/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files
--------------------------------------------------------------------------------
YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished
Termination w.r.t. Q of the given QTRS could be proven:
(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 215 ms]
(2) QTRS
(3) DependencyPairsProof [EQUIVALENT, 1137 ms]
(4) QDP
(5) QDPOrderProof [EQUIVALENT, 2086 ms]
(6) QDP
(7) DependencyGraphProof [EQUIVALENT, 3 ms]
(8) AND
(9) QDP
(10) QDPOrderProof [EQUIVALENT, 121 ms]
(11) QDP
(12) DependencyGraphProof [EQUIVALENT, 0 ms]
(13) QDP
(14) QDPOrderProof [EQUIVALENT, 74 ms]
(15) QDP
(16) SemLabProof [SOUND, 1149 ms]
(17) QDP
(18) DependencyGraphProof [EQUIVALENT, 0 ms]
(19) AND
(20) QDP
(21) QDPOrderProof [EQUIVALENT, 248 ms]
(22) QDP
(23) PisEmptyProof [EQUIVALENT, 0 ms]
(24) YES
(25) QDP
(26) QDPOrderProof [EQUIVALENT, 257 ms]
(27) QDP
(28) PisEmptyProof [EQUIVALENT, 0 ms]
(29) YES
(30) QDP
(31) QDPOrderProof [EQUIVALENT, 110 ms]
(32) QDP
(33) QDPOrderProof [EQUIVALENT, 0 ms]
(34) QDP
(35) PisEmptyProof [EQUIVALENT, 0 ms]
(36) YES
(37) QDP
(38) QDPOrderProof [EQUIVALENT, 106 ms]
(39) QDP
(40) QDPOrderProof [EQUIVALENT, 74 ms]
(41) QDP
(42) QDPOrderProof [EQUIVALENT, 0 ms]
(43) QDP
(44) PisEmptyProof [EQUIVALENT, 0 ms]
(45) YES
(46) QDP
(47) QDPOrderProof [EQUIVALENT, 111 ms]
(48) QDP
(49) QDPOrderProof [EQUIVALENT, 95 ms]
(50) QDP
(51) QDPOrderProof [EQUIVALENT, 12 ms]
(52) QDP
(53) PisEmptyProof [EQUIVALENT, 0 ms]
(54) YES
(55) QDP
(56) QDPOrderProof [EQUIVALENT, 103 ms]
(57) QDP
(58) QDPOrderProof [EQUIVALENT, 95 ms]
(59) QDP
(60) QDPOrderProof [EQUIVALENT, 0 ms]
(61) QDP
(62) QDPOrderProof [EQUIVALENT, 5769 ms]
(63) QDP
(64) PisEmptyProof [EQUIVALENT, 0 ms]
(65) YES
----------------------------------------
(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1)))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
3(2(3(1(2(2(2(4(3(5(x1)))))))))) -> 5(3(4(4(4(4(0(4(0(x1)))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
4(2(3(1(4(3(0(3(1(1(5(0(4(1(3(4(3(1(4(4(x1)))))))))))))))))))) -> 4(1(4(4(2(5(5(1(5(4(5(5(2(0(5(0(2(0(5(x1)))))))))))))))))))
1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(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:
3(2(3(1(2(2(2(4(3(5(x1)))))))))) -> 5(3(4(4(4(4(0(4(0(x1)))))))))
4(2(3(1(4(3(0(3(1(1(5(0(4(1(3(4(3(1(4(4(x1)))))))))))))))))))) -> 4(1(4(4(2(5(5(1(5(4(5(5(2(0(5(0(2(0(5(x1)))))))))))))))))))
----------------------------------------
(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1)))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(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(0(1(2(x1)))) -> 0^1(3(4(2(x1))))
0^1(0(1(2(x1)))) -> 3^1(4(2(x1)))
0^1(0(1(2(x1)))) -> 4^1(2(x1))
2^1(0(1(0(5(3(x1)))))) -> 2^1(0(3(1(3(3(x1))))))
2^1(0(1(0(5(3(x1)))))) -> 0^1(3(1(3(3(x1)))))
2^1(0(1(0(5(3(x1)))))) -> 3^1(1(3(3(x1))))
2^1(0(1(0(5(3(x1)))))) -> 1^1(3(3(x1)))
2^1(0(1(0(5(3(x1)))))) -> 3^1(3(x1))
4^1(1(1(5(0(4(x1)))))) -> 4^1(4(2(2(0(4(x1))))))
4^1(1(1(5(0(4(x1)))))) -> 4^1(2(2(0(4(x1)))))
4^1(1(1(5(0(4(x1)))))) -> 2^1(2(0(4(x1))))
4^1(1(1(5(0(4(x1)))))) -> 2^1(0(4(x1)))
2^1(3(2(5(5(0(0(0(x1)))))))) -> 0^1(2(1(4(2(5(3(2(x1))))))))
2^1(3(2(5(5(0(0(0(x1)))))))) -> 2^1(1(4(2(5(3(2(x1)))))))
2^1(3(2(5(5(0(0(0(x1)))))))) -> 1^1(4(2(5(3(2(x1))))))
2^1(3(2(5(5(0(0(0(x1)))))))) -> 4^1(2(5(3(2(x1)))))
2^1(3(2(5(5(0(0(0(x1)))))))) -> 2^1(5(3(2(x1))))
2^1(3(2(5(5(0(0(0(x1)))))))) -> 5^1(3(2(x1)))
2^1(3(2(5(5(0(0(0(x1)))))))) -> 3^1(2(x1))
2^1(3(2(5(5(0(0(0(x1)))))))) -> 2^1(x1)
5^1(0(0(5(3(3(4(3(x1)))))))) -> 5^1(3(1(3(0(3(0(3(x1))))))))
5^1(0(0(5(3(3(4(3(x1)))))))) -> 3^1(1(3(0(3(0(3(x1)))))))
5^1(0(0(5(3(3(4(3(x1)))))))) -> 1^1(3(0(3(0(3(x1))))))
5^1(0(0(5(3(3(4(3(x1)))))))) -> 3^1(0(3(0(3(x1)))))
5^1(0(0(5(3(3(4(3(x1)))))))) -> 0^1(3(0(3(x1))))
5^1(0(0(5(3(3(4(3(x1)))))))) -> 3^1(0(3(x1)))
5^1(0(0(5(3(3(4(3(x1)))))))) -> 0^1(3(x1))
1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 4^1(3(1(0(4(1(0(3(3(x1)))))))))
1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 3^1(1(0(4(1(0(3(3(x1))))))))
1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 1^1(0(4(1(0(3(3(x1)))))))
1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 0^1(4(1(0(3(3(x1))))))
1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 4^1(1(0(3(3(x1)))))
1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 1^1(0(3(3(x1))))
1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 0^1(3(3(x1)))
1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 3^1(3(x1))
1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 3^1(x1)
3^1(0(0(1(0(4(5(4(2(x1))))))))) -> 4^1(4(5(3(1(2(3(4(2(x1)))))))))
3^1(0(0(1(0(4(5(4(2(x1))))))))) -> 4^1(5(3(1(2(3(4(2(x1))))))))
3^1(0(0(1(0(4(5(4(2(x1))))))))) -> 5^1(3(1(2(3(4(2(x1)))))))
3^1(0(0(1(0(4(5(4(2(x1))))))))) -> 3^1(1(2(3(4(2(x1))))))
3^1(0(0(1(0(4(5(4(2(x1))))))))) -> 1^1(2(3(4(2(x1)))))
3^1(0(0(1(0(4(5(4(2(x1))))))))) -> 2^1(3(4(2(x1))))
3^1(0(0(1(0(4(5(4(2(x1))))))))) -> 3^1(4(2(x1)))
0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0^1(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0^1(0(4(1(1(3(5(3(4(2(x1))))))))))
0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0^1(4(1(1(3(5(3(4(2(x1)))))))))
0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 4^1(1(1(3(5(3(4(2(x1))))))))
0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 1^1(1(3(5(3(4(2(x1)))))))
0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 1^1(3(5(3(4(2(x1))))))
0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 3^1(5(3(4(2(x1)))))
0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 5^1(3(4(2(x1))))
0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 3^1(4(2(x1)))
0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 4^1(2(x1))
0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 2^1(x1)
1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1^1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 2^1(2(3(3(3(3(0(1(1(3(4(x1)))))))))))
1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 2^1(3(3(3(3(0(1(1(3(4(x1))))))))))
1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 3^1(3(3(3(0(1(1(3(4(x1)))))))))
1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 3^1(3(3(0(1(1(3(4(x1))))))))
1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 3^1(3(0(1(1(3(4(x1)))))))
1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 3^1(0(1(1(3(4(x1))))))
1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 0^1(1(1(3(4(x1)))))
1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1^1(1(3(4(x1))))
1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1^1(3(4(x1)))
1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 3^1(4(x1))
4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5^1(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 4^1(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))
4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 3^1(2(2(3(3(2(1(2(5(0(5(x1)))))))))))
4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 2^1(2(3(3(2(1(2(5(0(5(x1))))))))))
4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 2^1(3(3(2(1(2(5(0(5(x1)))))))))
4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 3^1(3(2(1(2(5(0(5(x1))))))))
4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 3^1(2(1(2(5(0(5(x1)))))))
4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 2^1(1(2(5(0(5(x1))))))
4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 1^1(2(5(0(5(x1)))))
4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 2^1(5(0(5(x1))))
4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5^1(0(5(x1)))
4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 0^1(5(x1))
4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5^1(x1)
1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1^1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))
1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 4^1(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))
1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 4^1(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))
1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 0^1(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))
1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 4^1(1(4(0(3(2(4(1(0(2(4(x1)))))))))))
1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1^1(4(0(3(2(4(1(0(2(4(x1))))))))))
1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 4^1(0(3(2(4(1(0(2(4(x1)))))))))
1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 0^1(3(2(4(1(0(2(4(x1))))))))
1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 3^1(2(4(1(0(2(4(x1)))))))
1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 2^1(4(1(0(2(4(x1))))))
1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 4^1(1(0(2(4(x1)))))
1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1^1(0(2(4(x1))))
1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 0^1(2(4(x1)))
1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 2^1(4(x1))
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2^1(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2^1(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 5^1(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 4^1(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2^1(0(1(5(1(1(4(1(0(3(4(x1)))))))))))
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 0^1(1(5(1(1(4(1(0(3(4(x1))))))))))
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 1^1(5(1(1(4(1(0(3(4(x1)))))))))
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 5^1(1(1(4(1(0(3(4(x1))))))))
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 1^1(1(4(1(0(3(4(x1)))))))
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 1^1(4(1(0(3(4(x1))))))
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 4^1(1(0(3(4(x1)))))
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 1^1(0(3(4(x1))))
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 0^1(3(4(x1)))
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 3^1(4(x1))
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 4^1(x1)
5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5^1(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5^1(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))
5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 3^1(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))
5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 3^1(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))
5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 1^1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))
5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5^1(0(3(1(0(4(3(1(3(2(x1))))))))))
5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 0^1(3(1(0(4(3(1(3(2(x1)))))))))
5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 3^1(1(0(4(3(1(3(2(x1))))))))
5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 1^1(0(4(3(1(3(2(x1)))))))
5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 0^1(4(3(1(3(2(x1))))))
5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 4^1(3(1(3(2(x1)))))
5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 3^1(1(3(2(x1))))
5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 1^1(3(2(x1)))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2^1(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 5^1(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2^1(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 3^1(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 3^1(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 5^1(4(3(5(4(5(5(0(0(3(1(x1)))))))))))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 4^1(3(5(4(5(5(0(0(3(1(x1))))))))))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 3^1(5(4(5(5(0(0(3(1(x1)))))))))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 5^1(4(5(5(0(0(3(1(x1))))))))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 4^1(5(5(0(0(3(1(x1)))))))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 5^1(5(0(0(3(1(x1))))))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 5^1(0(0(3(1(x1)))))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 0^1(0(3(1(x1))))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 0^1(3(1(x1)))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 3^1(1(x1))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 1^1(x1)
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0^1(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 4^1(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 4^1(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 3^1(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0^1(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 2^1(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 5^1(0(5(4(1(3(5(4(4(1(0(x1)))))))))))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0^1(5(4(1(3(5(4(4(1(0(x1))))))))))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 5^1(4(1(3(5(4(4(1(0(x1)))))))))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 4^1(1(3(5(4(4(1(0(x1))))))))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 1^1(3(5(4(4(1(0(x1)))))))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 3^1(5(4(4(1(0(x1))))))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 5^1(4(4(1(0(x1)))))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 4^1(4(1(0(x1))))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 4^1(1(0(x1)))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 1^1(0(x1))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0^1(x1)
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2^1(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))))
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2^1(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 4^1(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 3^1(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2^1(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 1^1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 3^1(2(2(1(4(3(0(2(2(5(5(x1)))))))))))
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2^1(2(1(4(3(0(2(2(5(5(x1))))))))))
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2^1(1(4(3(0(2(2(5(5(x1)))))))))
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 1^1(4(3(0(2(2(5(5(x1))))))))
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 4^1(3(0(2(2(5(5(x1)))))))
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 3^1(0(2(2(5(5(x1))))))
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 0^1(2(2(5(5(x1)))))
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2^1(2(5(5(x1))))
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2^1(5(5(x1)))
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 5^1(5(x1))
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 5^1(x1)
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5^1(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 3^1(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 1^1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 4^1(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 4^1(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5^1(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 0^1(5(0(3(1(2(1(1(2(4(3(x1)))))))))))
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5^1(0(3(1(2(1(1(2(4(3(x1))))))))))
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 0^1(3(1(2(1(1(2(4(3(x1)))))))))
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 3^1(1(2(1(1(2(4(3(x1))))))))
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 1^1(2(1(1(2(4(3(x1)))))))
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 2^1(1(1(2(4(3(x1))))))
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 1^1(1(2(4(3(x1)))))
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 1^1(2(4(3(x1))))
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 2^1(4(3(x1)))
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 4^1(3(x1))
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 3^1(x1)
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5^1(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5^1(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 0^1(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 0^1(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 1^1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 1^1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 1^1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 0^1(2(2(2(1(3(2(1(2(2(2(x1)))))))))))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 2^1(2(2(1(3(2(1(2(2(2(x1))))))))))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 2^1(2(1(3(2(1(2(2(2(x1)))))))))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 2^1(1(3(2(1(2(2(2(x1))))))))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 1^1(3(2(1(2(2(2(x1)))))))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 3^1(2(1(2(2(2(x1))))))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 2^1(1(2(2(2(x1)))))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 1^1(2(2(2(x1))))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 2^1(2(2(x1)))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 2^1(2(x1))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3^1(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 1^1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 0^1(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3^1(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 4^1(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 2^1(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 2^1(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 1^1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 0^1(1(5(2(0(5(4(4(2(1(4(x1)))))))))))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 1^1(5(2(0(5(4(4(2(1(4(x1))))))))))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 5^1(2(0(5(4(4(2(1(4(x1)))))))))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 2^1(0(5(4(4(2(1(4(x1))))))))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 0^1(5(4(4(2(1(4(x1)))))))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 5^1(4(4(2(1(4(x1))))))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 4^1(4(2(1(4(x1)))))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 4^1(2(1(4(x1))))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 2^1(1(4(x1)))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 1^1(4(x1))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 3^1(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 1^1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 4^1(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 2^1(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 2^1(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 4^1(5(0(2(3(0(5(2(1(5(1(x1)))))))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 5^1(0(2(3(0(5(2(1(5(1(x1))))))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(2(3(0(5(2(1(5(1(x1)))))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 2^1(3(0(5(2(1(5(1(x1))))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 3^1(0(5(2(1(5(1(x1)))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(5(2(1(5(1(x1))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 5^1(2(1(5(1(x1)))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 2^1(1(5(1(x1))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 1^1(5(1(x1)))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1^1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 5^1(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 5^1(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1^1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1^1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 3^1(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 0^1(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 3^1(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 5^1(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 5^1(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 3^1(2(4(2(0(3(5(1(0(3(2(x1)))))))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 2^1(4(2(0(3(5(1(0(3(2(x1))))))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 4^1(2(0(3(5(1(0(3(2(x1)))))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 2^1(0(3(5(1(0(3(2(x1))))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 0^1(3(5(1(0(3(2(x1)))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 3^1(5(1(0(3(2(x1))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 5^1(1(0(3(2(x1)))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1^1(0(3(2(x1))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 0^1(3(2(x1)))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 3^1(2(x1))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4^1(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 3^1(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 0^1(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 2^1(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4^1(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 0^1(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4^1(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 3^1(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 0^1(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4^1(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 2^1(0(5(1(4(2(3(2(3(0(0(x1)))))))))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 0^1(5(1(4(2(3(2(3(0(0(x1))))))))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 5^1(1(4(2(3(2(3(0(0(x1)))))))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 1^1(4(2(3(2(3(0(0(x1))))))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4^1(2(3(2(3(0(0(x1)))))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 2^1(3(2(3(0(0(x1))))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 3^1(2(3(0(0(x1)))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 2^1(3(0(0(x1))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 3^1(0(0(x1)))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 0^1(0(x1))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4^1(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 2^1(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4^1(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 5^1(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 1^1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 3^1(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 5^1(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 2^1(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4^1(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4^1(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 2^1(0(5(0(1(2(5(2(4(5(0(x1)))))))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 0^1(5(0(1(2(5(2(4(5(0(x1))))))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 5^1(0(1(2(5(2(4(5(0(x1)))))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 0^1(1(2(5(2(4(5(0(x1))))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 1^1(2(5(2(4(5(0(x1)))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 2^1(5(2(4(5(0(x1))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 5^1(2(4(5(0(x1)))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 2^1(4(5(0(x1))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4^1(5(0(x1)))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 5^1(0(x1))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 0^1(x1)
The TRS R consists of the following rules:
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1)))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(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(0(1(2(x1)))) -> 4^1(2(x1))
2^1(0(1(0(5(3(x1)))))) -> 0^1(3(1(3(3(x1)))))
2^1(0(1(0(5(3(x1)))))) -> 3^1(1(3(3(x1))))
2^1(0(1(0(5(3(x1)))))) -> 1^1(3(3(x1)))
2^1(0(1(0(5(3(x1)))))) -> 3^1(3(x1))
4^1(1(1(5(0(4(x1)))))) -> 4^1(2(2(0(4(x1)))))
4^1(1(1(5(0(4(x1)))))) -> 2^1(2(0(4(x1))))
4^1(1(1(5(0(4(x1)))))) -> 2^1(0(4(x1)))
2^1(3(2(5(5(0(0(0(x1)))))))) -> 0^1(2(1(4(2(5(3(2(x1))))))))
2^1(3(2(5(5(0(0(0(x1)))))))) -> 2^1(1(4(2(5(3(2(x1)))))))
2^1(3(2(5(5(0(0(0(x1)))))))) -> 1^1(4(2(5(3(2(x1))))))
2^1(3(2(5(5(0(0(0(x1)))))))) -> 4^1(2(5(3(2(x1)))))
2^1(3(2(5(5(0(0(0(x1)))))))) -> 2^1(5(3(2(x1))))
2^1(3(2(5(5(0(0(0(x1)))))))) -> 5^1(3(2(x1)))
2^1(3(2(5(5(0(0(0(x1)))))))) -> 3^1(2(x1))
2^1(3(2(5(5(0(0(0(x1)))))))) -> 2^1(x1)
5^1(0(0(5(3(3(4(3(x1)))))))) -> 1^1(3(0(3(0(3(x1))))))
5^1(0(0(5(3(3(4(3(x1)))))))) -> 3^1(0(3(0(3(x1)))))
5^1(0(0(5(3(3(4(3(x1)))))))) -> 0^1(3(0(3(x1))))
5^1(0(0(5(3(3(4(3(x1)))))))) -> 3^1(0(3(x1)))
5^1(0(0(5(3(3(4(3(x1)))))))) -> 0^1(3(x1))
1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 3^1(1(0(4(1(0(3(3(x1))))))))
1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 1^1(0(4(1(0(3(3(x1)))))))
1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 0^1(4(1(0(3(3(x1))))))
1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 4^1(1(0(3(3(x1)))))
1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 1^1(0(3(3(x1))))
1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 0^1(3(3(x1)))
1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 3^1(3(x1))
1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 3^1(x1)
3^1(0(0(1(0(4(5(4(2(x1))))))))) -> 4^1(5(3(1(2(3(4(2(x1))))))))
3^1(0(0(1(0(4(5(4(2(x1))))))))) -> 5^1(3(1(2(3(4(2(x1)))))))
3^1(0(0(1(0(4(5(4(2(x1))))))))) -> 3^1(1(2(3(4(2(x1))))))
3^1(0(0(1(0(4(5(4(2(x1))))))))) -> 1^1(2(3(4(2(x1)))))
3^1(0(0(1(0(4(5(4(2(x1))))))))) -> 2^1(3(4(2(x1))))
3^1(0(0(1(0(4(5(4(2(x1))))))))) -> 3^1(4(2(x1)))
0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0^1(0(4(1(1(3(5(3(4(2(x1))))))))))
0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0^1(4(1(1(3(5(3(4(2(x1)))))))))
0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 4^1(1(1(3(5(3(4(2(x1))))))))
0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 1^1(1(3(5(3(4(2(x1)))))))
0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 1^1(3(5(3(4(2(x1))))))
0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 3^1(5(3(4(2(x1)))))
0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 5^1(3(4(2(x1))))
0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 3^1(4(2(x1)))
0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 4^1(2(x1))
0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 2^1(x1)
1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 2^1(2(3(3(3(3(0(1(1(3(4(x1)))))))))))
1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 2^1(3(3(3(3(0(1(1(3(4(x1))))))))))
1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 3^1(3(3(3(0(1(1(3(4(x1)))))))))
1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 3^1(3(3(0(1(1(3(4(x1))))))))
1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 3^1(3(0(1(1(3(4(x1)))))))
1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 3^1(0(1(1(3(4(x1))))))
1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 0^1(1(1(3(4(x1)))))
1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1^1(1(3(4(x1))))
1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1^1(3(4(x1)))
1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 3^1(4(x1))
4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5^1(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 4^1(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))
4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 3^1(2(2(3(3(2(1(2(5(0(5(x1)))))))))))
4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 2^1(2(3(3(2(1(2(5(0(5(x1))))))))))
4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 2^1(3(3(2(1(2(5(0(5(x1)))))))))
4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 3^1(3(2(1(2(5(0(5(x1))))))))
4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 3^1(2(1(2(5(0(5(x1)))))))
4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 2^1(1(2(5(0(5(x1))))))
4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 1^1(2(5(0(5(x1)))))
4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 2^1(5(0(5(x1))))
4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5^1(0(5(x1)))
4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 0^1(5(x1))
4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5^1(x1)
1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 4^1(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))
1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 4^1(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))
1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 0^1(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))
1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 4^1(1(4(0(3(2(4(1(0(2(4(x1)))))))))))
1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1^1(4(0(3(2(4(1(0(2(4(x1))))))))))
1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 4^1(0(3(2(4(1(0(2(4(x1)))))))))
1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 0^1(3(2(4(1(0(2(4(x1))))))))
1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 3^1(2(4(1(0(2(4(x1)))))))
1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 2^1(4(1(0(2(4(x1))))))
1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 4^1(1(0(2(4(x1)))))
1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1^1(0(2(4(x1))))
1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 0^1(2(4(x1)))
1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 2^1(4(x1))
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2^1(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 5^1(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 4^1(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2^1(0(1(5(1(1(4(1(0(3(4(x1)))))))))))
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 0^1(1(5(1(1(4(1(0(3(4(x1))))))))))
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 1^1(5(1(1(4(1(0(3(4(x1)))))))))
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 5^1(1(1(4(1(0(3(4(x1))))))))
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 1^1(1(4(1(0(3(4(x1)))))))
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 1^1(4(1(0(3(4(x1))))))
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 4^1(1(0(3(4(x1)))))
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 1^1(0(3(4(x1))))
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 0^1(3(4(x1)))
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 3^1(4(x1))
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 4^1(x1)
5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5^1(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))
5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 3^1(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))
5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 3^1(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))
5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 1^1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))
5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5^1(0(3(1(0(4(3(1(3(2(x1))))))))))
5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 0^1(3(1(0(4(3(1(3(2(x1)))))))))
5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 3^1(1(0(4(3(1(3(2(x1))))))))
5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 1^1(0(4(3(1(3(2(x1)))))))
5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 0^1(4(3(1(3(2(x1))))))
5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 4^1(3(1(3(2(x1)))))
5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 3^1(1(3(2(x1))))
5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 1^1(3(2(x1)))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 5^1(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2^1(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 3^1(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 3^1(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 5^1(4(3(5(4(5(5(0(0(3(1(x1)))))))))))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 4^1(3(5(4(5(5(0(0(3(1(x1))))))))))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 3^1(5(4(5(5(0(0(3(1(x1)))))))))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 5^1(4(5(5(0(0(3(1(x1))))))))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 4^1(5(5(0(0(3(1(x1)))))))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 5^1(5(0(0(3(1(x1))))))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 5^1(0(0(3(1(x1)))))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 0^1(0(3(1(x1))))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 0^1(3(1(x1)))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 3^1(1(x1))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 1^1(x1)
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 4^1(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 3^1(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0^1(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 2^1(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 5^1(0(5(4(1(3(5(4(4(1(0(x1)))))))))))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0^1(5(4(1(3(5(4(4(1(0(x1))))))))))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 5^1(4(1(3(5(4(4(1(0(x1)))))))))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 4^1(1(3(5(4(4(1(0(x1))))))))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 1^1(3(5(4(4(1(0(x1)))))))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 3^1(5(4(4(1(0(x1))))))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 5^1(4(4(1(0(x1)))))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 4^1(4(1(0(x1))))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 4^1(1(0(x1)))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 1^1(0(x1))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0^1(x1)
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2^1(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 4^1(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 3^1(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2^1(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 1^1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 3^1(2(2(1(4(3(0(2(2(5(5(x1)))))))))))
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2^1(2(1(4(3(0(2(2(5(5(x1))))))))))
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2^1(1(4(3(0(2(2(5(5(x1)))))))))
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 1^1(4(3(0(2(2(5(5(x1))))))))
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 4^1(3(0(2(2(5(5(x1)))))))
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 3^1(0(2(2(5(5(x1))))))
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 0^1(2(2(5(5(x1)))))
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2^1(2(5(5(x1))))
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2^1(5(5(x1)))
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 5^1(5(x1))
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 5^1(x1)
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5^1(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 3^1(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 1^1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 4^1(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 4^1(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5^1(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 0^1(5(0(3(1(2(1(1(2(4(3(x1)))))))))))
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5^1(0(3(1(2(1(1(2(4(3(x1))))))))))
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 0^1(3(1(2(1(1(2(4(3(x1)))))))))
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 3^1(1(2(1(1(2(4(3(x1))))))))
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 1^1(2(1(1(2(4(3(x1)))))))
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 2^1(1(1(2(4(3(x1))))))
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 1^1(1(2(4(3(x1)))))
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 1^1(2(4(3(x1))))
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 2^1(4(3(x1)))
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 4^1(3(x1))
2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 3^1(x1)
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5^1(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 0^1(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 0^1(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 1^1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 1^1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 1^1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 0^1(2(2(2(1(3(2(1(2(2(2(x1)))))))))))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 2^1(2(2(1(3(2(1(2(2(2(x1))))))))))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 2^1(2(1(3(2(1(2(2(2(x1)))))))))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 2^1(1(3(2(1(2(2(2(x1))))))))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 1^1(3(2(1(2(2(2(x1)))))))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 3^1(2(1(2(2(2(x1))))))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 2^1(1(2(2(2(x1)))))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 1^1(2(2(2(x1))))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 2^1(2(2(x1)))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 2^1(2(x1))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 1^1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 0^1(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3^1(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 4^1(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 2^1(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 2^1(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 1^1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 0^1(1(5(2(0(5(4(4(2(1(4(x1)))))))))))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 1^1(5(2(0(5(4(4(2(1(4(x1))))))))))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 5^1(2(0(5(4(4(2(1(4(x1)))))))))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 2^1(0(5(4(4(2(1(4(x1))))))))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 0^1(5(4(4(2(1(4(x1)))))))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 5^1(4(4(2(1(4(x1))))))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 4^1(4(2(1(4(x1)))))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 4^1(2(1(4(x1))))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 2^1(1(4(x1)))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 1^1(4(x1))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 1^1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 4^1(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 2^1(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 2^1(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 4^1(5(0(2(3(0(5(2(1(5(1(x1)))))))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 5^1(0(2(3(0(5(2(1(5(1(x1))))))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(2(3(0(5(2(1(5(1(x1)))))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 2^1(3(0(5(2(1(5(1(x1))))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 3^1(0(5(2(1(5(1(x1)))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(5(2(1(5(1(x1))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 5^1(2(1(5(1(x1)))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 2^1(1(5(1(x1))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 1^1(5(1(x1)))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 5^1(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 5^1(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1^1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1^1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 3^1(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 0^1(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 3^1(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 5^1(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 5^1(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 3^1(2(4(2(0(3(5(1(0(3(2(x1)))))))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 2^1(4(2(0(3(5(1(0(3(2(x1))))))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 4^1(2(0(3(5(1(0(3(2(x1)))))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 2^1(0(3(5(1(0(3(2(x1))))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 0^1(3(5(1(0(3(2(x1)))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 3^1(5(1(0(3(2(x1))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 5^1(1(0(3(2(x1)))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1^1(0(3(2(x1))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 0^1(3(2(x1)))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 3^1(2(x1))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 3^1(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 0^1(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 2^1(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4^1(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 0^1(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4^1(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 3^1(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 0^1(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4^1(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 2^1(0(5(1(4(2(3(2(3(0(0(x1)))))))))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 0^1(5(1(4(2(3(2(3(0(0(x1))))))))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 5^1(1(4(2(3(2(3(0(0(x1)))))))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 1^1(4(2(3(2(3(0(0(x1))))))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4^1(2(3(2(3(0(0(x1)))))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 2^1(3(2(3(0(0(x1))))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 3^1(2(3(0(0(x1)))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 2^1(3(0(0(x1))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 3^1(0(0(x1)))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 0^1(0(x1))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 2^1(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4^1(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 5^1(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 1^1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 3^1(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 5^1(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 2^1(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4^1(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4^1(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 2^1(0(5(0(1(2(5(2(4(5(0(x1)))))))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 0^1(5(0(1(2(5(2(4(5(0(x1))))))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 5^1(0(1(2(5(2(4(5(0(x1)))))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 0^1(1(2(5(2(4(5(0(x1))))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 1^1(2(5(2(4(5(0(x1)))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 2^1(5(2(4(5(0(x1))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 5^1(2(4(5(0(x1)))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 2^1(4(5(0(x1))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4^1(5(0(x1)))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 5^1(0(x1))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 0^1(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)) = x_1
POL(1(x_1)) = 1 + x_1
POL(1^1(x_1)) = 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)) = 1 + x_1
POL(4(x_1)) = 1 + x_1
POL(4^1(x_1)) = 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:
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1)))))))))
1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))
1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
----------------------------------------
(6)
Obligation:
Q DP problem:
The TRS P consists of the following rules:
0^1(0(1(2(x1)))) -> 0^1(3(4(2(x1))))
0^1(0(1(2(x1)))) -> 3^1(4(2(x1)))
2^1(0(1(0(5(3(x1)))))) -> 2^1(0(3(1(3(3(x1))))))
4^1(1(1(5(0(4(x1)))))) -> 4^1(4(2(2(0(4(x1))))))
5^1(0(0(5(3(3(4(3(x1)))))))) -> 5^1(3(1(3(0(3(0(3(x1))))))))
5^1(0(0(5(3(3(4(3(x1)))))))) -> 3^1(1(3(0(3(0(3(x1)))))))
1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 4^1(3(1(0(4(1(0(3(3(x1)))))))))
3^1(0(0(1(0(4(5(4(2(x1))))))))) -> 4^1(4(5(3(1(2(3(4(2(x1)))))))))
0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0^1(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1^1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1^1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2^1(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5^1(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2^1(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0^1(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 4^1(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))
1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2^1(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5^1(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3^1(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 3^1(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1^1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4^1(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4^1(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
The TRS R consists of the following rules:
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1)))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(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 5 SCCs with 7 less nodes.
----------------------------------------
(8)
Complex Obligation (AND)
----------------------------------------
(9)
Obligation:
Q DP problem:
The TRS P consists of the following rules:
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3^1(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
3^1(0(0(1(0(4(5(4(2(x1))))))))) -> 4^1(4(5(3(1(2(3(4(2(x1)))))))))
4^1(1(1(5(0(4(x1)))))) -> 4^1(4(2(2(0(4(x1))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4^1(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
The TRS R consists of the following rules:
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1)))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(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.
4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3^1(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(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)) = x_1
POL(2(x_1)) = 1
POL(3(x_1)) = 0
POL(3^1(x_1)) = 0
POL(4(x_1)) = x_1
POL(4^1(x_1)) = x_1
POL(5(x_1)) = x_1
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
----------------------------------------
(11)
Obligation:
Q DP problem:
The TRS P consists of the following rules:
3^1(0(0(1(0(4(5(4(2(x1))))))))) -> 4^1(4(5(3(1(2(3(4(2(x1)))))))))
4^1(1(1(5(0(4(x1)))))) -> 4^1(4(2(2(0(4(x1))))))
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4^1(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
The TRS R consists of the following rules:
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1)))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------
(12) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------
(13)
Obligation:
Q DP problem:
The TRS P consists of the following rules:
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4^1(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
4^1(1(1(5(0(4(x1)))))) -> 4^1(4(2(2(0(4(x1))))))
The TRS R consists of the following rules:
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1)))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------
(14) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
4^1(1(1(5(0(4(x1)))))) -> 4^1(4(2(2(0(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)) = 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:
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
----------------------------------------
(15)
Obligation:
Q DP problem:
The TRS P consists of the following rules:
4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4^1(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
The TRS R consists of the following rules:
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1)))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------
(16) SemLabProof (SOUND)
We found the following model for the rules of the TRSs R and P.
Interpretation over the domain with elements from 0 to 1.
5: 1 + x0
4^1: 0
0: 1 + x0
1: 1 + x0
2: 1 + x0
3: 1 + x0
4: 1 + x0
By semantic labelling [SEMLAB] we obtain the following labelled QDP problem.
----------------------------------------
(17)
Obligation:
Q DP problem:
The TRS P consists of the following rules:
4^1.0(2.1(4.0(2.1(5.0(5.1(2.0(0.1(2.0(0.1(2.0(0.1(3.0(1.1(5.0(3.1(0.0(1.1(4.0(4.1(2.0(x1))))))))))))))))))))) -> 4^1.0(2.1(4.0(5.1(1.0(3.1(5.0(2.1(4.0(4.1(2.0(0.1(5.0(0.1(1.0(2.1(5.0(2.1(4.0(5.1(0.0(x1)))))))))))))))))))))
4^1.1(2.0(4.1(2.0(5.1(5.0(2.1(0.0(2.1(0.0(2.1(0.0(3.1(1.0(5.1(3.0(0.1(1.0(4.1(4.0(2.1(x1))))))))))))))))))))) -> 4^1.1(2.0(4.1(5.0(1.1(3.0(5.1(2.0(4.1(4.0(2.1(0.0(5.1(0.0(1.1(2.0(5.1(2.0(4.1(5.0(0.1(x1)))))))))))))))))))))
The TRS R consists of the following rules:
0.1(0.0(1.1(2.0(x1)))) -> 0.1(3.0(4.1(2.0(x1))))
0.0(0.1(1.0(2.1(x1)))) -> 0.0(3.1(4.0(2.1(x1))))
2.1(0.0(1.1(0.0(5.1(3.0(x1)))))) -> 2.1(0.0(3.1(1.0(3.1(3.0(x1))))))
2.0(0.1(1.0(0.1(5.0(3.1(x1)))))) -> 2.0(0.1(3.0(1.1(3.0(3.1(x1))))))
4.1(1.0(1.1(5.0(0.1(4.0(x1)))))) -> 4.1(4.0(2.1(2.0(0.1(4.0(x1))))))
4.0(1.1(1.0(5.1(0.0(4.1(x1)))))) -> 4.0(4.1(2.0(2.1(0.0(4.1(x1))))))
2.1(3.0(2.1(5.0(5.1(0.0(0.1(0.0(x1)))))))) -> 0.1(2.0(1.1(4.0(2.1(5.0(3.1(2.0(x1))))))))
2.0(3.1(2.0(5.1(5.0(0.1(0.0(0.1(x1)))))))) -> 0.0(2.1(1.0(4.1(2.0(5.1(3.0(2.1(x1))))))))
5.1(0.0(0.1(5.0(3.1(3.0(4.1(3.0(x1)))))))) -> 5.1(3.0(1.1(3.0(0.1(3.0(0.1(3.0(x1))))))))
5.0(0.1(0.0(5.1(3.0(3.1(4.0(3.1(x1)))))))) -> 5.0(3.1(1.0(3.1(0.0(3.1(0.0(3.1(x1))))))))
1.0(0.1(0.0(1.1(5.0(5.1(4.0(2.1(4.0(x1))))))))) -> 4.0(3.1(1.0(0.1(4.0(1.1(0.0(3.1(3.0(x1)))))))))
1.1(0.0(0.1(1.0(5.1(5.0(4.1(2.0(4.1(x1))))))))) -> 4.1(3.0(1.1(0.0(4.1(1.0(0.1(3.0(3.1(x1)))))))))
3.0(0.1(0.0(1.1(0.0(4.1(5.0(4.1(2.0(x1))))))))) -> 4.0(4.1(5.0(3.1(1.0(2.1(3.0(4.1(2.0(x1)))))))))
3.1(0.0(0.1(1.0(0.1(4.0(5.1(4.0(2.1(x1))))))))) -> 4.1(4.0(5.1(3.0(1.1(2.0(3.1(4.0(2.1(x1)))))))))
0.0(0.1(1.0(2.1(5.0(1.1(4.0(4.1(5.0(5.1(4.0(x1))))))))))) -> 0.0(0.1(0.0(4.1(1.0(1.1(3.0(5.1(3.0(4.1(2.0(x1)))))))))))
0.1(0.0(1.1(2.0(5.1(1.0(4.1(4.0(5.1(5.0(4.1(x1))))))))))) -> 0.1(0.0(0.1(4.0(1.1(1.0(3.1(5.0(3.1(4.0(2.1(x1)))))))))))
1.1(5.0(2.1(2.0(5.1(1.0(1.1(2.0(4.1(4.0(5.1(4.0(x1)))))))))))) -> 1.1(2.0(2.1(3.0(3.1(3.0(3.1(0.0(1.1(1.0(3.1(4.0(x1))))))))))))
1.0(5.1(2.0(2.1(5.0(1.1(1.0(2.1(4.0(4.1(5.0(4.1(x1)))))))))))) -> 1.0(2.1(2.0(3.1(3.0(3.1(3.0(0.1(1.0(1.1(3.0(4.1(x1))))))))))))
4.0(1.1(5.0(3.1(3.0(5.1(0.0(4.1(5.0(1.1(4.0(4.1(1.0(x1))))))))))))) -> 5.0(4.1(3.0(2.1(2.0(3.1(3.0(2.1(1.0(2.1(5.0(0.1(5.0(x1)))))))))))))
4.1(1.0(5.1(3.0(3.1(5.0(0.1(4.0(5.1(1.0(4.1(4.0(1.1(x1))))))))))))) -> 5.1(4.0(3.1(2.0(2.1(3.0(3.1(2.0(1.1(2.0(5.1(0.0(5.1(x1)))))))))))))
1.0(0.1(3.0(0.1(0.0(0.1(0.0(2.1(3.0(1.1(4.0(0.1(2.0(0.1(4.0(x1))))))))))))))) -> 1.0(4.1(4.0(0.1(4.0(1.1(4.0(0.1(3.0(2.1(4.0(1.1(0.0(2.1(4.0(x1)))))))))))))))
1.1(0.0(3.1(0.0(0.1(0.0(0.1(2.0(3.1(1.0(4.1(0.0(2.1(0.0(4.1(x1))))))))))))))) -> 1.1(4.0(4.1(0.0(4.1(1.0(4.1(0.0(3.1(2.0(4.1(1.0(0.1(2.0(4.1(x1)))))))))))))))
2.0(5.1(0.0(5.1(1.0(5.1(2.0(3.1(5.0(3.1(1.0(3.1(5.0(4.1(5.0(x1))))))))))))))) -> 2.0(2.1(5.0(4.1(2.0(0.1(1.0(5.1(1.0(1.1(4.0(1.1(0.0(3.1(4.0(x1)))))))))))))))
2.1(5.0(0.1(5.0(1.1(5.0(2.1(3.0(5.1(3.0(1.1(3.0(5.1(4.0(5.1(x1))))))))))))))) -> 2.1(2.0(5.1(4.0(2.1(0.0(1.1(5.0(1.1(1.0(4.1(1.0(0.1(3.0(4.1(x1)))))))))))))))
5.0(2.1(5.0(1.1(1.0(5.1(0.0(0.1(3.0(0.1(5.0(5.1(3.0(3.1(2.0(x1))))))))))))))) -> 5.0(5.1(3.0(3.1(1.0(5.1(0.0(3.1(1.0(0.1(4.0(3.1(1.0(3.1(2.0(x1)))))))))))))))
5.1(2.0(5.1(1.0(1.1(5.0(0.1(0.0(3.1(0.0(5.1(5.0(3.1(3.0(2.1(x1))))))))))))))) -> 5.1(5.0(3.1(3.0(1.1(5.0(0.1(3.0(1.1(0.0(4.1(3.0(1.1(3.0(2.1(x1)))))))))))))))
2.1(1.0(1.1(5.0(2.1(5.0(5.1(0.0(1.1(3.0(4.1(1.0(4.1(4.0(5.1(3.0(x1)))))))))))))))) -> 2.1(5.0(2.1(3.0(3.1(5.0(4.1(3.0(5.1(4.0(5.1(5.0(0.1(0.0(3.1(1.0(x1))))))))))))))))
2.0(1.1(1.0(5.1(2.0(5.1(5.0(0.1(1.0(3.1(4.0(1.1(4.0(4.1(5.0(3.1(x1)))))))))))))))) -> 2.0(5.1(2.0(3.1(3.0(5.1(4.0(3.1(5.0(4.1(5.0(5.1(0.0(0.1(3.0(1.1(x1))))))))))))))))
0.0(4.1(4.0(5.1(0.0(4.1(5.0(0.1(5.0(2.1(3.0(0.1(0.0(2.1(4.0(4.1(2.0(x1))))))))))))))))) -> 0.0(4.1(4.0(3.1(0.0(2.1(5.0(0.1(5.0(4.1(1.0(3.1(5.0(4.1(4.0(1.1(0.0(x1)))))))))))))))))
0.1(4.0(4.1(5.0(0.1(4.0(5.1(0.0(5.1(2.0(3.1(0.0(0.1(2.0(4.1(4.0(2.1(x1))))))))))))))))) -> 0.1(4.0(4.1(3.0(0.1(2.0(5.1(0.0(5.1(4.0(1.1(3.0(5.1(4.0(4.1(1.0(0.1(x1)))))))))))))))))
1.0(1.1(0.0(0.1(3.0(1.1(3.0(0.1(2.0(1.1(2.0(5.1(2.0(5.1(0.0(2.1(0.0(x1))))))))))))))))) -> 2.0(2.1(4.0(3.1(2.0(1.1(3.0(2.1(2.0(1.1(4.0(3.1(0.0(2.1(2.0(5.1(5.0(x1)))))))))))))))))
1.1(1.0(0.1(0.0(3.1(1.0(3.1(0.0(2.1(1.0(2.1(5.0(2.1(5.0(0.1(2.0(0.1(x1))))))))))))))))) -> 2.1(2.0(4.1(3.0(2.1(1.0(3.1(2.0(2.1(1.0(4.1(3.0(0.1(2.0(2.1(5.0(5.1(x1)))))))))))))))))
2.0(5.1(4.0(4.1(4.0(2.1(3.0(5.1(4.0(0.1(4.0(0.1(0.0(5.1(4.0(5.1(5.0(x1))))))))))))))))) -> 5.0(3.1(1.0(4.1(4.0(5.1(0.0(5.1(0.0(3.1(1.0(2.1(1.0(1.1(2.0(4.1(3.0(x1)))))))))))))))))
2.1(5.0(4.1(4.0(4.1(2.0(3.1(5.0(4.1(0.0(4.1(0.0(0.1(5.0(4.1(5.0(5.1(x1))))))))))))))))) -> 5.1(3.0(1.1(4.0(4.1(5.0(0.1(5.0(0.1(3.0(1.1(2.0(1.1(1.0(2.1(4.0(3.1(x1)))))))))))))))))
5.1(0.0(0.1(1.0(3.1(3.0(2.1(5.0(3.1(4.0(0.1(3.0(4.1(3.0(5.1(4.0(4.1(2.0(x1)))))))))))))))))) -> 5.1(5.0(0.1(0.0(1.1(1.0(1.1(0.0(2.1(2.0(2.1(1.0(3.1(2.0(1.1(2.0(2.1(2.0(x1))))))))))))))))))
5.0(0.1(0.0(1.1(3.0(3.1(2.0(5.1(3.0(4.1(0.0(3.1(4.0(3.1(5.0(4.1(4.0(2.1(x1)))))))))))))))))) -> 5.0(5.1(0.0(0.1(1.0(1.1(1.0(0.1(2.0(2.1(2.0(1.1(3.0(2.1(1.0(2.1(2.0(2.1(x1))))))))))))))))))
4.0(4.1(0.0(2.1(2.0(1.1(1.0(4.1(1.0(1.1(1.0(0.1(0.0(0.1(2.0(3.1(4.0(4.1(4.0(x1))))))))))))))))))) -> 3.0(1.1(0.0(3.1(4.0(2.1(2.0(1.1(0.0(1.1(5.0(2.1(0.0(5.1(4.0(4.1(2.0(1.1(4.0(x1)))))))))))))))))))
4.1(4.0(0.1(2.0(2.1(1.0(1.1(4.0(1.1(1.0(1.1(0.0(0.1(0.0(2.1(3.0(4.1(4.0(4.1(x1))))))))))))))))))) -> 3.1(1.0(0.1(3.0(4.1(2.0(2.1(1.0(0.1(1.0(5.1(2.0(0.1(5.0(4.1(4.0(2.1(1.0(4.1(x1)))))))))))))))))))
0.1(3.0(2.1(2.0(1.1(4.0(5.1(3.0(3.1(5.0(5.1(1.0(5.1(1.0(0.1(5.0(1.1(5.0(5.1(1.0(x1)))))))))))))))))))) -> 0.1(3.0(1.1(4.0(0.1(2.0(0.1(0.0(2.1(4.0(5.1(0.0(2.1(3.0(0.1(5.0(2.1(1.0(5.1(1.0(x1))))))))))))))))))))
0.0(3.1(2.0(2.1(1.0(4.1(5.0(3.1(3.0(5.1(5.0(1.1(5.0(1.1(0.0(5.1(1.0(5.1(5.0(1.1(x1)))))))))))))))))))) -> 0.0(3.1(1.0(4.1(0.0(2.1(0.0(0.1(2.0(4.1(5.0(0.1(2.0(3.1(0.0(5.1(2.0(1.1(5.0(1.1(x1))))))))))))))))))))
1.0(4.1(1.0(4.1(2.0(5.1(4.0(5.1(5.0(0.1(5.0(0.1(4.0(0.1(4.0(5.1(0.0(0.1(5.0(4.1(2.0(x1))))))))))))))))))))) -> 1.0(5.1(5.0(1.1(1.0(3.1(0.0(3.1(5.0(5.1(3.0(2.1(4.0(2.1(0.0(3.1(5.0(1.1(0.0(3.1(2.0(x1)))))))))))))))))))))
1.1(4.0(1.1(4.0(2.1(5.0(4.1(5.0(5.1(0.0(5.1(0.0(4.1(0.0(4.1(5.0(0.1(0.0(5.1(4.0(2.1(x1))))))))))))))))))))) -> 1.1(5.0(5.1(1.0(1.1(3.0(0.1(3.0(5.1(5.0(3.1(2.0(4.1(2.0(0.1(3.0(5.1(1.0(0.1(3.0(2.1(x1)))))))))))))))))))))
2.0(2.1(4.0(0.1(0.0(1.1(4.0(3.1(1.0(4.1(4.0(3.1(1.0(2.1(1.0(1.1(5.0(0.1(4.0(5.1(0.0(x1))))))))))))))))))))) -> 4.0(3.1(0.0(2.1(4.0(0.1(4.0(3.1(0.0(4.1(2.0(0.1(5.0(1.1(4.0(2.1(3.0(2.1(3.0(0.1(0.0(x1)))))))))))))))))))))
2.1(2.0(4.1(0.0(0.1(1.0(4.1(3.0(1.1(4.0(4.1(3.0(1.1(2.0(1.1(1.0(5.1(0.0(4.1(5.0(0.1(x1))))))))))))))))))))) -> 4.1(3.0(0.1(2.0(4.1(0.0(4.1(3.0(0.1(4.0(2.1(0.0(5.1(1.0(4.1(2.0(3.1(2.0(3.1(0.0(0.1(x1)))))))))))))))))))))
4.0(2.1(4.0(2.1(5.0(5.1(2.0(0.1(2.0(0.1(2.0(0.1(3.0(1.1(5.0(3.1(0.0(1.1(4.0(4.1(2.0(x1))))))))))))))))))))) -> 4.0(2.1(4.0(5.1(1.0(3.1(5.0(2.1(4.0(4.1(2.0(0.1(5.0(0.1(1.0(2.1(5.0(2.1(4.0(5.1(0.0(x1)))))))))))))))))))))
4.1(2.0(4.1(2.0(5.1(5.0(2.1(0.0(2.1(0.0(2.1(0.0(3.1(1.0(5.1(3.0(0.1(1.0(4.1(4.0(2.1(x1))))))))))))))))))))) -> 4.1(2.0(4.1(5.0(1.1(3.0(5.1(2.0(4.1(4.0(2.1(0.0(5.1(0.0(1.1(2.0(5.1(2.0(4.1(5.0(0.1(x1)))))))))))))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------
(18) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
----------------------------------------
(19)
Complex Obligation (AND)
----------------------------------------
(20)
Obligation:
Q DP problem:
The TRS P consists of the following rules:
4^1.1(2.0(4.1(2.0(5.1(5.0(2.1(0.0(2.1(0.0(2.1(0.0(3.1(1.0(5.1(3.0(0.1(1.0(4.1(4.0(2.1(x1))))))))))))))))))))) -> 4^1.1(2.0(4.1(5.0(1.1(3.0(5.1(2.0(4.1(4.0(2.1(0.0(5.1(0.0(1.1(2.0(5.1(2.0(4.1(5.0(0.1(x1)))))))))))))))))))))
The TRS R consists of the following rules:
0.1(0.0(1.1(2.0(x1)))) -> 0.1(3.0(4.1(2.0(x1))))
0.0(0.1(1.0(2.1(x1)))) -> 0.0(3.1(4.0(2.1(x1))))
2.1(0.0(1.1(0.0(5.1(3.0(x1)))))) -> 2.1(0.0(3.1(1.0(3.1(3.0(x1))))))
2.0(0.1(1.0(0.1(5.0(3.1(x1)))))) -> 2.0(0.1(3.0(1.1(3.0(3.1(x1))))))
4.1(1.0(1.1(5.0(0.1(4.0(x1)))))) -> 4.1(4.0(2.1(2.0(0.1(4.0(x1))))))
4.0(1.1(1.0(5.1(0.0(4.1(x1)))))) -> 4.0(4.1(2.0(2.1(0.0(4.1(x1))))))
2.1(3.0(2.1(5.0(5.1(0.0(0.1(0.0(x1)))))))) -> 0.1(2.0(1.1(4.0(2.1(5.0(3.1(2.0(x1))))))))
2.0(3.1(2.0(5.1(5.0(0.1(0.0(0.1(x1)))))))) -> 0.0(2.1(1.0(4.1(2.0(5.1(3.0(2.1(x1))))))))
5.1(0.0(0.1(5.0(3.1(3.0(4.1(3.0(x1)))))))) -> 5.1(3.0(1.1(3.0(0.1(3.0(0.1(3.0(x1))))))))
5.0(0.1(0.0(5.1(3.0(3.1(4.0(3.1(x1)))))))) -> 5.0(3.1(1.0(3.1(0.0(3.1(0.0(3.1(x1))))))))
1.0(0.1(0.0(1.1(5.0(5.1(4.0(2.1(4.0(x1))))))))) -> 4.0(3.1(1.0(0.1(4.0(1.1(0.0(3.1(3.0(x1)))))))))
1.1(0.0(0.1(1.0(5.1(5.0(4.1(2.0(4.1(x1))))))))) -> 4.1(3.0(1.1(0.0(4.1(1.0(0.1(3.0(3.1(x1)))))))))
3.0(0.1(0.0(1.1(0.0(4.1(5.0(4.1(2.0(x1))))))))) -> 4.0(4.1(5.0(3.1(1.0(2.1(3.0(4.1(2.0(x1)))))))))
3.1(0.0(0.1(1.0(0.1(4.0(5.1(4.0(2.1(x1))))))))) -> 4.1(4.0(5.1(3.0(1.1(2.0(3.1(4.0(2.1(x1)))))))))
0.0(0.1(1.0(2.1(5.0(1.1(4.0(4.1(5.0(5.1(4.0(x1))))))))))) -> 0.0(0.1(0.0(4.1(1.0(1.1(3.0(5.1(3.0(4.1(2.0(x1)))))))))))
0.1(0.0(1.1(2.0(5.1(1.0(4.1(4.0(5.1(5.0(4.1(x1))))))))))) -> 0.1(0.0(0.1(4.0(1.1(1.0(3.1(5.0(3.1(4.0(2.1(x1)))))))))))
1.1(5.0(2.1(2.0(5.1(1.0(1.1(2.0(4.1(4.0(5.1(4.0(x1)))))))))))) -> 1.1(2.0(2.1(3.0(3.1(3.0(3.1(0.0(1.1(1.0(3.1(4.0(x1))))))))))))
1.0(5.1(2.0(2.1(5.0(1.1(1.0(2.1(4.0(4.1(5.0(4.1(x1)))))))))))) -> 1.0(2.1(2.0(3.1(3.0(3.1(3.0(0.1(1.0(1.1(3.0(4.1(x1))))))))))))
4.0(1.1(5.0(3.1(3.0(5.1(0.0(4.1(5.0(1.1(4.0(4.1(1.0(x1))))))))))))) -> 5.0(4.1(3.0(2.1(2.0(3.1(3.0(2.1(1.0(2.1(5.0(0.1(5.0(x1)))))))))))))
4.1(1.0(5.1(3.0(3.1(5.0(0.1(4.0(5.1(1.0(4.1(4.0(1.1(x1))))))))))))) -> 5.1(4.0(3.1(2.0(2.1(3.0(3.1(2.0(1.1(2.0(5.1(0.0(5.1(x1)))))))))))))
1.0(0.1(3.0(0.1(0.0(0.1(0.0(2.1(3.0(1.1(4.0(0.1(2.0(0.1(4.0(x1))))))))))))))) -> 1.0(4.1(4.0(0.1(4.0(1.1(4.0(0.1(3.0(2.1(4.0(1.1(0.0(2.1(4.0(x1)))))))))))))))
1.1(0.0(3.1(0.0(0.1(0.0(0.1(2.0(3.1(1.0(4.1(0.0(2.1(0.0(4.1(x1))))))))))))))) -> 1.1(4.0(4.1(0.0(4.1(1.0(4.1(0.0(3.1(2.0(4.1(1.0(0.1(2.0(4.1(x1)))))))))))))))
2.0(5.1(0.0(5.1(1.0(5.1(2.0(3.1(5.0(3.1(1.0(3.1(5.0(4.1(5.0(x1))))))))))))))) -> 2.0(2.1(5.0(4.1(2.0(0.1(1.0(5.1(1.0(1.1(4.0(1.1(0.0(3.1(4.0(x1)))))))))))))))
2.1(5.0(0.1(5.0(1.1(5.0(2.1(3.0(5.1(3.0(1.1(3.0(5.1(4.0(5.1(x1))))))))))))))) -> 2.1(2.0(5.1(4.0(2.1(0.0(1.1(5.0(1.1(1.0(4.1(1.0(0.1(3.0(4.1(x1)))))))))))))))
5.0(2.1(5.0(1.1(1.0(5.1(0.0(0.1(3.0(0.1(5.0(5.1(3.0(3.1(2.0(x1))))))))))))))) -> 5.0(5.1(3.0(3.1(1.0(5.1(0.0(3.1(1.0(0.1(4.0(3.1(1.0(3.1(2.0(x1)))))))))))))))
5.1(2.0(5.1(1.0(1.1(5.0(0.1(0.0(3.1(0.0(5.1(5.0(3.1(3.0(2.1(x1))))))))))))))) -> 5.1(5.0(3.1(3.0(1.1(5.0(0.1(3.0(1.1(0.0(4.1(3.0(1.1(3.0(2.1(x1)))))))))))))))
2.1(1.0(1.1(5.0(2.1(5.0(5.1(0.0(1.1(3.0(4.1(1.0(4.1(4.0(5.1(3.0(x1)))))))))))))))) -> 2.1(5.0(2.1(3.0(3.1(5.0(4.1(3.0(5.1(4.0(5.1(5.0(0.1(0.0(3.1(1.0(x1))))))))))))))))
2.0(1.1(1.0(5.1(2.0(5.1(5.0(0.1(1.0(3.1(4.0(1.1(4.0(4.1(5.0(3.1(x1)))))))))))))))) -> 2.0(5.1(2.0(3.1(3.0(5.1(4.0(3.1(5.0(4.1(5.0(5.1(0.0(0.1(3.0(1.1(x1))))))))))))))))
0.0(4.1(4.0(5.1(0.0(4.1(5.0(0.1(5.0(2.1(3.0(0.1(0.0(2.1(4.0(4.1(2.0(x1))))))))))))))))) -> 0.0(4.1(4.0(3.1(0.0(2.1(5.0(0.1(5.0(4.1(1.0(3.1(5.0(4.1(4.0(1.1(0.0(x1)))))))))))))))))
0.1(4.0(4.1(5.0(0.1(4.0(5.1(0.0(5.1(2.0(3.1(0.0(0.1(2.0(4.1(4.0(2.1(x1))))))))))))))))) -> 0.1(4.0(4.1(3.0(0.1(2.0(5.1(0.0(5.1(4.0(1.1(3.0(5.1(4.0(4.1(1.0(0.1(x1)))))))))))))))))
1.0(1.1(0.0(0.1(3.0(1.1(3.0(0.1(2.0(1.1(2.0(5.1(2.0(5.1(0.0(2.1(0.0(x1))))))))))))))))) -> 2.0(2.1(4.0(3.1(2.0(1.1(3.0(2.1(2.0(1.1(4.0(3.1(0.0(2.1(2.0(5.1(5.0(x1)))))))))))))))))
1.1(1.0(0.1(0.0(3.1(1.0(3.1(0.0(2.1(1.0(2.1(5.0(2.1(5.0(0.1(2.0(0.1(x1))))))))))))))))) -> 2.1(2.0(4.1(3.0(2.1(1.0(3.1(2.0(2.1(1.0(4.1(3.0(0.1(2.0(2.1(5.0(5.1(x1)))))))))))))))))
2.0(5.1(4.0(4.1(4.0(2.1(3.0(5.1(4.0(0.1(4.0(0.1(0.0(5.1(4.0(5.1(5.0(x1))))))))))))))))) -> 5.0(3.1(1.0(4.1(4.0(5.1(0.0(5.1(0.0(3.1(1.0(2.1(1.0(1.1(2.0(4.1(3.0(x1)))))))))))))))))
2.1(5.0(4.1(4.0(4.1(2.0(3.1(5.0(4.1(0.0(4.1(0.0(0.1(5.0(4.1(5.0(5.1(x1))))))))))))))))) -> 5.1(3.0(1.1(4.0(4.1(5.0(0.1(5.0(0.1(3.0(1.1(2.0(1.1(1.0(2.1(4.0(3.1(x1)))))))))))))))))
5.1(0.0(0.1(1.0(3.1(3.0(2.1(5.0(3.1(4.0(0.1(3.0(4.1(3.0(5.1(4.0(4.1(2.0(x1)))))))))))))))))) -> 5.1(5.0(0.1(0.0(1.1(1.0(1.1(0.0(2.1(2.0(2.1(1.0(3.1(2.0(1.1(2.0(2.1(2.0(x1))))))))))))))))))
5.0(0.1(0.0(1.1(3.0(3.1(2.0(5.1(3.0(4.1(0.0(3.1(4.0(3.1(5.0(4.1(4.0(2.1(x1)))))))))))))))))) -> 5.0(5.1(0.0(0.1(1.0(1.1(1.0(0.1(2.0(2.1(2.0(1.1(3.0(2.1(1.0(2.1(2.0(2.1(x1))))))))))))))))))
4.0(4.1(0.0(2.1(2.0(1.1(1.0(4.1(1.0(1.1(1.0(0.1(0.0(0.1(2.0(3.1(4.0(4.1(4.0(x1))))))))))))))))))) -> 3.0(1.1(0.0(3.1(4.0(2.1(2.0(1.1(0.0(1.1(5.0(2.1(0.0(5.1(4.0(4.1(2.0(1.1(4.0(x1)))))))))))))))))))
4.1(4.0(0.1(2.0(2.1(1.0(1.1(4.0(1.1(1.0(1.1(0.0(0.1(0.0(2.1(3.0(4.1(4.0(4.1(x1))))))))))))))))))) -> 3.1(1.0(0.1(3.0(4.1(2.0(2.1(1.0(0.1(1.0(5.1(2.0(0.1(5.0(4.1(4.0(2.1(1.0(4.1(x1)))))))))))))))))))
0.1(3.0(2.1(2.0(1.1(4.0(5.1(3.0(3.1(5.0(5.1(1.0(5.1(1.0(0.1(5.0(1.1(5.0(5.1(1.0(x1)))))))))))))))))))) -> 0.1(3.0(1.1(4.0(0.1(2.0(0.1(0.0(2.1(4.0(5.1(0.0(2.1(3.0(0.1(5.0(2.1(1.0(5.1(1.0(x1))))))))))))))))))))
0.0(3.1(2.0(2.1(1.0(4.1(5.0(3.1(3.0(5.1(5.0(1.1(5.0(1.1(0.0(5.1(1.0(5.1(5.0(1.1(x1)))))))))))))))))))) -> 0.0(3.1(1.0(4.1(0.0(2.1(0.0(0.1(2.0(4.1(5.0(0.1(2.0(3.1(0.0(5.1(2.0(1.1(5.0(1.1(x1))))))))))))))))))))
1.0(4.1(1.0(4.1(2.0(5.1(4.0(5.1(5.0(0.1(5.0(0.1(4.0(0.1(4.0(5.1(0.0(0.1(5.0(4.1(2.0(x1))))))))))))))))))))) -> 1.0(5.1(5.0(1.1(1.0(3.1(0.0(3.1(5.0(5.1(3.0(2.1(4.0(2.1(0.0(3.1(5.0(1.1(0.0(3.1(2.0(x1)))))))))))))))))))))
1.1(4.0(1.1(4.0(2.1(5.0(4.1(5.0(5.1(0.0(5.1(0.0(4.1(0.0(4.1(5.0(0.1(0.0(5.1(4.0(2.1(x1))))))))))))))))))))) -> 1.1(5.0(5.1(1.0(1.1(3.0(0.1(3.0(5.1(5.0(3.1(2.0(4.1(2.0(0.1(3.0(5.1(1.0(0.1(3.0(2.1(x1)))))))))))))))))))))
2.0(2.1(4.0(0.1(0.0(1.1(4.0(3.1(1.0(4.1(4.0(3.1(1.0(2.1(1.0(1.1(5.0(0.1(4.0(5.1(0.0(x1))))))))))))))))))))) -> 4.0(3.1(0.0(2.1(4.0(0.1(4.0(3.1(0.0(4.1(2.0(0.1(5.0(1.1(4.0(2.1(3.0(2.1(3.0(0.1(0.0(x1)))))))))))))))))))))
2.1(2.0(4.1(0.0(0.1(1.0(4.1(3.0(1.1(4.0(4.1(3.0(1.1(2.0(1.1(1.0(5.1(0.0(4.1(5.0(0.1(x1))))))))))))))))))))) -> 4.1(3.0(0.1(2.0(4.1(0.0(4.1(3.0(0.1(4.0(2.1(0.0(5.1(1.0(4.1(2.0(3.1(2.0(3.1(0.0(0.1(x1)))))))))))))))))))))
4.0(2.1(4.0(2.1(5.0(5.1(2.0(0.1(2.0(0.1(2.0(0.1(3.0(1.1(5.0(3.1(0.0(1.1(4.0(4.1(2.0(x1))))))))))))))))))))) -> 4.0(2.1(4.0(5.1(1.0(3.1(5.0(2.1(4.0(4.1(2.0(0.1(5.0(0.1(1.0(2.1(5.0(2.1(4.0(5.1(0.0(x1)))))))))))))))))))))
4.1(2.0(4.1(2.0(5.1(5.0(2.1(0.0(2.1(0.0(2.1(0.0(3.1(1.0(5.1(3.0(0.1(1.0(4.1(4.0(2.1(x1))))))))))))))))))))) -> 4.1(2.0(4.1(5.0(1.1(3.0(5.1(2.0(4.1(4.0(2.1(0.0(5.1(0.0(1.1(2.0(5.1(2.0(4.1(5.0(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.
4^1.1(2.0(4.1(2.0(5.1(5.0(2.1(0.0(2.1(0.0(2.1(0.0(3.1(1.0(5.1(3.0(0.1(1.0(4.1(4.0(2.1(x1))))))))))))))))))))) -> 4^1.1(2.0(4.1(5.0(1.1(3.0(5.1(2.0(4.1(4.0(2.1(0.0(5.1(0.0(1.1(2.0(5.1(2.0(4.1(5.0(0.1(x1)))))))))))))))))))))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(0.0(x_1)) = 0
POL(0.1(x_1)) = 0
POL(1.0(x_1)) = x_1
POL(1.1(x_1)) = 1
POL(2.0(x_1)) = x_1
POL(2.1(x_1)) = 1
POL(3.0(x_1)) = 0
POL(3.1(x_1)) = 0
POL(4.0(x_1)) = 0
POL(4.1(x_1)) = x_1
POL(4^1.1(x_1)) = x_1
POL(5.0(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:
0.1(0.0(1.1(2.0(5.1(1.0(4.1(4.0(5.1(5.0(4.1(x1))))))))))) -> 0.1(0.0(0.1(4.0(1.1(1.0(3.1(5.0(3.1(4.0(2.1(x1)))))))))))
0.1(0.0(1.1(2.0(x1)))) -> 0.1(3.0(4.1(2.0(x1))))
0.1(4.0(4.1(5.0(0.1(4.0(5.1(0.0(5.1(2.0(3.1(0.0(0.1(2.0(4.1(4.0(2.1(x1))))))))))))))))) -> 0.1(4.0(4.1(3.0(0.1(2.0(5.1(0.0(5.1(4.0(1.1(3.0(5.1(4.0(4.1(1.0(0.1(x1)))))))))))))))))
0.1(3.0(2.1(2.0(1.1(4.0(5.1(3.0(3.1(5.0(5.1(1.0(5.1(1.0(0.1(5.0(1.1(5.0(5.1(1.0(x1)))))))))))))))))))) -> 0.1(3.0(1.1(4.0(0.1(2.0(0.1(0.0(2.1(4.0(5.1(0.0(2.1(3.0(0.1(5.0(2.1(1.0(5.1(1.0(x1))))))))))))))))))))
5.0(2.1(5.0(1.1(1.0(5.1(0.0(0.1(3.0(0.1(5.0(5.1(3.0(3.1(2.0(x1))))))))))))))) -> 5.0(5.1(3.0(3.1(1.0(5.1(0.0(3.1(1.0(0.1(4.0(3.1(1.0(3.1(2.0(x1)))))))))))))))
5.0(0.1(0.0(5.1(3.0(3.1(4.0(3.1(x1)))))))) -> 5.0(3.1(1.0(3.1(0.0(3.1(0.0(3.1(x1))))))))
5.0(0.1(0.0(1.1(3.0(3.1(2.0(5.1(3.0(4.1(0.0(3.1(4.0(3.1(5.0(4.1(4.0(2.1(x1)))))))))))))))))) -> 5.0(5.1(0.0(0.1(1.0(1.1(1.0(0.1(2.0(2.1(2.0(1.1(3.0(2.1(1.0(2.1(2.0(2.1(x1))))))))))))))))))
4.1(4.0(0.1(2.0(2.1(1.0(1.1(4.0(1.1(1.0(1.1(0.0(0.1(0.0(2.1(3.0(4.1(4.0(4.1(x1))))))))))))))))))) -> 3.1(1.0(0.1(3.0(4.1(2.0(2.1(1.0(0.1(1.0(5.1(2.0(0.1(5.0(4.1(4.0(2.1(1.0(4.1(x1)))))))))))))))))))
3.1(0.0(0.1(1.0(0.1(4.0(5.1(4.0(2.1(x1))))))))) -> 4.1(4.0(5.1(3.0(1.1(2.0(3.1(4.0(2.1(x1)))))))))
4.1(1.0(1.1(5.0(0.1(4.0(x1)))))) -> 4.1(4.0(2.1(2.0(0.1(4.0(x1))))))
4.1(2.0(4.1(2.0(5.1(5.0(2.1(0.0(2.1(0.0(2.1(0.0(3.1(1.0(5.1(3.0(0.1(1.0(4.1(4.0(2.1(x1))))))))))))))))))))) -> 4.1(2.0(4.1(5.0(1.1(3.0(5.1(2.0(4.1(4.0(2.1(0.0(5.1(0.0(1.1(2.0(5.1(2.0(4.1(5.0(0.1(x1)))))))))))))))))))))
4.1(1.0(5.1(3.0(3.1(5.0(0.1(4.0(5.1(1.0(4.1(4.0(1.1(x1))))))))))))) -> 5.1(4.0(3.1(2.0(2.1(3.0(3.1(2.0(1.1(2.0(5.1(0.0(5.1(x1)))))))))))))
2.0(5.1(0.0(5.1(1.0(5.1(2.0(3.1(5.0(3.1(1.0(3.1(5.0(4.1(5.0(x1))))))))))))))) -> 2.0(2.1(5.0(4.1(2.0(0.1(1.0(5.1(1.0(1.1(4.0(1.1(0.0(3.1(4.0(x1)))))))))))))))
2.0(0.1(1.0(0.1(5.0(3.1(x1)))))) -> 2.0(0.1(3.0(1.1(3.0(3.1(x1))))))
2.0(1.1(1.0(5.1(2.0(5.1(5.0(0.1(1.0(3.1(4.0(1.1(4.0(4.1(5.0(3.1(x1)))))))))))))))) -> 2.0(5.1(2.0(3.1(3.0(5.1(4.0(3.1(5.0(4.1(5.0(5.1(0.0(0.1(3.0(1.1(x1))))))))))))))))
2.0(3.1(2.0(5.1(5.0(0.1(0.0(0.1(x1)))))))) -> 0.0(2.1(1.0(4.1(2.0(5.1(3.0(2.1(x1))))))))
2.0(5.1(4.0(4.1(4.0(2.1(3.0(5.1(4.0(0.1(4.0(0.1(0.0(5.1(4.0(5.1(5.0(x1))))))))))))))))) -> 5.0(3.1(1.0(4.1(4.0(5.1(0.0(5.1(0.0(3.1(1.0(2.1(1.0(1.1(2.0(4.1(3.0(x1)))))))))))))))))
2.0(2.1(4.0(0.1(0.0(1.1(4.0(3.1(1.0(4.1(4.0(3.1(1.0(2.1(1.0(1.1(5.0(0.1(4.0(5.1(0.0(x1))))))))))))))))))))) -> 4.0(3.1(0.0(2.1(4.0(0.1(4.0(3.1(0.0(4.1(2.0(0.1(5.0(1.1(4.0(2.1(3.0(2.1(3.0(0.1(0.0(x1)))))))))))))))))))))
5.1(2.0(5.1(1.0(1.1(5.0(0.1(0.0(3.1(0.0(5.1(5.0(3.1(3.0(2.1(x1))))))))))))))) -> 5.1(5.0(3.1(3.0(1.1(5.0(0.1(3.0(1.1(0.0(4.1(3.0(1.1(3.0(2.1(x1)))))))))))))))
5.1(0.0(0.1(5.0(3.1(3.0(4.1(3.0(x1)))))))) -> 5.1(3.0(1.1(3.0(0.1(3.0(0.1(3.0(x1))))))))
5.1(0.0(0.1(1.0(3.1(3.0(2.1(5.0(3.1(4.0(0.1(3.0(4.1(3.0(5.1(4.0(4.1(2.0(x1)))))))))))))))))) -> 5.1(5.0(0.1(0.0(1.1(1.0(1.1(0.0(2.1(2.0(2.1(1.0(3.1(2.0(1.1(2.0(2.1(2.0(x1))))))))))))))))))
1.1(0.0(0.1(1.0(5.1(5.0(4.1(2.0(4.1(x1))))))))) -> 4.1(3.0(1.1(0.0(4.1(1.0(0.1(3.0(3.1(x1)))))))))
1.1(0.0(3.1(0.0(0.1(0.0(0.1(2.0(3.1(1.0(4.1(0.0(2.1(0.0(4.1(x1))))))))))))))) -> 1.1(4.0(4.1(0.0(4.1(1.0(4.1(0.0(3.1(2.0(4.1(1.0(0.1(2.0(4.1(x1)))))))))))))))
1.1(5.0(2.1(2.0(5.1(1.0(1.1(2.0(4.1(4.0(5.1(4.0(x1)))))))))))) -> 1.1(2.0(2.1(3.0(3.1(3.0(3.1(0.0(1.1(1.0(3.1(4.0(x1))))))))))))
1.1(4.0(1.1(4.0(2.1(5.0(4.1(5.0(5.1(0.0(5.1(0.0(4.1(0.0(4.1(5.0(0.1(0.0(5.1(4.0(2.1(x1))))))))))))))))))))) -> 1.1(5.0(5.1(1.0(1.1(3.0(0.1(3.0(5.1(5.0(3.1(2.0(4.1(2.0(0.1(3.0(5.1(1.0(0.1(3.0(2.1(x1)))))))))))))))))))))
1.1(1.0(0.1(0.0(3.1(1.0(3.1(0.0(2.1(1.0(2.1(5.0(2.1(5.0(0.1(2.0(0.1(x1))))))))))))))))) -> 2.1(2.0(4.1(3.0(2.1(1.0(3.1(2.0(2.1(1.0(4.1(3.0(0.1(2.0(2.1(5.0(5.1(x1)))))))))))))))))
0.0(0.1(1.0(2.1(5.0(1.1(4.0(4.1(5.0(5.1(4.0(x1))))))))))) -> 0.0(0.1(0.0(4.1(1.0(1.1(3.0(5.1(3.0(4.1(2.0(x1)))))))))))
0.0(0.1(1.0(2.1(x1)))) -> 0.0(3.1(4.0(2.1(x1))))
0.0(4.1(4.0(5.1(0.0(4.1(5.0(0.1(5.0(2.1(3.0(0.1(0.0(2.1(4.0(4.1(2.0(x1))))))))))))))))) -> 0.0(4.1(4.0(3.1(0.0(2.1(5.0(0.1(5.0(4.1(1.0(3.1(5.0(4.1(4.0(1.1(0.0(x1)))))))))))))))))
0.0(3.1(2.0(2.1(1.0(4.1(5.0(3.1(3.0(5.1(5.0(1.1(5.0(1.1(0.0(5.1(1.0(5.1(5.0(1.1(x1)))))))))))))))))))) -> 0.0(3.1(1.0(4.1(0.0(2.1(0.0(0.1(2.0(4.1(5.0(0.1(2.0(3.1(0.0(5.1(2.0(1.1(5.0(1.1(x1))))))))))))))))))))
2.1(5.0(0.1(5.0(1.1(5.0(2.1(3.0(5.1(3.0(1.1(3.0(5.1(4.0(5.1(x1))))))))))))))) -> 2.1(2.0(5.1(4.0(2.1(0.0(1.1(5.0(1.1(1.0(4.1(1.0(0.1(3.0(4.1(x1)))))))))))))))
2.1(0.0(1.1(0.0(5.1(3.0(x1)))))) -> 2.1(0.0(3.1(1.0(3.1(3.0(x1))))))
2.1(1.0(1.1(5.0(2.1(5.0(5.1(0.0(1.1(3.0(4.1(1.0(4.1(4.0(5.1(3.0(x1)))))))))))))))) -> 2.1(5.0(2.1(3.0(3.1(5.0(4.1(3.0(5.1(4.0(5.1(5.0(0.1(0.0(3.1(1.0(x1))))))))))))))))
2.1(3.0(2.1(5.0(5.1(0.0(0.1(0.0(x1)))))))) -> 0.1(2.0(1.1(4.0(2.1(5.0(3.1(2.0(x1))))))))
2.1(5.0(4.1(4.0(4.1(2.0(3.1(5.0(4.1(0.0(4.1(0.0(0.1(5.0(4.1(5.0(5.1(x1))))))))))))))))) -> 5.1(3.0(1.1(4.0(4.1(5.0(0.1(5.0(0.1(3.0(1.1(2.0(1.1(1.0(2.1(4.0(3.1(x1)))))))))))))))))
2.1(2.0(4.1(0.0(0.1(1.0(4.1(3.0(1.1(4.0(4.1(3.0(1.1(2.0(1.1(1.0(5.1(0.0(4.1(5.0(0.1(x1))))))))))))))))))))) -> 4.1(3.0(0.1(2.0(4.1(0.0(4.1(3.0(0.1(4.0(2.1(0.0(5.1(1.0(4.1(2.0(3.1(2.0(3.1(0.0(0.1(x1)))))))))))))))))))))
4.0(4.1(0.0(2.1(2.0(1.1(1.0(4.1(1.0(1.1(1.0(0.1(0.0(0.1(2.0(3.1(4.0(4.1(4.0(x1))))))))))))))))))) -> 3.0(1.1(0.0(3.1(4.0(2.1(2.0(1.1(0.0(1.1(5.0(2.1(0.0(5.1(4.0(4.1(2.0(1.1(4.0(x1)))))))))))))))))))
3.0(0.1(0.0(1.1(0.0(4.1(5.0(4.1(2.0(x1))))))))) -> 4.0(4.1(5.0(3.1(1.0(2.1(3.0(4.1(2.0(x1)))))))))
4.0(1.1(1.0(5.1(0.0(4.1(x1)))))) -> 4.0(4.1(2.0(2.1(0.0(4.1(x1))))))
4.0(2.1(4.0(2.1(5.0(5.1(2.0(0.1(2.0(0.1(2.0(0.1(3.0(1.1(5.0(3.1(0.0(1.1(4.0(4.1(2.0(x1))))))))))))))))))))) -> 4.0(2.1(4.0(5.1(1.0(3.1(5.0(2.1(4.0(4.1(2.0(0.1(5.0(0.1(1.0(2.1(5.0(2.1(4.0(5.1(0.0(x1)))))))))))))))))))))
4.0(1.1(5.0(3.1(3.0(5.1(0.0(4.1(5.0(1.1(4.0(4.1(1.0(x1))))))))))))) -> 5.0(4.1(3.0(2.1(2.0(3.1(3.0(2.1(1.0(2.1(5.0(0.1(5.0(x1)))))))))))))
1.0(0.1(0.0(1.1(5.0(5.1(4.0(2.1(4.0(x1))))))))) -> 4.0(3.1(1.0(0.1(4.0(1.1(0.0(3.1(3.0(x1)))))))))
1.0(0.1(3.0(0.1(0.0(0.1(0.0(2.1(3.0(1.1(4.0(0.1(2.0(0.1(4.0(x1))))))))))))))) -> 1.0(4.1(4.0(0.1(4.0(1.1(4.0(0.1(3.0(2.1(4.0(1.1(0.0(2.1(4.0(x1)))))))))))))))
1.0(5.1(2.0(2.1(5.0(1.1(1.0(2.1(4.0(4.1(5.0(4.1(x1)))))))))))) -> 1.0(2.1(2.0(3.1(3.0(3.1(3.0(0.1(1.0(1.1(3.0(4.1(x1))))))))))))
1.0(4.1(1.0(4.1(2.0(5.1(4.0(5.1(5.0(0.1(5.0(0.1(4.0(0.1(4.0(5.1(0.0(0.1(5.0(4.1(2.0(x1))))))))))))))))))))) -> 1.0(5.1(5.0(1.1(1.0(3.1(0.0(3.1(5.0(5.1(3.0(2.1(4.0(2.1(0.0(3.1(5.0(1.1(0.0(3.1(2.0(x1)))))))))))))))))))))
1.0(1.1(0.0(0.1(3.0(1.1(3.0(0.1(2.0(1.1(2.0(5.1(2.0(5.1(0.0(2.1(0.0(x1))))))))))))))))) -> 2.0(2.1(4.0(3.1(2.0(1.1(3.0(2.1(2.0(1.1(4.0(3.1(0.0(2.1(2.0(5.1(5.0(x1)))))))))))))))))
----------------------------------------
(22)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
0.1(0.0(1.1(2.0(x1)))) -> 0.1(3.0(4.1(2.0(x1))))
0.0(0.1(1.0(2.1(x1)))) -> 0.0(3.1(4.0(2.1(x1))))
2.1(0.0(1.1(0.0(5.1(3.0(x1)))))) -> 2.1(0.0(3.1(1.0(3.1(3.0(x1))))))
2.0(0.1(1.0(0.1(5.0(3.1(x1)))))) -> 2.0(0.1(3.0(1.1(3.0(3.1(x1))))))
4.1(1.0(1.1(5.0(0.1(4.0(x1)))))) -> 4.1(4.0(2.1(2.0(0.1(4.0(x1))))))
4.0(1.1(1.0(5.1(0.0(4.1(x1)))))) -> 4.0(4.1(2.0(2.1(0.0(4.1(x1))))))
2.1(3.0(2.1(5.0(5.1(0.0(0.1(0.0(x1)))))))) -> 0.1(2.0(1.1(4.0(2.1(5.0(3.1(2.0(x1))))))))
2.0(3.1(2.0(5.1(5.0(0.1(0.0(0.1(x1)))))))) -> 0.0(2.1(1.0(4.1(2.0(5.1(3.0(2.1(x1))))))))
5.1(0.0(0.1(5.0(3.1(3.0(4.1(3.0(x1)))))))) -> 5.1(3.0(1.1(3.0(0.1(3.0(0.1(3.0(x1))))))))
5.0(0.1(0.0(5.1(3.0(3.1(4.0(3.1(x1)))))))) -> 5.0(3.1(1.0(3.1(0.0(3.1(0.0(3.1(x1))))))))
1.0(0.1(0.0(1.1(5.0(5.1(4.0(2.1(4.0(x1))))))))) -> 4.0(3.1(1.0(0.1(4.0(1.1(0.0(3.1(3.0(x1)))))))))
1.1(0.0(0.1(1.0(5.1(5.0(4.1(2.0(4.1(x1))))))))) -> 4.1(3.0(1.1(0.0(4.1(1.0(0.1(3.0(3.1(x1)))))))))
3.0(0.1(0.0(1.1(0.0(4.1(5.0(4.1(2.0(x1))))))))) -> 4.0(4.1(5.0(3.1(1.0(2.1(3.0(4.1(2.0(x1)))))))))
3.1(0.0(0.1(1.0(0.1(4.0(5.1(4.0(2.1(x1))))))))) -> 4.1(4.0(5.1(3.0(1.1(2.0(3.1(4.0(2.1(x1)))))))))
0.0(0.1(1.0(2.1(5.0(1.1(4.0(4.1(5.0(5.1(4.0(x1))))))))))) -> 0.0(0.1(0.0(4.1(1.0(1.1(3.0(5.1(3.0(4.1(2.0(x1)))))))))))
0.1(0.0(1.1(2.0(5.1(1.0(4.1(4.0(5.1(5.0(4.1(x1))))))))))) -> 0.1(0.0(0.1(4.0(1.1(1.0(3.1(5.0(3.1(4.0(2.1(x1)))))))))))
1.1(5.0(2.1(2.0(5.1(1.0(1.1(2.0(4.1(4.0(5.1(4.0(x1)))))))))))) -> 1.1(2.0(2.1(3.0(3.1(3.0(3.1(0.0(1.1(1.0(3.1(4.0(x1))))))))))))
1.0(5.1(2.0(2.1(5.0(1.1(1.0(2.1(4.0(4.1(5.0(4.1(x1)))))))))))) -> 1.0(2.1(2.0(3.1(3.0(3.1(3.0(0.1(1.0(1.1(3.0(4.1(x1))))))))))))
4.0(1.1(5.0(3.1(3.0(5.1(0.0(4.1(5.0(1.1(4.0(4.1(1.0(x1))))))))))))) -> 5.0(4.1(3.0(2.1(2.0(3.1(3.0(2.1(1.0(2.1(5.0(0.1(5.0(x1)))))))))))))
4.1(1.0(5.1(3.0(3.1(5.0(0.1(4.0(5.1(1.0(4.1(4.0(1.1(x1))))))))))))) -> 5.1(4.0(3.1(2.0(2.1(3.0(3.1(2.0(1.1(2.0(5.1(0.0(5.1(x1)))))))))))))
1.0(0.1(3.0(0.1(0.0(0.1(0.0(2.1(3.0(1.1(4.0(0.1(2.0(0.1(4.0(x1))))))))))))))) -> 1.0(4.1(4.0(0.1(4.0(1.1(4.0(0.1(3.0(2.1(4.0(1.1(0.0(2.1(4.0(x1)))))))))))))))
1.1(0.0(3.1(0.0(0.1(0.0(0.1(2.0(3.1(1.0(4.1(0.0(2.1(0.0(4.1(x1))))))))))))))) -> 1.1(4.0(4.1(0.0(4.1(1.0(4.1(0.0(3.1(2.0(4.1(1.0(0.1(2.0(4.1(x1)))))))))))))))
2.0(5.1(0.0(5.1(1.0(5.1(2.0(3.1(5.0(3.1(1.0(3.1(5.0(4.1(5.0(x1))))))))))))))) -> 2.0(2.1(5.0(4.1(2.0(0.1(1.0(5.1(1.0(1.1(4.0(1.1(0.0(3.1(4.0(x1)))))))))))))))
2.1(5.0(0.1(5.0(1.1(5.0(2.1(3.0(5.1(3.0(1.1(3.0(5.1(4.0(5.1(x1))))))))))))))) -> 2.1(2.0(5.1(4.0(2.1(0.0(1.1(5.0(1.1(1.0(4.1(1.0(0.1(3.0(4.1(x1)))))))))))))))
5.0(2.1(5.0(1.1(1.0(5.1(0.0(0.1(3.0(0.1(5.0(5.1(3.0(3.1(2.0(x1))))))))))))))) -> 5.0(5.1(3.0(3.1(1.0(5.1(0.0(3.1(1.0(0.1(4.0(3.1(1.0(3.1(2.0(x1)))))))))))))))
5.1(2.0(5.1(1.0(1.1(5.0(0.1(0.0(3.1(0.0(5.1(5.0(3.1(3.0(2.1(x1))))))))))))))) -> 5.1(5.0(3.1(3.0(1.1(5.0(0.1(3.0(1.1(0.0(4.1(3.0(1.1(3.0(2.1(x1)))))))))))))))
2.1(1.0(1.1(5.0(2.1(5.0(5.1(0.0(1.1(3.0(4.1(1.0(4.1(4.0(5.1(3.0(x1)))))))))))))))) -> 2.1(5.0(2.1(3.0(3.1(5.0(4.1(3.0(5.1(4.0(5.1(5.0(0.1(0.0(3.1(1.0(x1))))))))))))))))
2.0(1.1(1.0(5.1(2.0(5.1(5.0(0.1(1.0(3.1(4.0(1.1(4.0(4.1(5.0(3.1(x1)))))))))))))))) -> 2.0(5.1(2.0(3.1(3.0(5.1(4.0(3.1(5.0(4.1(5.0(5.1(0.0(0.1(3.0(1.1(x1))))))))))))))))
0.0(4.1(4.0(5.1(0.0(4.1(5.0(0.1(5.0(2.1(3.0(0.1(0.0(2.1(4.0(4.1(2.0(x1))))))))))))))))) -> 0.0(4.1(4.0(3.1(0.0(2.1(5.0(0.1(5.0(4.1(1.0(3.1(5.0(4.1(4.0(1.1(0.0(x1)))))))))))))))))
0.1(4.0(4.1(5.0(0.1(4.0(5.1(0.0(5.1(2.0(3.1(0.0(0.1(2.0(4.1(4.0(2.1(x1))))))))))))))))) -> 0.1(4.0(4.1(3.0(0.1(2.0(5.1(0.0(5.1(4.0(1.1(3.0(5.1(4.0(4.1(1.0(0.1(x1)))))))))))))))))
1.0(1.1(0.0(0.1(3.0(1.1(3.0(0.1(2.0(1.1(2.0(5.1(2.0(5.1(0.0(2.1(0.0(x1))))))))))))))))) -> 2.0(2.1(4.0(3.1(2.0(1.1(3.0(2.1(2.0(1.1(4.0(3.1(0.0(2.1(2.0(5.1(5.0(x1)))))))))))))))))
1.1(1.0(0.1(0.0(3.1(1.0(3.1(0.0(2.1(1.0(2.1(5.0(2.1(5.0(0.1(2.0(0.1(x1))))))))))))))))) -> 2.1(2.0(4.1(3.0(2.1(1.0(3.1(2.0(2.1(1.0(4.1(3.0(0.1(2.0(2.1(5.0(5.1(x1)))))))))))))))))
2.0(5.1(4.0(4.1(4.0(2.1(3.0(5.1(4.0(0.1(4.0(0.1(0.0(5.1(4.0(5.1(5.0(x1))))))))))))))))) -> 5.0(3.1(1.0(4.1(4.0(5.1(0.0(5.1(0.0(3.1(1.0(2.1(1.0(1.1(2.0(4.1(3.0(x1)))))))))))))))))
2.1(5.0(4.1(4.0(4.1(2.0(3.1(5.0(4.1(0.0(4.1(0.0(0.1(5.0(4.1(5.0(5.1(x1))))))))))))))))) -> 5.1(3.0(1.1(4.0(4.1(5.0(0.1(5.0(0.1(3.0(1.1(2.0(1.1(1.0(2.1(4.0(3.1(x1)))))))))))))))))
5.1(0.0(0.1(1.0(3.1(3.0(2.1(5.0(3.1(4.0(0.1(3.0(4.1(3.0(5.1(4.0(4.1(2.0(x1)))))))))))))))))) -> 5.1(5.0(0.1(0.0(1.1(1.0(1.1(0.0(2.1(2.0(2.1(1.0(3.1(2.0(1.1(2.0(2.1(2.0(x1))))))))))))))))))
5.0(0.1(0.0(1.1(3.0(3.1(2.0(5.1(3.0(4.1(0.0(3.1(4.0(3.1(5.0(4.1(4.0(2.1(x1)))))))))))))))))) -> 5.0(5.1(0.0(0.1(1.0(1.1(1.0(0.1(2.0(2.1(2.0(1.1(3.0(2.1(1.0(2.1(2.0(2.1(x1))))))))))))))))))
4.0(4.1(0.0(2.1(2.0(1.1(1.0(4.1(1.0(1.1(1.0(0.1(0.0(0.1(2.0(3.1(4.0(4.1(4.0(x1))))))))))))))))))) -> 3.0(1.1(0.0(3.1(4.0(2.1(2.0(1.1(0.0(1.1(5.0(2.1(0.0(5.1(4.0(4.1(2.0(1.1(4.0(x1)))))))))))))))))))
4.1(4.0(0.1(2.0(2.1(1.0(1.1(4.0(1.1(1.0(1.1(0.0(0.1(0.0(2.1(3.0(4.1(4.0(4.1(x1))))))))))))))))))) -> 3.1(1.0(0.1(3.0(4.1(2.0(2.1(1.0(0.1(1.0(5.1(2.0(0.1(5.0(4.1(4.0(2.1(1.0(4.1(x1)))))))))))))))))))
0.1(3.0(2.1(2.0(1.1(4.0(5.1(3.0(3.1(5.0(5.1(1.0(5.1(1.0(0.1(5.0(1.1(5.0(5.1(1.0(x1)))))))))))))))))))) -> 0.1(3.0(1.1(4.0(0.1(2.0(0.1(0.0(2.1(4.0(5.1(0.0(2.1(3.0(0.1(5.0(2.1(1.0(5.1(1.0(x1))))))))))))))))))))
0.0(3.1(2.0(2.1(1.0(4.1(5.0(3.1(3.0(5.1(5.0(1.1(5.0(1.1(0.0(5.1(1.0(5.1(5.0(1.1(x1)))))))))))))))))))) -> 0.0(3.1(1.0(4.1(0.0(2.1(0.0(0.1(2.0(4.1(5.0(0.1(2.0(3.1(0.0(5.1(2.0(1.1(5.0(1.1(x1))))))))))))))))))))
1.0(4.1(1.0(4.1(2.0(5.1(4.0(5.1(5.0(0.1(5.0(0.1(4.0(0.1(4.0(5.1(0.0(0.1(5.0(4.1(2.0(x1))))))))))))))))))))) -> 1.0(5.1(5.0(1.1(1.0(3.1(0.0(3.1(5.0(5.1(3.0(2.1(4.0(2.1(0.0(3.1(5.0(1.1(0.0(3.1(2.0(x1)))))))))))))))))))))
1.1(4.0(1.1(4.0(2.1(5.0(4.1(5.0(5.1(0.0(5.1(0.0(4.1(0.0(4.1(5.0(0.1(0.0(5.1(4.0(2.1(x1))))))))))))))))))))) -> 1.1(5.0(5.1(1.0(1.1(3.0(0.1(3.0(5.1(5.0(3.1(2.0(4.1(2.0(0.1(3.0(5.1(1.0(0.1(3.0(2.1(x1)))))))))))))))))))))
2.0(2.1(4.0(0.1(0.0(1.1(4.0(3.1(1.0(4.1(4.0(3.1(1.0(2.1(1.0(1.1(5.0(0.1(4.0(5.1(0.0(x1))))))))))))))))))))) -> 4.0(3.1(0.0(2.1(4.0(0.1(4.0(3.1(0.0(4.1(2.0(0.1(5.0(1.1(4.0(2.1(3.0(2.1(3.0(0.1(0.0(x1)))))))))))))))))))))
2.1(2.0(4.1(0.0(0.1(1.0(4.1(3.0(1.1(4.0(4.1(3.0(1.1(2.0(1.1(1.0(5.1(0.0(4.1(5.0(0.1(x1))))))))))))))))))))) -> 4.1(3.0(0.1(2.0(4.1(0.0(4.1(3.0(0.1(4.0(2.1(0.0(5.1(1.0(4.1(2.0(3.1(2.0(3.1(0.0(0.1(x1)))))))))))))))))))))
4.0(2.1(4.0(2.1(5.0(5.1(2.0(0.1(2.0(0.1(2.0(0.1(3.0(1.1(5.0(3.1(0.0(1.1(4.0(4.1(2.0(x1))))))))))))))))))))) -> 4.0(2.1(4.0(5.1(1.0(3.1(5.0(2.1(4.0(4.1(2.0(0.1(5.0(0.1(1.0(2.1(5.0(2.1(4.0(5.1(0.0(x1)))))))))))))))))))))
4.1(2.0(4.1(2.0(5.1(5.0(2.1(0.0(2.1(0.0(2.1(0.0(3.1(1.0(5.1(3.0(0.1(1.0(4.1(4.0(2.1(x1))))))))))))))))))))) -> 4.1(2.0(4.1(5.0(1.1(3.0(5.1(2.0(4.1(4.0(2.1(0.0(5.1(0.0(1.1(2.0(5.1(2.0(4.1(5.0(0.1(x1)))))))))))))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------
(23) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------
(24)
YES
----------------------------------------
(25)
Obligation:
Q DP problem:
The TRS P consists of the following rules:
4^1.0(2.1(4.0(2.1(5.0(5.1(2.0(0.1(2.0(0.1(2.0(0.1(3.0(1.1(5.0(3.1(0.0(1.1(4.0(4.1(2.0(x1))))))))))))))))))))) -> 4^1.0(2.1(4.0(5.1(1.0(3.1(5.0(2.1(4.0(4.1(2.0(0.1(5.0(0.1(1.0(2.1(5.0(2.1(4.0(5.1(0.0(x1)))))))))))))))))))))
The TRS R consists of the following rules:
0.1(0.0(1.1(2.0(x1)))) -> 0.1(3.0(4.1(2.0(x1))))
0.0(0.1(1.0(2.1(x1)))) -> 0.0(3.1(4.0(2.1(x1))))
2.1(0.0(1.1(0.0(5.1(3.0(x1)))))) -> 2.1(0.0(3.1(1.0(3.1(3.0(x1))))))
2.0(0.1(1.0(0.1(5.0(3.1(x1)))))) -> 2.0(0.1(3.0(1.1(3.0(3.1(x1))))))
4.1(1.0(1.1(5.0(0.1(4.0(x1)))))) -> 4.1(4.0(2.1(2.0(0.1(4.0(x1))))))
4.0(1.1(1.0(5.1(0.0(4.1(x1)))))) -> 4.0(4.1(2.0(2.1(0.0(4.1(x1))))))
2.1(3.0(2.1(5.0(5.1(0.0(0.1(0.0(x1)))))))) -> 0.1(2.0(1.1(4.0(2.1(5.0(3.1(2.0(x1))))))))
2.0(3.1(2.0(5.1(5.0(0.1(0.0(0.1(x1)))))))) -> 0.0(2.1(1.0(4.1(2.0(5.1(3.0(2.1(x1))))))))
5.1(0.0(0.1(5.0(3.1(3.0(4.1(3.0(x1)))))))) -> 5.1(3.0(1.1(3.0(0.1(3.0(0.1(3.0(x1))))))))
5.0(0.1(0.0(5.1(3.0(3.1(4.0(3.1(x1)))))))) -> 5.0(3.1(1.0(3.1(0.0(3.1(0.0(3.1(x1))))))))
1.0(0.1(0.0(1.1(5.0(5.1(4.0(2.1(4.0(x1))))))))) -> 4.0(3.1(1.0(0.1(4.0(1.1(0.0(3.1(3.0(x1)))))))))
1.1(0.0(0.1(1.0(5.1(5.0(4.1(2.0(4.1(x1))))))))) -> 4.1(3.0(1.1(0.0(4.1(1.0(0.1(3.0(3.1(x1)))))))))
3.0(0.1(0.0(1.1(0.0(4.1(5.0(4.1(2.0(x1))))))))) -> 4.0(4.1(5.0(3.1(1.0(2.1(3.0(4.1(2.0(x1)))))))))
3.1(0.0(0.1(1.0(0.1(4.0(5.1(4.0(2.1(x1))))))))) -> 4.1(4.0(5.1(3.0(1.1(2.0(3.1(4.0(2.1(x1)))))))))
0.0(0.1(1.0(2.1(5.0(1.1(4.0(4.1(5.0(5.1(4.0(x1))))))))))) -> 0.0(0.1(0.0(4.1(1.0(1.1(3.0(5.1(3.0(4.1(2.0(x1)))))))))))
0.1(0.0(1.1(2.0(5.1(1.0(4.1(4.0(5.1(5.0(4.1(x1))))))))))) -> 0.1(0.0(0.1(4.0(1.1(1.0(3.1(5.0(3.1(4.0(2.1(x1)))))))))))
1.1(5.0(2.1(2.0(5.1(1.0(1.1(2.0(4.1(4.0(5.1(4.0(x1)))))))))))) -> 1.1(2.0(2.1(3.0(3.1(3.0(3.1(0.0(1.1(1.0(3.1(4.0(x1))))))))))))
1.0(5.1(2.0(2.1(5.0(1.1(1.0(2.1(4.0(4.1(5.0(4.1(x1)))))))))))) -> 1.0(2.1(2.0(3.1(3.0(3.1(3.0(0.1(1.0(1.1(3.0(4.1(x1))))))))))))
4.0(1.1(5.0(3.1(3.0(5.1(0.0(4.1(5.0(1.1(4.0(4.1(1.0(x1))))))))))))) -> 5.0(4.1(3.0(2.1(2.0(3.1(3.0(2.1(1.0(2.1(5.0(0.1(5.0(x1)))))))))))))
4.1(1.0(5.1(3.0(3.1(5.0(0.1(4.0(5.1(1.0(4.1(4.0(1.1(x1))))))))))))) -> 5.1(4.0(3.1(2.0(2.1(3.0(3.1(2.0(1.1(2.0(5.1(0.0(5.1(x1)))))))))))))
1.0(0.1(3.0(0.1(0.0(0.1(0.0(2.1(3.0(1.1(4.0(0.1(2.0(0.1(4.0(x1))))))))))))))) -> 1.0(4.1(4.0(0.1(4.0(1.1(4.0(0.1(3.0(2.1(4.0(1.1(0.0(2.1(4.0(x1)))))))))))))))
1.1(0.0(3.1(0.0(0.1(0.0(0.1(2.0(3.1(1.0(4.1(0.0(2.1(0.0(4.1(x1))))))))))))))) -> 1.1(4.0(4.1(0.0(4.1(1.0(4.1(0.0(3.1(2.0(4.1(1.0(0.1(2.0(4.1(x1)))))))))))))))
2.0(5.1(0.0(5.1(1.0(5.1(2.0(3.1(5.0(3.1(1.0(3.1(5.0(4.1(5.0(x1))))))))))))))) -> 2.0(2.1(5.0(4.1(2.0(0.1(1.0(5.1(1.0(1.1(4.0(1.1(0.0(3.1(4.0(x1)))))))))))))))
2.1(5.0(0.1(5.0(1.1(5.0(2.1(3.0(5.1(3.0(1.1(3.0(5.1(4.0(5.1(x1))))))))))))))) -> 2.1(2.0(5.1(4.0(2.1(0.0(1.1(5.0(1.1(1.0(4.1(1.0(0.1(3.0(4.1(x1)))))))))))))))
5.0(2.1(5.0(1.1(1.0(5.1(0.0(0.1(3.0(0.1(5.0(5.1(3.0(3.1(2.0(x1))))))))))))))) -> 5.0(5.1(3.0(3.1(1.0(5.1(0.0(3.1(1.0(0.1(4.0(3.1(1.0(3.1(2.0(x1)))))))))))))))
5.1(2.0(5.1(1.0(1.1(5.0(0.1(0.0(3.1(0.0(5.1(5.0(3.1(3.0(2.1(x1))))))))))))))) -> 5.1(5.0(3.1(3.0(1.1(5.0(0.1(3.0(1.1(0.0(4.1(3.0(1.1(3.0(2.1(x1)))))))))))))))
2.1(1.0(1.1(5.0(2.1(5.0(5.1(0.0(1.1(3.0(4.1(1.0(4.1(4.0(5.1(3.0(x1)))))))))))))))) -> 2.1(5.0(2.1(3.0(3.1(5.0(4.1(3.0(5.1(4.0(5.1(5.0(0.1(0.0(3.1(1.0(x1))))))))))))))))
2.0(1.1(1.0(5.1(2.0(5.1(5.0(0.1(1.0(3.1(4.0(1.1(4.0(4.1(5.0(3.1(x1)))))))))))))))) -> 2.0(5.1(2.0(3.1(3.0(5.1(4.0(3.1(5.0(4.1(5.0(5.1(0.0(0.1(3.0(1.1(x1))))))))))))))))
0.0(4.1(4.0(5.1(0.0(4.1(5.0(0.1(5.0(2.1(3.0(0.1(0.0(2.1(4.0(4.1(2.0(x1))))))))))))))))) -> 0.0(4.1(4.0(3.1(0.0(2.1(5.0(0.1(5.0(4.1(1.0(3.1(5.0(4.1(4.0(1.1(0.0(x1)))))))))))))))))
0.1(4.0(4.1(5.0(0.1(4.0(5.1(0.0(5.1(2.0(3.1(0.0(0.1(2.0(4.1(4.0(2.1(x1))))))))))))))))) -> 0.1(4.0(4.1(3.0(0.1(2.0(5.1(0.0(5.1(4.0(1.1(3.0(5.1(4.0(4.1(1.0(0.1(x1)))))))))))))))))
1.0(1.1(0.0(0.1(3.0(1.1(3.0(0.1(2.0(1.1(2.0(5.1(2.0(5.1(0.0(2.1(0.0(x1))))))))))))))))) -> 2.0(2.1(4.0(3.1(2.0(1.1(3.0(2.1(2.0(1.1(4.0(3.1(0.0(2.1(2.0(5.1(5.0(x1)))))))))))))))))
1.1(1.0(0.1(0.0(3.1(1.0(3.1(0.0(2.1(1.0(2.1(5.0(2.1(5.0(0.1(2.0(0.1(x1))))))))))))))))) -> 2.1(2.0(4.1(3.0(2.1(1.0(3.1(2.0(2.1(1.0(4.1(3.0(0.1(2.0(2.1(5.0(5.1(x1)))))))))))))))))
2.0(5.1(4.0(4.1(4.0(2.1(3.0(5.1(4.0(0.1(4.0(0.1(0.0(5.1(4.0(5.1(5.0(x1))))))))))))))))) -> 5.0(3.1(1.0(4.1(4.0(5.1(0.0(5.1(0.0(3.1(1.0(2.1(1.0(1.1(2.0(4.1(3.0(x1)))))))))))))))))
2.1(5.0(4.1(4.0(4.1(2.0(3.1(5.0(4.1(0.0(4.1(0.0(0.1(5.0(4.1(5.0(5.1(x1))))))))))))))))) -> 5.1(3.0(1.1(4.0(4.1(5.0(0.1(5.0(0.1(3.0(1.1(2.0(1.1(1.0(2.1(4.0(3.1(x1)))))))))))))))))
5.1(0.0(0.1(1.0(3.1(3.0(2.1(5.0(3.1(4.0(0.1(3.0(4.1(3.0(5.1(4.0(4.1(2.0(x1)))))))))))))))))) -> 5.1(5.0(0.1(0.0(1.1(1.0(1.1(0.0(2.1(2.0(2.1(1.0(3.1(2.0(1.1(2.0(2.1(2.0(x1))))))))))))))))))
5.0(0.1(0.0(1.1(3.0(3.1(2.0(5.1(3.0(4.1(0.0(3.1(4.0(3.1(5.0(4.1(4.0(2.1(x1)))))))))))))))))) -> 5.0(5.1(0.0(0.1(1.0(1.1(1.0(0.1(2.0(2.1(2.0(1.1(3.0(2.1(1.0(2.1(2.0(2.1(x1))))))))))))))))))
4.0(4.1(0.0(2.1(2.0(1.1(1.0(4.1(1.0(1.1(1.0(0.1(0.0(0.1(2.0(3.1(4.0(4.1(4.0(x1))))))))))))))))))) -> 3.0(1.1(0.0(3.1(4.0(2.1(2.0(1.1(0.0(1.1(5.0(2.1(0.0(5.1(4.0(4.1(2.0(1.1(4.0(x1)))))))))))))))))))
4.1(4.0(0.1(2.0(2.1(1.0(1.1(4.0(1.1(1.0(1.1(0.0(0.1(0.0(2.1(3.0(4.1(4.0(4.1(x1))))))))))))))))))) -> 3.1(1.0(0.1(3.0(4.1(2.0(2.1(1.0(0.1(1.0(5.1(2.0(0.1(5.0(4.1(4.0(2.1(1.0(4.1(x1)))))))))))))))))))
0.1(3.0(2.1(2.0(1.1(4.0(5.1(3.0(3.1(5.0(5.1(1.0(5.1(1.0(0.1(5.0(1.1(5.0(5.1(1.0(x1)))))))))))))))))))) -> 0.1(3.0(1.1(4.0(0.1(2.0(0.1(0.0(2.1(4.0(5.1(0.0(2.1(3.0(0.1(5.0(2.1(1.0(5.1(1.0(x1))))))))))))))))))))
0.0(3.1(2.0(2.1(1.0(4.1(5.0(3.1(3.0(5.1(5.0(1.1(5.0(1.1(0.0(5.1(1.0(5.1(5.0(1.1(x1)))))))))))))))))))) -> 0.0(3.1(1.0(4.1(0.0(2.1(0.0(0.1(2.0(4.1(5.0(0.1(2.0(3.1(0.0(5.1(2.0(1.1(5.0(1.1(x1))))))))))))))))))))
1.0(4.1(1.0(4.1(2.0(5.1(4.0(5.1(5.0(0.1(5.0(0.1(4.0(0.1(4.0(5.1(0.0(0.1(5.0(4.1(2.0(x1))))))))))))))))))))) -> 1.0(5.1(5.0(1.1(1.0(3.1(0.0(3.1(5.0(5.1(3.0(2.1(4.0(2.1(0.0(3.1(5.0(1.1(0.0(3.1(2.0(x1)))))))))))))))))))))
1.1(4.0(1.1(4.0(2.1(5.0(4.1(5.0(5.1(0.0(5.1(0.0(4.1(0.0(4.1(5.0(0.1(0.0(5.1(4.0(2.1(x1))))))))))))))))))))) -> 1.1(5.0(5.1(1.0(1.1(3.0(0.1(3.0(5.1(5.0(3.1(2.0(4.1(2.0(0.1(3.0(5.1(1.0(0.1(3.0(2.1(x1)))))))))))))))))))))
2.0(2.1(4.0(0.1(0.0(1.1(4.0(3.1(1.0(4.1(4.0(3.1(1.0(2.1(1.0(1.1(5.0(0.1(4.0(5.1(0.0(x1))))))))))))))))))))) -> 4.0(3.1(0.0(2.1(4.0(0.1(4.0(3.1(0.0(4.1(2.0(0.1(5.0(1.1(4.0(2.1(3.0(2.1(3.0(0.1(0.0(x1)))))))))))))))))))))
2.1(2.0(4.1(0.0(0.1(1.0(4.1(3.0(1.1(4.0(4.1(3.0(1.1(2.0(1.1(1.0(5.1(0.0(4.1(5.0(0.1(x1))))))))))))))))))))) -> 4.1(3.0(0.1(2.0(4.1(0.0(4.1(3.0(0.1(4.0(2.1(0.0(5.1(1.0(4.1(2.0(3.1(2.0(3.1(0.0(0.1(x1)))))))))))))))))))))
4.0(2.1(4.0(2.1(5.0(5.1(2.0(0.1(2.0(0.1(2.0(0.1(3.0(1.1(5.0(3.1(0.0(1.1(4.0(4.1(2.0(x1))))))))))))))))))))) -> 4.0(2.1(4.0(5.1(1.0(3.1(5.0(2.1(4.0(4.1(2.0(0.1(5.0(0.1(1.0(2.1(5.0(2.1(4.0(5.1(0.0(x1)))))))))))))))))))))
4.1(2.0(4.1(2.0(5.1(5.0(2.1(0.0(2.1(0.0(2.1(0.0(3.1(1.0(5.1(3.0(0.1(1.0(4.1(4.0(2.1(x1))))))))))))))))))))) -> 4.1(2.0(4.1(5.0(1.1(3.0(5.1(2.0(4.1(4.0(2.1(0.0(5.1(0.0(1.1(2.0(5.1(2.0(4.1(5.0(0.1(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.
4^1.0(2.1(4.0(2.1(5.0(5.1(2.0(0.1(2.0(0.1(2.0(0.1(3.0(1.1(5.0(3.1(0.0(1.1(4.0(4.1(2.0(x1))))))))))))))))))))) -> 4^1.0(2.1(4.0(5.1(1.0(3.1(5.0(2.1(4.0(4.1(2.0(0.1(5.0(0.1(1.0(2.1(5.0(2.1(4.0(5.1(0.0(x1)))))))))))))))))))))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(0.0(x_1)) = 0
POL(0.1(x_1)) = 0
POL(1.0(x_1)) = 1
POL(1.1(x_1)) = x_1
POL(2.0(x_1)) = 1
POL(2.1(x_1)) = x_1
POL(3.0(x_1)) = 0
POL(3.1(x_1)) = 0
POL(4.0(x_1)) = x_1
POL(4.1(x_1)) = 0
POL(4^1.0(x_1)) = x_1
POL(5.0(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:
0.0(0.1(1.0(2.1(5.0(1.1(4.0(4.1(5.0(5.1(4.0(x1))))))))))) -> 0.0(0.1(0.0(4.1(1.0(1.1(3.0(5.1(3.0(4.1(2.0(x1)))))))))))
0.0(0.1(1.0(2.1(x1)))) -> 0.0(3.1(4.0(2.1(x1))))
0.0(4.1(4.0(5.1(0.0(4.1(5.0(0.1(5.0(2.1(3.0(0.1(0.0(2.1(4.0(4.1(2.0(x1))))))))))))))))) -> 0.0(4.1(4.0(3.1(0.0(2.1(5.0(0.1(5.0(4.1(1.0(3.1(5.0(4.1(4.0(1.1(0.0(x1)))))))))))))))))
0.0(3.1(2.0(2.1(1.0(4.1(5.0(3.1(3.0(5.1(5.0(1.1(5.0(1.1(0.0(5.1(1.0(5.1(5.0(1.1(x1)))))))))))))))))))) -> 0.0(3.1(1.0(4.1(0.0(2.1(0.0(0.1(2.0(4.1(5.0(0.1(2.0(3.1(0.0(5.1(2.0(1.1(5.0(1.1(x1))))))))))))))))))))
5.1(2.0(5.1(1.0(1.1(5.0(0.1(0.0(3.1(0.0(5.1(5.0(3.1(3.0(2.1(x1))))))))))))))) -> 5.1(5.0(3.1(3.0(1.1(5.0(0.1(3.0(1.1(0.0(4.1(3.0(1.1(3.0(2.1(x1)))))))))))))))
5.1(0.0(0.1(5.0(3.1(3.0(4.1(3.0(x1)))))))) -> 5.1(3.0(1.1(3.0(0.1(3.0(0.1(3.0(x1))))))))
5.1(0.0(0.1(1.0(3.1(3.0(2.1(5.0(3.1(4.0(0.1(3.0(4.1(3.0(5.1(4.0(4.1(2.0(x1)))))))))))))))))) -> 5.1(5.0(0.1(0.0(1.1(1.0(1.1(0.0(2.1(2.0(2.1(1.0(3.1(2.0(1.1(2.0(2.1(2.0(x1))))))))))))))))))
4.0(4.1(0.0(2.1(2.0(1.1(1.0(4.1(1.0(1.1(1.0(0.1(0.0(0.1(2.0(3.1(4.0(4.1(4.0(x1))))))))))))))))))) -> 3.0(1.1(0.0(3.1(4.0(2.1(2.0(1.1(0.0(1.1(5.0(2.1(0.0(5.1(4.0(4.1(2.0(1.1(4.0(x1)))))))))))))))))))
3.0(0.1(0.0(1.1(0.0(4.1(5.0(4.1(2.0(x1))))))))) -> 4.0(4.1(5.0(3.1(1.0(2.1(3.0(4.1(2.0(x1)))))))))
4.0(1.1(1.0(5.1(0.0(4.1(x1)))))) -> 4.0(4.1(2.0(2.1(0.0(4.1(x1))))))
4.0(2.1(4.0(2.1(5.0(5.1(2.0(0.1(2.0(0.1(2.0(0.1(3.0(1.1(5.0(3.1(0.0(1.1(4.0(4.1(2.0(x1))))))))))))))))))))) -> 4.0(2.1(4.0(5.1(1.0(3.1(5.0(2.1(4.0(4.1(2.0(0.1(5.0(0.1(1.0(2.1(5.0(2.1(4.0(5.1(0.0(x1)))))))))))))))))))))
4.0(1.1(5.0(3.1(3.0(5.1(0.0(4.1(5.0(1.1(4.0(4.1(1.0(x1))))))))))))) -> 5.0(4.1(3.0(2.1(2.0(3.1(3.0(2.1(1.0(2.1(5.0(0.1(5.0(x1)))))))))))))
2.1(5.0(0.1(5.0(1.1(5.0(2.1(3.0(5.1(3.0(1.1(3.0(5.1(4.0(5.1(x1))))))))))))))) -> 2.1(2.0(5.1(4.0(2.1(0.0(1.1(5.0(1.1(1.0(4.1(1.0(0.1(3.0(4.1(x1)))))))))))))))
2.1(0.0(1.1(0.0(5.1(3.0(x1)))))) -> 2.1(0.0(3.1(1.0(3.1(3.0(x1))))))
2.1(1.0(1.1(5.0(2.1(5.0(5.1(0.0(1.1(3.0(4.1(1.0(4.1(4.0(5.1(3.0(x1)))))))))))))))) -> 2.1(5.0(2.1(3.0(3.1(5.0(4.1(3.0(5.1(4.0(5.1(5.0(0.1(0.0(3.1(1.0(x1))))))))))))))))
2.1(3.0(2.1(5.0(5.1(0.0(0.1(0.0(x1)))))))) -> 0.1(2.0(1.1(4.0(2.1(5.0(3.1(2.0(x1))))))))
2.1(5.0(4.1(4.0(4.1(2.0(3.1(5.0(4.1(0.0(4.1(0.0(0.1(5.0(4.1(5.0(5.1(x1))))))))))))))))) -> 5.1(3.0(1.1(4.0(4.1(5.0(0.1(5.0(0.1(3.0(1.1(2.0(1.1(1.0(2.1(4.0(3.1(x1)))))))))))))))))
2.1(2.0(4.1(0.0(0.1(1.0(4.1(3.0(1.1(4.0(4.1(3.0(1.1(2.0(1.1(1.0(5.1(0.0(4.1(5.0(0.1(x1))))))))))))))))))))) -> 4.1(3.0(0.1(2.0(4.1(0.0(4.1(3.0(0.1(4.0(2.1(0.0(5.1(1.0(4.1(2.0(3.1(2.0(3.1(0.0(0.1(x1)))))))))))))))))))))
5.0(2.1(5.0(1.1(1.0(5.1(0.0(0.1(3.0(0.1(5.0(5.1(3.0(3.1(2.0(x1))))))))))))))) -> 5.0(5.1(3.0(3.1(1.0(5.1(0.0(3.1(1.0(0.1(4.0(3.1(1.0(3.1(2.0(x1)))))))))))))))
5.0(0.1(0.0(5.1(3.0(3.1(4.0(3.1(x1)))))))) -> 5.0(3.1(1.0(3.1(0.0(3.1(0.0(3.1(x1))))))))
5.0(0.1(0.0(1.1(3.0(3.1(2.0(5.1(3.0(4.1(0.0(3.1(4.0(3.1(5.0(4.1(4.0(2.1(x1)))))))))))))))))) -> 5.0(5.1(0.0(0.1(1.0(1.1(1.0(0.1(2.0(2.1(2.0(1.1(3.0(2.1(1.0(2.1(2.0(2.1(x1))))))))))))))))))
1.0(0.1(0.0(1.1(5.0(5.1(4.0(2.1(4.0(x1))))))))) -> 4.0(3.1(1.0(0.1(4.0(1.1(0.0(3.1(3.0(x1)))))))))
1.0(0.1(3.0(0.1(0.0(0.1(0.0(2.1(3.0(1.1(4.0(0.1(2.0(0.1(4.0(x1))))))))))))))) -> 1.0(4.1(4.0(0.1(4.0(1.1(4.0(0.1(3.0(2.1(4.0(1.1(0.0(2.1(4.0(x1)))))))))))))))
1.0(5.1(2.0(2.1(5.0(1.1(1.0(2.1(4.0(4.1(5.0(4.1(x1)))))))))))) -> 1.0(2.1(2.0(3.1(3.0(3.1(3.0(0.1(1.0(1.1(3.0(4.1(x1))))))))))))
1.0(4.1(1.0(4.1(2.0(5.1(4.0(5.1(5.0(0.1(5.0(0.1(4.0(0.1(4.0(5.1(0.0(0.1(5.0(4.1(2.0(x1))))))))))))))))))))) -> 1.0(5.1(5.0(1.1(1.0(3.1(0.0(3.1(5.0(5.1(3.0(2.1(4.0(2.1(0.0(3.1(5.0(1.1(0.0(3.1(2.0(x1)))))))))))))))))))))
1.0(1.1(0.0(0.1(3.0(1.1(3.0(0.1(2.0(1.1(2.0(5.1(2.0(5.1(0.0(2.1(0.0(x1))))))))))))))))) -> 2.0(2.1(4.0(3.1(2.0(1.1(3.0(2.1(2.0(1.1(4.0(3.1(0.0(2.1(2.0(5.1(5.0(x1)))))))))))))))))
0.1(0.0(1.1(2.0(5.1(1.0(4.1(4.0(5.1(5.0(4.1(x1))))))))))) -> 0.1(0.0(0.1(4.0(1.1(1.0(3.1(5.0(3.1(4.0(2.1(x1)))))))))))
0.1(0.0(1.1(2.0(x1)))) -> 0.1(3.0(4.1(2.0(x1))))
0.1(4.0(4.1(5.0(0.1(4.0(5.1(0.0(5.1(2.0(3.1(0.0(0.1(2.0(4.1(4.0(2.1(x1))))))))))))))))) -> 0.1(4.0(4.1(3.0(0.1(2.0(5.1(0.0(5.1(4.0(1.1(3.0(5.1(4.0(4.1(1.0(0.1(x1)))))))))))))))))
0.1(3.0(2.1(2.0(1.1(4.0(5.1(3.0(3.1(5.0(5.1(1.0(5.1(1.0(0.1(5.0(1.1(5.0(5.1(1.0(x1)))))))))))))))))))) -> 0.1(3.0(1.1(4.0(0.1(2.0(0.1(0.0(2.1(4.0(5.1(0.0(2.1(3.0(0.1(5.0(2.1(1.0(5.1(1.0(x1))))))))))))))))))))
2.0(5.1(0.0(5.1(1.0(5.1(2.0(3.1(5.0(3.1(1.0(3.1(5.0(4.1(5.0(x1))))))))))))))) -> 2.0(2.1(5.0(4.1(2.0(0.1(1.0(5.1(1.0(1.1(4.0(1.1(0.0(3.1(4.0(x1)))))))))))))))
2.0(0.1(1.0(0.1(5.0(3.1(x1)))))) -> 2.0(0.1(3.0(1.1(3.0(3.1(x1))))))
2.0(1.1(1.0(5.1(2.0(5.1(5.0(0.1(1.0(3.1(4.0(1.1(4.0(4.1(5.0(3.1(x1)))))))))))))))) -> 2.0(5.1(2.0(3.1(3.0(5.1(4.0(3.1(5.0(4.1(5.0(5.1(0.0(0.1(3.0(1.1(x1))))))))))))))))
2.0(3.1(2.0(5.1(5.0(0.1(0.0(0.1(x1)))))))) -> 0.0(2.1(1.0(4.1(2.0(5.1(3.0(2.1(x1))))))))
2.0(5.1(4.0(4.1(4.0(2.1(3.0(5.1(4.0(0.1(4.0(0.1(0.0(5.1(4.0(5.1(5.0(x1))))))))))))))))) -> 5.0(3.1(1.0(4.1(4.0(5.1(0.0(5.1(0.0(3.1(1.0(2.1(1.0(1.1(2.0(4.1(3.0(x1)))))))))))))))))
2.0(2.1(4.0(0.1(0.0(1.1(4.0(3.1(1.0(4.1(4.0(3.1(1.0(2.1(1.0(1.1(5.0(0.1(4.0(5.1(0.0(x1))))))))))))))))))))) -> 4.0(3.1(0.0(2.1(4.0(0.1(4.0(3.1(0.0(4.1(2.0(0.1(5.0(1.1(4.0(2.1(3.0(2.1(3.0(0.1(0.0(x1)))))))))))))))))))))
4.1(4.0(0.1(2.0(2.1(1.0(1.1(4.0(1.1(1.0(1.1(0.0(0.1(0.0(2.1(3.0(4.1(4.0(4.1(x1))))))))))))))))))) -> 3.1(1.0(0.1(3.0(4.1(2.0(2.1(1.0(0.1(1.0(5.1(2.0(0.1(5.0(4.1(4.0(2.1(1.0(4.1(x1)))))))))))))))))))
3.1(0.0(0.1(1.0(0.1(4.0(5.1(4.0(2.1(x1))))))))) -> 4.1(4.0(5.1(3.0(1.1(2.0(3.1(4.0(2.1(x1)))))))))
4.1(1.0(1.1(5.0(0.1(4.0(x1)))))) -> 4.1(4.0(2.1(2.0(0.1(4.0(x1))))))
4.1(2.0(4.1(2.0(5.1(5.0(2.1(0.0(2.1(0.0(2.1(0.0(3.1(1.0(5.1(3.0(0.1(1.0(4.1(4.0(2.1(x1))))))))))))))))))))) -> 4.1(2.0(4.1(5.0(1.1(3.0(5.1(2.0(4.1(4.0(2.1(0.0(5.1(0.0(1.1(2.0(5.1(2.0(4.1(5.0(0.1(x1)))))))))))))))))))))
4.1(1.0(5.1(3.0(3.1(5.0(0.1(4.0(5.1(1.0(4.1(4.0(1.1(x1))))))))))))) -> 5.1(4.0(3.1(2.0(2.1(3.0(3.1(2.0(1.1(2.0(5.1(0.0(5.1(x1)))))))))))))
1.1(0.0(0.1(1.0(5.1(5.0(4.1(2.0(4.1(x1))))))))) -> 4.1(3.0(1.1(0.0(4.1(1.0(0.1(3.0(3.1(x1)))))))))
1.1(0.0(3.1(0.0(0.1(0.0(0.1(2.0(3.1(1.0(4.1(0.0(2.1(0.0(4.1(x1))))))))))))))) -> 1.1(4.0(4.1(0.0(4.1(1.0(4.1(0.0(3.1(2.0(4.1(1.0(0.1(2.0(4.1(x1)))))))))))))))
1.1(5.0(2.1(2.0(5.1(1.0(1.1(2.0(4.1(4.0(5.1(4.0(x1)))))))))))) -> 1.1(2.0(2.1(3.0(3.1(3.0(3.1(0.0(1.1(1.0(3.1(4.0(x1))))))))))))
1.1(4.0(1.1(4.0(2.1(5.0(4.1(5.0(5.1(0.0(5.1(0.0(4.1(0.0(4.1(5.0(0.1(0.0(5.1(4.0(2.1(x1))))))))))))))))))))) -> 1.1(5.0(5.1(1.0(1.1(3.0(0.1(3.0(5.1(5.0(3.1(2.0(4.1(2.0(0.1(3.0(5.1(1.0(0.1(3.0(2.1(x1)))))))))))))))))))))
1.1(1.0(0.1(0.0(3.1(1.0(3.1(0.0(2.1(1.0(2.1(5.0(2.1(5.0(0.1(2.0(0.1(x1))))))))))))))))) -> 2.1(2.0(4.1(3.0(2.1(1.0(3.1(2.0(2.1(1.0(4.1(3.0(0.1(2.0(2.1(5.0(5.1(x1)))))))))))))))))
----------------------------------------
(27)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
0.1(0.0(1.1(2.0(x1)))) -> 0.1(3.0(4.1(2.0(x1))))
0.0(0.1(1.0(2.1(x1)))) -> 0.0(3.1(4.0(2.1(x1))))
2.1(0.0(1.1(0.0(5.1(3.0(x1)))))) -> 2.1(0.0(3.1(1.0(3.1(3.0(x1))))))
2.0(0.1(1.0(0.1(5.0(3.1(x1)))))) -> 2.0(0.1(3.0(1.1(3.0(3.1(x1))))))
4.1(1.0(1.1(5.0(0.1(4.0(x1)))))) -> 4.1(4.0(2.1(2.0(0.1(4.0(x1))))))
4.0(1.1(1.0(5.1(0.0(4.1(x1)))))) -> 4.0(4.1(2.0(2.1(0.0(4.1(x1))))))
2.1(3.0(2.1(5.0(5.1(0.0(0.1(0.0(x1)))))))) -> 0.1(2.0(1.1(4.0(2.1(5.0(3.1(2.0(x1))))))))
2.0(3.1(2.0(5.1(5.0(0.1(0.0(0.1(x1)))))))) -> 0.0(2.1(1.0(4.1(2.0(5.1(3.0(2.1(x1))))))))
5.1(0.0(0.1(5.0(3.1(3.0(4.1(3.0(x1)))))))) -> 5.1(3.0(1.1(3.0(0.1(3.0(0.1(3.0(x1))))))))
5.0(0.1(0.0(5.1(3.0(3.1(4.0(3.1(x1)))))))) -> 5.0(3.1(1.0(3.1(0.0(3.1(0.0(3.1(x1))))))))
1.0(0.1(0.0(1.1(5.0(5.1(4.0(2.1(4.0(x1))))))))) -> 4.0(3.1(1.0(0.1(4.0(1.1(0.0(3.1(3.0(x1)))))))))
1.1(0.0(0.1(1.0(5.1(5.0(4.1(2.0(4.1(x1))))))))) -> 4.1(3.0(1.1(0.0(4.1(1.0(0.1(3.0(3.1(x1)))))))))
3.0(0.1(0.0(1.1(0.0(4.1(5.0(4.1(2.0(x1))))))))) -> 4.0(4.1(5.0(3.1(1.0(2.1(3.0(4.1(2.0(x1)))))))))
3.1(0.0(0.1(1.0(0.1(4.0(5.1(4.0(2.1(x1))))))))) -> 4.1(4.0(5.1(3.0(1.1(2.0(3.1(4.0(2.1(x1)))))))))
0.0(0.1(1.0(2.1(5.0(1.1(4.0(4.1(5.0(5.1(4.0(x1))))))))))) -> 0.0(0.1(0.0(4.1(1.0(1.1(3.0(5.1(3.0(4.1(2.0(x1)))))))))))
0.1(0.0(1.1(2.0(5.1(1.0(4.1(4.0(5.1(5.0(4.1(x1))))))))))) -> 0.1(0.0(0.1(4.0(1.1(1.0(3.1(5.0(3.1(4.0(2.1(x1)))))))))))
1.1(5.0(2.1(2.0(5.1(1.0(1.1(2.0(4.1(4.0(5.1(4.0(x1)))))))))))) -> 1.1(2.0(2.1(3.0(3.1(3.0(3.1(0.0(1.1(1.0(3.1(4.0(x1))))))))))))
1.0(5.1(2.0(2.1(5.0(1.1(1.0(2.1(4.0(4.1(5.0(4.1(x1)))))))))))) -> 1.0(2.1(2.0(3.1(3.0(3.1(3.0(0.1(1.0(1.1(3.0(4.1(x1))))))))))))
4.0(1.1(5.0(3.1(3.0(5.1(0.0(4.1(5.0(1.1(4.0(4.1(1.0(x1))))))))))))) -> 5.0(4.1(3.0(2.1(2.0(3.1(3.0(2.1(1.0(2.1(5.0(0.1(5.0(x1)))))))))))))
4.1(1.0(5.1(3.0(3.1(5.0(0.1(4.0(5.1(1.0(4.1(4.0(1.1(x1))))))))))))) -> 5.1(4.0(3.1(2.0(2.1(3.0(3.1(2.0(1.1(2.0(5.1(0.0(5.1(x1)))))))))))))
1.0(0.1(3.0(0.1(0.0(0.1(0.0(2.1(3.0(1.1(4.0(0.1(2.0(0.1(4.0(x1))))))))))))))) -> 1.0(4.1(4.0(0.1(4.0(1.1(4.0(0.1(3.0(2.1(4.0(1.1(0.0(2.1(4.0(x1)))))))))))))))
1.1(0.0(3.1(0.0(0.1(0.0(0.1(2.0(3.1(1.0(4.1(0.0(2.1(0.0(4.1(x1))))))))))))))) -> 1.1(4.0(4.1(0.0(4.1(1.0(4.1(0.0(3.1(2.0(4.1(1.0(0.1(2.0(4.1(x1)))))))))))))))
2.0(5.1(0.0(5.1(1.0(5.1(2.0(3.1(5.0(3.1(1.0(3.1(5.0(4.1(5.0(x1))))))))))))))) -> 2.0(2.1(5.0(4.1(2.0(0.1(1.0(5.1(1.0(1.1(4.0(1.1(0.0(3.1(4.0(x1)))))))))))))))
2.1(5.0(0.1(5.0(1.1(5.0(2.1(3.0(5.1(3.0(1.1(3.0(5.1(4.0(5.1(x1))))))))))))))) -> 2.1(2.0(5.1(4.0(2.1(0.0(1.1(5.0(1.1(1.0(4.1(1.0(0.1(3.0(4.1(x1)))))))))))))))
5.0(2.1(5.0(1.1(1.0(5.1(0.0(0.1(3.0(0.1(5.0(5.1(3.0(3.1(2.0(x1))))))))))))))) -> 5.0(5.1(3.0(3.1(1.0(5.1(0.0(3.1(1.0(0.1(4.0(3.1(1.0(3.1(2.0(x1)))))))))))))))
5.1(2.0(5.1(1.0(1.1(5.0(0.1(0.0(3.1(0.0(5.1(5.0(3.1(3.0(2.1(x1))))))))))))))) -> 5.1(5.0(3.1(3.0(1.1(5.0(0.1(3.0(1.1(0.0(4.1(3.0(1.1(3.0(2.1(x1)))))))))))))))
2.1(1.0(1.1(5.0(2.1(5.0(5.1(0.0(1.1(3.0(4.1(1.0(4.1(4.0(5.1(3.0(x1)))))))))))))))) -> 2.1(5.0(2.1(3.0(3.1(5.0(4.1(3.0(5.1(4.0(5.1(5.0(0.1(0.0(3.1(1.0(x1))))))))))))))))
2.0(1.1(1.0(5.1(2.0(5.1(5.0(0.1(1.0(3.1(4.0(1.1(4.0(4.1(5.0(3.1(x1)))))))))))))))) -> 2.0(5.1(2.0(3.1(3.0(5.1(4.0(3.1(5.0(4.1(5.0(5.1(0.0(0.1(3.0(1.1(x1))))))))))))))))
0.0(4.1(4.0(5.1(0.0(4.1(5.0(0.1(5.0(2.1(3.0(0.1(0.0(2.1(4.0(4.1(2.0(x1))))))))))))))))) -> 0.0(4.1(4.0(3.1(0.0(2.1(5.0(0.1(5.0(4.1(1.0(3.1(5.0(4.1(4.0(1.1(0.0(x1)))))))))))))))))
0.1(4.0(4.1(5.0(0.1(4.0(5.1(0.0(5.1(2.0(3.1(0.0(0.1(2.0(4.1(4.0(2.1(x1))))))))))))))))) -> 0.1(4.0(4.1(3.0(0.1(2.0(5.1(0.0(5.1(4.0(1.1(3.0(5.1(4.0(4.1(1.0(0.1(x1)))))))))))))))))
1.0(1.1(0.0(0.1(3.0(1.1(3.0(0.1(2.0(1.1(2.0(5.1(2.0(5.1(0.0(2.1(0.0(x1))))))))))))))))) -> 2.0(2.1(4.0(3.1(2.0(1.1(3.0(2.1(2.0(1.1(4.0(3.1(0.0(2.1(2.0(5.1(5.0(x1)))))))))))))))))
1.1(1.0(0.1(0.0(3.1(1.0(3.1(0.0(2.1(1.0(2.1(5.0(2.1(5.0(0.1(2.0(0.1(x1))))))))))))))))) -> 2.1(2.0(4.1(3.0(2.1(1.0(3.1(2.0(2.1(1.0(4.1(3.0(0.1(2.0(2.1(5.0(5.1(x1)))))))))))))))))
2.0(5.1(4.0(4.1(4.0(2.1(3.0(5.1(4.0(0.1(4.0(0.1(0.0(5.1(4.0(5.1(5.0(x1))))))))))))))))) -> 5.0(3.1(1.0(4.1(4.0(5.1(0.0(5.1(0.0(3.1(1.0(2.1(1.0(1.1(2.0(4.1(3.0(x1)))))))))))))))))
2.1(5.0(4.1(4.0(4.1(2.0(3.1(5.0(4.1(0.0(4.1(0.0(0.1(5.0(4.1(5.0(5.1(x1))))))))))))))))) -> 5.1(3.0(1.1(4.0(4.1(5.0(0.1(5.0(0.1(3.0(1.1(2.0(1.1(1.0(2.1(4.0(3.1(x1)))))))))))))))))
5.1(0.0(0.1(1.0(3.1(3.0(2.1(5.0(3.1(4.0(0.1(3.0(4.1(3.0(5.1(4.0(4.1(2.0(x1)))))))))))))))))) -> 5.1(5.0(0.1(0.0(1.1(1.0(1.1(0.0(2.1(2.0(2.1(1.0(3.1(2.0(1.1(2.0(2.1(2.0(x1))))))))))))))))))
5.0(0.1(0.0(1.1(3.0(3.1(2.0(5.1(3.0(4.1(0.0(3.1(4.0(3.1(5.0(4.1(4.0(2.1(x1)))))))))))))))))) -> 5.0(5.1(0.0(0.1(1.0(1.1(1.0(0.1(2.0(2.1(2.0(1.1(3.0(2.1(1.0(2.1(2.0(2.1(x1))))))))))))))))))
4.0(4.1(0.0(2.1(2.0(1.1(1.0(4.1(1.0(1.1(1.0(0.1(0.0(0.1(2.0(3.1(4.0(4.1(4.0(x1))))))))))))))))))) -> 3.0(1.1(0.0(3.1(4.0(2.1(2.0(1.1(0.0(1.1(5.0(2.1(0.0(5.1(4.0(4.1(2.0(1.1(4.0(x1)))))))))))))))))))
4.1(4.0(0.1(2.0(2.1(1.0(1.1(4.0(1.1(1.0(1.1(0.0(0.1(0.0(2.1(3.0(4.1(4.0(4.1(x1))))))))))))))))))) -> 3.1(1.0(0.1(3.0(4.1(2.0(2.1(1.0(0.1(1.0(5.1(2.0(0.1(5.0(4.1(4.0(2.1(1.0(4.1(x1)))))))))))))))))))
0.1(3.0(2.1(2.0(1.1(4.0(5.1(3.0(3.1(5.0(5.1(1.0(5.1(1.0(0.1(5.0(1.1(5.0(5.1(1.0(x1)))))))))))))))))))) -> 0.1(3.0(1.1(4.0(0.1(2.0(0.1(0.0(2.1(4.0(5.1(0.0(2.1(3.0(0.1(5.0(2.1(1.0(5.1(1.0(x1))))))))))))))))))))
0.0(3.1(2.0(2.1(1.0(4.1(5.0(3.1(3.0(5.1(5.0(1.1(5.0(1.1(0.0(5.1(1.0(5.1(5.0(1.1(x1)))))))))))))))))))) -> 0.0(3.1(1.0(4.1(0.0(2.1(0.0(0.1(2.0(4.1(5.0(0.1(2.0(3.1(0.0(5.1(2.0(1.1(5.0(1.1(x1))))))))))))))))))))
1.0(4.1(1.0(4.1(2.0(5.1(4.0(5.1(5.0(0.1(5.0(0.1(4.0(0.1(4.0(5.1(0.0(0.1(5.0(4.1(2.0(x1))))))))))))))))))))) -> 1.0(5.1(5.0(1.1(1.0(3.1(0.0(3.1(5.0(5.1(3.0(2.1(4.0(2.1(0.0(3.1(5.0(1.1(0.0(3.1(2.0(x1)))))))))))))))))))))
1.1(4.0(1.1(4.0(2.1(5.0(4.1(5.0(5.1(0.0(5.1(0.0(4.1(0.0(4.1(5.0(0.1(0.0(5.1(4.0(2.1(x1))))))))))))))))))))) -> 1.1(5.0(5.1(1.0(1.1(3.0(0.1(3.0(5.1(5.0(3.1(2.0(4.1(2.0(0.1(3.0(5.1(1.0(0.1(3.0(2.1(x1)))))))))))))))))))))
2.0(2.1(4.0(0.1(0.0(1.1(4.0(3.1(1.0(4.1(4.0(3.1(1.0(2.1(1.0(1.1(5.0(0.1(4.0(5.1(0.0(x1))))))))))))))))))))) -> 4.0(3.1(0.0(2.1(4.0(0.1(4.0(3.1(0.0(4.1(2.0(0.1(5.0(1.1(4.0(2.1(3.0(2.1(3.0(0.1(0.0(x1)))))))))))))))))))))
2.1(2.0(4.1(0.0(0.1(1.0(4.1(3.0(1.1(4.0(4.1(3.0(1.1(2.0(1.1(1.0(5.1(0.0(4.1(5.0(0.1(x1))))))))))))))))))))) -> 4.1(3.0(0.1(2.0(4.1(0.0(4.1(3.0(0.1(4.0(2.1(0.0(5.1(1.0(4.1(2.0(3.1(2.0(3.1(0.0(0.1(x1)))))))))))))))))))))
4.0(2.1(4.0(2.1(5.0(5.1(2.0(0.1(2.0(0.1(2.0(0.1(3.0(1.1(5.0(3.1(0.0(1.1(4.0(4.1(2.0(x1))))))))))))))))))))) -> 4.0(2.1(4.0(5.1(1.0(3.1(5.0(2.1(4.0(4.1(2.0(0.1(5.0(0.1(1.0(2.1(5.0(2.1(4.0(5.1(0.0(x1)))))))))))))))))))))
4.1(2.0(4.1(2.0(5.1(5.0(2.1(0.0(2.1(0.0(2.1(0.0(3.1(1.0(5.1(3.0(0.1(1.0(4.1(4.0(2.1(x1))))))))))))))))))))) -> 4.1(2.0(4.1(5.0(1.1(3.0(5.1(2.0(4.1(4.0(2.1(0.0(5.1(0.0(1.1(2.0(5.1(2.0(4.1(5.0(0.1(x1)))))))))))))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------
(28) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------
(29)
YES
----------------------------------------
(30)
Obligation:
Q DP problem:
The TRS P consists of the following rules:
5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5^1(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
5^1(0(0(5(3(3(4(3(x1)))))))) -> 5^1(3(1(3(0(3(0(3(x1))))))))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5^1(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
The TRS R consists of the following rules:
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1)))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------
(31) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
5^1(0(0(5(3(3(4(3(x1)))))))) -> 5^1(3(1(3(0(3(0(3(x1))))))))
5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5^1(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(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)) = 0
POL(4(x_1)) = 0
POL(5(x_1)) = 0
POL(5^1(x_1)) = x_1
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
----------------------------------------
(32)
Obligation:
Q DP problem:
The TRS P consists of the following rules:
5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5^1(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
The TRS R consists of the following rules:
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1)))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(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.
5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5^1(5(3(3(1(5(0(3(1(0(4(3(1(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 + x_1
POL(3(x_1)) = 0
POL(4(x_1)) = 0
POL(5(x_1)) = 1
POL(5^1(x_1)) = x_1
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
----------------------------------------
(34)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1)))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------
(35) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------
(36)
YES
----------------------------------------
(37)
Obligation:
Q DP problem:
The TRS P consists of the following rules:
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2^1(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
2^1(0(1(0(5(3(x1)))))) -> 2^1(0(3(1(3(3(x1))))))
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2^1(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
The TRS R consists of the following rules:
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1)))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------
(38) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2^1(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(0(x_1)) = 0
POL(1(x_1)) = 1
POL(2(x_1)) = 0
POL(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(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
----------------------------------------
(39)
Obligation:
Q DP problem:
The TRS P consists of the following rules:
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2^1(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
2^1(0(1(0(5(3(x1)))))) -> 2^1(0(3(1(3(3(x1))))))
The TRS R consists of the following rules:
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1)))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(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.
2^1(0(1(0(5(3(x1)))))) -> 2^1(0(3(1(3(3(x1))))))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(0(x_1)) = x_1
POL(1(x_1)) = 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(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
----------------------------------------
(41)
Obligation:
Q DP problem:
The TRS P consists of the following rules:
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2^1(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
The TRS R consists of the following rules:
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1)))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(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.
2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2^1(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(0(x_1)) = x_1
POL(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)) = x_1
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1)))))))))
1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))
1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
----------------------------------------
(43)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1)))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------
(44) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------
(45)
YES
----------------------------------------
(46)
Obligation:
Q DP problem:
The TRS P consists of the following rules:
1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1^1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))
1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1^1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1^1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
The TRS R consists of the following rules:
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1)))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------
(47) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1^1(4(4(0(4(1(4(0(3(2(4(1(0(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)) = x_1
POL(1^1(x_1)) = x_1
POL(2(x_1)) = 1
POL(3(x_1)) = 0
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:
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1)))))))))
1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))
1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
----------------------------------------
(48)
Obligation:
Q DP problem:
The TRS P consists of the following rules:
1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1^1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1^1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
The TRS R consists of the following rules:
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1)))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(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(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1^1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(0(x_1)) = 1
POL(1(x_1)) = x_1
POL(1^1(x_1)) = x_1
POL(2(x_1)) = 1
POL(3(x_1)) = 0
POL(4(x_1)) = x_1
POL(5(x_1)) = x_1
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1)))))))))
1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))
1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
----------------------------------------
(50)
Obligation:
Q DP problem:
The TRS P consists of the following rules:
1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1^1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
The TRS R consists of the following rules:
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1)))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(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(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1^1(2(2(3(3(3(3(0(1(1(3(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)) = 1
POL(1^1(x_1)) = x_1
POL(2(x_1)) = x_1
POL(3(x_1)) = 0
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:
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
----------------------------------------
(52)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1)))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------
(53) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------
(54)
YES
----------------------------------------
(55)
Obligation:
Q DP problem:
The TRS P consists of the following rules:
0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0^1(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
0^1(0(1(2(x1)))) -> 0^1(3(4(2(x1))))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0^1(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
The TRS R consists of the following rules:
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1)))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------
(56) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
0^1(0(1(2(x1)))) -> 0^1(3(4(2(x1))))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(0(x_1)) = 1
POL(0^1(x_1)) = x_1
POL(1(x_1)) = 1 + x_1
POL(2(x_1)) = 0
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(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
----------------------------------------
(57)
Obligation:
Q DP problem:
The TRS P consists of the following rules:
0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0^1(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0^1(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
The TRS R consists of the following rules:
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1)))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------
(58) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0^1(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(0(x_1)) = x_1
POL(0^1(x_1)) = x_1
POL(1(x_1)) = 1
POL(2(x_1)) = 0
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(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
----------------------------------------
(59)
Obligation:
Q DP problem:
The TRS P consists of the following rules:
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0^1(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
The TRS R consists of the following rules:
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1)))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------
(60) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0^1(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(0(x_1)) = 1
POL(0^1(x_1)) = x_1
POL(1(x_1)) = 1
POL(2(x_1)) = 1
POL(3(x_1)) = 0
POL(4(x_1)) = x_1
POL(5(x_1)) = x_1
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
----------------------------------------
(61)
Obligation:
Q DP problem:
The TRS P consists of the following rules:
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
The TRS R consists of the following rules:
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1)))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------
(62) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( 0^1_1(x_1) ) = max{0, 2x_1 - 2}
POL( 3_1(x_1) ) = max{0, x_1 - 2}
POL( 0_1(x_1) ) = 2
POL( 4_1(x_1) ) = max{0, -2}
POL( 5_1(x_1) ) = max{0, 2x_1 - 2}
POL( 2_1(x_1) ) = 2x_1
POL( 1_1(x_1) ) = 1
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1)))))))))
1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))
1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
----------------------------------------
(63)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
0(0(1(2(x1)))) -> 0(3(4(2(x1))))
2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1))))))
4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1))))))
2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1))))))))
5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1))))))))
1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1)))))))))
3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1)))))))))
0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))
1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))
4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))
1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))
2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))
5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))
2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))
0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))
1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))))
2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))
5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))
4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))
0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))
1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))
2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))
4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------
(64) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------
(65)
YES