YES
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) QTRSRRRProof [EQUIVALENT, 226 ms]
(2) QTRS
(3) DependencyPairsProof [EQUIVALENT, 985 ms]
(4) QDP
(5) QDPOrderProof [EQUIVALENT, 1976 ms]
(6) QDP
(7) DependencyGraphProof [EQUIVALENT, 0 ms]
(8) AND
    (9) QDP
        (10) QDPOrderProof [EQUIVALENT, 61 ms]
        (11) QDP
        (12) QDPOrderProof [EQUIVALENT, 5305 ms]
        (13) QDP
        (14) PisEmptyProof [EQUIVALENT, 0 ms]
        (15) YES
    (16) QDP
        (17) QDPOrderProof [EQUIVALENT, 96 ms]
        (18) QDP
        (19) QDPOrderProof [EQUIVALENT, 0 ms]
        (20) QDP
        (21) PisEmptyProof [EQUIVALENT, 0 ms]
        (22) YES
    (23) QDP
        (24) QDPOrderProof [EQUIVALENT, 62 ms]
        (25) QDP
        (26) QDPOrderProof [EQUIVALENT, 74 ms]
        (27) QDP
        (28) QDPOrderProof [EQUIVALENT, 40 ms]
        (29) QDP
        (30) PisEmptyProof [EQUIVALENT, 0 ms]
        (31) YES
    (32) QDP
        (33) QDPOrderProof [EQUIVALENT, 56 ms]
        (34) QDP
        (35) QDPOrderProof [EQUIVALENT, 73 ms]
        (36) QDP
        (37) PisEmptyProof [EQUIVALENT, 0 ms]
        (38) YES
    (39) QDP
        (40) QDPOrderProof [EQUIVALENT, 63 ms]
        (41) QDP
        (42) QDPOrderProof [EQUIVALENT, 38 ms]
        (43) QDP
        (44) QDPOrderProof [EQUIVALENT, 51 ms]
        (45) QDP
        (46) PisEmptyProof [EQUIVALENT, 0 ms]
        (47) YES
    (48) QDP
        (49) QDPOrderProof [EQUIVALENT, 57 ms]
        (50) QDP
        (51) QDPOrderProof [EQUIVALENT, 23 ms]
        (52) QDP
        (53) PisEmptyProof [EQUIVALENT, 0 ms]
        (54) YES


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

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

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

Q is empty.

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

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

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

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




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

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

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

Q is empty.

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

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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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


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

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

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

The TRS R consists of the following rules:

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

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

(7) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 6 SCCs with 9 less nodes.
----------------------------------------

(8)
Complex Obligation (AND)

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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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

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

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


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

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

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

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

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

(15)
YES

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

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

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

The TRS R consists of the following rules:

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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

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


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

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

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

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

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

(22)
YES

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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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


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

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

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

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

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

(31)
YES

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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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


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

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

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

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

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

(38)
YES

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

(39)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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


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

(41)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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


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

(43)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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


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

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

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

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

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

(47)
YES

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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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


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

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

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(53) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
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(54)
YES