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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) QTRS
(1) QTRS Reverse [EQUIVALENT, 0 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 284 ms]
(4) QTRS
(5) DependencyPairsProof [EQUIVALENT, 804 ms]
(6) QDP
(7) QDPOrderProof [EQUIVALENT, 1244 ms]
(8) QDP
(9) DependencyGraphProof [EQUIVALENT, 0 ms]
(10) AND
    (11) QDP
        (12) QDPOrderProof [EQUIVALENT, 308 ms]
        (13) QDP
        (14) DependencyGraphProof [EQUIVALENT, 0 ms]
        (15) AND
            (16) QDP
                (17) QDPOrderProof [EQUIVALENT, 59 ms]
                (18) QDP
                (19) QDPOrderProof [EQUIVALENT, 72 ms]
                (20) QDP
                (21) QDPOrderProof [EQUIVALENT, 46 ms]
                (22) QDP
                (23) MRRProof [EQUIVALENT, 226 ms]
                (24) QDP
                (25) PisEmptyProof [EQUIVALENT, 0 ms]
                (26) YES
            (27) QDP
                (28) QDPOrderProof [EQUIVALENT, 74 ms]
                (29) QDP
                (30) QDPOrderProof [EQUIVALENT, 49 ms]
                (31) QDP
                (32) MRRProof [EQUIVALENT, 1435 ms]
                (33) QDP
                (34) QDPOrderProof [EQUIVALENT, 0 ms]
                (35) QDP
                (36) PisEmptyProof [EQUIVALENT, 0 ms]
                (37) YES
            (38) QDP
                (39) QDPOrderProof [EQUIVALENT, 83 ms]
                (40) QDP
                (41) QDPOrderProof [EQUIVALENT, 59 ms]
                (42) QDP
                (43) QDPOrderProof [EQUIVALENT, 58 ms]
                (44) QDP
                (45) QDPOrderProof [EQUIVALENT, 91 ms]
                (46) QDP
                (47) DependencyGraphProof [EQUIVALENT, 0 ms]
                (48) AND
                    (49) QDP
                        (50) QDPOrderProof [EQUIVALENT, 43 ms]
                        (51) QDP
                        (52) QDPOrderProof [EQUIVALENT, 44 ms]
                        (53) QDP
                        (54) QDPOrderProof [EQUIVALENT, 42 ms]
                        (55) QDP
                        (56) QDPOrderProof [EQUIVALENT, 61 ms]
                        (57) QDP
                        (58) QDPOrderProof [EQUIVALENT, 42 ms]
                        (59) QDP
                        (60) QDPOrderProof [EQUIVALENT, 1434 ms]
                        (61) QDP
                        (62) MRRProof [EQUIVALENT, 3277 ms]
                        (63) QDP
                        (64) MRRProof [EQUIVALENT, 338 ms]
                        (65) QDP
                        (66) MRRProof [EQUIVALENT, 185 ms]
                        (67) QDP
                        (68) MRRProof [EQUIVALENT, 126 ms]
                        (69) QDP
                        (70) DependencyGraphProof [EQUIVALENT, 0 ms]
                        (71) TRUE
                    (72) QDP
                        (73) QDPOrderProof [EQUIVALENT, 43 ms]
                        (74) QDP
                        (75) QDPOrderProof [EQUIVALENT, 47 ms]
                        (76) QDP
                        (77) QDPOrderProof [EQUIVALENT, 43 ms]
                        (78) QDP
                        (79) QDPOrderProof [EQUIVALENT, 43 ms]
                        (80) QDP
                        (81) QDPOrderProof [EQUIVALENT, 16 ms]
                        (82) QDP
                        (83) QDPOrderProof [EQUIVALENT, 50 ms]
                        (84) QDP
                        (85) MRRProof [EQUIVALENT, 1421 ms]
                        (86) QDP
                        (87) QDPOrderProof [EQUIVALENT, 16 ms]
                        (88) QDP
                        (89) MRRProof [EQUIVALENT, 95 ms]
                        (90) QDP
                        (91) DependencyGraphProof [EQUIVALENT, 0 ms]
                        (92) AND
                            (93) QDP
                                (94) MRRProof [EQUIVALENT, 85 ms]
                                (95) QDP
                                (96) MRRProof [EQUIVALENT, 67 ms]
                                (97) QDP
                                (98) MRRProof [EQUIVALENT, 64 ms]
                                (99) QDP
                                (100) MRRProof [EQUIVALENT, 56 ms]
                                (101) QDP
                                (102) QDPOrderProof [EQUIVALENT, 1385 ms]
                                (103) QDP
                                (104) PisEmptyProof [EQUIVALENT, 0 ms]
                                (105) YES
                            (106) QDP
                                (107) QDPOrderProof [EQUIVALENT, 165 ms]
                                (108) QDP
                                (109) PisEmptyProof [EQUIVALENT, 0 ms]
                                (110) YES
    (111) QDP
        (112) QDPOrderProof [EQUIVALENT, 71 ms]
        (113) QDP
        (114) QDPOrderProof [EQUIVALENT, 62 ms]
        (115) QDP
        (116) MRRProof [EQUIVALENT, 344 ms]
        (117) QDP
        (118) QDPOrderProof [EQUIVALENT, 5 ms]
        (119) QDP
        (120) PisEmptyProof [EQUIVALENT, 0 ms]
        (121) YES
    (122) QDP
        (123) QDPOrderProof [EQUIVALENT, 0 ms]
        (124) QDP
        (125) QDPOrderProof [EQUIVALENT, 72 ms]
        (126) QDP
        (127) QDPOrderProof [EQUIVALENT, 65 ms]
        (128) QDP
        (129) MRRProof [EQUIVALENT, 3096 ms]
        (130) QDP
        (131) MRRProof [EQUIVALENT, 310 ms]
        (132) QDP
        (133) MRRProof [EQUIVALENT, 82 ms]
        (134) QDP
        (135) MRRProof [EQUIVALENT, 146 ms]
        (136) QDP
        (137) DependencyGraphProof [EQUIVALENT, 0 ms]
        (138) QDP
        (139) MRRProof [EQUIVALENT, 51 ms]
        (140) QDP
        (141) MRRProof [EQUIVALENT, 97 ms]
        (142) QDP
        (143) QDPOrderProof [EQUIVALENT, 255 ms]
        (144) QDP
        (145) QDPOrderProof [EQUIVALENT, 145 ms]
        (146) QDP
        (147) PisEmptyProof [EQUIVALENT, 0 ms]
        (148) YES


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

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

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

Q is empty.

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

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

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

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

Q is empty.

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

(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(0(x_1)) = 9 + x_1
   POL(1(x_1)) = 5 + x_1
   POL(2(x_1)) = 8 + x_1
   POL(3(x_1)) = 9 + x_1
   POL(4(x_1)) = 9 + x_1
   POL(5(x_1)) = 8 + x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

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




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

(4)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

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

Q is empty.

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

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

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

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

The TRS R consists of the following rules:

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

(9) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 30 less nodes.
----------------------------------------

(10)
Complex Obligation (AND)

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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

(13)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

(14) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 10 less nodes.
----------------------------------------

(15)
Complex Obligation (AND)

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

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

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

The TRS R consists of the following rules:

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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

Strictly oriented dependency pairs:

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

Strictly oriented rules of the TRS R:

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

Used ordering: Polynomial interpretation [POLO]:

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


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

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

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

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

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

(26)
YES

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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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

Strictly oriented dependency pairs:

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

Strictly oriented rules of the TRS R:

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

Used ordering: Polynomial interpretation [POLO]:

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


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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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


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

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

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

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

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

(37)
YES

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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

(47) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
----------------------------------------

(48)
Complex Obligation (AND)

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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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


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

(51)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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


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

(53)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

(55)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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


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

(57)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

(59)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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


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

(61)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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

Strictly oriented dependency pairs:

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

Strictly oriented rules of the TRS R:

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

Used ordering: Polynomial interpretation [POLO]:

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


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

(63)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


Strictly oriented rules of the TRS R:

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

Used ordering: Polynomial interpretation [POLO]:

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


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

(65)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


Strictly oriented rules of the TRS R:

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

Used ordering: Polynomial interpretation [POLO]:

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


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

(67)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


Strictly oriented rules of the TRS R:

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

Used ordering: Polynomial interpretation [POLO]:

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


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

(69)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

(70) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
----------------------------------------

(71)
TRUE

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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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


Strictly oriented rules of the TRS R:

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

Used ordering: Polynomial interpretation [POLO]:

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


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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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


Strictly oriented rules of the TRS R:

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

Used ordering: Polynomial interpretation [POLO]:

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


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

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

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

The TRS R consists of the following rules:

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

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

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

(92)
Complex Obligation (AND)

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

(93)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


Strictly oriented rules of the TRS R:

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

Used ordering: Polynomial interpretation [POLO]:

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


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

(95)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


Strictly oriented rules of the TRS R:

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

Used ordering: Polynomial interpretation [POLO]:

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


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

(97)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


Strictly oriented rules of the TRS R:

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

Used ordering: Polynomial interpretation [POLO]:

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


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

(99)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


Strictly oriented rules of the TRS R:

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

Used ordering: Polynomial interpretation [POLO]:

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


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

(101)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

(103)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

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

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

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

(105)
YES

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

(106)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

(108)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

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

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

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

(110)
YES

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

(111)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

(113)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

(115)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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

Strictly oriented dependency pairs:

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

Strictly oriented rules of the TRS R:

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

Used ordering: Polynomial interpretation [POLO]:

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


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

(117)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

(119)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

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

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

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

(121)
YES

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

(122)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

(124)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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


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

(126)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

(128)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


Strictly oriented rules of the TRS R:

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

Used ordering: Polynomial interpretation [POLO]:

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


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

(130)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


Strictly oriented rules of the TRS R:

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

Used ordering: Polynomial interpretation [POLO]:

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


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

(132)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


Strictly oriented rules of the TRS R:

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

Used ordering: Polynomial interpretation [POLO]:

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


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

(134)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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

Strictly oriented dependency pairs:

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

Strictly oriented rules of the TRS R:

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

Used ordering: Polynomial interpretation [POLO]:

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


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

(136)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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

(138)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


Strictly oriented rules of the TRS R:

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

Used ordering: Polynomial interpretation [POLO]:

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


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

(140)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


Strictly oriented rules of the TRS R:

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

Used ordering: Polynomial interpretation [POLO]:

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


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

(142)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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


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

(144)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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(146)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(147) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
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(148)
YES