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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) DependencyPairsProof [EQUIVALENT, 222 ms]
(2) QDP
(3) DependencyGraphProof [EQUIVALENT, 1 ms]
(4) QDP
(5) QDPOrderProof [EQUIVALENT, 415 ms]
(6) QDP
(7) QDPOrderProof [EQUIVALENT, 179 ms]
(8) QDP
(9) QDPOrderProof [EQUIVALENT, 154 ms]
(10) QDP
(11) QDPOrderProof [EQUIVALENT, 154 ms]
(12) QDP
(13) QDPOrderProof [EQUIVALENT, 100 ms]
(14) QDP
(15) QDPOrderProof [EQUIVALENT, 814 ms]
(16) QDP
(17) QDPOrderProof [EQUIVALENT, 1936 ms]
(18) QDP
(19) TransformationProof [EQUIVALENT, 0 ms]
(20) QDP
(21) DependencyGraphProof [EQUIVALENT, 0 ms]
(22) QDP
(23) TransformationProof [EQUIVALENT, 0 ms]
(24) QDP
(25) DependencyGraphProof [EQUIVALENT, 0 ms]
(26) QDP
(27) TransformationProof [EQUIVALENT, 0 ms]
(28) QDP
(29) DependencyGraphProof [EQUIVALENT, 0 ms]
(30) QDP
(31) TransformationProof [EQUIVALENT, 0 ms]
(32) QDP
(33) DependencyGraphProof [EQUIVALENT, 0 ms]
(34) QDP
(35) TransformationProof [EQUIVALENT, 0 ms]
(36) QDP
(37) DependencyGraphProof [EQUIVALENT, 0 ms]
(38) QDP
(39) TransformationProof [EQUIVALENT, 0 ms]
(40) QDP
(41) DependencyGraphProof [EQUIVALENT, 0 ms]
(42) QDP
(43) TransformationProof [EQUIVALENT, 0 ms]
(44) QDP
(45) DependencyGraphProof [EQUIVALENT, 0 ms]
(46) QDP
(47) TransformationProof [EQUIVALENT, 14 ms]
(48) QDP
(49) DependencyGraphProof [EQUIVALENT, 0 ms]
(50) QDP
(51) TransformationProof [EQUIVALENT, 0 ms]
(52) QDP
(53) DependencyGraphProof [EQUIVALENT, 0 ms]
(54) QDP
(55) TransformationProof [EQUIVALENT, 0 ms]
(56) QDP
(57) DependencyGraphProof [EQUIVALENT, 0 ms]
(58) QDP
(59) TransformationProof [EQUIVALENT, 37 ms]
(60) QDP
(61) DependencyGraphProof [EQUIVALENT, 0 ms]
(62) QDP
(63) TransformationProof [EQUIVALENT, 23 ms]
(64) QDP
(65) DependencyGraphProof [EQUIVALENT, 0 ms]
(66) QDP
(67) TransformationProof [EQUIVALENT, 0 ms]
(68) QDP
(69) DependencyGraphProof [EQUIVALENT, 0 ms]
(70) QDP
(71) TransformationProof [EQUIVALENT, 21 ms]
(72) QDP
(73) DependencyGraphProof [EQUIVALENT, 0 ms]
(74) QDP
(75) TransformationProof [EQUIVALENT, 35 ms]
(76) QDP
(77) DependencyGraphProof [EQUIVALENT, 0 ms]
(78) QDP
(79) TransformationProof [EQUIVALENT, 28 ms]
(80) QDP
(81) DependencyGraphProof [EQUIVALENT, 0 ms]
(82) QDP
(83) TransformationProof [EQUIVALENT, 4 ms]
(84) QDP
(85) DependencyGraphProof [EQUIVALENT, 0 ms]
(86) QDP
(87) TransformationProof [EQUIVALENT, 0 ms]
(88) QDP
(89) DependencyGraphProof [EQUIVALENT, 0 ms]
(90) QDP
(91) TransformationProof [EQUIVALENT, 26 ms]
(92) QDP
(93) DependencyGraphProof [EQUIVALENT, 0 ms]
(94) QDP
(95) TransformationProof [EQUIVALENT, 21 ms]
(96) QDP
(97) DependencyGraphProof [EQUIVALENT, 0 ms]
(98) QDP
(99) TransformationProof [EQUIVALENT, 0 ms]
(100) QDP
(101) DependencyGraphProof [EQUIVALENT, 0 ms]
(102) QDP
(103) QDPOrderProof [EQUIVALENT, 1208 ms]
(104) QDP
(105) DependencyGraphProof [EQUIVALENT, 0 ms]
(106) AND
    (107) QDP
        (108) QDPOrderProof [EQUIVALENT, 956 ms]
        (109) QDP
        (110) PisEmptyProof [EQUIVALENT, 0 ms]
        (111) YES
    (112) QDP
        (113) QDPOrderProof [EQUIVALENT, 24 ms]
        (114) QDP
        (115) QDPOrderProof [EQUIVALENT, 19 ms]
        (116) QDP
        (117) DependencyGraphProof [EQUIVALENT, 0 ms]
        (118) QDP
        (119) QDPOrderProof [EQUIVALENT, 30 ms]
        (120) QDP
        (121) PisEmptyProof [EQUIVALENT, 0 ms]
        (122) YES


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

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

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

Q is empty.

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

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

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

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

The TRS R consists of the following rules:

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

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

(3) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 86 less nodes.
----------------------------------------

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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


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

(8)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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


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

(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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


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

(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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


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

(14)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

(19) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule 4^1(5(2(4(x1)))) -> 4^1(1(5(5(2(0(3(1(3(3(x1)))))))))) at position [0] we obtained the following new rules [LPAR04]:

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


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

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

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

The TRS R consists of the following rules:

   5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1))))))))))
   5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1))))))))))
   2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1))))))))))
   5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1))))))))))
   5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1))))))))))
   5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1))))))))))
   5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1))))))))))
   2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1))))))))))
   4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1))))))))))
   4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1))))))))))
   0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1))))))))))
   2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1))))))))))
   5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1))))))))))
   0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1))))))))))
   1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1))))))))))
   4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1))))))))))
   4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1))))))))))
   5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1))))))))))
   5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1))))))))))
   0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1))))))))))
   4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(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 1 SCC with 3 less nodes.
----------------------------------------

(22)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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

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


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

(24)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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

(26)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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

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

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

The TRS R consists of the following rules:

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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

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

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

The TRS R consists of the following rules:

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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

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

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

The TRS R consists of the following rules:

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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

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

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

The TRS R consists of the following rules:

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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

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

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

The TRS R consists of the following rules:

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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

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

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

The TRS R consists of the following rules:

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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

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

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

The TRS R consists of the following rules:

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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

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

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

The TRS R consists of the following rules:

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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

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

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

The TRS R consists of the following rules:

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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

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

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

The TRS R consists of the following rules:

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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

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

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

The TRS R consists of the following rules:

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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

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

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

The TRS R consists of the following rules:

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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

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

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

The TRS R consists of the following rules:

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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

(82)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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

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


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

(84)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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

(86)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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

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


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

(88)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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

(90)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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

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


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

(92)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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

(94)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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

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


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

(96)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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

(98)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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

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


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

(100)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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

(102)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

(104)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

(105) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 7 less nodes.
----------------------------------------

(106)
Complex Obligation (AND)

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

(107)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

(109)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

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

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

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

(111)
YES

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

(112)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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


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

(114)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

(116)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

(117) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
----------------------------------------

(118)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

(120)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

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

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

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

(122)
YES