YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) QTRS
(1) QTRS Reverse [EQUIVALENT, 0 ms]
(2) QTRS
(3) DependencyPairsProof [EQUIVALENT, 362 ms]
(4) QDP
(5) DependencyGraphProof [EQUIVALENT, 0 ms]
(6) AND
    (7) QDP
        (8) UsableRulesProof [EQUIVALENT, 0 ms]
        (9) QDP
        (10) MRRProof [EQUIVALENT, 0 ms]
        (11) QDP
        (12) PisEmptyProof [EQUIVALENT, 0 ms]
        (13) YES
    (14) QDP
        (15) QDPOrderProof [EQUIVALENT, 336 ms]
        (16) QDP
        (17) DependencyGraphProof [EQUIVALENT, 0 ms]
        (18) QDP
        (19) QDPOrderProof [EQUIVALENT, 169 ms]
        (20) QDP
        (21) DependencyGraphProof [EQUIVALENT, 0 ms]
        (22) AND
            (23) QDP
                (24) QDPOrderProof [EQUIVALENT, 6875 ms]
                (25) QDP
                (26) PisEmptyProof [EQUIVALENT, 0 ms]
                (27) YES
            (28) QDP
                (29) TransformationProof [EQUIVALENT, 0 ms]
                (30) QDP
                (31) DependencyGraphProof [EQUIVALENT, 0 ms]
                (32) QDP
                (33) TransformationProof [EQUIVALENT, 0 ms]
                (34) QDP
                (35) DependencyGraphProof [EQUIVALENT, 0 ms]
                (36) QDP
                (37) TransformationProof [EQUIVALENT, 0 ms]
                (38) QDP
                (39) DependencyGraphProof [EQUIVALENT, 0 ms]
                (40) QDP
                (41) TransformationProof [EQUIVALENT, 0 ms]
                (42) QDP
                (43) DependencyGraphProof [EQUIVALENT, 0 ms]
                (44) QDP
                (45) TransformationProof [EQUIVALENT, 0 ms]
                (46) QDP
                (47) DependencyGraphProof [EQUIVALENT, 0 ms]
                (48) QDP
                (49) TransformationProof [EQUIVALENT, 0 ms]
                (50) QDP
                (51) DependencyGraphProof [EQUIVALENT, 0 ms]
                (52) QDP
                (53) TransformationProof [EQUIVALENT, 0 ms]
                (54) QDP
                (55) DependencyGraphProof [EQUIVALENT, 0 ms]
                (56) QDP
                (57) TransformationProof [EQUIVALENT, 0 ms]
                (58) QDP
                (59) DependencyGraphProof [EQUIVALENT, 0 ms]
                (60) QDP
                (61) TransformationProof [EQUIVALENT, 0 ms]
                (62) QDP
                (63) DependencyGraphProof [EQUIVALENT, 0 ms]
                (64) QDP
                (65) TransformationProof [EQUIVALENT, 0 ms]
                (66) QDP
                (67) DependencyGraphProof [EQUIVALENT, 0 ms]
                (68) QDP
                (69) TransformationProof [EQUIVALENT, 0 ms]
                (70) QDP
                (71) DependencyGraphProof [EQUIVALENT, 0 ms]
                (72) QDP
                (73) TransformationProof [EQUIVALENT, 18 ms]
                (74) QDP
                (75) DependencyGraphProof [EQUIVALENT, 0 ms]
                (76) QDP
                (77) QDPOrderProof [EQUIVALENT, 981 ms]
                (78) QDP
                (79) UsableRulesProof [EQUIVALENT, 0 ms]
                (80) QDP
                (81) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (82) YES


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

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

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

Q is empty.

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

(1) QTRS Reverse (EQUIVALENT)
We applied the QTRS Reverse Processor [REVERSE].
----------------------------------------

(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

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

Q is empty.

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

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

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

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

The TRS R consists of the following rules:

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

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

(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 137 less nodes.
----------------------------------------

(6)
Complex Obligation (AND)

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

(7)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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

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

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

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

(10) MRRProof (EQUIVALENT)
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented dependency pairs:

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


Used ordering: Polynomial interpretation [POLO]:

   POL(0(x_1)) = 2*x_1
   POL(1(x_1)) = 1 + x_1
   POL(2^1(x_1)) = x_1
   POL(4(x_1)) = x_1


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

(11)
Obligation:
Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(12) PisEmptyProof (EQUIVALENT)
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:

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

The TRS R consists of the following rules:

   2(1(1(0(x1)))) -> 2(3(1(0(1(x1)))))
   2(1(1(0(x1)))) -> 2(1(0(1(4(x1)))))
   2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1))))))
   2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1))))))
   2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1))))))
   2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1))))))
   5(1(1(0(x1)))) -> 5(1(0(1(4(x1)))))
   5(1(1(0(x1)))) -> 1(0(1(4(5(x1)))))
   5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1))))))
   5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1))))))
   5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1))))))
   5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1))))))
   5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1))))))
   5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1))))))
   5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1))))))
   5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1))))))
   5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1))))))
   5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1))))))
   5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1))))))
   5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1))))))
   5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1))))))
   5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1))))))
   0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1))))))
   5(3(1(0(x1)))) -> 1(4(5(3(0(x1)))))
   5(4(1(0(x1)))) -> 1(4(5(3(0(x1)))))
   5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1))))))
   5(4(2(0(x1)))) -> 5(3(2(0(4(x1)))))
   5(4(2(0(x1)))) -> 5(2(0(4(4(x1)))))
   5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1))))))
   5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1))))))
   5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1))))))
   5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1))))))
   0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1))))))
   5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1))))))
   0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1))))))
   0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1))))))
   5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1))))))
   5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1))))))
   2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1))))))
   2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1))))))
   2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1))))))
   5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1))))))
   5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1))))))
   5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1))))))
   2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1))))))
   5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1))))))
   0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1))))))
   5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1))))))
   5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1))))))
   5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1))))))
   0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1))))))
   5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1))))))
   5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1))))))
   5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1))))))
   5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1))))))
   5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1))))))
   5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1))))))
   5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1))))))
   5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1))))))
   5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1))))))
   5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1))))))
   2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1))))))
   5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1))))))
   5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1))))))
   5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1))))))
   5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1))))))
   5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1))))))
   5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1))))))
   2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1))))))
   5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1))))))
   5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1))))))
   5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1))))))
   5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1))))))
   5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1))))))
   5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1))))))
   5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1))))))
   5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1))))))
   5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1))))))
   5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(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(1(0(x1)))) -> 2^1(0(x1))
   5^1(4(1(1(0(x1))))) -> 2^1(x1)
   5^1(4(0(2(0(x1))))) -> 2^1(0(0(x1)))
   5^1(4(0(2(0(x1))))) -> 0^1(0(x1))
   5^1(1(3(2(0(x1))))) -> 0^1(0(x1))
   5^1(1(3(2(0(x1))))) -> 2^1(5(3(0(x1))))
   5^1(1(3(2(0(x1))))) -> 2^1(x1)
   5^1(1(5(2(0(x1))))) -> 2^1(5(3(0(x1))))
   5^1(1(5(2(0(x1))))) -> 2^1(x1)
   5^1(1(2(4(0(x1))))) -> 2^1(0(x1))
   5^1(1(2(4(0(x1))))) -> 2^1(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)) = 0
   POL(2(x_1)) = 1
   POL(2^1(x_1)) = 0
   POL(3(x_1)) = 0
   POL(4(x_1)) = 0
   POL(5(x_1)) = 1
   POL(5^1(x_1)) = 1

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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

(18)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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


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

(20)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

(21) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes.
----------------------------------------

(22)
Complex Obligation (AND)

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

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

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

The TRS R consists of the following rules:

   2(1(1(0(x1)))) -> 2(3(1(0(1(x1)))))
   2(1(1(0(x1)))) -> 2(1(0(1(4(x1)))))
   2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1))))))
   2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1))))))
   2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1))))))
   2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1))))))
   5(1(1(0(x1)))) -> 5(1(0(1(4(x1)))))
   5(1(1(0(x1)))) -> 1(0(1(4(5(x1)))))
   5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1))))))
   5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1))))))
   5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1))))))
   5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1))))))
   5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1))))))
   5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1))))))
   5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1))))))
   5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1))))))
   5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1))))))
   5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1))))))
   5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1))))))
   5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1))))))
   5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1))))))
   5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1))))))
   0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1))))))
   5(3(1(0(x1)))) -> 1(4(5(3(0(x1)))))
   5(4(1(0(x1)))) -> 1(4(5(3(0(x1)))))
   5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1))))))
   5(4(2(0(x1)))) -> 5(3(2(0(4(x1)))))
   5(4(2(0(x1)))) -> 5(2(0(4(4(x1)))))
   5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1))))))
   5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1))))))
   5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1))))))
   5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1))))))
   0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1))))))
   5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1))))))
   0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1))))))
   0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1))))))
   5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1))))))
   5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1))))))
   2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1))))))
   2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1))))))
   2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1))))))
   5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1))))))
   5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1))))))
   5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1))))))
   2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1))))))
   5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1))))))
   0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1))))))
   5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1))))))
   5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1))))))
   5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1))))))
   0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1))))))
   5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1))))))
   5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1))))))
   5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1))))))
   5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1))))))
   5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1))))))
   5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1))))))
   5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1))))))
   5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1))))))
   5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1))))))
   5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1))))))
   2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1))))))
   5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1))))))
   5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1))))))
   5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1))))))
   5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1))))))
   5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1))))))
   5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1))))))
   2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1))))))
   5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1))))))
   5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1))))))
   5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1))))))
   5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1))))))
   5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1))))))
   5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1))))))
   5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1))))))
   5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1))))))
   5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1))))))
   5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(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(5(0(2(0(x1))))) -> 0^1(5(2(0(x1))))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(0^1(x_1)) =  	[[-I]] 	 +  	[[0A, -I, -I]] 	* 	x_1
>>>

   <<<
 POL(5(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
>>>

   <<<
 POL(0(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, -I, -I], [0A, 0A, 0A]] 	* 	x_1
>>>

   <<<
 POL(2(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, -I, 0A], [0A, -I, 0A], [-I, -I, -I]] 	* 	x_1
>>>

   <<<
 POL(1(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, -I, -I], [0A, 0A, 0A], [-I, 0A, -I]] 	* 	x_1
>>>

   <<<
 POL(3(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, -I, -I], [0A, -I, -I], [-I, -I, -I]] 	* 	x_1
>>>

   <<<
 POL(4(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, -I], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
>>>


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

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


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

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

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

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

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

(27)
YES

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

(28)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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

   (5^1(1(1(0(2(1(0(x0))))))) -> 5^1(3(0(1(4(2(0(x0))))))),5^1(1(1(0(2(1(0(x0))))))) -> 5^1(3(0(1(4(2(0(x0))))))))
   (5^1(1(1(0(5(0(1(0(x0)))))))) -> 5^1(0(0(0(5(1(4(x0))))))),5^1(1(1(0(5(0(1(0(x0)))))))) -> 5^1(0(0(0(5(1(4(x0))))))))
   (5^1(1(1(0(2(1(1(0(x0)))))))) -> 5^1(0(1(2(1(4(0(x0))))))),5^1(1(1(0(2(1(1(0(x0)))))))) -> 5^1(0(1(2(1(4(0(x0))))))))
   (5^1(1(1(0(2(1(1(0(x0)))))))) -> 5^1(0(0(1(2(1(4(x0))))))),5^1(1(1(0(2(1(1(0(x0)))))))) -> 5^1(0(0(1(2(1(4(x0))))))))
   (5^1(1(1(0(5(4(1(0(x0)))))))) -> 5^1(3(0(1(4(5(0(x0))))))),5^1(1(1(0(5(4(1(0(x0)))))))) -> 5^1(3(0(1(4(5(0(x0))))))))
   (5^1(1(1(0(5(0(2(0(x0)))))))) -> 5^1(0(3(0(5(2(0(x0))))))),5^1(1(1(0(5(0(2(0(x0)))))))) -> 5^1(0(3(0(5(2(0(x0))))))))


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

(30)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   5^1(1(1(0(x1)))) -> 5^1(x1)
   5^1(1(1(0(x1)))) -> 5^1(3(x1))
   5^1(3(1(0(x1)))) -> 5^1(3(0(x1)))
   5^1(3(1(1(0(x1))))) -> 5^1(3(1(0(1(4(x1))))))
   5^1(3(1(1(0(x1))))) -> 5^1(x1)
   5^1(4(1(0(x1)))) -> 5^1(3(0(x1)))
   5^1(1(2(0(0(x1))))) -> 5^1(2(0(0(x1))))
   5^1(4(1(1(0(x1))))) -> 5^1(0(x1))
   5^1(1(3(2(0(x1))))) -> 5^1(3(2(1(0(0(x1))))))
   5^1(1(3(2(0(x1))))) -> 5^1(2(0(x1)))
   5^1(1(3(2(0(x1))))) -> 5^1(3(0(x1)))
   5^1(1(3(2(0(x1))))) -> 5^1(3(2(x1)))
   5^1(1(3(2(0(x1))))) -> 5^1(2(x1))
   5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1))))))
   5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1))))
   5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1)))
   5^1(1(1(0(2(1(0(x0))))))) -> 5^1(3(0(1(4(2(0(x0)))))))
   5^1(1(1(0(5(0(1(0(x0)))))))) -> 5^1(0(0(0(5(1(4(x0)))))))
   5^1(1(1(0(2(1(1(0(x0)))))))) -> 5^1(0(1(2(1(4(0(x0)))))))
   5^1(1(1(0(2(1(1(0(x0)))))))) -> 5^1(0(0(1(2(1(4(x0)))))))
   5^1(1(1(0(5(4(1(0(x0)))))))) -> 5^1(3(0(1(4(5(0(x0)))))))
   5^1(1(1(0(5(0(2(0(x0)))))))) -> 5^1(0(3(0(5(2(0(x0)))))))

The TRS R consists of the following rules:

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

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

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

(32)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

(33) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule 5^1(3(1(0(x1)))) -> 5^1(3(0(x1))) at position [0,0] we obtained the following new rules [LPAR04]:

   (5^1(3(1(0(2(1(0(x0))))))) -> 5^1(3(3(0(1(4(2(0(x0)))))))),5^1(3(1(0(2(1(0(x0))))))) -> 5^1(3(3(0(1(4(2(0(x0)))))))))
   (5^1(3(1(0(5(0(1(0(x0)))))))) -> 5^1(3(0(0(0(5(1(4(x0)))))))),5^1(3(1(0(5(0(1(0(x0)))))))) -> 5^1(3(0(0(0(5(1(4(x0)))))))))
   (5^1(3(1(0(2(1(1(0(x0)))))))) -> 5^1(3(0(1(2(1(4(0(x0)))))))),5^1(3(1(0(2(1(1(0(x0)))))))) -> 5^1(3(0(1(2(1(4(0(x0)))))))))
   (5^1(3(1(0(2(1(1(0(x0)))))))) -> 5^1(3(0(0(1(2(1(4(x0)))))))),5^1(3(1(0(2(1(1(0(x0)))))))) -> 5^1(3(0(0(1(2(1(4(x0)))))))))
   (5^1(3(1(0(5(4(1(0(x0)))))))) -> 5^1(3(3(0(1(4(5(0(x0)))))))),5^1(3(1(0(5(4(1(0(x0)))))))) -> 5^1(3(3(0(1(4(5(0(x0)))))))))
   (5^1(3(1(0(5(0(2(0(x0)))))))) -> 5^1(3(0(3(0(5(2(0(x0)))))))),5^1(3(1(0(5(0(2(0(x0)))))))) -> 5^1(3(0(3(0(5(2(0(x0)))))))))


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

(34)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   5^1(1(1(0(x1)))) -> 5^1(3(x1))
   5^1(3(1(1(0(x1))))) -> 5^1(3(1(0(1(4(x1))))))
   5^1(3(1(1(0(x1))))) -> 5^1(x1)
   5^1(1(1(0(x1)))) -> 5^1(x1)
   5^1(4(1(0(x1)))) -> 5^1(3(0(x1)))
   5^1(1(2(0(0(x1))))) -> 5^1(2(0(0(x1))))
   5^1(4(1(1(0(x1))))) -> 5^1(0(x1))
   5^1(1(3(2(0(x1))))) -> 5^1(3(2(1(0(0(x1))))))
   5^1(1(3(2(0(x1))))) -> 5^1(2(0(x1)))
   5^1(1(3(2(0(x1))))) -> 5^1(3(0(x1)))
   5^1(1(3(2(0(x1))))) -> 5^1(3(2(x1)))
   5^1(1(3(2(0(x1))))) -> 5^1(2(x1))
   5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1))))))
   5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1))))
   5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1)))
   5^1(3(1(0(2(1(0(x0))))))) -> 5^1(3(3(0(1(4(2(0(x0))))))))
   5^1(3(1(0(5(0(1(0(x0)))))))) -> 5^1(3(0(0(0(5(1(4(x0))))))))
   5^1(3(1(0(2(1(1(0(x0)))))))) -> 5^1(3(0(1(2(1(4(0(x0))))))))
   5^1(3(1(0(2(1(1(0(x0)))))))) -> 5^1(3(0(0(1(2(1(4(x0))))))))
   5^1(3(1(0(5(4(1(0(x0)))))))) -> 5^1(3(3(0(1(4(5(0(x0))))))))
   5^1(3(1(0(5(0(2(0(x0)))))))) -> 5^1(3(0(3(0(5(2(0(x0))))))))

The TRS R consists of the following rules:

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

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

(35) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 7 less nodes.
----------------------------------------

(36)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

(37) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule 5^1(4(1(0(x1)))) -> 5^1(3(0(x1))) at position [0,0] we obtained the following new rules [LPAR04]:

   (5^1(4(1(0(2(1(0(x0))))))) -> 5^1(3(3(0(1(4(2(0(x0)))))))),5^1(4(1(0(2(1(0(x0))))))) -> 5^1(3(3(0(1(4(2(0(x0)))))))))
   (5^1(4(1(0(5(0(1(0(x0)))))))) -> 5^1(3(0(0(0(5(1(4(x0)))))))),5^1(4(1(0(5(0(1(0(x0)))))))) -> 5^1(3(0(0(0(5(1(4(x0)))))))))
   (5^1(4(1(0(2(1(1(0(x0)))))))) -> 5^1(3(0(1(2(1(4(0(x0)))))))),5^1(4(1(0(2(1(1(0(x0)))))))) -> 5^1(3(0(1(2(1(4(0(x0)))))))))
   (5^1(4(1(0(2(1(1(0(x0)))))))) -> 5^1(3(0(0(1(2(1(4(x0)))))))),5^1(4(1(0(2(1(1(0(x0)))))))) -> 5^1(3(0(0(1(2(1(4(x0)))))))))
   (5^1(4(1(0(5(4(1(0(x0)))))))) -> 5^1(3(3(0(1(4(5(0(x0)))))))),5^1(4(1(0(5(4(1(0(x0)))))))) -> 5^1(3(3(0(1(4(5(0(x0)))))))))
   (5^1(4(1(0(5(0(2(0(x0)))))))) -> 5^1(3(0(3(0(5(2(0(x0)))))))),5^1(4(1(0(5(0(2(0(x0)))))))) -> 5^1(3(0(3(0(5(2(0(x0)))))))))


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

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

   5^1(3(1(1(0(x1))))) -> 5^1(x1)
   5^1(1(1(0(x1)))) -> 5^1(x1)
   5^1(1(1(0(x1)))) -> 5^1(3(x1))
   5^1(1(2(0(0(x1))))) -> 5^1(2(0(0(x1))))
   5^1(4(1(1(0(x1))))) -> 5^1(0(x1))
   5^1(1(3(2(0(x1))))) -> 5^1(3(2(1(0(0(x1))))))
   5^1(1(3(2(0(x1))))) -> 5^1(2(0(x1)))
   5^1(1(3(2(0(x1))))) -> 5^1(3(0(x1)))
   5^1(1(3(2(0(x1))))) -> 5^1(3(2(x1)))
   5^1(1(3(2(0(x1))))) -> 5^1(2(x1))
   5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1))))))
   5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1))))
   5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1)))
   5^1(4(1(0(2(1(0(x0))))))) -> 5^1(3(3(0(1(4(2(0(x0))))))))
   5^1(4(1(0(5(0(1(0(x0)))))))) -> 5^1(3(0(0(0(5(1(4(x0))))))))
   5^1(4(1(0(2(1(1(0(x0)))))))) -> 5^1(3(0(1(2(1(4(0(x0))))))))
   5^1(4(1(0(2(1(1(0(x0)))))))) -> 5^1(3(0(0(1(2(1(4(x0))))))))
   5^1(4(1(0(5(4(1(0(x0)))))))) -> 5^1(3(3(0(1(4(5(0(x0))))))))
   5^1(4(1(0(5(0(2(0(x0)))))))) -> 5^1(3(0(3(0(5(2(0(x0))))))))

The TRS R consists of the following rules:

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

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

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

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

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

The TRS R consists of the following rules:

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

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

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

   (5^1(1(2(0(0(2(1(0(x0)))))))) -> 5^1(2(0(3(0(1(4(2(0(x0))))))))),5^1(1(2(0(0(2(1(0(x0)))))))) -> 5^1(2(0(3(0(1(4(2(0(x0))))))))))
   (5^1(1(2(0(0(5(0(1(0(x0))))))))) -> 5^1(2(0(0(0(0(5(1(4(x0))))))))),5^1(1(2(0(0(5(0(1(0(x0))))))))) -> 5^1(2(0(0(0(0(5(1(4(x0))))))))))
   (5^1(1(2(0(0(2(1(1(0(x0))))))))) -> 5^1(2(0(0(1(2(1(4(0(x0))))))))),5^1(1(2(0(0(2(1(1(0(x0))))))))) -> 5^1(2(0(0(1(2(1(4(0(x0))))))))))
   (5^1(1(2(0(0(2(1(1(0(x0))))))))) -> 5^1(2(0(0(0(1(2(1(4(x0))))))))),5^1(1(2(0(0(2(1(1(0(x0))))))))) -> 5^1(2(0(0(0(1(2(1(4(x0))))))))))
   (5^1(1(2(0(0(5(4(1(0(x0))))))))) -> 5^1(2(0(3(0(1(4(5(0(x0))))))))),5^1(1(2(0(0(5(4(1(0(x0))))))))) -> 5^1(2(0(3(0(1(4(5(0(x0))))))))))
   (5^1(1(2(0(0(5(0(2(0(x0))))))))) -> 5^1(2(0(0(3(0(5(2(0(x0))))))))),5^1(1(2(0(0(5(0(2(0(x0))))))))) -> 5^1(2(0(0(3(0(5(2(0(x0))))))))))


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

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

   5^1(1(1(0(x1)))) -> 5^1(x1)
   5^1(1(1(0(x1)))) -> 5^1(3(x1))
   5^1(3(1(1(0(x1))))) -> 5^1(x1)
   5^1(4(1(1(0(x1))))) -> 5^1(0(x1))
   5^1(1(3(2(0(x1))))) -> 5^1(3(2(1(0(0(x1))))))
   5^1(1(3(2(0(x1))))) -> 5^1(2(0(x1)))
   5^1(1(3(2(0(x1))))) -> 5^1(3(0(x1)))
   5^1(1(3(2(0(x1))))) -> 5^1(3(2(x1)))
   5^1(1(3(2(0(x1))))) -> 5^1(2(x1))
   5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1))))))
   5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1))))
   5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1)))
   5^1(1(2(0(0(2(1(0(x0)))))))) -> 5^1(2(0(3(0(1(4(2(0(x0)))))))))
   5^1(1(2(0(0(5(0(1(0(x0))))))))) -> 5^1(2(0(0(0(0(5(1(4(x0)))))))))
   5^1(1(2(0(0(2(1(1(0(x0))))))))) -> 5^1(2(0(0(1(2(1(4(0(x0)))))))))
   5^1(1(2(0(0(2(1(1(0(x0))))))))) -> 5^1(2(0(0(0(1(2(1(4(x0)))))))))
   5^1(1(2(0(0(5(4(1(0(x0))))))))) -> 5^1(2(0(3(0(1(4(5(0(x0)))))))))
   5^1(1(2(0(0(5(0(2(0(x0))))))))) -> 5^1(2(0(0(3(0(5(2(0(x0)))))))))

The TRS R consists of the following rules:

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

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

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

(44)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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

   (5^1(4(1(1(0(2(1(0(x0)))))))) -> 5^1(3(0(1(4(2(0(x0))))))),5^1(4(1(1(0(2(1(0(x0)))))))) -> 5^1(3(0(1(4(2(0(x0))))))))
   (5^1(4(1(1(0(5(0(1(0(x0))))))))) -> 5^1(0(0(0(5(1(4(x0))))))),5^1(4(1(1(0(5(0(1(0(x0))))))))) -> 5^1(0(0(0(5(1(4(x0))))))))
   (5^1(4(1(1(0(2(1(1(0(x0))))))))) -> 5^1(0(1(2(1(4(0(x0))))))),5^1(4(1(1(0(2(1(1(0(x0))))))))) -> 5^1(0(1(2(1(4(0(x0))))))))
   (5^1(4(1(1(0(2(1(1(0(x0))))))))) -> 5^1(0(0(1(2(1(4(x0))))))),5^1(4(1(1(0(2(1(1(0(x0))))))))) -> 5^1(0(0(1(2(1(4(x0))))))))
   (5^1(4(1(1(0(5(4(1(0(x0))))))))) -> 5^1(3(0(1(4(5(0(x0))))))),5^1(4(1(1(0(5(4(1(0(x0))))))))) -> 5^1(3(0(1(4(5(0(x0))))))))
   (5^1(4(1(1(0(5(0(2(0(x0))))))))) -> 5^1(0(3(0(5(2(0(x0))))))),5^1(4(1(1(0(5(0(2(0(x0))))))))) -> 5^1(0(3(0(5(2(0(x0))))))))


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

(46)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   5^1(1(1(0(x1)))) -> 5^1(3(x1))
   5^1(3(1(1(0(x1))))) -> 5^1(x1)
   5^1(1(1(0(x1)))) -> 5^1(x1)
   5^1(1(3(2(0(x1))))) -> 5^1(3(2(1(0(0(x1))))))
   5^1(1(3(2(0(x1))))) -> 5^1(2(0(x1)))
   5^1(1(3(2(0(x1))))) -> 5^1(3(0(x1)))
   5^1(1(3(2(0(x1))))) -> 5^1(3(2(x1)))
   5^1(1(3(2(0(x1))))) -> 5^1(2(x1))
   5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1))))))
   5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1))))
   5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1)))
   5^1(4(1(1(0(2(1(0(x0)))))))) -> 5^1(3(0(1(4(2(0(x0)))))))
   5^1(4(1(1(0(5(0(1(0(x0))))))))) -> 5^1(0(0(0(5(1(4(x0)))))))
   5^1(4(1(1(0(2(1(1(0(x0))))))))) -> 5^1(0(1(2(1(4(0(x0)))))))
   5^1(4(1(1(0(2(1(1(0(x0))))))))) -> 5^1(0(0(1(2(1(4(x0)))))))
   5^1(4(1(1(0(5(4(1(0(x0))))))))) -> 5^1(3(0(1(4(5(0(x0)))))))
   5^1(4(1(1(0(5(0(2(0(x0))))))))) -> 5^1(0(3(0(5(2(0(x0)))))))

The TRS R consists of the following rules:

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

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

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

(48)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

(49) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule 5^1(1(3(2(0(x1))))) -> 5^1(3(2(1(0(0(x1)))))) at position [0,0] we obtained the following new rules [LPAR04]:

   (5^1(1(3(2(0(2(1(0(x0)))))))) -> 5^1(3(2(1(0(3(0(1(4(2(0(x0))))))))))),5^1(1(3(2(0(2(1(0(x0)))))))) -> 5^1(3(2(1(0(3(0(1(4(2(0(x0))))))))))))
   (5^1(1(3(2(0(5(0(1(0(x0))))))))) -> 5^1(3(2(1(0(0(0(0(5(1(4(x0))))))))))),5^1(1(3(2(0(5(0(1(0(x0))))))))) -> 5^1(3(2(1(0(0(0(0(5(1(4(x0))))))))))))
   (5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(3(2(1(0(0(1(2(1(4(0(x0))))))))))),5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(3(2(1(0(0(1(2(1(4(0(x0))))))))))))
   (5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(3(2(1(0(0(0(1(2(1(4(x0))))))))))),5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(3(2(1(0(0(0(1(2(1(4(x0))))))))))))
   (5^1(1(3(2(0(5(4(1(0(x0))))))))) -> 5^1(3(2(1(0(3(0(1(4(5(0(x0))))))))))),5^1(1(3(2(0(5(4(1(0(x0))))))))) -> 5^1(3(2(1(0(3(0(1(4(5(0(x0))))))))))))
   (5^1(1(3(2(0(5(0(2(0(x0))))))))) -> 5^1(3(2(1(0(0(3(0(5(2(0(x0))))))))))),5^1(1(3(2(0(5(0(2(0(x0))))))))) -> 5^1(3(2(1(0(0(3(0(5(2(0(x0))))))))))))


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

(50)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   5^1(3(1(1(0(x1))))) -> 5^1(x1)
   5^1(1(1(0(x1)))) -> 5^1(x1)
   5^1(1(1(0(x1)))) -> 5^1(3(x1))
   5^1(1(3(2(0(x1))))) -> 5^1(2(0(x1)))
   5^1(1(3(2(0(x1))))) -> 5^1(3(0(x1)))
   5^1(1(3(2(0(x1))))) -> 5^1(3(2(x1)))
   5^1(1(3(2(0(x1))))) -> 5^1(2(x1))
   5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1))))))
   5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1))))
   5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1)))
   5^1(1(3(2(0(2(1(0(x0)))))))) -> 5^1(3(2(1(0(3(0(1(4(2(0(x0)))))))))))
   5^1(1(3(2(0(5(0(1(0(x0))))))))) -> 5^1(3(2(1(0(0(0(0(5(1(4(x0)))))))))))
   5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(3(2(1(0(0(1(2(1(4(0(x0)))))))))))
   5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(3(2(1(0(0(0(1(2(1(4(x0)))))))))))
   5^1(1(3(2(0(5(4(1(0(x0))))))))) -> 5^1(3(2(1(0(3(0(1(4(5(0(x0)))))))))))
   5^1(1(3(2(0(5(0(2(0(x0))))))))) -> 5^1(3(2(1(0(0(3(0(5(2(0(x0)))))))))))

The TRS R consists of the following rules:

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

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

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

(52)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

(53) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule 5^1(1(3(2(0(x1))))) -> 5^1(2(0(x1))) at position [0] we obtained the following new rules [LPAR04]:

   (5^1(1(3(2(0(2(1(0(x0)))))))) -> 5^1(2(2(1(0(4(0(x0))))))),5^1(1(3(2(0(2(1(0(x0)))))))) -> 5^1(2(2(1(0(4(0(x0))))))))
   (5^1(1(3(2(0(2(4(0(x0)))))))) -> 5^1(2(2(3(4(0(0(x0))))))),5^1(1(3(2(0(2(4(0(x0)))))))) -> 5^1(2(2(3(4(0(0(x0))))))))
   (5^1(1(3(2(0(2(1(0(x0)))))))) -> 5^1(2(3(0(1(4(2(0(x0)))))))),5^1(1(3(2(0(2(1(0(x0)))))))) -> 5^1(2(3(0(1(4(2(0(x0)))))))))
   (5^1(1(3(2(0(5(0(1(0(x0))))))))) -> 5^1(2(0(0(0(5(1(4(x0)))))))),5^1(1(3(2(0(5(0(1(0(x0))))))))) -> 5^1(2(0(0(0(5(1(4(x0)))))))))
   (5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(2(0(1(2(1(4(0(x0)))))))),5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(2(0(1(2(1(4(0(x0)))))))))
   (5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(2(0(0(1(2(1(4(x0)))))))),5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(2(0(0(1(2(1(4(x0)))))))))
   (5^1(1(3(2(0(5(4(1(0(x0))))))))) -> 5^1(2(3(0(1(4(5(0(x0)))))))),5^1(1(3(2(0(5(4(1(0(x0))))))))) -> 5^1(2(3(0(1(4(5(0(x0)))))))))
   (5^1(1(3(2(0(5(0(2(0(x0))))))))) -> 5^1(2(0(3(0(5(2(0(x0)))))))),5^1(1(3(2(0(5(0(2(0(x0))))))))) -> 5^1(2(0(3(0(5(2(0(x0)))))))))


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

(54)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   5^1(1(1(0(x1)))) -> 5^1(x1)
   5^1(1(1(0(x1)))) -> 5^1(3(x1))
   5^1(3(1(1(0(x1))))) -> 5^1(x1)
   5^1(1(3(2(0(x1))))) -> 5^1(3(0(x1)))
   5^1(1(3(2(0(x1))))) -> 5^1(3(2(x1)))
   5^1(1(3(2(0(x1))))) -> 5^1(2(x1))
   5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1))))))
   5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1))))
   5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1)))
   5^1(1(3(2(0(2(1(0(x0)))))))) -> 5^1(2(2(1(0(4(0(x0)))))))
   5^1(1(3(2(0(2(4(0(x0)))))))) -> 5^1(2(2(3(4(0(0(x0)))))))
   5^1(1(3(2(0(2(1(0(x0)))))))) -> 5^1(2(3(0(1(4(2(0(x0))))))))
   5^1(1(3(2(0(5(0(1(0(x0))))))))) -> 5^1(2(0(0(0(5(1(4(x0))))))))
   5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(2(0(1(2(1(4(0(x0))))))))
   5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(2(0(0(1(2(1(4(x0))))))))
   5^1(1(3(2(0(5(4(1(0(x0))))))))) -> 5^1(2(3(0(1(4(5(0(x0))))))))
   5^1(1(3(2(0(5(0(2(0(x0))))))))) -> 5^1(2(0(3(0(5(2(0(x0))))))))

The TRS R consists of the following rules:

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

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

(55) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 8 less nodes.
----------------------------------------

(56)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

(57) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule 5^1(1(3(2(0(x1))))) -> 5^1(3(0(x1))) at position [0,0] we obtained the following new rules [LPAR04]:

   (5^1(1(3(2(0(2(1(0(x0)))))))) -> 5^1(3(3(0(1(4(2(0(x0)))))))),5^1(1(3(2(0(2(1(0(x0)))))))) -> 5^1(3(3(0(1(4(2(0(x0)))))))))
   (5^1(1(3(2(0(5(0(1(0(x0))))))))) -> 5^1(3(0(0(0(5(1(4(x0)))))))),5^1(1(3(2(0(5(0(1(0(x0))))))))) -> 5^1(3(0(0(0(5(1(4(x0)))))))))
   (5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(3(0(1(2(1(4(0(x0)))))))),5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(3(0(1(2(1(4(0(x0)))))))))
   (5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(3(0(0(1(2(1(4(x0)))))))),5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(3(0(0(1(2(1(4(x0)))))))))
   (5^1(1(3(2(0(5(4(1(0(x0))))))))) -> 5^1(3(3(0(1(4(5(0(x0)))))))),5^1(1(3(2(0(5(4(1(0(x0))))))))) -> 5^1(3(3(0(1(4(5(0(x0)))))))))
   (5^1(1(3(2(0(5(0(2(0(x0))))))))) -> 5^1(3(0(3(0(5(2(0(x0)))))))),5^1(1(3(2(0(5(0(2(0(x0))))))))) -> 5^1(3(0(3(0(5(2(0(x0)))))))))


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

(58)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   5^1(1(1(0(x1)))) -> 5^1(3(x1))
   5^1(3(1(1(0(x1))))) -> 5^1(x1)
   5^1(1(1(0(x1)))) -> 5^1(x1)
   5^1(1(3(2(0(x1))))) -> 5^1(3(2(x1)))
   5^1(1(3(2(0(x1))))) -> 5^1(2(x1))
   5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1))))))
   5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1))))
   5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1)))
   5^1(1(3(2(0(2(1(0(x0)))))))) -> 5^1(3(3(0(1(4(2(0(x0))))))))
   5^1(1(3(2(0(5(0(1(0(x0))))))))) -> 5^1(3(0(0(0(5(1(4(x0))))))))
   5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(3(0(1(2(1(4(0(x0))))))))
   5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(3(0(0(1(2(1(4(x0))))))))
   5^1(1(3(2(0(5(4(1(0(x0))))))))) -> 5^1(3(3(0(1(4(5(0(x0))))))))
   5^1(1(3(2(0(5(0(2(0(x0))))))))) -> 5^1(3(0(3(0(5(2(0(x0))))))))

The TRS R consists of the following rules:

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

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

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

(60)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

(61) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule 5^1(1(3(2(0(x1))))) -> 5^1(3(2(x1))) at position [0,0] we obtained the following new rules [LPAR04]:

   (5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(3(1(0(1(x0))))))),5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(3(1(0(1(x0))))))))
   (5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(1(0(1(4(x0))))))),5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(1(0(1(4(x0))))))))
   (5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(3(1(4(1(0(x0)))))))),5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(3(1(4(1(0(x0)))))))))
   (5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(1(4(1(4(0(x0)))))))),5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(1(4(1(4(0(x0)))))))))
   (5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(1(3(0(1(4(x0)))))))),5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(1(3(0(1(4(x0)))))))))
   (5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(0(1(3(1(4(x0)))))))),5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(0(1(3(1(4(x0)))))))))
   (5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(3(2(1(4(1(4(0(x0)))))))),5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(3(2(1(4(1(4(0(x0)))))))))
   (5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(3(0(2(1(3(1(4(x0)))))))),5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(3(0(2(1(3(1(4(x0)))))))))
   (5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(3(1(0(1(4(2(4(x0)))))))),5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(3(1(0(1(4(2(4(x0)))))))))
   (5^1(1(3(2(0(0(2(1(0(x0))))))))) -> 5^1(3(2(2(1(0(4(0(x0)))))))),5^1(1(3(2(0(0(2(1(0(x0))))))))) -> 5^1(3(2(2(1(0(4(0(x0)))))))))
   (5^1(1(3(2(0(1(5(2(0(x0))))))))) -> 5^1(3(5(1(2(3(2(0(x0)))))))),5^1(1(3(2(0(1(5(2(0(x0))))))))) -> 5^1(3(5(1(2(3(2(0(x0)))))))))
   (5^1(1(3(2(0(0(2(4(0(x0))))))))) -> 5^1(3(2(2(3(4(0(0(x0)))))))),5^1(1(3(2(0(0(2(4(0(x0))))))))) -> 5^1(3(2(2(3(4(0(0(x0)))))))))


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

(62)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   5^1(3(1(1(0(x1))))) -> 5^1(x1)
   5^1(1(1(0(x1)))) -> 5^1(x1)
   5^1(1(1(0(x1)))) -> 5^1(3(x1))
   5^1(1(3(2(0(x1))))) -> 5^1(2(x1))
   5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1))))))
   5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1))))
   5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1)))
   5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(3(1(0(1(x0)))))))
   5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(1(0(1(4(x0)))))))
   5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(3(1(4(1(0(x0))))))))
   5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(1(4(1(4(0(x0))))))))
   5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(1(3(0(1(4(x0))))))))
   5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(0(1(3(1(4(x0))))))))
   5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(3(2(1(4(1(4(0(x0))))))))
   5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(3(0(2(1(3(1(4(x0))))))))
   5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(3(1(0(1(4(2(4(x0))))))))
   5^1(1(3(2(0(0(2(1(0(x0))))))))) -> 5^1(3(2(2(1(0(4(0(x0))))))))
   5^1(1(3(2(0(1(5(2(0(x0))))))))) -> 5^1(3(5(1(2(3(2(0(x0))))))))
   5^1(1(3(2(0(0(2(4(0(x0))))))))) -> 5^1(3(2(2(3(4(0(0(x0))))))))

The TRS R consists of the following rules:

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

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

(63) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 12 less nodes.
----------------------------------------

(64)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

(65) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule 5^1(1(3(2(0(x1))))) -> 5^1(2(x1)) at position [0] we obtained the following new rules [LPAR04]:

   (5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(3(1(0(1(x0)))))),5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(3(1(0(1(x0)))))))
   (5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(1(0(1(4(x0)))))),5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(1(0(1(4(x0)))))))
   (5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(3(1(4(1(0(x0))))))),5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(3(1(4(1(0(x0))))))))
   (5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(1(4(1(4(0(x0))))))),5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(1(4(1(4(0(x0))))))))
   (5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(1(3(0(1(4(x0))))))),5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(1(3(0(1(4(x0))))))))
   (5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(0(1(3(1(4(x0))))))),5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(0(1(3(1(4(x0))))))))
   (5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(2(1(4(1(4(0(x0))))))),5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(2(1(4(1(4(0(x0))))))))
   (5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(0(2(1(3(1(4(x0))))))),5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(0(2(1(3(1(4(x0))))))))
   (5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(1(0(1(4(2(4(x0))))))),5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(1(0(1(4(2(4(x0))))))))
   (5^1(1(3(2(0(0(2(1(0(x0))))))))) -> 5^1(2(2(1(0(4(0(x0))))))),5^1(1(3(2(0(0(2(1(0(x0))))))))) -> 5^1(2(2(1(0(4(0(x0))))))))
   (5^1(1(3(2(0(1(5(2(0(x0))))))))) -> 5^1(5(1(2(3(2(0(x0))))))),5^1(1(3(2(0(1(5(2(0(x0))))))))) -> 5^1(5(1(2(3(2(0(x0))))))))
   (5^1(1(3(2(0(0(2(4(0(x0))))))))) -> 5^1(2(2(3(4(0(0(x0))))))),5^1(1(3(2(0(0(2(4(0(x0))))))))) -> 5^1(2(2(3(4(0(0(x0))))))))


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

(66)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   5^1(1(1(0(x1)))) -> 5^1(x1)
   5^1(1(1(0(x1)))) -> 5^1(3(x1))
   5^1(3(1(1(0(x1))))) -> 5^1(x1)
   5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1))))))
   5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1))))
   5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1)))
   5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(3(1(0(1(x0))))))
   5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(1(0(1(4(x0))))))
   5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(3(1(4(1(0(x0)))))))
   5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(1(4(1(4(0(x0)))))))
   5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(1(3(0(1(4(x0)))))))
   5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(0(1(3(1(4(x0)))))))
   5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(2(1(4(1(4(0(x0)))))))
   5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(0(2(1(3(1(4(x0)))))))
   5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(1(0(1(4(2(4(x0)))))))
   5^1(1(3(2(0(0(2(1(0(x0))))))))) -> 5^1(2(2(1(0(4(0(x0)))))))
   5^1(1(3(2(0(1(5(2(0(x0))))))))) -> 5^1(5(1(2(3(2(0(x0)))))))
   5^1(1(3(2(0(0(2(4(0(x0))))))))) -> 5^1(2(2(3(4(0(0(x0)))))))

The TRS R consists of the following rules:

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

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

(67) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 12 less nodes.
----------------------------------------

(68)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

(69) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule 5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1)))))) at position [0,0] we obtained the following new rules [LPAR04]:

   (5^1(1(5(2(0(2(1(0(x0)))))))) -> 5^1(1(2(5(3(3(0(1(4(2(0(x0))))))))))),5^1(1(5(2(0(2(1(0(x0)))))))) -> 5^1(1(2(5(3(3(0(1(4(2(0(x0))))))))))))
   (5^1(1(5(2(0(5(0(1(0(x0))))))))) -> 5^1(1(2(5(3(0(0(0(5(1(4(x0))))))))))),5^1(1(5(2(0(5(0(1(0(x0))))))))) -> 5^1(1(2(5(3(0(0(0(5(1(4(x0))))))))))))
   (5^1(1(5(2(0(2(1(1(0(x0))))))))) -> 5^1(1(2(5(3(0(1(2(1(4(0(x0))))))))))),5^1(1(5(2(0(2(1(1(0(x0))))))))) -> 5^1(1(2(5(3(0(1(2(1(4(0(x0))))))))))))
   (5^1(1(5(2(0(2(1(1(0(x0))))))))) -> 5^1(1(2(5(3(0(0(1(2(1(4(x0))))))))))),5^1(1(5(2(0(2(1(1(0(x0))))))))) -> 5^1(1(2(5(3(0(0(1(2(1(4(x0))))))))))))
   (5^1(1(5(2(0(5(4(1(0(x0))))))))) -> 5^1(1(2(5(3(3(0(1(4(5(0(x0))))))))))),5^1(1(5(2(0(5(4(1(0(x0))))))))) -> 5^1(1(2(5(3(3(0(1(4(5(0(x0))))))))))))
   (5^1(1(5(2(0(5(0(2(0(x0))))))))) -> 5^1(1(2(5(3(0(3(0(5(2(0(x0))))))))))),5^1(1(5(2(0(5(0(2(0(x0))))))))) -> 5^1(1(2(5(3(0(3(0(5(2(0(x0))))))))))))


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

(70)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   5^1(1(1(0(x1)))) -> 5^1(3(x1))
   5^1(3(1(1(0(x1))))) -> 5^1(x1)
   5^1(1(1(0(x1)))) -> 5^1(x1)
   5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1))))
   5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1)))
   5^1(1(5(2(0(2(1(0(x0)))))))) -> 5^1(1(2(5(3(3(0(1(4(2(0(x0)))))))))))
   5^1(1(5(2(0(5(0(1(0(x0))))))))) -> 5^1(1(2(5(3(0(0(0(5(1(4(x0)))))))))))
   5^1(1(5(2(0(2(1(1(0(x0))))))))) -> 5^1(1(2(5(3(0(1(2(1(4(0(x0)))))))))))
   5^1(1(5(2(0(2(1(1(0(x0))))))))) -> 5^1(1(2(5(3(0(0(1(2(1(4(x0)))))))))))
   5^1(1(5(2(0(5(4(1(0(x0))))))))) -> 5^1(1(2(5(3(3(0(1(4(5(0(x0)))))))))))
   5^1(1(5(2(0(5(0(2(0(x0))))))))) -> 5^1(1(2(5(3(0(3(0(5(2(0(x0)))))))))))

The TRS R consists of the following rules:

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

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

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

(72)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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

   (5^1(1(2(4(0(2(1(0(x0)))))))) -> 5^1(2(2(1(0(4(0(x0))))))),5^1(1(2(4(0(2(1(0(x0)))))))) -> 5^1(2(2(1(0(4(0(x0))))))))
   (5^1(1(2(4(0(2(4(0(x0)))))))) -> 5^1(2(2(3(4(0(0(x0))))))),5^1(1(2(4(0(2(4(0(x0)))))))) -> 5^1(2(2(3(4(0(0(x0))))))))
   (5^1(1(2(4(0(2(1(0(x0)))))))) -> 5^1(2(3(0(1(4(2(0(x0)))))))),5^1(1(2(4(0(2(1(0(x0)))))))) -> 5^1(2(3(0(1(4(2(0(x0)))))))))
   (5^1(1(2(4(0(5(0(1(0(x0))))))))) -> 5^1(2(0(0(0(5(1(4(x0)))))))),5^1(1(2(4(0(5(0(1(0(x0))))))))) -> 5^1(2(0(0(0(5(1(4(x0)))))))))
   (5^1(1(2(4(0(2(1(1(0(x0))))))))) -> 5^1(2(0(1(2(1(4(0(x0)))))))),5^1(1(2(4(0(2(1(1(0(x0))))))))) -> 5^1(2(0(1(2(1(4(0(x0)))))))))
   (5^1(1(2(4(0(2(1(1(0(x0))))))))) -> 5^1(2(0(0(1(2(1(4(x0)))))))),5^1(1(2(4(0(2(1(1(0(x0))))))))) -> 5^1(2(0(0(1(2(1(4(x0)))))))))
   (5^1(1(2(4(0(5(4(1(0(x0))))))))) -> 5^1(2(3(0(1(4(5(0(x0)))))))),5^1(1(2(4(0(5(4(1(0(x0))))))))) -> 5^1(2(3(0(1(4(5(0(x0)))))))))
   (5^1(1(2(4(0(5(0(2(0(x0))))))))) -> 5^1(2(0(3(0(5(2(0(x0)))))))),5^1(1(2(4(0(5(0(2(0(x0))))))))) -> 5^1(2(0(3(0(5(2(0(x0)))))))))


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

(74)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   5^1(3(1(1(0(x1))))) -> 5^1(x1)
   5^1(1(1(0(x1)))) -> 5^1(x1)
   5^1(1(1(0(x1)))) -> 5^1(3(x1))
   5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1))))
   5^1(1(2(4(0(2(1(0(x0)))))))) -> 5^1(2(2(1(0(4(0(x0)))))))
   5^1(1(2(4(0(2(4(0(x0)))))))) -> 5^1(2(2(3(4(0(0(x0)))))))
   5^1(1(2(4(0(2(1(0(x0)))))))) -> 5^1(2(3(0(1(4(2(0(x0))))))))
   5^1(1(2(4(0(5(0(1(0(x0))))))))) -> 5^1(2(0(0(0(5(1(4(x0))))))))
   5^1(1(2(4(0(2(1(1(0(x0))))))))) -> 5^1(2(0(1(2(1(4(0(x0))))))))
   5^1(1(2(4(0(2(1(1(0(x0))))))))) -> 5^1(2(0(0(1(2(1(4(x0))))))))
   5^1(1(2(4(0(5(4(1(0(x0))))))))) -> 5^1(2(3(0(1(4(5(0(x0))))))))
   5^1(1(2(4(0(5(0(2(0(x0))))))))) -> 5^1(2(0(3(0(5(2(0(x0))))))))

The TRS R consists of the following rules:

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

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

(75) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 8 less nodes.
----------------------------------------

(76)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

(78)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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

(80)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

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

From the DPs we obtained the following set of size-change graphs:
*5^1(3(1(1(0(x1))))) -> 5^1(x1)
The graph contains the following edges 1 > 1


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

(82)
YES