5(5(2(x1)))) -> 5(0(1(5(2(x1)))))
   1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1)))))))
   1(0(1(2(x1)))) -> 1(1(3(0(2(x1)))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1))))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1))))))
   1(0(5(4(x1)))) -> 0(1(1(5(4(x1)))))
   1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1))))))
   1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1))))))
   1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1)))))))
   1(5(1(2(x1)))) -> 0(1(1(5(2(x1)))))
   1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1))))))
   5(0(0(2(x1)))) -> 5(0(3(0(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1))))))
   0(0(0(1(2(x1))))) -> 0(2(0(1(0(3(3(4(x1))))))))
   0(0(2(5(2(x1))))) -> 0(3(2(0(5(2(x1))))))
   0(1(2(5(0(x1))))) -> 3(3(2(2(0(0(1(5(x1))))))))
   0(1(2(5(2(x1))))) -> 0(3(2(1(5(3(2(x1)))))))
   0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1))))))
   0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1)))))))
   0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1)))))))
   0(5(0(2(2(x1))))) -> 0(2(5(0(3(2(x1))))))
   0(5(0(5(1(x1))))) -> 0(1(0(3(5(5(x1))))))
   0(5(1(3(0(x1))))) -> 0(0(1(1(5(3(x1))))))
   0(5(2(2(4(x1))))) -> 0(5(3(2(2(4(x1))))))
   0(5(2(3(1(x1))))) -> 0(1(5(3(2(2(2(x1)))))))
   0(5(2(4(1(x1))))) -> 0(4(3(2(5(1(x1))))))
   0(5(3(5(2(x1))))) -> 0(0(3(5(5(2(x1))))))
   0(5(5(3(1(x1))))) -> 5(0(1(5(3(3(2(x1)))))))
   1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1))))))
   1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1))))))
   1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1))))))
   1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1))))))
   1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1)))))))
   1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1))))))))
   5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1)))))))
   5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1))))))
   5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1))))))
   0(0(0(5(1(2(x1)))))) -> 0(0(1(5(0(2(4(x1)))))))
   0(0(1(2(4(1(x1)))))) -> 1(3(0(2(3(0(4(1(x1))))))))
   0(0(2(1(2(0(x1)))))) -> 0(1(0(3(2(2(2(0(x1))))))))
   0(0(2(3(0(5(x1)))))) -> 0(0(3(0(3(2(5(x1)))))))
   0(0(5(2(3(4(x1)))))) -> 0(4(0(3(1(2(5(x1)))))))
   0(0(5(5(3(4(x1)))))) -> 1(4(1(0(0(3(5(5(x1))))))))
   0(1(2(0(1(2(x1)))))) -> 1(0(3(2(2(1(1(0(x1))))))))
   0(1(2(2(0(5(x1)))))) -> 0(4(1(5(0(3(2(2(x1))))))))
   0(1(2(5(5(5(x1)))))) -> 0(2(5(1(5(3(5(x1)))))))
   0(1(3(1(5(2(x1)))))) -> 1(0(1(5(3(3(2(x1)))))))
   0(1(4(4(0(5(x1)))))) -> 4(3(0(0(1(5(4(x1)))))))
   0(2(5(3(5(1(x1)))))) -> 0(3(3(2(2(1(5(5(x1))))))))
   0(5(0(0(5(4(x1)))))) -> 0(1(0(1(0(4(5(5(x1))))))))
   0(5(1(2(1(4(x1)))))) -> 1(1(5(3(2(2(0(4(x1))))))))
   0(5(5(1(2(5(x1)))))) -> 0(2(1(5(5(4(5(x1)))))))
   1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1)))))))
   1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1))))))))
   1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1)))))))
   1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1))))))))
   1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1)))))))
   1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1)))))))
   1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1)))))))
   0(0(1(2(3(5(5(x1))))))) -> 5(1(5(0(3(0(2(1(x1))))))))
   0(0(2(3(4(2(1(x1))))))) -> 0(0(4(1(3(2(3(2(x1))))))))
   0(1(0(0(5(3(4(x1))))))) -> 0(3(0(5(0(4(3(1(x1))))))))
   0(1(2(4(4(0(5(x1))))))) -> 5(4(0(1(0(3(2(4(x1))))))))
   0(5(2(5(1(3(4(x1))))))) -> 1(4(5(2(0(3(1(5(x1))))))))
   0(5(5(0(2(5(1(x1))))))) -> 5(3(0(0(1(5(2(5(x1))))))))
   0(5(5(2(5(3(4(x1))))))) -> 0(3(2(1(4(5(5(5(x1))))))))
   1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1))))))))
   1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1))))))))
   1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1))))))))
   1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(1(5(4(x1))))))))

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

(12) PisEmptyProof (EQUIVALENT)
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:

   0^1(2(5(3(5(1(x1)))))) -> 1^1(5(5(x1)))
   1^1(0(1(2(x1)))) -> 0^1(2(x1))

The TRS R consists of the following rules:

   0(0(1(2(x1)))) -> 0(3(1(0(2(x1)))))
   0(1(2(2(x1)))) -> 1(2(0(3(2(2(x1))))))
   0(1(2(4(x1)))) -> 0(3(2(3(1(4(x1))))))
   0(5(0(5(x1)))) -> 0(3(0(5(5(x1)))))
   0(5(1(2(x1)))) -> 1(0(1(5(2(x1)))))
   0(5(1(2(x1)))) -> 0(1(0(1(5(2(x1))))))
   0(5(1(2(x1)))) -> 0(3(2(3(1(5(x1))))))
   0(5(4(2(x1)))) -> 0(4(5(3(2(x1)))))
   0(5(5(2(x1)))) -> 5(0(1(5(2(x1)))))
   1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1)))))))
   1(0(1(2(x1)))) -> 1(1(3(0(2(x1)))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1))))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1))))))
   1(0(5(4(x1)))) -> 0(1(1(5(4(x1)))))
   1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1))))))
   1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1))))))
   1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1)))))))
   1(5(1(2(x1)))) -> 0(1(1(5(2(x1)))))
   1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1))))))
   5(0(0(2(x1)))) -> 5(0(3(0(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1))))))
   0(0(0(1(2(x1))))) -> 0(2(0(1(0(3(3(4(x1))))))))
   0(0(2(5(2(x1))))) -> 0(3(2(0(5(2(x1))))))
   0(1(2(5(0(x1))))) -> 3(3(2(2(0(0(1(5(x1))))))))
   0(1(2(5(2(x1))))) -> 0(3(2(1(5(3(2(x1)))))))
   0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1))))))
   0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1)))))))
   0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1)))))))
   0(5(0(2(2(x1))))) -> 0(2(5(0(3(2(x1))))))
   0(5(0(5(1(x1))))) -> 0(1(0(3(5(5(x1))))))
   0(5(1(3(0(x1))))) -> 0(0(1(1(5(3(x1))))))
   0(5(2(2(4(x1))))) -> 0(5(3(2(2(4(x1))))))
   0(5(2(3(1(x1))))) -> 0(1(5(3(2(2(2(x1)))))))
   0(5(2(4(1(x1))))) -> 0(4(3(2(5(1(x1))))))
   0(5(3(5(2(x1))))) -> 0(0(3(5(5(2(x1))))))
   0(5(5(3(1(x1))))) -> 5(0(1(5(3(3(2(x1)))))))
   1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1))))))
   1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1))))))
   1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1))))))
   1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1))))))
   1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1)))))))
   1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1))))))))
   5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1)))))))
   5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1))))))
   5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1))))))
   0(0(0(5(1(2(x1)))))) -> 0(0(1(5(0(2(4(x1)))))))
   0(0(1(2(4(1(x1)))))) -> 1(3(0(2(3(0(4(1(x1))))))))
   0(0(2(1(2(0(x1)))))) -> 0(1(0(3(2(2(2(0(x1))))))))
   0(0(2(3(0(5(x1)))))) -> 0(0(3(0(3(2(5(x1)))))))
   0(0(5(2(3(4(x1)))))) -> 0(4(0(3(1(2(5(x1)))))))
   0(0(5(5(3(4(x1)))))) -> 1(4(1(0(0(3(5(5(x1))))))))
   0(1(2(0(1(2(x1)))))) -> 1(0(3(2(2(1(1(0(x1))))))))
   0(1(2(2(0(5(x1)))))) -> 0(4(1(5(0(3(2(2(x1))))))))
   0(1(2(5(5(5(x1)))))) -> 0(2(5(1(5(3(5(x1)))))))
   0(1(3(1(5(2(x1)))))) -> 1(0(1(5(3(3(2(x1)))))))
   0(1(4(4(0(5(x1)))))) -> 4(3(0(0(1(5(4(x1)))))))
   0(2(5(3(5(1(x1)))))) -> 0(3(3(2(2(1(5(5(x1))))))))
   0(5(0(0(5(4(x1)))))) -> 0(1(0(1(0(4(5(5(x1))))))))
   0(5(1(2(1(4(x1)))))) -> 1(1(5(3(2(2(0(4(x1))))))))
   0(5(5(1(2(5(x1)))))) -> 0(2(1(5(5(4(5(x1)))))))
   1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1)))))))
   1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1))))))))
   1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1)))))))
   1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1))))))))
   1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1)))))))
   1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1)))))))
   1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1)))))))
   0(0(1(2(3(5(5(x1))))))) -> 5(1(5(0(3(0(2(1(x1))))))))
   0(0(2(3(4(2(1(x1))))))) -> 0(0(4(1(3(2(3(2(x1))))))))
   0(1(0(0(5(3(4(x1))))))) -> 0(3(0(5(0(4(3(1(x1))))))))
   0(1(2(4(4(0(5(x1))))))) -> 5(4(0(1(0(3(2(4(x1))))))))
   0(5(2(5(1(3(4(x1))))))) -> 1(4(5(2(0(3(1(5(x1))))))))
   0(5(5(0(2(5(1(x1))))))) -> 5(3(0(0(1(5(2(5(x1))))))))
   0(5(5(2(5(3(4(x1))))))) -> 0(3(2(1(4(5(5(5(x1))))))))
   1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1))))))))
   1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1))))))))
   1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1))))))))
   1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(1(5(4(x1))))))))

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

(15) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   0^1(2(5(3(5(1(x1)))))) -> 1^1(5(5(x1)))
   1^1(0(1(2(x1)))) -> 0^1(2(x1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(0(x_1)) = 1 + x_1
   POL(0^1(x_1)) = 1
   POL(1(x_1)) = 1
   POL(1^1(x_1)) = x_1
   POL(2(x_1)) = 0
   POL(3(x_1)) = 0
   POL(4(x_1)) = 0
   POL(5(x_1)) = 0

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

   5(0(0(2(x1)))) -> 5(0(3(0(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1))))))
   5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1)))))))
   5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1))))))
   5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1))))))


----------------------------------------

(16)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

   0(0(1(2(x1)))) -> 0(3(1(0(2(x1)))))
   0(1(2(2(x1)))) -> 1(2(0(3(2(2(x1))))))
   0(1(2(4(x1)))) -> 0(3(2(3(1(4(x1))))))
   0(5(0(5(x1)))) -> 0(3(0(5(5(x1)))))
   0(5(1(2(x1)))) -> 1(0(1(5(2(x1)))))
   0(5(1(2(x1)))) -> 0(1(0(1(5(2(x1))))))
   0(5(1(2(x1)))) -> 0(3(2(3(1(5(x1))))))
   0(5(4(2(x1)))) -> 0(4(5(3(2(x1)))))
   0(5(5(2(x1)))) -> 5(0(1(5(2(x1)))))
   1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1)))))))
   1(0(1(2(x1)))) -> 1(1(3(0(2(x1)))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1))))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1))))))
   1(0(5(4(x1)))) -> 0(1(1(5(4(x1)))))
   1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1))))))
   1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1))))))
   1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1)))))))
   1(5(1(2(x1)))) -> 0(1(1(5(2(x1)))))
   1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1))))))
   5(0(0(2(x1)))) -> 5(0(3(0(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1))))))
   0(0(0(1(2(x1))))) -> 0(2(0(1(0(3(3(4(x1))))))))
   0(0(2(5(2(x1))))) -> 0(3(2(0(5(2(x1))))))
   0(1(2(5(0(x1))))) -> 3(3(2(2(0(0(1(5(x1))))))))
   0(1(2(5(2(x1))))) -> 0(3(2(1(5(3(2(x1)))))))
   0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1))))))
   0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1)))))))
   0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1)))))))
   0(5(0(2(2(x1))))) -> 0(2(5(0(3(2(x1))))))
   0(5(0(5(1(x1))))) -> 0(1(0(3(5(5(x1))))))
   0(5(1(3(0(x1))))) -> 0(0(1(1(5(3(x1))))))
   0(5(2(2(4(x1))))) -> 0(5(3(2(2(4(x1))))))
   0(5(2(3(1(x1))))) -> 0(1(5(3(2(2(2(x1)))))))
   0(5(2(4(1(x1))))) -> 0(4(3(2(5(1(x1))))))
   0(5(3(5(2(x1))))) -> 0(0(3(5(5(2(x1))))))
   0(5(5(3(1(x1))))) -> 5(0(1(5(3(3(2(x1)))))))
   1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1))))))
   1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1))))))
   1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1))))))
   1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1))))))
   1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1)))))))
   1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1))))))))
   5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1)))))))
   5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1))))))
   5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1))))))
   0(0(0(5(1(2(x1)))))) -> 0(0(1(5(0(2(4(x1)))))))
   0(0(1(2(4(1(x1)))))) -> 1(3(0(2(3(0(4(1(x1))))))))
   0(0(2(1(2(0(x1)))))) -> 0(1(0(3(2(2(2(0(x1))))))))
   0(0(2(3(0(5(x1)))))) -> 0(0(3(0(3(2(5(x1)))))))
   0(0(5(2(3(4(x1)))))) -> 0(4(0(3(1(2(5(x1)))))))
   0(0(5(5(3(4(x1)))))) -> 1(4(1(0(0(3(5(5(x1))))))))
   0(1(2(0(1(2(x1)))))) -> 1(0(3(2(2(1(1(0(x1))))))))
   0(1(2(2(0(5(x1)))))) -> 0(4(1(5(0(3(2(2(x1))))))))
   0(1(2(5(5(5(x1)))))) -> 0(2(5(1(5(3(5(x1)))))))
   0(1(3(1(5(2(x1)))))) -> 1(0(1(5(3(3(2(x1)))))))
   0(1(4(4(0(5(x1)))))) -> 4(3(0(0(1(5(4(x1)))))))
   0(2(5(3(5(1(x1)))))) -> 0(3(3(2(2(1(5(5(x1))))))))
   0(5(0(0(5(4(x1)))))) -> 0(1(0(1(0(4(5(5(x1))))))))
   0(5(1(2(1(4(x1)))))) -> 1(1(5(3(2(2(0(4(x1))))))))
   0(5(5(1(2(5(x1)))))) -> 0(2(1(5(5(4(5(x1)))))))
   1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1)))))))
   1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1))))))))
   1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1)))))))
   1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1))))))))
   1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1)))))))
   1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1)))))))
   1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1)))))))
   0(0(1(2(3(5(5(x1))))))) -> 5(1(5(0(3(0(2(1(x1))))))))
   0(0(2(3(4(2(1(x1))))))) -> 0(0(4(1(3(2(3(2(x1))))))))
   0(1(0(0(5(3(4(x1))))))) -> 0(3(0(5(0(4(3(1(x1))))))))
   0(1(2(4(4(0(5(x1))))))) -> 5(4(0(1(0(3(2(4(x1))))))))
   0(5(2(5(1(3(4(x1))))))) -> 1(4(5(2(0(3(1(5(x1))))))))
   0(5(5(0(2(5(1(x1))))))) -> 5(3(0(0(1(5(2(5(x1))))))))
   0(5(5(2(5(3(4(x1))))))) -> 0(3(2(1(4(5(5(5(x1))))))))
   1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1))))))))
   1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1))))))))
   1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1))))))))
   1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(1(5(4(x1))))))))

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

(17) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------

(18)
YES

----------------------------------------

(19)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   0^1(5(0(5(x1)))) -> 0^1(5(5(x1)))
   0^1(1(2(5(0(x1))))) -> 0^1(0(1(5(x1))))
   0^1(1(2(5(0(x1))))) -> 0^1(1(5(x1)))
   0^1(5(0(5(1(x1))))) -> 0^1(1(0(3(5(5(x1))))))
   0^1(0(0(5(1(2(x1)))))) -> 0^1(0(1(5(0(2(4(x1)))))))
   0^1(0(0(5(1(2(x1)))))) -> 0^1(1(5(0(2(4(x1))))))
   0^1(0(5(5(3(4(x1)))))) -> 0^1(0(3(5(5(x1)))))
   0^1(1(2(0(1(2(x1)))))) -> 0^1(x1)
   0^1(0(5(2(3(4(x1)))))) -> 1^1(2(5(x1)))
   1^1(2(1(2(0(0(x1)))))) -> 0^1(1(x1))
   0^1(5(0(0(5(4(x1)))))) -> 0^1(1(0(1(0(4(5(5(x1))))))))
   0^1(5(0(0(5(4(x1)))))) -> 0^1(1(0(4(5(5(x1))))))
   1^1(2(1(2(0(0(x1)))))) -> 1^1(x1)

The TRS R consists of the following rules:

   0(0(1(2(x1)))) -> 0(3(1(0(2(x1)))))
   0(1(2(2(x1)))) -> 1(2(0(3(2(2(x1))))))
   0(1(2(4(x1)))) -> 0(3(2(3(1(4(x1))))))
   0(5(0(5(x1)))) -> 0(3(0(5(5(x1)))))
   0(5(1(2(x1)))) -> 1(0(1(5(2(x1)))))
   0(5(1(2(x1)))) -> 0(1(0(1(5(2(x1))))))
   0(5(1(2(x1)))) -> 0(3(2(3(1(5(x1))))))
   0(5(4(2(x1)))) -> 0(4(5(3(2(x1)))))
   0(5(5(2(x1)))) -> 5(0(1(5(2(x1)))))
   1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1)))))))
   1(0(1(2(x1)))) -> 1(1(3(0(2(x1)))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1))))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1))))))
   1(0(5(4(x1)))) -> 0(1(1(5(4(x1)))))
   1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1))))))
   1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1))))))
   1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1)))))))
   1(5(1(2(x1)))) -> 0(1(1(5(2(x1)))))
   1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1))))))
   5(0(0(2(x1)))) -> 5(0(3(0(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1))))))
   0(0(0(1(2(x1))))) -> 0(2(0(1(0(3(3(4(x1))))))))
   0(0(2(5(2(x1))))) -> 0(3(2(0(5(2(x1))))))
   0(1(2(5(0(x1))))) -> 3(3(2(2(0(0(1(5(x1))))))))
   0(1(2(5(2(x1))))) -> 0(3(2(1(5(3(2(x1)))))))
   0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1))))))
   0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1)))))))
   0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1)))))))
   0(5(0(2(2(x1))))) -> 0(2(5(0(3(2(x1))))))
   0(5(0(5(1(x1))))) -> 0(1(0(3(5(5(x1))))))
   0(5(1(3(0(x1))))) -> 0(0(1(1(5(3(x1))))))
   0(5(2(2(4(x1))))) -> 0(5(3(2(2(4(x1))))))
   0(5(2(3(1(x1))))) -> 0(1(5(3(2(2(2(x1)))))))
   0(5(2(4(1(x1))))) -> 0(4(3(2(5(1(x1))))))
   0(5(3(5(2(x1))))) -> 0(0(3(5(5(2(x1))))))
   0(5(5(3(1(x1))))) -> 5(0(1(5(3(3(2(x1)))))))
   1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1))))))
   1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1))))))
   1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1))))))
   1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1))))))
   1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1)))))))
   1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1))))))))
   5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1)))))))
   5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1))))))
   5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1))))))
   0(0(0(5(1(2(x1)))))) -> 0(0(1(5(0(2(4(x1)))))))
   0(0(1(2(4(1(x1)))))) -> 1(3(0(2(3(0(4(1(x1))))))))
   0(0(2(1(2(0(x1)))))) -> 0(1(0(3(2(2(2(0(x1))))))))
   0(0(2(3(0(5(x1)))))) -> 0(0(3(0(3(2(5(x1)))))))
   0(0(5(2(3(4(x1)))))) -> 0(4(0(3(1(2(5(x1)))))))
   0(0(5(5(3(4(x1)))))) -> 1(4(1(0(0(3(5(5(x1))))))))
   0(1(2(0(1(2(x1)))))) -> 1(0(3(2(2(1(1(0(x1))))))))
   0(1(2(2(0(5(x1)))))) -> 0(4(1(5(0(3(2(2(x1))))))))
   0(1(2(5(5(5(x1)))))) -> 0(2(5(1(5(3(5(x1)))))))
   0(1(3(1(5(2(x1)))))) -> 1(0(1(5(3(3(2(x1)))))))
   0(1(4(4(0(5(x1)))))) -> 4(3(0(0(1(5(4(x1)))))))
   0(2(5(3(5(1(x1)))))) -> 0(3(3(2(2(1(5(5(x1))))))))
   0(5(0(0(5(4(x1)))))) -> 0(1(0(1(0(4(5(5(x1))))))))
   0(5(1(2(1(4(x1)))))) -> 1(1(5(3(2(2(0(4(x1))))))))
   0(5(5(1(2(5(x1)))))) -> 0(2(1(5(5(4(5(x1)))))))
   1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1)))))))
   1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1))))))))
   1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1)))))))
   1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1))))))))
   1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1)))))))
   1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1)))))))
   1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1)))))))
   0(0(1(2(3(5(5(x1))))))) -> 5(1(5(0(3(0(2(1(x1))))))))
   0(0(2(3(4(2(1(x1))))))) -> 0(0(4(1(3(2(3(2(x1))))))))
   0(1(0(0(5(3(4(x1))))))) -> 0(3(0(5(0(4(3(1(x1))))))))
   0(1(2(4(4(0(5(x1))))))) -> 5(4(0(1(0(3(2(4(x1))))))))
   0(5(2(5(1(3(4(x1))))))) -> 1(4(5(2(0(3(1(5(x1))))))))
   0(5(5(0(2(5(1(x1))))))) -> 5(3(0(0(1(5(2(5(x1))))))))
   0(5(5(2(5(3(4(x1))))))) -> 0(3(2(1(4(5(5(5(x1))))))))
   1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1))))))))
   1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1))))))))
   1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1))))))))
   1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(1(5(4(x1))))))))

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

(20) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   1^1(2(1(2(0(0(x1)))))) -> 0^1(1(x1))
   1^1(2(1(2(0(0(x1)))))) -> 1^1(x1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(0(x_1)) = 1 + x_1
   POL(0^1(x_1)) = 0
   POL(1(x_1)) = x_1
   POL(1^1(x_1)) = x_1
   POL(2(x_1)) = x_1
   POL(3(x_1)) = 0
   POL(4(x_1)) = 0
   POL(5(x_1)) = 0

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

   5(0(0(2(x1)))) -> 5(0(3(0(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1))))))
   5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1)))))))
   5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1))))))
   5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1))))))


----------------------------------------

(21)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   0^1(5(0(5(x1)))) -> 0^1(5(5(x1)))
   0^1(1(2(5(0(x1))))) -> 0^1(0(1(5(x1))))
   0^1(1(2(5(0(x1))))) -> 0^1(1(5(x1)))
   0^1(5(0(5(1(x1))))) -> 0^1(1(0(3(5(5(x1))))))
   0^1(0(0(5(1(2(x1)))))) -> 0^1(0(1(5(0(2(4(x1)))))))
   0^1(0(0(5(1(2(x1)))))) -> 0^1(1(5(0(2(4(x1))))))
   0^1(0(5(5(3(4(x1)))))) -> 0^1(0(3(5(5(x1)))))
   0^1(1(2(0(1(2(x1)))))) -> 0^1(x1)
   0^1(0(5(2(3(4(x1)))))) -> 1^1(2(5(x1)))
   0^1(5(0(0(5(4(x1)))))) -> 0^1(1(0(1(0(4(5(5(x1))))))))
   0^1(5(0(0(5(4(x1)))))) -> 0^1(1(0(4(5(5(x1))))))

The TRS R consists of the following rules:

   0(0(1(2(x1)))) -> 0(3(1(0(2(x1)))))
   0(1(2(2(x1)))) -> 1(2(0(3(2(2(x1))))))
   0(1(2(4(x1)))) -> 0(3(2(3(1(4(x1))))))
   0(5(0(5(x1)))) -> 0(3(0(5(5(x1)))))
   0(5(1(2(x1)))) -> 1(0(1(5(2(x1)))))
   0(5(1(2(x1)))) -> 0(1(0(1(5(2(x1))))))
   0(5(1(2(x1)))) -> 0(3(2(3(1(5(x1))))))
   0(5(4(2(x1)))) -> 0(4(5(3(2(x1)))))
   0(5(5(2(x1)))) -> 5(0(1(5(2(x1)))))
   1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1)))))))
   1(0(1(2(x1)))) -> 1(1(3(0(2(x1)))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1))))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1))))))
   1(0(5(4(x1)))) -> 0(1(1(5(4(x1)))))
   1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1))))))
   1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1))))))
   1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1)))))))
   1(5(1(2(x1)))) -> 0(1(1(5(2(x1)))))
   1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1))))))
   5(0(0(2(x1)))) -> 5(0(3(0(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1))))))
   0(0(0(1(2(x1))))) -> 0(2(0(1(0(3(3(4(x1))))))))
   0(0(2(5(2(x1))))) -> 0(3(2(0(5(2(x1))))))
   0(1(2(5(0(x1))))) -> 3(3(2(2(0(0(1(5(x1))))))))
   0(1(2(5(2(x1))))) -> 0(3(2(1(5(3(2(x1)))))))
   0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1))))))
   0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1)))))))
   0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1)))))))
   0(5(0(2(2(x1))))) -> 0(2(5(0(3(2(x1))))))
   0(5(0(5(1(x1))))) -> 0(1(0(3(5(5(x1))))))
   0(5(1(3(0(x1))))) -> 0(0(1(1(5(3(x1))))))
   0(5(2(2(4(x1))))) -> 0(5(3(2(2(4(x1))))))
   0(5(2(3(1(x1))))) -> 0(1(5(3(2(2(2(x1)))))))
   0(5(2(4(1(x1))))) -> 0(4(3(2(5(1(x1))))))
   0(5(3(5(2(x1))))) -> 0(0(3(5(5(2(x1))))))
   0(5(5(3(1(x1))))) -> 5(0(1(5(3(3(2(x1)))))))
   1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1))))))
   1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1))))))
   1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1))))))
   1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1))))))
   1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1)))))))
   1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1))))))))
   5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1)))))))
   5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1))))))
   5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1))))))
   0(0(0(5(1(2(x1)))))) -> 0(0(1(5(0(2(4(x1)))))))
   0(0(1(2(4(1(x1)))))) -> 1(3(0(2(3(0(4(1(x1))))))))
   0(0(2(1(2(0(x1)))))) -> 0(1(0(3(2(2(2(0(x1))))))))
   0(0(2(3(0(5(x1)))))) -> 0(0(3(0(3(2(5(x1)))))))
   0(0(5(2(3(4(x1)))))) -> 0(4(0(3(1(2(5(x1)))))))
   0(0(5(5(3(4(x1)))))) -> 1(4(1(0(0(3(5(5(x1))))))))
   0(1(2(0(1(2(x1)))))) -> 1(0(3(2(2(1(1(0(x1))))))))
   0(1(2(2(0(5(x1)))))) -> 0(4(1(5(0(3(2(2(x1))))))))
   0(1(2(5(5(5(x1)))))) -> 0(2(5(1(5(3(5(x1)))))))
   0(1(3(1(5(2(x1)))))) -> 1(0(1(5(3(3(2(x1)))))))
   0(1(4(4(0(5(x1)))))) -> 4(3(0(0(1(5(4(x1)))))))
   0(2(5(3(5(1(x1)))))) -> 0(3(3(2(2(1(5(5(x1))))))))
   0(5(0(0(5(4(x1)))))) -> 0(1(0(1(0(4(5(5(x1))))))))
   0(5(1(2(1(4(x1)))))) -> 1(1(5(3(2(2(0(4(x1))))))))
   0(5(5(1(2(5(x1)))))) -> 0(2(1(5(5(4(5(x1)))))))
   1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1)))))))
   1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1))))))))
   1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1)))))))
   1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1))))))))
   1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1)))))))
   1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1)))))))
   1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1)))))))
   0(0(1(2(3(5(5(x1))))))) -> 5(1(5(0(3(0(2(1(x1))))))))
   0(0(2(3(4(2(1(x1))))))) -> 0(0(4(1(3(2(3(2(x1))))))))
   0(1(0(0(5(3(4(x1))))))) -> 0(3(0(5(0(4(3(1(x1))))))))
   0(1(2(4(4(0(5(x1))))))) -> 5(4(0(1(0(3(2(4(x1))))))))
   0(5(2(5(1(3(4(x1))))))) -> 1(4(5(2(0(3(1(5(x1))))))))
   0(5(5(0(2(5(1(x1))))))) -> 5(3(0(0(1(5(2(5(x1))))))))
   0(5(5(2(5(3(4(x1))))))) -> 0(3(2(1(4(5(5(5(x1))))))))
   1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1))))))))
   1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1))))))))
   1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1))))))))
   1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(1(5(4(x1))))))))

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

(22) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(23)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   0^1(1(2(5(0(x1))))) -> 0^1(0(1(5(x1))))
   0^1(5(0(5(x1)))) -> 0^1(5(5(x1)))
   0^1(1(2(5(0(x1))))) -> 0^1(1(5(x1)))
   0^1(5(0(5(1(x1))))) -> 0^1(1(0(3(5(5(x1))))))
   0^1(0(0(5(1(2(x1)))))) -> 0^1(0(1(5(0(2(4(x1)))))))
   0^1(0(0(5(1(2(x1)))))) -> 0^1(1(5(0(2(4(x1))))))
   0^1(0(5(5(3(4(x1)))))) -> 0^1(0(3(5(5(x1)))))
   0^1(1(2(0(1(2(x1)))))) -> 0^1(x1)
   0^1(5(0(0(5(4(x1)))))) -> 0^1(1(0(1(0(4(5(5(x1))))))))
   0^1(5(0(0(5(4(x1)))))) -> 0^1(1(0(4(5(5(x1))))))

The TRS R consists of the following rules:

   0(0(1(2(x1)))) -> 0(3(1(0(2(x1)))))
   0(1(2(2(x1)))) -> 1(2(0(3(2(2(x1))))))
   0(1(2(4(x1)))) -> 0(3(2(3(1(4(x1))))))
   0(5(0(5(x1)))) -> 0(3(0(5(5(x1)))))
   0(5(1(2(x1)))) -> 1(0(1(5(2(x1)))))
   0(5(1(2(x1)))) -> 0(1(0(1(5(2(x1))))))
   0(5(1(2(x1)))) -> 0(3(2(3(1(5(x1))))))
   0(5(4(2(x1)))) -> 0(4(5(3(2(x1)))))
   0(5(5(2(x1)))) -> 5(0(1(5(2(x1)))))
   1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1)))))))
   1(0(1(2(x1)))) -> 1(1(3(0(2(x1)))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1))))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1))))))
   1(0(5(4(x1)))) -> 0(1(1(5(4(x1)))))
   1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1))))))
   1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1))))))
   1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1)))))))
   1(5(1(2(x1)))) -> 0(1(1(5(2(x1)))))
   1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1))))))
   5(0(0(2(x1)))) -> 5(0(3(0(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1))))))
   0(0(0(1(2(x1))))) -> 0(2(0(1(0(3(3(4(x1))))))))
   0(0(2(5(2(x1))))) -> 0(3(2(0(5(2(x1))))))
   0(1(2(5(0(x1))))) -> 3(3(2(2(0(0(1(5(x1))))))))
   0(1(2(5(2(x1))))) -> 0(3(2(1(5(3(2(x1)))))))
   0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1))))))
   0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1)))))))
   0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1)))))))
   0(5(0(2(2(x1))))) -> 0(2(5(0(3(2(x1))))))
   0(5(0(5(1(x1))))) -> 0(1(0(3(5(5(x1))))))
   0(5(1(3(0(x1))))) -> 0(0(1(1(5(3(x1))))))
   0(5(2(2(4(x1))))) -> 0(5(3(2(2(4(x1))))))
   0(5(2(3(1(x1))))) -> 0(1(5(3(2(2(2(x1)))))))
   0(5(2(4(1(x1))))) -> 0(4(3(2(5(1(x1))))))
   0(5(3(5(2(x1))))) -> 0(0(3(5(5(2(x1))))))
   0(5(5(3(1(x1))))) -> 5(0(1(5(3(3(2(x1)))))))
   1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1))))))
   1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1))))))
   1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1))))))
   1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1))))))
   1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1)))))))
   1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1))))))))
   5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1)))))))
   5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1))))))
   5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1))))))
   0(0(0(5(1(2(x1)))))) -> 0(0(1(5(0(2(4(x1)))))))
   0(0(1(2(4(1(x1)))))) -> 1(3(0(2(3(0(4(1(x1))))))))
   0(0(2(1(2(0(x1)))))) -> 0(1(0(3(2(2(2(0(x1))))))))
   0(0(2(3(0(5(x1)))))) -> 0(0(3(0(3(2(5(x1)))))))
   0(0(5(2(3(4(x1)))))) -> 0(4(0(3(1(2(5(x1)))))))
   0(0(5(5(3(4(x1)))))) -> 1(4(1(0(0(3(5(5(x1))))))))
   0(1(2(0(1(2(x1)))))) -> 1(0(3(2(2(1(1(0(x1))))))))
   0(1(2(2(0(5(x1)))))) -> 0(4(1(5(0(3(2(2(x1))))))))
   0(1(2(5(5(5(x1)))))) -> 0(2(5(1(5(3(5(x1)))))))
   0(1(3(1(5(2(x1)))))) -> 1(0(1(5(3(3(2(x1)))))))
   0(1(4(4(0(5(x1)))))) -> 4(3(0(0(1(5(4(x1)))))))
   0(2(5(3(5(1(x1)))))) -> 0(3(3(2(2(1(5(5(x1))))))))
   0(5(0(0(5(4(x1)))))) -> 0(1(0(1(0(4(5(5(x1))))))))
   0(5(1(2(1(4(x1)))))) -> 1(1(5(3(2(2(0(4(x1))))))))
   0(5(5(1(2(5(x1)))))) -> 0(2(1(5(5(4(5(x1)))))))
   1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1)))))))
   1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1))))))))
   1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1)))))))
   1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1))))))))
   1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1)))))))
   1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1)))))))
   1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1)))))))
   0(0(1(2(3(5(5(x1))))))) -> 5(1(5(0(3(0(2(1(x1))))))))
   0(0(2(3(4(2(1(x1))))))) -> 0(0(4(1(3(2(3(2(x1))))))))
   0(1(0(0(5(3(4(x1))))))) -> 0(3(0(5(0(4(3(1(x1))))))))
   0(1(2(4(4(0(5(x1))))))) -> 5(4(0(1(0(3(2(4(x1))))))))
   0(5(2(5(1(3(4(x1))))))) -> 1(4(5(2(0(3(1(5(x1))))))))
   0(5(5(0(2(5(1(x1))))))) -> 5(3(0(0(1(5(2(5(x1))))))))
   0(5(5(2(5(3(4(x1))))))) -> 0(3(2(1(4(5(5(5(x1))))))))
   1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1))))))))
   1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1))))))))
   1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1))))))))
   1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(1(5(4(x1))))))))

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

(24) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   0^1(1(2(5(0(x1))))) -> 0^1(0(1(5(x1))))
   0^1(1(2(5(0(x1))))) -> 0^1(1(5(x1)))
   0^1(1(2(0(1(2(x1)))))) -> 0^1(x1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(0(x_1)) = x_1
   POL(0^1(x_1)) = x_1
   POL(1(x_1)) = x_1
   POL(2(x_1)) = 1 + x_1
   POL(3(x_1)) = 0
   POL(4(x_1)) = 0
   POL(5(x_1)) = x_1

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

   5(0(0(2(x1)))) -> 5(0(3(0(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1))))))
   5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1)))))))
   5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1))))))
   5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1))))))
   1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1)))))))
   1(0(1(2(x1)))) -> 1(1(3(0(2(x1)))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1))))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1))))))
   1(0(5(4(x1)))) -> 0(1(1(5(4(x1)))))
   1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1))))))
   1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1))))))
   1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1)))))))
   1(5(1(2(x1)))) -> 0(1(1(5(2(x1)))))
   1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1))))))
   1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1))))))
   1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1))))))
   1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1))))))
   1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1))))))
   1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1)))))))
   1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1))))))))
   1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1)))))))
   1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1))))))))
   1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1)))))))
   1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1))))))))
   1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1)))))))
   1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1)))))))
   1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1)))))))
   1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1))))))))
   1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1))))))))
   1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1))))))))
   1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(1(5(4(x1))))))))
   0(0(1(2(x1)))) -> 0(3(1(0(2(x1)))))
   0(1(2(2(x1)))) -> 1(2(0(3(2(2(x1))))))
   0(1(2(4(x1)))) -> 0(3(2(3(1(4(x1))))))
   0(5(0(5(x1)))) -> 0(3(0(5(5(x1)))))
   0(5(1(2(x1)))) -> 1(0(1(5(2(x1)))))
   0(5(1(2(x1)))) -> 0(1(0(1(5(2(x1))))))
   0(5(1(2(x1)))) -> 0(3(2(3(1(5(x1))))))
   0(5(4(2(x1)))) -> 0(4(5(3(2(x1)))))
   0(5(5(2(x1)))) -> 5(0(1(5(2(x1)))))
   0(0(0(1(2(x1))))) -> 0(2(0(1(0(3(3(4(x1))))))))
   0(0(2(5(2(x1))))) -> 0(3(2(0(5(2(x1))))))
   0(1(2(5(0(x1))))) -> 3(3(2(2(0(0(1(5(x1))))))))
   0(1(2(5(2(x1))))) -> 0(3(2(1(5(3(2(x1)))))))
   0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1))))))
   0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1)))))))
   0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1)))))))
   0(5(0(2(2(x1))))) -> 0(2(5(0(3(2(x1))))))
   0(0(0(5(1(2(x1)))))) -> 0(0(1(5(0(2(4(x1)))))))
   0(5(0(5(1(x1))))) -> 0(1(0(3(5(5(x1))))))
   0(5(0(0(5(4(x1)))))) -> 0(1(0(1(0(4(5(5(x1))))))))
   0(5(1(3(0(x1))))) -> 0(0(1(1(5(3(x1))))))
   0(5(2(2(4(x1))))) -> 0(5(3(2(2(4(x1))))))
   0(5(2(3(1(x1))))) -> 0(1(5(3(2(2(2(x1)))))))
   0(5(2(4(1(x1))))) -> 0(4(3(2(5(1(x1))))))
   0(5(3(5(2(x1))))) -> 0(0(3(5(5(2(x1))))))
   0(5(5(3(1(x1))))) -> 5(0(1(5(3(3(2(x1)))))))
   0(0(1(2(4(1(x1)))))) -> 1(3(0(2(3(0(4(1(x1))))))))
   0(0(2(1(2(0(x1)))))) -> 0(1(0(3(2(2(2(0(x1))))))))
   0(0(2(3(0(5(x1)))))) -> 0(0(3(0(3(2(5(x1)))))))
   0(0(5(2(3(4(x1)))))) -> 0(4(0(3(1(2(5(x1)))))))
   0(0(5(5(3(4(x1)))))) -> 1(4(1(0(0(3(5(5(x1))))))))
   0(1(2(0(1(2(x1)))))) -> 1(0(3(2(2(1(1(0(x1))))))))
   0(1(2(2(0(5(x1)))))) -> 0(4(1(5(0(3(2(2(x1))))))))
   0(1(2(5(5(5(x1)))))) -> 0(2(5(1(5(3(5(x1)))))))
   0(1(3(1(5(2(x1)))))) -> 1(0(1(5(3(3(2(x1)))))))
   0(1(4(4(0(5(x1)))))) -> 4(3(0(0(1(5(4(x1)))))))
   0(2(5(3(5(1(x1)))))) -> 0(3(3(2(2(1(5(5(x1))))))))
   0(5(1(2(1(4(x1)))))) -> 1(1(5(3(2(2(0(4(x1))))))))
   0(5(5(1(2(5(x1)))))) -> 0(2(1(5(5(4(5(x1)))))))
   0(0(1(2(3(5(5(x1))))))) -> 5(1(5(0(3(0(2(1(x1))))))))
   0(0(2(3(4(2(1(x1))))))) -> 0(0(4(1(3(2(3(2(x1))))))))
   0(1(0(0(5(3(4(x1))))))) -> 0(3(0(5(0(4(3(1(x1))))))))
   0(1(2(4(4(0(5(x1))))))) -> 5(4(0(1(0(3(2(4(x1))))))))
   0(5(2(5(1(3(4(x1))))))) -> 1(4(5(2(0(3(1(5(x1))))))))
   0(5(5(0(2(5(1(x1))))))) -> 5(3(0(0(1(5(2(5(x1))))))))
   0(5(5(2(5(3(4(x1))))))) -> 0(3(2(1(4(5(5(5(x1))))))))


----------------------------------------

(25)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   0^1(5(0(5(x1)))) -> 0^1(5(5(x1)))
   0^1(5(0(5(1(x1))))) -> 0^1(1(0(3(5(5(x1))))))
   0^1(0(0(5(1(2(x1)))))) -> 0^1(0(1(5(0(2(4(x1)))))))
   0^1(0(0(5(1(2(x1)))))) -> 0^1(1(5(0(2(4(x1))))))
   0^1(0(5(5(3(4(x1)))))) -> 0^1(0(3(5(5(x1)))))
   0^1(5(0(0(5(4(x1)))))) -> 0^1(1(0(1(0(4(5(5(x1))))))))
   0^1(5(0(0(5(4(x1)))))) -> 0^1(1(0(4(5(5(x1))))))

The TRS R consists of the following rules:

   0(0(1(2(x1)))) -> 0(3(1(0(2(x1)))))
   0(1(2(2(x1)))) -> 1(2(0(3(2(2(x1))))))
   0(1(2(4(x1)))) -> 0(3(2(3(1(4(x1))))))
   0(5(0(5(x1)))) -> 0(3(0(5(5(x1)))))
   0(5(1(2(x1)))) -> 1(0(1(5(2(x1)))))
   0(5(1(2(x1)))) -> 0(1(0(1(5(2(x1))))))
   0(5(1(2(x1)))) -> 0(3(2(3(1(5(x1))))))
   0(5(4(2(x1)))) -> 0(4(5(3(2(x1)))))
   0(5(5(2(x1)))) -> 5(0(1(5(2(x1)))))
   1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1)))))))
   1(0(1(2(x1)))) -> 1(1(3(0(2(x1)))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1))))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1))))))
   1(0(5(4(x1)))) -> 0(1(1(5(4(x1)))))
   1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1))))))
   1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1))))))
   1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1)))))))
   1(5(1(2(x1)))) -> 0(1(1(5(2(x1)))))
   1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1))))))
   5(0(0(2(x1)))) -> 5(0(3(0(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1))))))
   0(0(0(1(2(x1))))) -> 0(2(0(1(0(3(3(4(x1))))))))
   0(0(2(5(2(x1))))) -> 0(3(2(0(5(2(x1))))))
   0(1(2(5(0(x1))))) -> 3(3(2(2(0(0(1(5(x1))))))))
   0(1(2(5(2(x1))))) -> 0(3(2(1(5(3(2(x1)))))))
   0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1))))))
   0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1)))))))
   0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1)))))))
   0(5(0(2(2(x1))))) -> 0(2(5(0(3(2(x1))))))
   0(5(0(5(1(x1))))) -> 0(1(0(3(5(5(x1))))))
   0(5(1(3(0(x1))))) -> 0(0(1(1(5(3(x1))))))
   0(5(2(2(4(x1))))) -> 0(5(3(2(2(4(x1))))))
   0(5(2(3(1(x1))))) -> 0(1(5(3(2(2(2(x1)))))))
   0(5(2(4(1(x1))))) -> 0(4(3(2(5(1(x1))))))
   0(5(3(5(2(x1))))) -> 0(0(3(5(5(2(x1))))))
   0(5(5(3(1(x1))))) -> 5(0(1(5(3(3(2(x1)))))))
   1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1))))))
   1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1))))))
   1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1))))))
   1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1))))))
   1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1)))))))
   1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1))))))))
   5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1)))))))
   5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1))))))
   5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1))))))
   0(0(0(5(1(2(x1)))))) -> 0(0(1(5(0(2(4(x1)))))))
   0(0(1(2(4(1(x1)))))) -> 1(3(0(2(3(0(4(1(x1))))))))
   0(0(2(1(2(0(x1)))))) -> 0(1(0(3(2(2(2(0(x1))))))))
   0(0(2(3(0(5(x1)))))) -> 0(0(3(0(3(2(5(x1)))))))
   0(0(5(2(3(4(x1)))))) -> 0(4(0(3(1(2(5(x1)))))))
   0(0(5(5(3(4(x1)))))) -> 1(4(1(0(0(3(5(5(x1))))))))
   0(1(2(0(1(2(x1)))))) -> 1(0(3(2(2(1(1(0(x1))))))))
   0(1(2(2(0(5(x1)))))) -> 0(4(1(5(0(3(2(2(x1))))))))
   0(1(2(5(5(5(x1)))))) -> 0(2(5(1(5(3(5(x1)))))))
   0(1(3(1(5(2(x1)))))) -> 1(0(1(5(3(3(2(x1)))))))
   0(1(4(4(0(5(x1)))))) -> 4(3(0(0(1(5(4(x1)))))))
   0(2(5(3(5(1(x1)))))) -> 0(3(3(2(2(1(5(5(x1))))))))
   0(5(0(0(5(4(x1)))))) -> 0(1(0(1(0(4(5(5(x1))))))))
   0(5(1(2(1(4(x1)))))) -> 1(1(5(3(2(2(0(4(x1))))))))
   0(5(5(1(2(5(x1)))))) -> 0(2(1(5(5(4(5(x1)))))))
   1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1)))))))
   1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1))))))))
   1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1)))))))
   1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1))))))))
   1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1)))))))
   1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1)))))))
   1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1)))))))
   0(0(1(2(3(5(5(x1))))))) -> 5(1(5(0(3(0(2(1(x1))))))))
   0(0(2(3(4(2(1(x1))))))) -> 0(0(4(1(3(2(3(2(x1))))))))
   0(1(0(0(5(3(4(x1))))))) -> 0(3(0(5(0(4(3(1(x1))))))))
   0(1(2(4(4(0(5(x1))))))) -> 5(4(0(1(0(3(2(4(x1))))))))
   0(5(2(5(1(3(4(x1))))))) -> 1(4(5(2(0(3(1(5(x1))))))))
   0(5(5(0(2(5(1(x1))))))) -> 5(3(0(0(1(5(2(5(x1))))))))
   0(5(5(2(5(3(4(x1))))))) -> 0(3(2(1(4(5(5(5(x1))))))))
   1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1))))))))
   1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1))))))))
   1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1))))))))
   1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(1(5(4(x1))))))))

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

(26) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule 0^1(0(0(5(1(2(x1)))))) -> 0^1(0(1(5(0(2(4(x1))))))) at position [0] we obtained the following new rules [LPAR04]:

   (0^1(0(0(5(1(2(y0)))))) -> 0^1(0(1(1(0(1(1(5(2(4(y0)))))))))),0^1(0(0(5(1(2(y0)))))) -> 0^1(0(1(1(0(1(1(5(2(4(y0)))))))))))


----------------------------------------

(27)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   0^1(5(0(5(x1)))) -> 0^1(5(5(x1)))
   0^1(5(0(5(1(x1))))) -> 0^1(1(0(3(5(5(x1))))))
   0^1(0(0(5(1(2(x1)))))) -> 0^1(1(5(0(2(4(x1))))))
   0^1(0(5(5(3(4(x1)))))) -> 0^1(0(3(5(5(x1)))))
   0^1(5(0(0(5(4(x1)))))) -> 0^1(1(0(1(0(4(5(5(x1))))))))
   0^1(5(0(0(5(4(x1)))))) -> 0^1(1(0(4(5(5(x1))))))
   0^1(0(0(5(1(2(y0)))))) -> 0^1(0(1(1(0(1(1(5(2(4(y0))))))))))

The TRS R consists of the following rules:

   0(0(1(2(x1)))) -> 0(3(1(0(2(x1)))))
   0(1(2(2(x1)))) -> 1(2(0(3(2(2(x1))))))
   0(1(2(4(x1)))) -> 0(3(2(3(1(4(x1))))))
   0(5(0(5(x1)))) -> 0(3(0(5(5(x1)))))
   0(5(1(2(x1)))) -> 1(0(1(5(2(x1)))))
   0(5(1(2(x1)))) -> 0(1(0(1(5(2(x1))))))
   0(5(1(2(x1)))) -> 0(3(2(3(1(5(x1))))))
   0(5(4(2(x1)))) -> 0(4(5(3(2(x1)))))
   0(5(5(2(x1)))) -> 5(0(1(5(2(x1)))))
   1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1)))))))
   1(0(1(2(x1)))) -> 1(1(3(0(2(x1)))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1))))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1))))))
   1(0(5(4(x1)))) -> 0(1(1(5(4(x1)))))
   1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1))))))
   1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1))))))
   1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1)))))))
   1(5(1(2(x1)))) -> 0(1(1(5(2(x1)))))
   1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1))))))
   5(0(0(2(x1)))) -> 5(0(3(0(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1))))))
   0(0(0(1(2(x1))))) -> 0(2(0(1(0(3(3(4(x1))))))))
   0(0(2(5(2(x1))))) -> 0(3(2(0(5(2(x1))))))
   0(1(2(5(0(x1))))) -> 3(3(2(2(0(0(1(5(x1))))))))
   0(1(2(5(2(x1))))) -> 0(3(2(1(5(3(2(x1)))))))
   0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1))))))
   0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1)))))))
   0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1)))))))
   0(5(0(2(2(x1))))) -> 0(2(5(0(3(2(x1))))))
   0(5(0(5(1(x1))))) -> 0(1(0(3(5(5(x1))))))
   0(5(1(3(0(x1))))) -> 0(0(1(1(5(3(x1))))))
   0(5(2(2(4(x1))))) -> 0(5(3(2(2(4(x1))))))
   0(5(2(3(1(x1))))) -> 0(1(5(3(2(2(2(x1)))))))
   0(5(2(4(1(x1))))) -> 0(4(3(2(5(1(x1))))))
   0(5(3(5(2(x1))))) -> 0(0(3(5(5(2(x1))))))
   0(5(5(3(1(x1))))) -> 5(0(1(5(3(3(2(x1)))))))
   1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1))))))
   1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1))))))
   1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1))))))
   1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1))))))
   1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1)))))))
   1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1))))))))
   5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1)))))))
   5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1))))))
   5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1))))))
   0(0(0(5(1(2(x1)))))) -> 0(0(1(5(0(2(4(x1)))))))
   0(0(1(2(4(1(x1)))))) -> 1(3(0(2(3(0(4(1(x1))))))))
   0(0(2(1(2(0(x1)))))) -> 0(1(0(3(2(2(2(0(x1))))))))
   0(0(2(3(0(5(x1)))))) -> 0(0(3(0(3(2(5(x1)))))))
   0(0(5(2(3(4(x1)))))) -> 0(4(0(3(1(2(5(x1)))))))
   0(0(5(5(3(4(x1)))))) -> 1(4(1(0(0(3(5(5(x1))))))))
   0(1(2(0(1(2(x1)))))) -> 1(0(3(2(2(1(1(0(x1))))))))
   0(1(2(2(0(5(x1)))))) -> 0(4(1(5(0(3(2(2(x1))))))))
   0(1(2(5(5(5(x1)))))) -> 0(2(5(1(5(3(5(x1)))))))
   0(1(3(1(5(2(x1)))))) -> 1(0(1(5(3(3(2(x1)))))))
   0(1(4(4(0(5(x1)))))) -> 4(3(0(0(1(5(4(x1)))))))
   0(2(5(3(5(1(x1)))))) -> 0(3(3(2(2(1(5(5(x1))))))))
   0(5(0(0(5(4(x1)))))) -> 0(1(0(1(0(4(5(5(x1))))))))
   0(5(1(2(1(4(x1)))))) -> 1(1(5(3(2(2(0(4(x1))))))))
   0(5(5(1(2(5(x1)))))) -> 0(2(1(5(5(4(5(x1)))))))
   1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1)))))))
   1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1))))))))
   1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1)))))))
   1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1))))))))
   1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1)))))))
   1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1)))))))
   1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1)))))))
   0(0(1(2(3(5(5(x1))))))) -> 5(1(5(0(3(0(2(1(x1))))))))
   0(0(2(3(4(2(1(x1))))))) -> 0(0(4(1(3(2(3(2(x1))))))))
   0(1(0(0(5(3(4(x1))))))) -> 0(3(0(5(0(4(3(1(x1))))))))
   0(1(2(4(4(0(5(x1))))))) -> 5(4(0(1(0(3(2(4(x1))))))))
   0(5(2(5(1(3(4(x1))))))) -> 1(4(5(2(0(3(1(5(x1))))))))
   0(5(5(0(2(5(1(x1))))))) -> 5(3(0(0(1(5(2(5(x1))))))))
   0(5(5(2(5(3(4(x1))))))) -> 0(3(2(1(4(5(5(5(x1))))))))
   1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1))))))))
   1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1))))))))
   1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1))))))))
   1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(1(5(4(x1))))))))

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

(28) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(29)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   0^1(5(0(5(1(x1))))) -> 0^1(1(0(3(5(5(x1))))))
   0^1(5(0(5(x1)))) -> 0^1(5(5(x1)))
   0^1(0(0(5(1(2(x1)))))) -> 0^1(1(5(0(2(4(x1))))))
   0^1(0(5(5(3(4(x1)))))) -> 0^1(0(3(5(5(x1)))))
   0^1(5(0(0(5(4(x1)))))) -> 0^1(1(0(1(0(4(5(5(x1))))))))
   0^1(5(0(0(5(4(x1)))))) -> 0^1(1(0(4(5(5(x1))))))

The TRS R consists of the following rules:

   0(0(1(2(x1)))) -> 0(3(1(0(2(x1)))))
   0(1(2(2(x1)))) -> 1(2(0(3(2(2(x1))))))
   0(1(2(4(x1)))) -> 0(3(2(3(1(4(x1))))))
   0(5(0(5(x1)))) -> 0(3(0(5(5(x1)))))
   0(5(1(2(x1)))) -> 1(0(1(5(2(x1)))))
   0(5(1(2(x1)))) -> 0(1(0(1(5(2(x1))))))
   0(5(1(2(x1)))) -> 0(3(2(3(1(5(x1))))))
   0(5(4(2(x1)))) -> 0(4(5(3(2(x1)))))
   0(5(5(2(x1)))) -> 5(0(1(5(2(x1)))))
   1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1)))))))
   1(0(1(2(x1)))) -> 1(1(3(0(2(x1)))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1))))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1))))))
   1(0(5(4(x1)))) -> 0(1(1(5(4(x1)))))
   1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1))))))
   1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1))))))
   1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1)))))))
   1(5(1(2(x1)))) -> 0(1(1(5(2(x1)))))
   1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1))))))
   5(0(0(2(x1)))) -> 5(0(3(0(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1))))))
   0(0(0(1(2(x1))))) -> 0(2(0(1(0(3(3(4(x1))))))))
   0(0(2(5(2(x1))))) -> 0(3(2(0(5(2(x1))))))
   0(1(2(5(0(x1))))) -> 3(3(2(2(0(0(1(5(x1))))))))
   0(1(2(5(2(x1))))) -> 0(3(2(1(5(3(2(x1)))))))
   0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1))))))
   0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1)))))))
   0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1)))))))
   0(5(0(2(2(x1))))) -> 0(2(5(0(3(2(x1))))))
   0(5(0(5(1(x1))))) -> 0(1(0(3(5(5(x1))))))
   0(5(1(3(0(x1))))) -> 0(0(1(1(5(3(x1))))))
   0(5(2(2(4(x1))))) -> 0(5(3(2(2(4(x1))))))
   0(5(2(3(1(x1))))) -> 0(1(5(3(2(2(2(x1)))))))
   0(5(2(4(1(x1))))) -> 0(4(3(2(5(1(x1))))))
   0(5(3(5(2(x1))))) -> 0(0(3(5(5(2(x1))))))
   0(5(5(3(1(x1))))) -> 5(0(1(5(3(3(2(x1)))))))
   1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1))))))
   1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1))))))
   1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1))))))
   1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1))))))
   1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1)))))))
   1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1))))))))
   5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1)))))))
   5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1))))))
   5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1))))))
   0(0(0(5(1(2(x1)))))) -> 0(0(1(5(0(2(4(x1)))))))
   0(0(1(2(4(1(x1)))))) -> 1(3(0(2(3(0(4(1(x1))))))))
   0(0(2(1(2(0(x1)))))) -> 0(1(0(3(2(2(2(0(x1))))))))
   0(0(2(3(0(5(x1)))))) -> 0(0(3(0(3(2(5(x1)))))))
   0(0(5(2(3(4(x1)))))) -> 0(4(0(3(1(2(5(x1)))))))
   0(0(5(5(3(4(x1)))))) -> 1(4(1(0(0(3(5(5(x1))))))))
   0(1(2(0(1(2(x1)))))) -> 1(0(3(2(2(1(1(0(x1))))))))
   0(1(2(2(0(5(x1)))))) -> 0(4(1(5(0(3(2(2(x1))))))))
   0(1(2(5(5(5(x1)))))) -> 0(2(5(1(5(3(5(x1)))))))
   0(1(3(1(5(2(x1)))))) -> 1(0(1(5(3(3(2(x1)))))))
   0(1(4(4(0(5(x1)))))) -> 4(3(0(0(1(5(4(x1)))))))
   0(2(5(3(5(1(x1)))))) -> 0(3(3(2(2(1(5(5(x1))))))))
   0(5(0(0(5(4(x1)))))) -> 0(1(0(1(0(4(5(5(x1))))))))
   0(5(1(2(1(4(x1)))))) -> 1(1(5(3(2(2(0(4(x1))))))))
   0(5(5(1(2(5(x1)))))) -> 0(2(1(5(5(4(5(x1)))))))
   1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1)))))))
   1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1))))))))
   1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1)))))))
   1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1))))))))
   1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1)))))))
   1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1)))))))
   1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1)))))))
   0(0(1(2(3(5(5(x1))))))) -> 5(1(5(0(3(0(2(1(x1))))))))
   0(0(2(3(4(2(1(x1))))))) -> 0(0(4(1(3(2(3(2(x1))))))))
   0(1(0(0(5(3(4(x1))))))) -> 0(3(0(5(0(4(3(1(x1))))))))
   0(1(2(4(4(0(5(x1))))))) -> 5(4(0(1(0(3(2(4(x1))))))))
   0(5(2(5(1(3(4(x1))))))) -> 1(4(5(2(0(3(1(5(x1))))))))
   0(5(5(0(2(5(1(x1))))))) -> 5(3(0(0(1(5(2(5(x1))))))))
   0(5(5(2(5(3(4(x1))))))) -> 0(3(2(1(4(5(5(5(x1))))))))
   1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1))))))))
   1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1))))))))
   1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1))))))))
   1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(1(5(4(x1))))))))

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

(30) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule 0^1(0(0(5(1(2(x1)))))) -> 0^1(1(5(0(2(4(x1)))))) at position [0] we obtained the following new rules [LPAR04]:

   (0^1(0(0(5(1(2(y0)))))) -> 0^1(1(1(0(1(1(5(2(4(y0))))))))),0^1(0(0(5(1(2(y0)))))) -> 0^1(1(1(0(1(1(5(2(4(y0))))))))))


----------------------------------------

(31)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   0^1(5(0(5(1(x1))))) -> 0^1(1(0(3(5(5(x1))))))
   0^1(5(0(5(x1)))) -> 0^1(5(5(x1)))
   0^1(0(5(5(3(4(x1)))))) -> 0^1(0(3(5(5(x1)))))
   0^1(5(0(0(5(4(x1)))))) -> 0^1(1(0(1(0(4(5(5(x1))))))))
   0^1(5(0(0(5(4(x1)))))) -> 0^1(1(0(4(5(5(x1))))))
   0^1(0(0(5(1(2(y0)))))) -> 0^1(1(1(0(1(1(5(2(4(y0)))))))))

The TRS R consists of the following rules:

   0(0(1(2(x1)))) -> 0(3(1(0(2(x1)))))
   0(1(2(2(x1)))) -> 1(2(0(3(2(2(x1))))))
   0(1(2(4(x1)))) -> 0(3(2(3(1(4(x1))))))
   0(5(0(5(x1)))) -> 0(3(0(5(5(x1)))))
   0(5(1(2(x1)))) -> 1(0(1(5(2(x1)))))
   0(5(1(2(x1)))) -> 0(1(0(1(5(2(x1))))))
   0(5(1(2(x1)))) -> 0(3(2(3(1(5(x1))))))
   0(5(4(2(x1)))) -> 0(4(5(3(2(x1)))))
   0(5(5(2(x1)))) -> 5(0(1(5(2(x1)))))
   1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1)))))))
   1(0(1(2(x1)))) -> 1(1(3(0(2(x1)))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1))))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1))))))
   1(0(5(4(x1)))) -> 0(1(1(5(4(x1)))))
   1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1))))))
   1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1))))))
   1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1)))))))
   1(5(1(2(x1)))) -> 0(1(1(5(2(x1)))))
   1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1))))))
   5(0(0(2(x1)))) -> 5(0(3(0(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1))))))
   0(0(0(1(2(x1))))) -> 0(2(0(1(0(3(3(4(x1))))))))
   0(0(2(5(2(x1))))) -> 0(3(2(0(5(2(x1))))))
   0(1(2(5(0(x1))))) -> 3(3(2(2(0(0(1(5(x1))))))))
   0(1(2(5(2(x1))))) -> 0(3(2(1(5(3(2(x1)))))))
   0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1))))))
   0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1)))))))
   0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1)))))))
   0(5(0(2(2(x1))))) -> 0(2(5(0(3(2(x1))))))
   0(5(0(5(1(x1))))) -> 0(1(0(3(5(5(x1))))))
   0(5(1(3(0(x1))))) -> 0(0(1(1(5(3(x1))))))
   0(5(2(2(4(x1))))) -> 0(5(3(2(2(4(x1))))))
   0(5(2(3(1(x1))))) -> 0(1(5(3(2(2(2(x1)))))))
   0(5(2(4(1(x1))))) -> 0(4(3(2(5(1(x1))))))
   0(5(3(5(2(x1))))) -> 0(0(3(5(5(2(x1))))))
   0(5(5(3(1(x1))))) -> 5(0(1(5(3(3(2(x1)))))))
   1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1))))))
   1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1))))))
   1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1))))))
   1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1))))))
   1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1)))))))
   1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1))))))))
   5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1)))))))
   5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1))))))
   5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1))))))
   0(0(0(5(1(2(x1)))))) -> 0(0(1(5(0(2(4(x1)))))))
   0(0(1(2(4(1(x1)))))) -> 1(3(0(2(3(0(4(1(x1))))))))
   0(0(2(1(2(0(x1)))))) -> 0(1(0(3(2(2(2(0(x1))))))))
   0(0(2(3(0(5(x1)))))) -> 0(0(3(0(3(2(5(x1)))))))
   0(0(5(2(3(4(x1)))))) -> 0(4(0(3(1(2(5(x1)))))))
   0(0(5(5(3(4(x1)))))) -> 1(4(1(0(0(3(5(5(x1))))))))
   0(1(2(0(1(2(x1)))))) -> 1(0(3(2(2(1(1(0(x1))))))))
   0(1(2(2(0(5(x1)))))) -> 0(4(1(5(0(3(2(2(x1))))))))
   0(1(2(5(5(5(x1)))))) -> 0(2(5(1(5(3(5(x1)))))))
   0(1(3(1(5(2(x1)))))) -> 1(0(1(5(3(3(2(x1)))))))
   0(1(4(4(0(5(x1)))))) -> 4(3(0(0(1(5(4(x1)))))))
   0(2(5(3(5(1(x1)))))) -> 0(3(3(2(2(1(5(5(x1))))))))
   0(5(0(0(5(4(x1)))))) -> 0(1(0(1(0(4(5(5(x1))))))))
   0(5(1(2(1(4(x1)))))) -> 1(1(5(3(2(2(0(4(x1))))))))
   0(5(5(1(2(5(x1)))))) -> 0(2(1(5(5(4(5(x1)))))))
   1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1)))))))
   1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1))))))))
   1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1)))))))
   1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1))))))))
   1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1)))))))
   1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1)))))))
   1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1)))))))
   0(0(1(2(3(5(5(x1))))))) -> 5(1(5(0(3(0(2(1(x1))))))))
   0(0(2(3(4(2(1(x1))))))) -> 0(0(4(1(3(2(3(2(x1))))))))
   0(1(0(0(5(3(4(x1))))))) -> 0(3(0(5(0(4(3(1(x1))))))))
   0(1(2(4(4(0(5(x1))))))) -> 5(4(0(1(0(3(2(4(x1))))))))
   0(5(2(5(1(3(4(x1))))))) -> 1(4(5(2(0(3(1(5(x1))))))))
   0(5(5(0(2(5(1(x1))))))) -> 5(3(0(0(1(5(2(5(x1))))))))
   0(5(5(2(5(3(4(x1))))))) -> 0(3(2(1(4(5(5(5(x1))))))))
   1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1))))))))
   1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1))))))))
   1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1))))))))
   1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(1(5(4(x1))))))))

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

(32) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(33)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   0^1(5(0(5(x1)))) -> 0^1(5(5(x1)))
   0^1(5(0(5(1(x1))))) -> 0^1(1(0(3(5(5(x1))))))
   0^1(0(5(5(3(4(x1)))))) -> 0^1(0(3(5(5(x1)))))
   0^1(5(0(0(5(4(x1)))))) -> 0^1(1(0(1(0(4(5(5(x1))))))))
   0^1(5(0(0(5(4(x1)))))) -> 0^1(1(0(4(5(5(x1))))))

The TRS R consists of the following rules:

   0(0(1(2(x1)))) -> 0(3(1(0(2(x1)))))
   0(1(2(2(x1)))) -> 1(2(0(3(2(2(x1))))))
   0(1(2(4(x1)))) -> 0(3(2(3(1(4(x1))))))
   0(5(0(5(x1)))) -> 0(3(0(5(5(x1)))))
   0(5(1(2(x1)))) -> 1(0(1(5(2(x1)))))
   0(5(1(2(x1)))) -> 0(1(0(1(5(2(x1))))))
   0(5(1(2(x1)))) -> 0(3(2(3(1(5(x1))))))
   0(5(4(2(x1)))) -> 0(4(5(3(2(x1)))))
   0(5(5(2(x1)))) -> 5(0(1(5(2(x1)))))
   1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1)))))))
   1(0(1(2(x1)))) -> 1(1(3(0(2(x1)))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1))))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1))))))
   1(0(5(4(x1)))) -> 0(1(1(5(4(x1)))))
   1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1))))))
   1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1))))))
   1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1)))))))
   1(5(1(2(x1)))) -> 0(1(1(5(2(x1)))))
   1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1))))))
   5(0(0(2(x1)))) -> 5(0(3(0(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1))))))
   0(0(0(1(2(x1))))) -> 0(2(0(1(0(3(3(4(x1))))))))
   0(0(2(5(2(x1))))) -> 0(3(2(0(5(2(x1))))))
   0(1(2(5(0(x1))))) -> 3(3(2(2(0(0(1(5(x1))))))))
   0(1(2(5(2(x1))))) -> 0(3(2(1(5(3(2(x1)))))))
   0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1))))))
   0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1)))))))
   0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1)))))))
   0(5(0(2(2(x1))))) -> 0(2(5(0(3(2(x1))))))
   0(5(0(5(1(x1))))) -> 0(1(0(3(5(5(x1))))))
   0(5(1(3(0(x1))))) -> 0(0(1(1(5(3(x1))))))
   0(5(2(2(4(x1))))) -> 0(5(3(2(2(4(x1))))))
   0(5(2(3(1(x1))))) -> 0(1(5(3(2(2(2(x1)))))))
   0(5(2(4(1(x1))))) -> 0(4(3(2(5(1(x1))))))
   0(5(3(5(2(x1))))) -> 0(0(3(5(5(2(x1))))))
   0(5(5(3(1(x1))))) -> 5(0(1(5(3(3(2(x1)))))))
   1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1))))))
   1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1))))))
   1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1))))))
   1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1))))))
   1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1)))))))
   1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1))))))))
   5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1)))))))
   5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1))))))
   5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1))))))
   0(0(0(5(1(2(x1)))))) -> 0(0(1(5(0(2(4(x1)))))))
   0(0(1(2(4(1(x1)))))) -> 1(3(0(2(3(0(4(1(x1))))))))
   0(0(2(1(2(0(x1)))))) -> 0(1(0(3(2(2(2(0(x1))))))))
   0(0(2(3(0(5(x1)))))) -> 0(0(3(0(3(2(5(x1)))))))
   0(0(5(2(3(4(x1)))))) -> 0(4(0(3(1(2(5(x1)))))))
   0(0(5(5(3(4(x1)))))) -> 1(4(1(0(0(3(5(5(x1))))))))
   0(1(2(0(1(2(x1)))))) -> 1(0(3(2(2(1(1(0(x1))))))))
   0(1(2(2(0(5(x1)))))) -> 0(4(1(5(0(3(2(2(x1))))))))
   0(1(2(5(5(5(x1)))))) -> 0(2(5(1(5(3(5(x1)))))))
   0(1(3(1(5(2(x1)))))) -> 1(0(1(5(3(3(2(x1)))))))
   0(1(4(4(0(5(x1)))))) -> 4(3(0(0(1(5(4(x1)))))))
   0(2(5(3(5(1(x1)))))) -> 0(3(3(2(2(1(5(5(x1))))))))
   0(5(0(0(5(4(x1)))))) -> 0(1(0(1(0(4(5(5(x1))))))))
   0(5(1(2(1(4(x1)))))) -> 1(1(5(3(2(2(0(4(x1))))))))
   0(5(5(1(2(5(x1)))))) -> 0(2(1(5(5(4(5(x1)))))))
   1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1)))))))
   1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1))))))))
   1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1)))))))
   1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1))))))))
   1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1)))))))
   1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1)))))))
   1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1)))))))
   0(0(1(2(3(5(5(x1))))))) -> 5(1(5(0(3(0(2(1(x1))))))))
   0(0(2(3(4(2(1(x1))))))) -> 0(0(4(1(3(2(3(2(x1))))))))
   0(1(0(0(5(3(4(x1))))))) -> 0(3(0(5(0(4(3(1(x1))))))))
   0(1(2(4(4(0(5(x1))))))) -> 5(4(0(1(0(3(2(4(x1))))))))
   0(5(2(5(1(3(4(x1))))))) -> 1(4(5(2(0(3(1(5(x1))))))))
   0(5(5(0(2(5(1(x1))))))) -> 5(3(0(0(1(5(2(5(x1))))))))
   0(5(5(2(5(3(4(x1))))))) -> 0(3(2(1(4(5(5(5(x1))))))))
   1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1))))))))
   1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1))))))))
   1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1))))))))
   1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(1(5(4(x1))))))))

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

(34) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   0^1(0(5(5(3(4(x1)))))) -> 0^1(0(3(5(5(x1)))))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( 0^1_1(x_1) ) = x_1
POL( 5_1(x_1) ) = 1
POL( 0_1(x_1) ) = max{0, 2x_1 - 1}
POL( 2_1(x_1) ) = 2
POL( 3_1(x_1) ) = max{0, -2}
POL( 1_1(x_1) ) = 1
POL( 4_1(x_1) ) = max{0, x_1 - 1}

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

   5(0(0(2(x1)))) -> 5(0(3(0(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1))))))
   5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1)))))))
   5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1))))))
   5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1))))))
   0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1))))))
   1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1)))))))
   1(0(1(2(x1)))) -> 1(1(3(0(2(x1)))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1))))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1))))))
   1(0(5(4(x1)))) -> 0(1(1(5(4(x1)))))
   1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1))))))
   1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1))))))
   1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1)))))))
   1(5(1(2(x1)))) -> 0(1(1(5(2(x1)))))
   1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1))))))
   1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1))))))
   1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1))))))
   1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1))))))
   1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1))))))
   1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1)))))))
   1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1))))))))
   1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1)))))))
   1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1))))))))
   1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1)))))))
   1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1))))))))
   1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1)))))))
   1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1)))))))
   1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1)))))))
   1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1))))))))
   1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1))))))))
   1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1))))))))
   1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(1(5(4(x1))))))))
   0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1)))))))
   0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1)))))))


----------------------------------------

(35)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   0^1(5(0(5(x1)))) -> 0^1(5(5(x1)))
   0^1(5(0(5(1(x1))))) -> 0^1(1(0(3(5(5(x1))))))
   0^1(5(0(0(5(4(x1)))))) -> 0^1(1(0(1(0(4(5(5(x1))))))))
   0^1(5(0(0(5(4(x1)))))) -> 0^1(1(0(4(5(5(x1))))))

The TRS R consists of the following rules:

   0(0(1(2(x1)))) -> 0(3(1(0(2(x1)))))
   0(1(2(2(x1)))) -> 1(2(0(3(2(2(x1))))))
   0(1(2(4(x1)))) -> 0(3(2(3(1(4(x1))))))
   0(5(0(5(x1)))) -> 0(3(0(5(5(x1)))))
   0(5(1(2(x1)))) -> 1(0(1(5(2(x1)))))
   0(5(1(2(x1)))) -> 0(1(0(1(5(2(x1))))))
   0(5(1(2(x1)))) -> 0(3(2(3(1(5(x1))))))
   0(5(4(2(x1)))) -> 0(4(5(3(2(x1)))))
   0(5(5(2(x1)))) -> 5(0(1(5(2(x1)))))
   1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1)))))))
   1(0(1(2(x1)))) -> 1(1(3(0(2(x1)))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1))))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1))))))
   1(0(5(4(x1)))) -> 0(1(1(5(4(x1)))))
   1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1))))))
   1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1))))))
   1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1)))))))
   1(5(1(2(x1)))) -> 0(1(1(5(2(x1)))))
   1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1))))))
   5(0(0(2(x1)))) -> 5(0(3(0(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1))))))
   0(0(0(1(2(x1))))) -> 0(2(0(1(0(3(3(4(x1))))))))
   0(0(2(5(2(x1))))) -> 0(3(2(0(5(2(x1))))))
   0(1(2(5(0(x1))))) -> 3(3(2(2(0(0(1(5(x1))))))))
   0(1(2(5(2(x1))))) -> 0(3(2(1(5(3(2(x1)))))))
   0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1))))))
   0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1)))))))
   0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1)))))))
   0(5(0(2(2(x1))))) -> 0(2(5(0(3(2(x1))))))
   0(5(0(5(1(x1))))) -> 0(1(0(3(5(5(x1))))))
   0(5(1(3(0(x1))))) -> 0(0(1(1(5(3(x1))))))
   0(5(2(2(4(x1))))) -> 0(5(3(2(2(4(x1))))))
   0(5(2(3(1(x1))))) -> 0(1(5(3(2(2(2(x1)))))))
   0(5(2(4(1(x1))))) -> 0(4(3(2(5(1(x1))))))
   0(5(3(5(2(x1))))) -> 0(0(3(5(5(2(x1))))))
   0(5(5(3(1(x1))))) -> 5(0(1(5(3(3(2(x1)))))))
   1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1))))))
   1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1))))))
   1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1))))))
   1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1))))))
   1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1)))))))
   1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1))))))))
   5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1)))))))
   5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1))))))
   5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1))))))
   0(0(0(5(1(2(x1)))))) -> 0(0(1(5(0(2(4(x1)))))))
   0(0(1(2(4(1(x1)))))) -> 1(3(0(2(3(0(4(1(x1))))))))
   0(0(2(1(2(0(x1)))))) -> 0(1(0(3(2(2(2(0(x1))))))))
   0(0(2(3(0(5(x1)))))) -> 0(0(3(0(3(2(5(x1)))))))
   0(0(5(2(3(4(x1)))))) -> 0(4(0(3(1(2(5(x1)))))))
   0(0(5(5(3(4(x1)))))) -> 1(4(1(0(0(3(5(5(x1))))))))
   0(1(2(0(1(2(x1)))))) -> 1(0(3(2(2(1(1(0(x1))))))))
   0(1(2(2(0(5(x1)))))) -> 0(4(1(5(0(3(2(2(x1))))))))
   0(1(2(5(5(5(x1)))))) -> 0(2(5(1(5(3(5(x1)))))))
   0(1(3(1(5(2(x1)))))) -> 1(0(1(5(3(3(2(x1)))))))
   0(1(4(4(0(5(x1)))))) -> 4(3(0(0(1(5(4(x1)))))))
   0(2(5(3(5(1(x1)))))) -> 0(3(3(2(2(1(5(5(x1))))))))
   0(5(0(0(5(4(x1)))))) -> 0(1(0(1(0(4(5(5(x1))))))))
   0(5(1(2(1(4(x1)))))) -> 1(1(5(3(2(2(0(4(x1))))))))
   0(5(5(1(2(5(x1)))))) -> 0(2(1(5(5(4(5(x1)))))))
   1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1)))))))
   1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1))))))))
   1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1)))))))
   1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1))))))))
   1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1)))))))
   1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1)))))))
   1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1)))))))
   0(0(1(2(3(5(5(x1))))))) -> 5(1(5(0(3(0(2(1(x1))))))))
   0(0(2(3(4(2(1(x1))))))) -> 0(0(4(1(3(2(3(2(x1))))))))
   0(1(0(0(5(3(4(x1))))))) -> 0(3(0(5(0(4(3(1(x1))))))))
   0(1(2(4(4(0(5(x1))))))) -> 5(4(0(1(0(3(2(4(x1))))))))
   0(5(2(5(1(3(4(x1))))))) -> 1(4(5(2(0(3(1(5(x1))))))))
   0(5(5(0(2(5(1(x1))))))) -> 5(3(0(0(1(5(2(5(x1))))))))
   0(5(5(2(5(3(4(x1))))))) -> 0(3(2(1(4(5(5(5(x1))))))))
   1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1))))))))
   1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1))))))))
   1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1))))))))
   1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(1(5(4(x1))))))))

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

(36) SplitQDPProof (EQUIVALENT)
We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem

----------------------------------------

(37)
Complex Obligation (AND)

----------------------------------------

(38)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   0^1(5(0(5(x1)))) -> 0^1(5(5(x1)))
   0^1(5(0(5(1(x1))))) -> 0^1(1(0(3(5(5(x1))))))
   0^1(5(0(0(5(4(x1)))))) -> 0^1(1(0(1(0(4(5(5(x1))))))))
   0^1(5(0(0(5(4(x1)))))) -> 0^1(1(0(4(5(5(x1))))))

The TRS R consists of the following rules:

   0(0(1(2(x1)))) -> 0(3(1(0(2(x1)))))
   0(1(2(2(x1)))) -> 1(2(0(3(2(2(x1))))))
   0(1(2(4(x1)))) -> 0(3(2(3(1(4(x1))))))
   0(5(0(5(x1)))) -> 0(3(0(5(5(x1)))))
   0(5(1(2(x1)))) -> 1(0(1(5(2(x1)))))
   0(5(1(2(x1)))) -> 0(1(0(1(5(2(x1))))))
   0(5(1(2(x1)))) -> 0(3(2(3(1(5(x1))))))
   0(5(4(2(x1)))) -> 0(4(5(3(2(x1)))))
   0(5(5(2(x1)))) -> 5(0(1(5(2(x1)))))
   1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1)))))))
   1(0(1(2(x1)))) -> 1(1(3(0(2(x1)))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1))))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1))))))
   1(0(5(4(x1)))) -> 0(1(1(5(4(x1)))))
   1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1))))))
   1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1))))))
   1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1)))))))
   1(5(1(2(x1)))) -> 0(1(1(5(2(x1)))))
   1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1))))))
   5(0(0(2(x1)))) -> 5(0(3(0(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1))))))
   0(0(0(1(2(x1))))) -> 0(2(0(1(0(3(3(4(x1))))))))
   0(0(2(5(2(x1))))) -> 0(3(2(0(5(2(x1))))))
   0(1(2(5(0(x1))))) -> 3(3(2(2(0(0(1(5(x1))))))))
   0(1(2(5(2(x1))))) -> 0(3(2(1(5(3(2(x1)))))))
   0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1))))))
   0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1)))))))
   0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1)))))))
   0(5(0(2(2(x1))))) -> 0(2(5(0(3(2(x1))))))
   0(5(0(5(1(x1))))) -> 0(1(0(3(5(5(x1))))))
   0(5(1(3(0(x1))))) -> 0(0(1(1(5(3(x1))))))
   0(5(2(2(4(x1))))) -> 0(5(3(2(2(4(x1))))))
   0(5(2(3(1(x1))))) -> 0(1(5(3(2(2(2(x1)))))))
   0(5(2(4(1(x1))))) -> 0(4(3(2(5(1(x1))))))
   0(5(3(5(2(x1))))) -> 0(0(3(5(5(2(x1))))))
   0(5(5(3(1(x1))))) -> 5(0(1(5(3(3(2(x1)))))))
   1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1))))))
   1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1))))))
   1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1))))))
   1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1))))))
   1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1)))))))
   1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1))))))))
   5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1)))))))
   5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1))))))
   5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1))))))
   0(0(0(5(1(2(x1)))))) -> 0(0(1(5(0(2(4(x1)))))))
   0(0(1(2(4(1(x1)))))) -> 1(3(0(2(3(0(4(1(x1))))))))
   0(0(2(1(2(0(x1)))))) -> 0(1(0(3(2(2(2(0(x1))))))))
   0(0(2(3(0(5(x1)))))) -> 0(0(3(0(3(2(5(x1)))))))
   0(0(5(2(3(4(x1)))))) -> 0(4(0(3(1(2(5(x1)))))))
   0(0(5(5(3(4(x1)))))) -> 1(4(1(0(0(3(5(5(x1))))))))
   0(1(2(0(1(2(x1)))))) -> 1(0(3(2(2(1(1(0(x1))))))))
   0(1(2(2(0(5(x1)))))) -> 0(4(1(5(0(3(2(2(x1))))))))
   0(1(2(5(5(5(x1)))))) -> 0(2(5(1(5(3(5(x1)))))))
   0(1(3(1(5(2(x1)))))) -> 1(0(1(5(3(3(2(x1)))))))
   0(1(4(4(0(5(x1)))))) -> 4(3(0(0(1(5(4(x1)))))))
   0(2(5(3(5(1(x1)))))) -> 0(3(3(2(2(1(5(5(x1))))))))
   0(5(0(0(5(4(x1)))))) -> 0(1(0(1(0(4(5(5(x1))))))))
   0(5(1(2(1(4(x1)))))) -> 1(1(5(3(2(2(0(4(x1))))))))
   0(5(5(1(2(5(x1)))))) -> 0(2(1(5(5(4(5(x1)))))))
   1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1)))))))
   1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1))))))))
   1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1)))))))
   1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1))))))))
   1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1)))))))
   1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1)))))))
   1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1)))))))
   0(0(1(2(3(5(5(x1))))))) -> 5(1(5(0(3(0(2(1(x1))))))))
   0(0(2(3(4(2(1(x1))))))) -> 0(0(4(1(3(2(3(2(x1))))))))
   0(1(0(0(5(3(4(x1))))))) -> 0(3(0(5(0(4(3(1(x1))))))))
   0(1(2(4(4(0(5(x1))))))) -> 5(4(0(1(0(3(2(4(x1))))))))
   0(5(2(5(1(3(4(x1))))))) -> 1(4(5(2(0(3(1(5(x1))))))))
   0(5(5(0(2(5(1(x1))))))) -> 5(3(0(0(1(5(2(5(x1))))))))
   0(5(5(2(5(3(4(x1))))))) -> 0(3(2(1(4(5(5(5(x1))))))))
   1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1))))))))
   1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1))))))))
   1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1))))))))
   1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(1(5(4(x1))))))))

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

(39) SemLabProof (SOUND)
We found the following model for the rules of the TRSs R and P.
Interpretation over the domain with elements from 0 to 1.
5: 0
0^1: 0
0: 0
1: 0
2: 1
3: 0
4: 0
By semantic labelling [SEMLAB] we obtain the following labelled QDP problem.
----------------------------------------

(40)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   0^1.0(5.0(0.0(5.0(x1)))) -> 0^1.0(5.0(5.0(x1)))
   0^1.0(5.0(0.0(5.0(1.0(x1))))) -> 0^1.0(1.0(0.0(3.0(5.0(5.0(x1))))))
   0^1.0(5.0(0.0(5.0(1.1(x1))))) -> 0^1.0(1.0(0.0(3.0(5.0(5.1(x1))))))
   0^1.0(5.0(0.0(5.1(x1)))) -> 0^1.0(5.0(5.1(x1)))
   0^1.0(5.0(0.0(0.0(5.0(4.0(x1)))))) -> 0^1.0(1.0(0.0(1.0(0.0(4.0(5.0(5.0(x1))))))))
   0^1.0(5.0(0.0(0.0(5.0(4.1(x1)))))) -> 0^1.0(1.0(0.0(1.0(0.0(4.0(5.0(5.1(x1))))))))
   0^1.0(5.0(0.0(0.0(5.0(4.0(x1)))))) -> 0^1.0(1.0(0.0(4.0(5.0(5.0(x1))))))
   0^1.0(5.0(0.0(0.0(5.0(4.1(x1)))))) -> 0^1.0(1.0(0.0(4.0(5.0(5.1(x1))))))

The TRS R consists of the following rules:

   0.0(0.0(1.1(2.0(x1)))) -> 0.0(3.0(1.0(0.1(2.0(x1)))))
   0.0(0.0(1.1(2.1(x1)))) -> 0.0(3.0(1.0(0.1(2.1(x1)))))
   0.0(1.1(2.1(2.0(x1)))) -> 1.1(2.0(0.0(3.1(2.1(2.0(x1))))))
   0.0(1.1(2.1(2.1(x1)))) -> 1.1(2.0(0.0(3.1(2.1(2.1(x1))))))
   0.0(1.1(2.0(4.0(x1)))) -> 0.0(3.1(2.0(3.0(1.0(4.0(x1))))))
   0.0(1.1(2.0(4.1(x1)))) -> 0.0(3.1(2.0(3.0(1.0(4.1(x1))))))
   0.0(5.0(0.0(5.0(x1)))) -> 0.0(3.0(0.0(5.0(5.0(x1)))))
   0.0(5.0(0.0(5.1(x1)))) -> 0.0(3.0(0.0(5.0(5.1(x1)))))
   0.0(5.0(1.1(2.0(x1)))) -> 1.0(0.0(1.0(5.1(2.0(x1)))))
   0.0(5.0(1.1(2.1(x1)))) -> 1.0(0.0(1.0(5.1(2.1(x1)))))
   0.0(5.0(1.1(2.0(x1)))) -> 0.0(1.0(0.0(1.0(5.1(2.0(x1))))))
   0.0(5.0(1.1(2.1(x1)))) -> 0.0(1.0(0.0(1.0(5.1(2.1(x1))))))
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   0.0(5.0(5.1(2.0(x1)))) -> 5.0(0.0(1.0(5.1(2.0(x1)))))
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   5.0(0.0(1.1(2.1(x1)))) -> 5.0(1.0(0.0(3.1(2.1(x1)))))
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   0.0(5.1(2.0(5.0(1.0(3.0(4.1(x1))))))) -> 1.0(4.0(5.1(2.0(0.0(3.0(1.0(5.1(x1))))))))
   0.0(5.0(5.0(0.1(2.0(5.0(1.0(x1))))))) -> 5.0(3.0(0.0(0.0(1.0(5.1(2.0(5.0(x1))))))))
   0.0(5.0(5.0(0.1(2.0(5.0(1.1(x1))))))) -> 5.0(3.0(0.0(0.0(1.0(5.1(2.0(5.1(x1))))))))
   0.0(5.0(5.1(2.0(5.0(3.0(4.0(x1))))))) -> 0.0(3.1(2.0(1.0(4.0(5.0(5.0(5.0(x1))))))))
   0.0(5.0(5.1(2.0(5.0(3.0(4.1(x1))))))) -> 0.0(3.1(2.0(1.0(4.0(5.0(5.0(5.1(x1))))))))
   1.0(0.0(1.1(2.0(3.0(4.0(5.0(x1))))))) -> 3.0(0.1(2.0(1.0(5.0(1.0(3.0(4.0(x1))))))))
   1.0(0.0(1.1(2.0(3.0(4.0(5.1(x1))))))) -> 3.0(0.1(2.0(1.0(5.0(1.0(3.0(4.1(x1))))))))
   1.0(1.0(0.1(2.0(0.1(2.1(2.0(x1))))))) -> 1.0(1.0(0.1(2.0(0.0(3.1(2.1(2.0(x1))))))))
   1.0(1.0(0.1(2.0(0.1(2.1(2.1(x1))))))) -> 1.0(1.0(0.1(2.0(0.0(3.1(2.1(2.1(x1))))))))
   1.1(2.0(4.0(3.0(5.0(3.0(5.0(x1))))))) -> 5.0(1.0(3.1(2.1(2.0(4.0(3.0(5.0(x1))))))))
   1.1(2.0(4.0(3.0(5.0(3.0(5.1(x1))))))) -> 5.0(1.0(3.1(2.1(2.0(4.0(3.0(5.1(x1))))))))
   1.0(4.0(3.0(5.1(2.0(5.1(2.0(x1))))))) -> 5.0(1.0(3.1(2.1(2.0(1.0(5.0(4.0(x1))))))))
   1.0(4.0(3.0(5.1(2.0(5.1(2.1(x1))))))) -> 5.0(1.0(3.1(2.1(2.0(1.0(5.0(4.1(x1))))))))

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

(41) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes.
----------------------------------------

(42)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   0^1.0(5.0(0.0(5.0(1.0(x1))))) -> 0^1.0(1.0(0.0(3.0(5.0(5.0(x1))))))
   0^1.0(5.0(0.0(5.0(x1)))) -> 0^1.0(5.0(5.0(x1)))

The TRS R consists of the following rules:

   0.0(0.0(1.1(2.0(x1)))) -> 0.0(3.0(1.0(0.1(2.0(x1)))))
   0.0(0.0(1.1(2.1(x1)))) -> 0.0(3.0(1.0(0.1(2.1(x1)))))
   0.0(1.1(2.1(2.0(x1)))) -> 1.1(2.0(0.0(3.1(2.1(2.0(x1))))))
   0.0(1.1(2.1(2.1(x1)))) -> 1.1(2.0(0.0(3.1(2.1(2.1(x1))))))
   0.0(1.1(2.0(4.0(x1)))) -> 0.0(3.1(2.0(3.0(1.0(4.0(x1))))))
   0.0(1.1(2.0(4.1(x1)))) -> 0.0(3.1(2.0(3.0(1.0(4.1(x1))))))
   0.0(5.0(0.0(5.0(x1)))) -> 0.0(3.0(0.0(5.0(5.0(x1)))))
   0.0(5.0(0.0(5.1(x1)))) -> 0.0(3.0(0.0(5.0(5.1(x1)))))
   0.0(5.0(1.1(2.0(x1)))) -> 1.0(0.0(1.0(5.1(2.0(x1)))))
   0.0(5.0(1.1(2.1(x1)))) -> 1.0(0.0(1.0(5.1(2.1(x1)))))
   0.0(5.0(1.1(2.0(x1)))) -> 0.0(1.0(0.0(1.0(5.1(2.0(x1))))))
   0.0(5.0(1.1(2.1(x1)))) -> 0.0(1.0(0.0(1.0(5.1(2.1(x1))))))
   0.0(5.0(1.1(2.0(x1)))) -> 0.0(3.1(2.0(3.0(1.0(5.0(x1))))))
   0.0(5.0(1.1(2.1(x1)))) -> 0.0(3.1(2.0(3.0(1.0(5.1(x1))))))
   0.0(5.0(4.1(2.0(x1)))) -> 0.0(4.0(5.0(3.1(2.0(x1)))))
   0.0(5.0(4.1(2.1(x1)))) -> 0.0(4.0(5.0(3.1(2.1(x1)))))
   0.0(5.0(5.1(2.0(x1)))) -> 5.0(0.0(1.0(5.1(2.0(x1)))))
   0.0(5.0(5.1(2.1(x1)))) -> 5.0(0.0(1.0(5.1(2.1(x1)))))
   1.0(0.0(0.0(5.0(x1)))) -> 1.0(1.0(0.0(0.0(1.0(5.0(4.0(x1)))))))
   1.0(0.0(0.0(5.1(x1)))) -> 1.0(1.0(0.0(0.0(1.0(5.0(4.1(x1)))))))
   1.0(0.0(1.1(2.0(x1)))) -> 1.0(1.0(3.0(0.1(2.0(x1)))))
   1.0(0.0(1.1(2.1(x1)))) -> 1.0(1.0(3.0(0.1(2.1(x1)))))
   1.0(0.0(1.1(2.0(x1)))) -> 1.0(1.0(0.0(3.1(2.1(2.0(x1))))))
   1.0(0.0(1.1(2.1(x1)))) -> 1.0(1.0(0.0(3.1(2.1(2.1(x1))))))
   1.0(0.0(1.1(2.0(x1)))) -> 1.0(1.0(0.0(3.1(2.0(3.0(x1))))))
   1.0(0.0(1.1(2.1(x1)))) -> 1.0(1.0(0.0(3.1(2.0(3.1(x1))))))
   1.0(0.0(5.0(4.0(x1)))) -> 0.0(1.0(1.0(5.0(4.0(x1)))))
   1.0(0.0(5.0(4.1(x1)))) -> 0.0(1.0(1.0(5.0(4.1(x1)))))
   1.1(2.0(0.0(5.0(x1)))) -> 0.0(3.1(2.0(3.0(1.0(5.0(x1))))))
   1.1(2.0(0.0(5.1(x1)))) -> 0.0(3.1(2.0(3.0(1.0(5.1(x1))))))
   1.1(2.0(0.0(5.0(x1)))) -> 5.0(0.0(3.0(3.1(2.0(1.0(x1))))))
   1.1(2.0(0.0(5.1(x1)))) -> 5.0(0.0(3.0(3.1(2.0(1.1(x1))))))
   1.0(5.0(0.1(2.0(x1)))) -> 1.0(1.0(0.0(1.0(1.0(5.1(2.0(x1)))))))
   1.0(5.0(0.1(2.1(x1)))) -> 1.0(1.0(0.0(1.0(1.0(5.1(2.1(x1)))))))
   1.0(5.0(1.1(2.0(x1)))) -> 0.0(1.0(1.0(5.1(2.0(x1)))))
   1.0(5.0(1.1(2.1(x1)))) -> 0.0(1.0(1.0(5.1(2.1(x1)))))
   1.0(5.0(1.1(2.0(x1)))) -> 1.0(0.0(1.0(5.0(3.1(2.0(x1))))))
   1.0(5.0(1.1(2.1(x1)))) -> 1.0(0.0(1.0(5.0(3.1(2.1(x1))))))
   5.0(0.0(0.1(2.0(x1)))) -> 5.0(0.0(3.0(0.1(2.0(x1)))))
   5.0(0.0(0.1(2.1(x1)))) -> 5.0(0.0(3.0(0.1(2.1(x1)))))
   5.0(0.0(1.1(2.0(x1)))) -> 5.0(1.0(0.0(3.1(2.0(x1)))))
   5.0(0.0(1.1(2.1(x1)))) -> 5.0(1.0(0.0(3.1(2.1(x1)))))
   5.0(0.0(1.1(2.0(x1)))) -> 5.0(1.0(0.0(3.1(2.0(3.0(x1))))))
   5.0(0.0(1.1(2.1(x1)))) -> 5.0(1.0(0.0(3.1(2.0(3.1(x1))))))
   0.0(0.0(0.0(1.1(2.0(x1))))) -> 0.1(2.0(0.0(1.0(0.0(3.0(3.0(4.0(x1))))))))
   0.0(0.0(0.0(1.1(2.1(x1))))) -> 0.1(2.0(0.0(1.0(0.0(3.0(3.0(4.1(x1))))))))
   0.0(0.1(2.0(5.1(2.0(x1))))) -> 0.0(3.1(2.0(0.0(5.1(2.0(x1))))))
   0.0(0.1(2.0(5.1(2.1(x1))))) -> 0.0(3.1(2.0(0.0(5.1(2.1(x1))))))
   0.0(1.1(2.0(5.0(0.0(x1))))) -> 3.0(3.1(2.1(2.0(0.0(0.0(1.0(5.0(x1))))))))
   0.0(1.1(2.0(5.0(0.1(x1))))) -> 3.0(3.1(2.1(2.0(0.0(0.0(1.0(5.1(x1))))))))
   0.0(1.1(2.0(5.1(2.0(x1))))) -> 0.0(3.1(2.0(1.0(5.0(3.1(2.0(x1)))))))
   0.0(1.1(2.0(5.1(2.1(x1))))) -> 0.0(3.1(2.0(1.0(5.0(3.1(2.1(x1)))))))
   0.0(3.0(5.1(2.1(2.0(x1))))) -> 0.0(4.0(5.0(3.1(2.1(2.0(x1))))))
   0.0(3.0(5.1(2.1(2.1(x1))))) -> 0.0(4.0(5.0(3.1(2.1(2.1(x1))))))
   0.0(4.1(2.0(0.0(5.0(x1))))) -> 0.0(4.0(0.0(3.1(2.0(1.0(5.0(x1)))))))
   0.0(4.1(2.0(0.0(5.1(x1))))) -> 0.0(4.0(0.0(3.1(2.0(1.0(5.1(x1)))))))
   0.0(4.1(2.0(5.1(2.0(x1))))) -> 0.0(5.0(4.0(3.0(3.1(2.1(2.0(x1)))))))
   0.0(4.1(2.0(5.1(2.1(x1))))) -> 0.0(5.0(4.0(3.0(3.1(2.1(2.1(x1)))))))
   0.0(5.0(0.1(2.1(2.0(x1))))) -> 0.1(2.0(5.0(0.0(3.1(2.0(x1))))))
   0.0(5.0(0.1(2.1(2.1(x1))))) -> 0.1(2.0(5.0(0.0(3.1(2.1(x1))))))
   0.0(5.0(0.0(5.0(1.0(x1))))) -> 0.0(1.0(0.0(3.0(5.0(5.0(x1))))))
   0.0(5.0(0.0(5.0(1.1(x1))))) -> 0.0(1.0(0.0(3.0(5.0(5.1(x1))))))
   0.0(5.0(1.0(3.0(0.0(x1))))) -> 0.0(0.0(1.0(1.0(5.0(3.0(x1))))))
   0.0(5.0(1.0(3.0(0.1(x1))))) -> 0.0(0.0(1.0(1.0(5.0(3.1(x1))))))
   0.0(5.1(2.1(2.0(4.0(x1))))) -> 0.0(5.0(3.1(2.1(2.0(4.0(x1))))))
   0.0(5.1(2.1(2.0(4.1(x1))))) -> 0.0(5.0(3.1(2.1(2.0(4.1(x1))))))
   0.0(5.1(2.0(3.0(1.0(x1))))) -> 0.0(1.0(5.0(3.1(2.1(2.1(2.0(x1)))))))
   0.0(5.1(2.0(3.0(1.1(x1))))) -> 0.0(1.0(5.0(3.1(2.1(2.1(2.1(x1)))))))
   0.0(5.1(2.0(4.0(1.0(x1))))) -> 0.0(4.0(3.1(2.0(5.0(1.0(x1))))))
   0.0(5.1(2.0(4.0(1.1(x1))))) -> 0.0(4.0(3.1(2.0(5.0(1.1(x1))))))
   0.0(5.0(3.0(5.1(2.0(x1))))) -> 0.0(0.0(3.0(5.0(5.1(2.0(x1))))))
   0.0(5.0(3.0(5.1(2.1(x1))))) -> 0.0(0.0(3.0(5.0(5.1(2.1(x1))))))
   0.0(5.0(5.0(3.0(1.0(x1))))) -> 5.0(0.0(1.0(5.0(3.0(3.1(2.0(x1)))))))
   0.0(5.0(5.0(3.0(1.1(x1))))) -> 5.0(0.0(1.0(5.0(3.0(3.1(2.1(x1)))))))
   1.0(0.0(5.0(5.0(1.0(x1))))) -> 0.0(4.0(5.0(1.0(5.0(1.0(x1))))))
   1.0(0.0(5.0(5.0(1.1(x1))))) -> 0.0(4.0(5.0(1.0(5.0(1.1(x1))))))
   1.0(1.1(2.1(2.0(0.0(x1))))) -> 1.0(1.0(3.1(2.1(2.0(0.0(x1))))))
   1.0(1.1(2.1(2.0(0.1(x1))))) -> 1.0(1.0(3.1(2.1(2.0(0.1(x1))))))
   1.0(1.1(2.0(3.0(4.0(x1))))) -> 1.0(1.0(3.1(2.1(2.0(4.0(x1))))))
   1.0(1.1(2.0(3.0(4.1(x1))))) -> 1.0(1.0(3.1(2.1(2.0(4.1(x1))))))
   1.0(1.0(3.0(5.1(2.0(x1))))) -> 1.0(1.0(5.0(3.0(3.1(2.0(x1))))))
   1.0(1.0(3.0(5.1(2.1(x1))))) -> 1.0(1.0(5.0(3.0(3.1(2.1(x1))))))
   1.0(5.0(0.0(5.0(0.0(x1))))) -> 0.0(1.0(5.0(3.0(5.0(1.0(0.0(x1)))))))
   1.0(5.0(0.0(5.0(0.1(x1))))) -> 0.0(1.0(5.0(3.0(5.0(1.0(0.1(x1)))))))
   1.0(5.0(5.0(1.1(2.0(x1))))) -> 1.0(5.0(1.0(1.0(5.0(3.1(2.1(2.0(x1))))))))
   1.0(5.0(5.0(1.1(2.1(x1))))) -> 1.0(5.0(1.0(1.0(5.0(3.1(2.1(2.1(x1))))))))
   5.0(0.1(2.0(0.0(5.0(x1))))) -> 5.0(0.0(3.0(3.1(2.0(0.0(5.0(x1)))))))
   5.0(0.1(2.0(0.0(5.1(x1))))) -> 5.0(0.0(3.0(3.1(2.0(0.0(5.1(x1)))))))
   5.0(0.1(2.0(3.0(4.0(x1))))) -> 5.0(0.0(3.1(2.0(3.0(4.0(x1))))))
   5.0(0.1(2.0(3.0(4.1(x1))))) -> 5.0(0.0(3.1(2.0(3.0(4.1(x1))))))
   5.0(5.0(0.0(1.1(2.0(x1))))) -> 5.0(5.0(3.0(0.1(2.0(1.0(x1))))))
   5.0(5.0(0.0(1.1(2.1(x1))))) -> 5.0(5.0(3.0(0.1(2.0(1.1(x1))))))
   0.0(0.0(0.0(5.0(1.1(2.0(x1)))))) -> 0.0(0.0(1.0(5.0(0.1(2.0(4.0(x1)))))))
   0.0(0.0(0.0(5.0(1.1(2.1(x1)))))) -> 0.0(0.0(1.0(5.0(0.1(2.0(4.1(x1)))))))
   0.0(0.0(1.1(2.0(4.0(1.0(x1)))))) -> 1.0(3.0(0.1(2.0(3.0(0.0(4.0(1.0(x1))))))))
   0.0(0.0(1.1(2.0(4.0(1.1(x1)))))) -> 1.0(3.0(0.1(2.0(3.0(0.0(4.0(1.1(x1))))))))
   0.0(0.1(2.0(1.1(2.0(0.0(x1)))))) -> 0.0(1.0(0.0(3.1(2.1(2.1(2.0(0.0(x1))))))))
   0.0(0.1(2.0(1.1(2.0(0.1(x1)))))) -> 0.0(1.0(0.0(3.1(2.1(2.1(2.0(0.1(x1))))))))
   0.0(0.1(2.0(3.0(0.0(5.0(x1)))))) -> 0.0(0.0(3.0(0.0(3.1(2.0(5.0(x1)))))))
   0.0(0.1(2.0(3.0(0.0(5.1(x1)))))) -> 0.0(0.0(3.0(0.0(3.1(2.0(5.1(x1)))))))
   0.0(0.0(5.1(2.0(3.0(4.0(x1)))))) -> 0.0(4.0(0.0(3.0(1.1(2.0(5.0(x1)))))))
   0.0(0.0(5.1(2.0(3.0(4.1(x1)))))) -> 0.0(4.0(0.0(3.0(1.1(2.0(5.1(x1)))))))
   0.0(0.0(5.0(5.0(3.0(4.0(x1)))))) -> 1.0(4.0(1.0(0.0(0.0(3.0(5.0(5.0(x1))))))))
   0.0(0.0(5.0(5.0(3.0(4.1(x1)))))) -> 1.0(4.0(1.0(0.0(0.0(3.0(5.0(5.1(x1))))))))
   0.0(1.1(2.0(0.0(1.1(2.0(x1)))))) -> 1.0(0.0(3.1(2.1(2.0(1.0(1.0(0.0(x1))))))))
   0.0(1.1(2.0(0.0(1.1(2.1(x1)))))) -> 1.0(0.0(3.1(2.1(2.0(1.0(1.0(0.1(x1))))))))
   0.0(1.1(2.1(2.0(0.0(5.0(x1)))))) -> 0.0(4.0(1.0(5.0(0.0(3.1(2.1(2.0(x1))))))))
   0.0(1.1(2.1(2.0(0.0(5.1(x1)))))) -> 0.0(4.0(1.0(5.0(0.0(3.1(2.1(2.1(x1))))))))
   0.0(1.1(2.0(5.0(5.0(5.0(x1)))))) -> 0.1(2.0(5.0(1.0(5.0(3.0(5.0(x1)))))))
   0.0(1.1(2.0(5.0(5.0(5.1(x1)))))) -> 0.1(2.0(5.0(1.0(5.0(3.0(5.1(x1)))))))
   0.0(1.0(3.0(1.0(5.1(2.0(x1)))))) -> 1.0(0.0(1.0(5.0(3.0(3.1(2.0(x1)))))))
   0.0(1.0(3.0(1.0(5.1(2.1(x1)))))) -> 1.0(0.0(1.0(5.0(3.0(3.1(2.1(x1)))))))
   0.0(1.0(4.0(4.0(0.0(5.0(x1)))))) -> 4.0(3.0(0.0(0.0(1.0(5.0(4.0(x1)))))))
   0.0(1.0(4.0(4.0(0.0(5.1(x1)))))) -> 4.0(3.0(0.0(0.0(1.0(5.0(4.1(x1)))))))
   0.1(2.0(5.0(3.0(5.0(1.0(x1)))))) -> 0.0(3.0(3.1(2.1(2.0(1.0(5.0(5.0(x1))))))))
   0.1(2.0(5.0(3.0(5.0(1.1(x1)))))) -> 0.0(3.0(3.1(2.1(2.0(1.0(5.0(5.1(x1))))))))
   0.0(5.0(0.0(0.0(5.0(4.0(x1)))))) -> 0.0(1.0(0.0(1.0(0.0(4.0(5.0(5.0(x1))))))))
   0.0(5.0(0.0(0.0(5.0(4.1(x1)))))) -> 0.0(1.0(0.0(1.0(0.0(4.0(5.0(5.1(x1))))))))
   0.0(5.0(1.1(2.0(1.0(4.0(x1)))))) -> 1.0(1.0(5.0(3.1(2.1(2.0(0.0(4.0(x1))))))))
   0.0(5.0(1.1(2.0(1.0(4.1(x1)))))) -> 1.0(1.0(5.0(3.1(2.1(2.0(0.0(4.1(x1))))))))
   0.0(5.0(5.0(1.1(2.0(5.0(x1)))))) -> 0.1(2.0(1.0(5.0(5.0(4.0(5.0(x1)))))))
   0.0(5.0(5.0(1.1(2.0(5.1(x1)))))) -> 0.1(2.0(1.0(5.0(5.0(4.0(5.1(x1)))))))
   1.0(0.0(0.1(2.0(3.0(4.0(x1)))))) -> 1.0(4.0(0.0(0.0(3.0(3.1(2.0(x1)))))))
   1.0(0.0(0.1(2.0(3.0(4.1(x1)))))) -> 1.0(4.0(0.0(0.0(3.0(3.1(2.1(x1)))))))
   1.0(0.0(1.0(3.0(5.0(1.0(x1)))))) -> 0.0(1.0(1.0(1.0(5.0(3.1(2.1(2.0(x1))))))))
   1.0(0.0(1.0(3.0(5.0(1.1(x1)))))) -> 0.0(1.0(1.0(1.0(5.0(3.1(2.1(2.1(x1))))))))
   1.0(0.0(5.0(4.1(2.0(1.0(x1)))))) -> 0.0(1.0(1.0(5.0(3.0(4.1(2.0(x1)))))))
   1.0(0.0(5.0(4.1(2.0(1.1(x1)))))) -> 0.0(1.0(1.0(5.0(3.0(4.1(2.1(x1)))))))
   1.0(1.0(0.0(1.1(2.1(2.0(x1)))))) -> 1.0(0.0(1.0(1.0(3.0(1.1(2.1(2.0(x1))))))))
   1.0(1.0(0.0(1.1(2.1(2.1(x1)))))) -> 1.0(0.0(1.0(1.0(3.0(1.1(2.1(2.1(x1))))))))
   1.1(2.0(1.1(2.0(0.0(0.0(x1)))))) -> 0.0(3.1(2.1(2.0(1.0(0.0(1.0(x1)))))))
   1.1(2.0(1.1(2.0(0.0(0.1(x1)))))) -> 0.0(3.1(2.1(2.0(1.0(0.0(1.1(x1)))))))
   1.0(4.0(1.0(0.0(0.0(5.0(x1)))))) -> 0.0(0.0(1.0(5.0(4.1(2.0(1.0(x1)))))))
   1.0(4.0(1.0(0.0(0.0(5.1(x1)))))) -> 0.0(0.0(1.0(5.0(4.1(2.0(1.1(x1)))))))
   1.0(4.0(3.0(5.0(0.1(2.0(x1)))))) -> 0.0(3.0(3.1(2.0(1.0(5.0(4.0(x1)))))))
   1.0(4.0(3.0(5.0(0.1(2.1(x1)))))) -> 0.0(3.0(3.1(2.0(1.0(5.0(4.1(x1)))))))
   0.0(0.0(1.1(2.0(3.0(5.0(5.0(x1))))))) -> 5.0(1.0(5.0(0.0(3.0(0.1(2.0(1.0(x1))))))))
   0.0(0.0(1.1(2.0(3.0(5.0(5.1(x1))))))) -> 5.0(1.0(5.0(0.0(3.0(0.1(2.0(1.1(x1))))))))
   0.0(0.1(2.0(3.0(4.1(2.0(1.0(x1))))))) -> 0.0(0.0(4.0(1.0(3.1(2.0(3.1(2.0(x1))))))))
   0.0(0.1(2.0(3.0(4.1(2.0(1.1(x1))))))) -> 0.0(0.0(4.0(1.0(3.1(2.0(3.1(2.1(x1))))))))
   0.0(1.0(0.0(0.0(5.0(3.0(4.0(x1))))))) -> 0.0(3.0(0.0(5.0(0.0(4.0(3.0(1.0(x1))))))))
   0.0(1.0(0.0(0.0(5.0(3.0(4.1(x1))))))) -> 0.0(3.0(0.0(5.0(0.0(4.0(3.0(1.1(x1))))))))
   0.0(1.1(2.0(4.0(4.0(0.0(5.0(x1))))))) -> 5.0(4.0(0.0(1.0(0.0(3.1(2.0(4.0(x1))))))))
   0.0(1.1(2.0(4.0(4.0(0.0(5.1(x1))))))) -> 5.0(4.0(0.0(1.0(0.0(3.1(2.0(4.1(x1))))))))
   0.0(5.1(2.0(5.0(1.0(3.0(4.0(x1))))))) -> 1.0(4.0(5.1(2.0(0.0(3.0(1.0(5.0(x1))))))))
   0.0(5.1(2.0(5.0(1.0(3.0(4.1(x1))))))) -> 1.0(4.0(5.1(2.0(0.0(3.0(1.0(5.1(x1))))))))
   0.0(5.0(5.0(0.1(2.0(5.0(1.0(x1))))))) -> 5.0(3.0(0.0(0.0(1.0(5.1(2.0(5.0(x1))))))))
   0.0(5.0(5.0(0.1(2.0(5.0(1.1(x1))))))) -> 5.0(3.0(0.0(0.0(1.0(5.1(2.0(5.1(x1))))))))
   0.0(5.0(5.1(2.0(5.0(3.0(4.0(x1))))))) -> 0.0(3.1(2.0(1.0(4.0(5.0(5.0(5.0(x1))))))))
   0.0(5.0(5.1(2.0(5.0(3.0(4.1(x1))))))) -> 0.0(3.1(2.0(1.0(4.0(5.0(5.0(5.1(x1))))))))
   1.0(0.0(1.1(2.0(3.0(4.0(5.0(x1))))))) -> 3.0(0.1(2.0(1.0(5.0(1.0(3.0(4.0(x1))))))))
   1.0(0.0(1.1(2.0(3.0(4.0(5.1(x1))))))) -> 3.0(0.1(2.0(1.0(5.0(1.0(3.0(4.1(x1))))))))
   1.0(1.0(0.1(2.0(0.1(2.1(2.0(x1))))))) -> 1.0(1.0(0.1(2.0(0.0(3.1(2.1(2.0(x1))))))))
   1.0(1.0(0.1(2.0(0.1(2.1(2.1(x1))))))) -> 1.0(1.0(0.1(2.0(0.0(3.1(2.1(2.1(x1))))))))
   1.1(2.0(4.0(3.0(5.0(3.0(5.0(x1))))))) -> 5.0(1.0(3.1(2.1(2.0(4.0(3.0(5.0(x1))))))))
   1.1(2.0(4.0(3.0(5.0(3.0(5.1(x1))))))) -> 5.0(1.0(3.1(2.1(2.0(4.0(3.0(5.1(x1))))))))
   1.0(4.0(3.0(5.1(2.0(5.1(2.0(x1))))))) -> 5.0(1.0(3.1(2.1(2.0(1.0(5.0(4.0(x1))))))))
   1.0(4.0(3.0(5.1(2.0(5.1(2.1(x1))))))) -> 5.0(1.0(3.1(2.1(2.0(1.0(5.0(4.1(x1))))))))

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

(43) PisEmptyProof (SOUND)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------

(44)
TRUE

----------------------------------------

(45)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   0^1(5(0(5(1(x1))))) -> 0^1(1(0(3(5(5(x1))))))
   0^1(5(0(5(x1)))) -> 0^1(5(5(x1)))

The TRS R consists of the following rules:

   0(0(1(2(x1)))) -> 0(3(1(0(2(x1)))))
   0(1(2(2(x1)))) -> 1(2(0(3(2(2(x1))))))
   0(1(2(4(x1)))) -> 0(3(2(3(1(4(x1))))))
   0(5(0(5(x1)))) -> 0(3(0(5(5(x1)))))
   0(5(1(2(x1)))) -> 1(0(1(5(2(x1)))))
   0(5(1(2(x1)))) -> 0(1(0(1(5(2(x1))))))
   0(5(1(2(x1)))) -> 0(3(2(3(1(5(x1))))))
   0(5(4(2(x1)))) -> 0(4(5(3(2(x1)))))
   0(5(5(2(x1)))) -> 5(0(1(5(2(x1)))))
   1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1)))))))
   1(0(1(2(x1)))) -> 1(1(3(0(2(x1)))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1))))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1))))))
   1(0(5(4(x1)))) -> 0(1(1(5(4(x1)))))
   1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1))))))
   1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1))))))
   1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1)))))))
   1(5(1(2(x1)))) -> 0(1(1(5(2(x1)))))
   1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1))))))
   5(0(0(2(x1)))) -> 5(0(3(0(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1))))))
   0(0(0(1(2(x1))))) -> 0(2(0(1(0(3(3(4(x1))))))))
   0(0(2(5(2(x1))))) -> 0(3(2(0(5(2(x1))))))
   0(1(2(5(0(x1))))) -> 3(3(2(2(0(0(1(5(x1))))))))
   0(1(2(5(2(x1))))) -> 0(3(2(1(5(3(2(x1)))))))
   0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1))))))
   0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1)))))))
   0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1)))))))
   0(5(0(2(2(x1))))) -> 0(2(5(0(3(2(x1))))))
   0(5(0(5(1(x1))))) -> 0(1(0(3(5(5(x1))))))
   0(5(1(3(0(x1))))) -> 0(0(1(1(5(3(x1))))))
   0(5(2(2(4(x1))))) -> 0(5(3(2(2(4(x1))))))
   0(5(2(3(1(x1))))) -> 0(1(5(3(2(2(2(x1)))))))
   0(5(2(4(1(x1))))) -> 0(4(3(2(5(1(x1))))))
   0(5(3(5(2(x1))))) -> 0(0(3(5(5(2(x1))))))
   0(5(5(3(1(x1))))) -> 5(0(1(5(3(3(2(x1)))))))
   1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1))))))
   1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1))))))
   1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1))))))
   1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1))))))
   1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1)))))))
   1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1))))))))
   5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1)))))))
   5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1))))))
   5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1))))))
   0(0(0(5(1(2(x1)))))) -> 0(0(1(5(0(2(4(x1)))))))
   0(0(1(2(4(1(x1)))))) -> 1(3(0(2(3(0(4(1(x1))))))))
   0(0(2(1(2(0(x1)))))) -> 0(1(0(3(2(2(2(0(x1))))))))
   0(0(2(3(0(5(x1)))))) -> 0(0(3(0(3(2(5(x1)))))))
   0(0(5(2(3(4(x1)))))) -> 0(4(0(3(1(2(5(x1)))))))
   0(0(5(5(3(4(x1)))))) -> 1(4(1(0(0(3(5(5(x1))))))))
   0(1(2(0(1(2(x1)))))) -> 1(0(3(2(2(1(1(0(x1))))))))
   0(1(2(2(0(5(x1)))))) -> 0(4(1(5(0(3(2(2(x1))))))))
   0(1(2(5(5(5(x1)))))) -> 0(2(5(1(5(3(5(x1)))))))
   0(1(3(1(5(2(x1)))))) -> 1(0(1(5(3(3(2(x1)))))))
   0(1(4(4(0(5(x1)))))) -> 4(3(0(0(1(5(4(x1)))))))
   0(2(5(3(5(1(x1)))))) -> 0(3(3(2(2(1(5(5(x1))))))))
   0(5(0(0(5(4(x1)))))) -> 0(1(0(1(0(4(5(5(x1))))))))
   0(5(1(2(1(4(x1)))))) -> 1(1(5(3(2(2(0(4(x1))))))))
   0(5(5(1(2(5(x1)))))) -> 0(2(1(5(5(4(5(x1)))))))
   1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1)))))))
   1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1))))))))
   1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1)))))))
   1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1))))))))
   1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1)))))))
   1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1)))))))
   1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1)))))))
   0(0(1(2(3(5(5(x1))))))) -> 5(1(5(0(3(0(2(1(x1))))))))
   0(0(2(3(4(2(1(x1))))))) -> 0(0(4(1(3(2(3(2(x1))))))))
   0(1(0(0(5(3(4(x1))))))) -> 0(3(0(5(0(4(3(1(x1))))))))
   0(1(2(4(4(0(5(x1))))))) -> 5(4(0(1(0(3(2(4(x1))))))))
   0(5(2(5(1(3(4(x1))))))) -> 1(4(5(2(0(3(1(5(x1))))))))
   0(5(5(0(2(5(1(x1))))))) -> 5(3(0(0(1(5(2(5(x1))))))))
   0(5(5(2(5(3(4(x1))))))) -> 0(3(2(1(4(5(5(5(x1))))))))
   1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1))))))))
   1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1))))))))
   1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1))))))))
   1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(1(5(4(x1))))))))

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

(46) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   0^1(5(0(5(1(x1))))) -> 0^1(1(0(3(5(5(x1))))))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(0(x_1)) = 0
   POL(0^1(x_1)) = x_1
   POL(1(x_1)) = x_1
   POL(2(x_1)) = 1
   POL(3(x_1)) = x_1
   POL(4(x_1)) = x_1
   POL(5(x_1)) = 1

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

   5(0(0(2(x1)))) -> 5(0(3(0(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1))))))
   5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1)))))))
   5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1))))))
   5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1))))))
   0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1))))))
   1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1)))))))
   1(0(1(2(x1)))) -> 1(1(3(0(2(x1)))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1))))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1))))))
   1(0(5(4(x1)))) -> 0(1(1(5(4(x1)))))
   1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1))))))
   1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1))))))
   1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1)))))))
   1(5(1(2(x1)))) -> 0(1(1(5(2(x1)))))
   1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1))))))
   1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1))))))
   1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1))))))
   1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1))))))
   1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1))))))
   1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1)))))))
   1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1))))))))
   1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1)))))))
   1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1))))))))
   1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1)))))))
   1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1))))))))
   1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1)))))))
   1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1)))))))
   1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1)))))))
   1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1))))))))
   1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1))))))))
   1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1))))))))
   1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(1(5(4(x1))))))))
   0(2(5(3(5(1(x1)))))) -> 0(3(3(2(2(1(5(5(x1))))))))
   0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1)))))))
   0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1)))))))


----------------------------------------

(47)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   0^1(5(0(5(x1)))) -> 0^1(5(5(x1)))

The TRS R consists of the following rules:

   0(0(1(2(x1)))) -> 0(3(1(0(2(x1)))))
   0(1(2(2(x1)))) -> 1(2(0(3(2(2(x1))))))
   0(1(2(4(x1)))) -> 0(3(2(3(1(4(x1))))))
   0(5(0(5(x1)))) -> 0(3(0(5(5(x1)))))
   0(5(1(2(x1)))) -> 1(0(1(5(2(x1)))))
   0(5(1(2(x1)))) -> 0(1(0(1(5(2(x1))))))
   0(5(1(2(x1)))) -> 0(3(2(3(1(5(x1))))))
   0(5(4(2(x1)))) -> 0(4(5(3(2(x1)))))
   0(5(5(2(x1)))) -> 5(0(1(5(2(x1)))))
   1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1)))))))
   1(0(1(2(x1)))) -> 1(1(3(0(2(x1)))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1))))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1))))))
   1(0(5(4(x1)))) -> 0(1(1(5(4(x1)))))
   1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1))))))
   1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1))))))
   1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1)))))))
   1(5(1(2(x1)))) -> 0(1(1(5(2(x1)))))
   1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1))))))
   5(0(0(2(x1)))) -> 5(0(3(0(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1))))))
   0(0(0(1(2(x1))))) -> 0(2(0(1(0(3(3(4(x1))))))))
   0(0(2(5(2(x1))))) -> 0(3(2(0(5(2(x1))))))
   0(1(2(5(0(x1))))) -> 3(3(2(2(0(0(1(5(x1))))))))
   0(1(2(5(2(x1))))) -> 0(3(2(1(5(3(2(x1)))))))
   0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1))))))
   0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1)))))))
   0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1)))))))
   0(5(0(2(2(x1))))) -> 0(2(5(0(3(2(x1))))))
   0(5(0(5(1(x1))))) -> 0(1(0(3(5(5(x1))))))
   0(5(1(3(0(x1))))) -> 0(0(1(1(5(3(x1))))))
   0(5(2(2(4(x1))))) -> 0(5(3(2(2(4(x1))))))
   0(5(2(3(1(x1))))) -> 0(1(5(3(2(2(2(x1)))))))
   0(5(2(4(1(x1))))) -> 0(4(3(2(5(1(x1))))))
   0(5(3(5(2(x1))))) -> 0(0(3(5(5(2(x1))))))
   0(5(5(3(1(x1))))) -> 5(0(1(5(3(3(2(x1)))))))
   1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1))))))
   1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1))))))
   1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1))))))
   1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1))))))
   1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1)))))))
   1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1))))))))
   5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1)))))))
   5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1))))))
   5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1))))))
   0(0(0(5(1(2(x1)))))) -> 0(0(1(5(0(2(4(x1)))))))
   0(0(1(2(4(1(x1)))))) -> 1(3(0(2(3(0(4(1(x1))))))))
   0(0(2(1(2(0(x1)))))) -> 0(1(0(3(2(2(2(0(x1))))))))
   0(0(2(3(0(5(x1)))))) -> 0(0(3(0(3(2(5(x1)))))))
   0(0(5(2(3(4(x1)))))) -> 0(4(0(3(1(2(5(x1)))))))
   0(0(5(5(3(4(x1)))))) -> 1(4(1(0(0(3(5(5(x1))))))))
   0(1(2(0(1(2(x1)))))) -> 1(0(3(2(2(1(1(0(x1))))))))
   0(1(2(2(0(5(x1)))))) -> 0(4(1(5(0(3(2(2(x1))))))))
   0(1(2(5(5(5(x1)))))) -> 0(2(5(1(5(3(5(x1)))))))
   0(1(3(1(5(2(x1)))))) -> 1(0(1(5(3(3(2(x1)))))))
   0(1(4(4(0(5(x1)))))) -> 4(3(0(0(1(5(4(x1)))))))
   0(2(5(3(5(1(x1)))))) -> 0(3(3(2(2(1(5(5(x1))))))))
   0(5(0(0(5(4(x1)))))) -> 0(1(0(1(0(4(5(5(x1))))))))
   0(5(1(2(1(4(x1)))))) -> 1(1(5(3(2(2(0(4(x1))))))))
   0(5(5(1(2(5(x1)))))) -> 0(2(1(5(5(4(5(x1)))))))
   1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1)))))))
   1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1))))))))
   1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1)))))))
   1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1))))))))
   1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1)))))))
   1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1)))))))
   1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1)))))))
   0(0(1(2(3(5(5(x1))))))) -> 5(1(5(0(3(0(2(1(x1))))))))
   0(0(2(3(4(2(1(x1))))))) -> 0(0(4(1(3(2(3(2(x1))))))))
   0(1(0(0(5(3(4(x1))))))) -> 0(3(0(5(0(4(3(1(x1))))))))
   0(1(2(4(4(0(5(x1))))))) -> 5(4(0(1(0(3(2(4(x1))))))))
   0(5(2(5(1(3(4(x1))))))) -> 1(4(5(2(0(3(1(5(x1))))))))
   0(5(5(0(2(5(1(x1))))))) -> 5(3(0(0(1(5(2(5(x1))))))))
   0(5(5(2(5(3(4(x1))))))) -> 0(3(2(1(4(5(5(5(x1))))))))
   1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1))))))))
   1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1))))))))
   1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1))))))))
   1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(1(5(4(x1))))))))

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

(48) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   0^1(5(0(5(x1)))) -> 0^1(5(5(x1)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(0(x_1)) = 1 + x_1
   POL(0^1(x_1)) = x_1
   POL(1(x_1)) = 0
   POL(2(x_1)) = x_1
   POL(3(x_1)) = 1
   POL(4(x_1)) = 0
   POL(5(x_1)) = 1 + x_1

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

   5(0(0(2(x1)))) -> 5(0(3(0(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1))))))
   5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1)))))))
   5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1))))))
   5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1))))))
   0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1))))))


----------------------------------------

(49)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

   0(0(1(2(x1)))) -> 0(3(1(0(2(x1)))))
   0(1(2(2(x1)))) -> 1(2(0(3(2(2(x1))))))
   0(1(2(4(x1)))) -> 0(3(2(3(1(4(x1))))))
   0(5(0(5(x1)))) -> 0(3(0(5(5(x1)))))
   0(5(1(2(x1)))) -> 1(0(1(5(2(x1)))))
   0(5(1(2(x1)))) -> 0(1(0(1(5(2(x1))))))
   0(5(1(2(x1)))) -> 0(3(2(3(1(5(x1))))))
   0(5(4(2(x1)))) -> 0(4(5(3(2(x1)))))
   0(5(5(2(x1)))) -> 5(0(1(5(2(x1)))))
   1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1)))))))
   1(0(1(2(x1)))) -> 1(1(3(0(2(x1)))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1))))))
   1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1))))))
   1(0(5(4(x1)))) -> 0(1(1(5(4(x1)))))
   1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1))))))
   1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1))))))
   1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1)))))))
   1(5(1(2(x1)))) -> 0(1(1(5(2(x1)))))
   1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1))))))
   5(0(0(2(x1)))) -> 5(0(3(0(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(x1)))))
   5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1))))))
   0(0(0(1(2(x1))))) -> 0(2(0(1(0(3(3(4(x1))))))))
   0(0(2(5(2(x1))))) -> 0(3(2(0(5(2(x1))))))
   0(1(2(5(0(x1))))) -> 3(3(2(2(0(0(1(5(x1))))))))
   0(1(2(5(2(x1))))) -> 0(3(2(1(5(3(2(x1)))))))
   0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1))))))
   0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1)))))))
   0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1)))))))
   0(5(0(2(2(x1))))) -> 0(2(5(0(3(2(x1))))))
   0(5(0(5(1(x1))))) -> 0(1(0(3(5(5(x1))))))
   0(5(1(3(0(x1))))) -> 0(0(1(1(5(3(x1))))))
   0(5(2(2(4(x1))))) -> 0(5(3(2(2(4(x1))))))
   0(5(2(3(1(x1))))) -> 0(1(5(3(2(2(2(x1)))))))
   0(5(2(4(1(x1))))) -> 0(4(3(2(5(1(x1))))))
   0(5(3(5(2(x1))))) -> 0(0(3(5(5(2(x1))))))
   0(5(5(3(1(x1))))) -> 5(0(1(5(3(3(2(x1)))))))
   1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1))))))
   1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1))))))
   1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1))))))
   1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1))))))
   1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1)))))))
   1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1))))))))
   5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1)))))))
   5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1))))))
   5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1))))))
   0(0(0(5(1(2(x1)))))) -> 0(0(1(5(0(2(4(x1)))))))
   0(0(1(2(4(1(x1)))))) -> 1(3(0(2(3(0(4(1(x1))))))))
   0(0(2(1(2(0(x1)))))) -> 0(1(0(3(2(2(2(0(x1))))))))
   0(0(2(3(0(5(x1)))))) -> 0(0(3(0(3(2(5(x1)))))))
   0(0(5(2(3(4(x1)))))) -> 0(4(0(3(1(2(5(x1)))))))
   0(0(5(5(3(4(x1)))))) -> 1(4(1(0(0(3(5(5(x1))))))))
   0(1(2(0(1(2(x1)))))) -> 1(0(3(2(2(1(1(0(x1))))))))
   0(1(2(2(0(5(x1)))))) -> 0(4(1(5(0(3(2(2(x1))))))))
   0(1(2(5(5(5(x1)))))) -> 0(2(5(1(5(3(5(x1)))))))
   0(1(3(1(5(2(x1)))))) -> 1(0(1(5(3(3(2(x1)))))))
   0(1(4(4(0(5(x1)))))) -> 4(3(0(0(1(5(4(x1)))))))
   0(2(5(3(5(1(x1)))))) -> 0(3(3(2(2(1(5(5(x1))))))))
   0(5(0(0(5(4(x1)))))) -> 0(1(0(1(0(4(5(5(x1))))))))
   0(5(1(2(1(4(x1)))))) -> 1(1(5(3(2(2(0(4(x1))))))))
   0(5(5(1(2(5(x1)))))) -> 0(2(1(5(5(4(5(x1)))))))
   1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1)))))))
   1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1))))))))
   1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1)))))))
   1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1))))))))
   1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1)))))))
   1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1)))))))
   1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1)))))))
   0(0(1(2(3(5(5(x1))))))) -> 5(1(5(0(3(0(2(1(x1))))))))
   0(0(2(3(4(2(1(x1))))))) -> 0(0(4(1(3(2(3(2(x1))))))))
   0(1(0(0(5(3(4(x1))))))) -> 0(3(0(5(0(4(3(1(x1))))))))
   0(1(2(4(4(0(5(x1))))))) -> 5(4(0(1(0(3(2(4(x1))))))))
   0(5(2(5(1(3(4(x1))))))) -> 1(4(5(2(0(3(1(5(x1))))))))
   0(5(5(0(2(5(1(x1))))))) -> 5(3(0(0(1(5(2(5(x1))))))))
   0(5(5(2(5(3(4(x1))))))) -> 0(3(2(1(4(5(5(5(x1))))))))
   1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1))))))))
   1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1))))))))
   1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1))))))))
   1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(1(5(4(x1))))))))

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

(50) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------

(51)
YES