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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) QTRS
(1) QTRS Reverse [EQUIVALENT, 0 ms]
(2) QTRS
(3) DependencyPairsProof [EQUIVALENT, 225 ms]
(4) QDP
(5) DependencyGraphProof [EQUIVALENT, 0 ms]
(6) AND
    (7) QDP
        (8) UsableRulesProof [EQUIVALENT, 0 ms]
        (9) QDP
        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (11) YES
    (12) QDP
        (13) QDPOrderProof [EQUIVALENT, 274 ms]
        (14) QDP
        (15) DependencyGraphProof [EQUIVALENT, 0 ms]
        (16) QDP
        (17) QDPOrderProof [EQUIVALENT, 174 ms]
        (18) QDP
        (19) DependencyGraphProof [EQUIVALENT, 0 ms]
        (20) QDP
        (21) QDPOrderProof [EQUIVALENT, 106 ms]
        (22) QDP
        (23) QDPOrderProof [EQUIVALENT, 83 ms]
        (24) QDP
        (25) DependencyGraphProof [EQUIVALENT, 0 ms]
        (26) QDP
        (27) QDPOrderProof [EQUIVALENT, 73 ms]
        (28) QDP
        (29) QDPOrderProof [EQUIVALENT, 54 ms]
        (30) QDP
        (31) DependencyGraphProof [EQUIVALENT, 0 ms]
        (32) AND
            (33) QDP
                (34) QDPOrderProof [EQUIVALENT, 3321 ms]
                (35) QDP
                (36) DependencyGraphProof [EQUIVALENT, 0 ms]
                (37) QDP
                (38) UsableRulesProof [EQUIVALENT, 0 ms]
                (39) QDP
                (40) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (41) YES
            (42) QDP
                (43) QDPOrderProof [EQUIVALENT, 42 ms]
                (44) QDP
                (45) PisEmptyProof [EQUIVALENT, 0 ms]
                (46) YES


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

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

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

Q is empty.

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

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

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

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

Q is empty.

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

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

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

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

The TRS R consists of the following rules:

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

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

(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 115 less nodes.
----------------------------------------

(6)
Complex Obligation (AND)

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

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

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

The TRS R consists of the following rules:

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

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

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

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

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

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

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

From the DPs we obtained the following set of size-change graphs:
*4^1(1(4(5(0(x1))))) -> 4^1(x1)
The graph contains the following edges 1 > 1


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

(11)
YES

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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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

(32)
Complex Obligation (AND)

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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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

(37)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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

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

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

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

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

From the DPs we obtained the following set of size-change graphs:
*5^1(4(0(3(0(x1))))) -> 5^1(x1)
The graph contains the following edges 1 > 1


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

(41)
YES

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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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


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

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

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

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

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

(46)
YES