/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


WORST_CASE(?, O(n^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 44 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxTRS
(7) CpxTrsMatchBoundsProof [FINISHED, 617 ms]
(8) BOUNDS(1, n^1)


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

(0)
Obligation:
The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

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

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(2(x_1)) -> 2(encArg(x_1))
   encArg(4(x_1)) -> 4(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(encArg(x_1))
   encArg(cons_5(x_1)) -> 5(encArg(x_1))
   encArg(cons_1(x_1)) -> 1(encArg(x_1))
   encArg(cons_3(x_1)) -> 3(encArg(x_1))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_1(x_1) -> 1(encArg(x_1))
   encode_2(x_1) -> 2(encArg(x_1))
   encode_3(x_1) -> 3(encArg(x_1))
   encode_4(x_1) -> 4(encArg(x_1))
   encode_5(x_1) -> 5(encArg(x_1))


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

(2)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

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

The (relative) TRS S consists of the following rules:

   encArg(2(x_1)) -> 2(encArg(x_1))
   encArg(4(x_1)) -> 4(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(encArg(x_1))
   encArg(cons_5(x_1)) -> 5(encArg(x_1))
   encArg(cons_1(x_1)) -> 1(encArg(x_1))
   encArg(cons_3(x_1)) -> 3(encArg(x_1))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_1(x_1) -> 1(encArg(x_1))
   encode_2(x_1) -> 2(encArg(x_1))
   encode_3(x_1) -> 3(encArg(x_1))
   encode_4(x_1) -> 4(encArg(x_1))
   encode_5(x_1) -> 5(encArg(x_1))

Rewrite Strategy: FULL
----------------------------------------

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

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

The (relative) TRS S consists of the following rules:

   encArg(2(x_1)) -> 2(encArg(x_1))
   encArg(4(x_1)) -> 4(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(encArg(x_1))
   encArg(cons_5(x_1)) -> 5(encArg(x_1))
   encArg(cons_1(x_1)) -> 1(encArg(x_1))
   encArg(cons_3(x_1)) -> 3(encArg(x_1))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_1(x_1) -> 1(encArg(x_1))
   encode_2(x_1) -> 2(encArg(x_1))
   encode_3(x_1) -> 3(encArg(x_1))
   encode_4(x_1) -> 4(encArg(x_1))
   encode_5(x_1) -> 5(encArg(x_1))

Rewrite Strategy: FULL
----------------------------------------

(5) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(6)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   0(0(1(2(x1)))) -> 3(1(0(1(0(2(x1))))))
   0(3(4(4(x1)))) -> 0(4(4(3(1(x1)))))
   5(1(3(4(x1)))) -> 3(1(1(5(4(x1)))))
   0(0(4(2(5(x1))))) -> 0(4(1(0(2(5(x1))))))
   0(1(4(3(2(x1))))) -> 1(4(3(1(0(2(x1))))))
   0(3(4(5(2(x1))))) -> 4(3(5(1(0(2(x1))))))
   0(3(5(1(4(x1))))) -> 0(3(1(5(4(1(x1))))))
   1(2(3(0(5(x1))))) -> 1(0(2(5(3(1(x1))))))
   1(4(4(3(0(x1))))) -> 3(1(1(0(4(4(x1))))))
   1(4(5(1(5(x1))))) -> 1(5(4(1(1(5(x1))))))
   5(0(1(2(0(x1))))) -> 2(1(0(5(1(0(x1))))))
   5(3(0(3(4(x1))))) -> 5(3(1(0(4(3(x1))))))
   0(1(2(0(3(0(x1)))))) -> 0(2(0(4(3(1(0(x1)))))))
   0(3(3(4(0(0(x1)))))) -> 3(0(3(1(0(4(0(x1)))))))
   1(2(3(3(5(2(x1)))))) -> 5(3(1(1(2(3(2(x1)))))))
   1(2(4(4(5(5(x1)))))) -> 2(1(5(4(5(4(4(x1)))))))
   1(3(1(4(2(5(x1)))))) -> 3(2(1(5(4(1(1(x1)))))))
   1(4(0(3(4(2(x1)))))) -> 0(4(4(3(2(1(2(x1)))))))
   3(0(1(3(4(4(x1)))))) -> 4(3(1(1(0(4(3(x1)))))))
   3(5(0(0(1(4(x1)))))) -> 4(3(1(0(1(0(5(x1)))))))
   3(5(2(4(0(5(x1)))))) -> 4(3(5(1(0(2(5(x1)))))))
   5(2(1(4(3(4(x1)))))) -> 2(1(1(5(4(4(3(x1)))))))
   5(3(1(4(0(2(x1)))))) -> 4(3(1(1(0(2(5(x1)))))))
   5(3(4(0(0(0(x1)))))) -> 0(3(5(0(4(0(4(x1)))))))
   0(0(3(4(5(1(4(x1))))))) -> 5(4(0(4(3(1(0(3(x1))))))))
   0(0(5(2(2(3(4(x1))))))) -> 3(1(0(2(2(5(0(4(3(x1)))))))))
   0(1(0(3(3(4(2(x1))))))) -> 0(4(3(3(1(0(2(1(1(2(x1))))))))))
   0(1(2(5(5(0(0(x1))))))) -> 1(0(2(0(5(5(1(0(x1))))))))
   0(1(3(0(4(1(2(x1))))))) -> 0(3(1(0(4(1(1(2(x1))))))))
   0(1(4(5(0(3(4(x1))))))) -> 0(5(1(0(4(3(4(1(x1))))))))
   0(1(4(5(1(2(4(x1))))))) -> 5(1(3(2(1(1(0(4(4(x1)))))))))
   0(1(5(1(3(3(4(x1))))))) -> 0(1(3(1(1(5(4(3(x1))))))))
   0(5(2(1(4(3(3(x1))))))) -> 5(4(1(0(2(3(3(1(x1))))))))
   0(5(4(0(1(3(0(x1))))))) -> 3(1(5(0(4(0(1(1(0(x1)))))))))
   1(2(3(3(4(2(2(x1))))))) -> 1(2(2(4(1(3(2(3(x1))))))))
   1(2(3(5(4(1(4(x1))))))) -> 3(2(3(1(1(1(5(4(4(x1)))))))))
   1(3(0(2(2(3(3(x1))))))) -> 3(1(3(1(2(3(1(0(2(x1)))))))))
   1(3(1(2(2(3(4(x1))))))) -> 1(3(2(1(3(2(1(4(x1))))))))
   1(3(3(4(0(1(4(x1))))))) -> 5(3(1(3(1(1(0(4(4(x1)))))))))
   1(4(4(3(0(3(2(x1))))))) -> 0(4(4(1(3(3(1(2(x1))))))))
   1(5(0(5(4(2(2(x1))))))) -> 5(5(4(2(1(1(0(2(x1))))))))
   3(2(2(3(0(2(3(x1))))))) -> 3(3(2(2(3(1(0(2(x1))))))))
   3(5(1(2(4(5(0(x1))))))) -> 5(2(3(1(1(5(4(0(x1))))))))
   5(0(3(3(5(4(3(x1))))))) -> 1(5(3(0(4(3(5(3(x1))))))))
   5(1(4(0(2(4(5(x1))))))) -> 1(0(2(5(4(4(1(5(x1))))))))
   5(2(1(0(0(3(0(x1))))))) -> 5(0(0(3(1(1(0(2(x1))))))))
   5(2(4(0(5(3(4(x1))))))) -> 1(5(4(5(4(3(0(2(x1))))))))
   0(0(1(4(2(3(5(4(x1)))))))) -> 3(2(0(4(4(1(0(1(5(x1)))))))))
   0(0(3(1(1(4(2(4(x1)))))))) -> 0(4(4(1(1(1(3(0(2(x1)))))))))
   0(1(1(2(4(2(0(3(x1)))))))) -> 1(4(0(3(1(1(2(1(0(2(x1))))))))))
   0(2(3(4(2(3(0(5(x1)))))))) -> 3(2(1(0(4(5(3(1(0(2(x1))))))))))
   1(3(0(1(3(3(4(5(x1)))))))) -> 5(3(3(2(3(1(1(0(4(x1)))))))))
   1(3(3(2(5(1(5(2(x1)))))))) -> 3(1(1(1(5(2(3(2(5(x1)))))))))
   1(3(3(5(0(0(2(3(x1)))))))) -> 1(5(3(3(0(1(3(1(0(2(x1))))))))))
   1(3(4(0(5(2(5(2(x1)))))))) -> 5(3(0(2(5(4(1(1(2(x1)))))))))
   1(3(4(5(0(0(5(0(x1)))))))) -> 0(0(4(5(4(0(3(1(5(x1)))))))))
   1(4(2(3(3(0(5(2(x1)))))))) -> 3(1(0(2(2(4(4(3(5(x1)))))))))
   1(4(3(4(0(3(4(0(x1)))))))) -> 1(0(4(4(0(3(1(5(4(3(x1))))))))))
   1(4(3(5(5(1(4(5(x1)))))))) -> 5(1(5(4(1(5(4(3(5(x1)))))))))
   5(0(1(4(5(4(2(2(x1)))))))) -> 2(1(5(4(4(1(0(2(5(1(x1))))))))))
   5(1(3(0(1(3(4(2(x1)))))))) -> 3(1(0(2(2(1(5(4(3(x1)))))))))
   5(1(3(0(1(4(4(5(x1)))))))) -> 3(0(1(1(1(5(4(5(4(x1)))))))))
   5(1(4(4(0(2(1(2(x1)))))))) -> 2(5(4(4(1(1(1(0(2(x1)))))))))
   5(5(5(0(3(4(1(2(x1)))))))) -> 5(1(0(2(4(5(5(3(1(x1)))))))))
   0(0(1(0(5(4(2(4(5(x1))))))))) -> 0(4(4(2(1(0(1(0(5(5(x1))))))))))
   1(2(5(2(2(3(3(5(5(x1))))))))) -> 2(2(5(3(5(1(3(2(1(5(x1))))))))))
   1(3(0(1(4(2(2(5(5(x1))))))))) -> 3(2(1(5(2(1(5(4(1(0(x1))))))))))
   1(3(0(1(4(2(4(2(5(x1))))))))) -> 1(0(4(4(3(2(5(5(1(2(x1))))))))))
   1(3(3(4(2(5(0(1(2(x1))))))))) -> 1(3(2(1(1(5(2(0(4(1(3(x1)))))))))))
   3(0(2(5(2(4(1(3(4(x1))))))))) -> 3(2(2(0(3(1(1(5(4(4(x1))))))))))
   3(0(5(3(4(1(2(0(5(x1))))))))) -> 4(0(5(1(3(1(1(3(2(0(5(x1)))))))))))
   3(2(0(1(1(0(5(4(2(x1))))))))) -> 5(4(3(2(1(1(0(2(0(2(x1))))))))))
   5(1(4(4(0(5(2(0(3(x1))))))))) -> 1(5(5(0(2(1(0(4(4(3(x1))))))))))
   0(0(2(4(5(1(4(0(1(4(x1)))))))))) -> 1(5(4(4(4(0(0(2(0(3(1(x1)))))))))))
   0(0(2(4(5(2(4(3(2(4(x1)))))))))) -> 3(2(0(4(5(4(4(2(0(2(1(x1)))))))))))
   0(0(5(5(4(1(4(4(5(2(x1)))))))))) -> 1(5(4(4(2(0(5(4(1(0(5(x1)))))))))))
   0(1(1(3(4(2(1(5(1(2(x1)))))))))) -> 1(2(5(3(0(2(4(1(1(1(1(x1)))))))))))
   0(1(4(3(0(5(5(1(5(3(x1)))))))))) -> 5(3(5(5(4(1(1(0(3(0(1(x1)))))))))))
   0(1(5(2(0(3(3(4(5(2(x1)))))))))) -> 3(2(0(5(1(1(0(3(4(5(2(x1)))))))))))
   0(5(1(0(0(1(2(2(0(4(x1)))))))))) -> 0(1(0(2(2(0(1(0(1(5(4(x1)))))))))))
   0(5(2(0(1(5(3(2(4(0(x1)))))))))) -> 0(2(4(5(5(3(1(0(2(1(0(x1)))))))))))
   1(2(0(0(1(1(3(1(0(0(x1)))))))))) -> 0(1(1(3(1(0(2(0(1(1(0(x1)))))))))))
   1(2(4(5(0(0(5(0(1(2(x1)))))))))) -> 0(2(1(1(5(4(0(2(5(0(3(x1)))))))))))
   1(2(4(5(3(0(0(0(4(4(x1)))))))))) -> 0(2(1(3(1(5(0(4(0(4(4(x1)))))))))))
   1(3(4(5(1(4(4(4(2(5(x1)))))))))) -> 5(2(4(1(5(4(4(4(1(3(4(x1)))))))))))
   1(4(0(0(4(2(2(0(5(5(x1)))))))))) -> 5(0(0(4(4(1(1(0(2(5(2(x1)))))))))))
   1(4(5(5(2(5(0(0(4(2(x1)))))))))) -> 4(4(1(0(5(2(5(1(0(2(5(x1)))))))))))
   1(5(1(4(2(3(0(0(2(0(x1)))))))))) -> 1(1(4(0(3(0(2(5(1(0(2(x1)))))))))))
   5(1(3(0(4(2(3(1(5(0(x1)))))))))) -> 5(3(2(1(1(1(5(4(0(3(0(x1)))))))))))
   5(2(5(3(2(2(5(1(3(0(x1)))))))))) -> 3(5(5(2(1(3(2(1(0(2(5(x1)))))))))))
   encArg(2(x_1)) -> 2(encArg(x_1))
   encArg(4(x_1)) -> 4(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(encArg(x_1))
   encArg(cons_5(x_1)) -> 5(encArg(x_1))
   encArg(cons_1(x_1)) -> 1(encArg(x_1))
   encArg(cons_3(x_1)) -> 3(encArg(x_1))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_1(x_1) -> 1(encArg(x_1))
   encode_2(x_1) -> 2(encArg(x_1))
   encode_3(x_1) -> 3(encArg(x_1))
   encode_4(x_1) -> 4(encArg(x_1))
   encode_5(x_1) -> 5(encArg(x_1))

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(7) CpxTrsMatchBoundsProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 3. 
The certificate found is represented by the following graph.

"[71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 464, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495, 496, 497, 498, 499, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 510, 511, 512, 513, 514, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 528, 529, 530, 531, 532, 533, 534, 535, 536, 537, 538, 539, 540, 541, 542, 543, 544, 545, 546, 547, 548, 549, 550, 551, 552, 553, 554, 555, 556, 557, 558, 559, 560, 561, 562, 563, 564, 565, 566, 567, 568, 569, 570, 571, 572, 573, 574, 575, 576, 577, 578, 579, 580, 581, 582, 583, 584, 585, 586, 587, 588, 589, 590, 591, 592, 593, 594, 595, 596, 597, 598, 599, 600, 601, 602, 603, 604, 605, 606, 607, 608, 609, 610, 611, 612, 613, 614, 615, 616, 617, 618, 619, 620, 621, 622, 623, 624, 625, 626, 627, 628, 629, 630, 631, 632, 633, 634, 635, 636, 637, 638, 639, 640, 641, 642, 643, 644, 645, 646, 647, 648, 649, 650, 651, 652, 653, 654, 655, 656, 657, 658, 659, 660, 661, 662, 663, 664, 665, 666, 667, 668, 669, 670, 671, 672, 673, 674, 675, 676, 677, 678, 679, 680, 681, 682, 683, 684, 685, 686, 687, 688, 689, 690, 691, 692, 693, 694, 695, 696, 697, 698, 699, 700, 701, 702, 703, 704, 705, 706, 707, 708, 709, 710, 711, 712, 713, 714, 715, 716, 717, 718, 719, 720, 721, 722, 723, 724, 725, 726, 727, 728, 729, 730, 731, 732, 733, 734, 735, 736, 737, 738, 739, 740, 741, 742, 743, 744, 745, 746, 747, 748, 749, 750, 751, 752, 753, 754, 755, 756, 757, 758, 759, 760, 761, 762, 763, 764, 765, 766, 767, 768, 769, 770, 771, 772, 773, 774, 775, 776, 777, 778, 779, 780]
{(71,72,[0_1|0, 5_1|0, 1_1|0, 3_1|0, encArg_1|0, encode_0_1|0, encode_1_1|0, encode_2_1|0, encode_3_1|0, encode_4_1|0, encode_5_1|0]), (71,73,[2_1|1, 4_1|1, 0_1|1, 5_1|1, 1_1|1, 3_1|1]), (71,74,[3_1|2]), (71,79,[3_1|2]), (71,87,[0_1|2]), (71,96,[0_1|2]), (71,101,[5_1|2]), (71,108,[0_1|2]), (71,116,[3_1|2]), (71,124,[1_1|2]), (71,134,[1_1|2]), (71,144,[3_1|2]), (71,154,[0_1|2]), (71,158,[4_1|2]), (71,163,[0_1|2]), (71,168,[3_1|2]), (71,174,[1_1|2]), (71,179,[5_1|2]), (71,189,[0_1|2]), (71,196,[5_1|2]), (71,204,[0_1|2]), (71,210,[1_1|2]), (71,217,[0_1|2]), (71,226,[0_1|2]), (71,233,[0_1|2]), (71,240,[3_1|2]), (71,250,[1_1|2]), (71,259,[1_1|2]), (71,269,[5_1|2]), (71,276,[0_1|2]), (71,286,[3_1|2]), (71,294,[0_1|2]), (71,304,[3_1|2]), (71,313,[3_1|2]), (71,317,[3_1|2]), (71,325,[3_1|2]), (71,333,[5_1|2]), (71,343,[1_1|2]), (71,350,[2_1|2]), (71,358,[1_1|2]), (71,367,[2_1|2]), (71,372,[2_1|2]), (71,381,[1_1|2]), (71,388,[5_1|2]), (71,393,[4_1|2]), (71,399,[0_1|2]), (71,405,[2_1|2]), (71,411,[5_1|2]), (71,418,[1_1|2]), (71,425,[3_1|2]), (71,435,[5_1|2]), (71,443,[1_1|2]), (71,448,[5_1|2]), (71,454,[1_1|2]), (71,461,[3_1|2]), (71,469,[2_1|2]), (71,475,[0_1|2]), (71,485,[0_1|2]), (71,495,[2_1|2]), (71,504,[0_1|2]), (71,514,[3_1|2]), (71,519,[0_1|2]), (71,526,[1_1|2]), (71,531,[4_1|2]), (71,541,[0_1|2]), (71,547,[5_1|2]), (71,557,[3_1|2]), (71,565,[1_1|2]), (71,574,[5_1|2]), (71,582,[3_1|2]), (71,588,[1_1|2]), (71,595,[3_1|2]), (71,603,[5_1|2]), (71,611,[3_1|2]), (71,620,[1_1|2]), (71,629,[5_1|2]), (71,637,[1_1|2]), (71,647,[3_1|2]), (71,655,[1_1|2]), (71,664,[5_1|2]), (71,672,[0_1|2]), (71,680,[5_1|2]), (71,690,[5_1|2]), (71,697,[1_1|2]), (71,707,[4_1|2]), (71,713,[3_1|2]), (71,722,[4_1|2]), (71,732,[4_1|2]), (71,738,[4_1|2]), (71,744,[5_1|2]), (71,751,[3_1|2]), (71,758,[5_1|2]), (72,72,[2_1|0, 4_1|0, cons_0_1|0, cons_5_1|0, cons_1_1|0, cons_3_1|0]), (73,72,[encArg_1|1]), (73,73,[2_1|1, 4_1|1, 0_1|1, 5_1|1, 1_1|1, 3_1|1]), (73,74,[3_1|2]), (73,79,[3_1|2]), (73,87,[0_1|2]), (73,96,[0_1|2]), (73,101,[5_1|2]), (73,108,[0_1|2]), (73,116,[3_1|2]), (73,124,[1_1|2]), (73,134,[1_1|2]), (73,144,[3_1|2]), (73,154,[0_1|2]), (73,158,[4_1|2]), (73,163,[0_1|2]), (73,168,[3_1|2]), (73,174,[1_1|2]), (73,179,[5_1|2]), (73,189,[0_1|2]), (73,196,[5_1|2]), (73,204,[0_1|2]), (73,210,[1_1|2]), (73,217,[0_1|2]), (73,226,[0_1|2]), (73,233,[0_1|2]), (73,240,[3_1|2]), (73,250,[1_1|2]), (73,259,[1_1|2]), (73,269,[5_1|2]), (73,276,[0_1|2]), (73,286,[3_1|2]), (73,294,[0_1|2]), (73,304,[3_1|2]), (73,313,[3_1|2]), (73,317,[3_1|2]), (73,325,[3_1|2]), (73,333,[5_1|2]), (73,343,[1_1|2]), (73,350,[2_1|2]), (73,358,[1_1|2]), (73,367,[2_1|2]), (73,372,[2_1|2]), (73,381,[1_1|2]), (73,388,[5_1|2]), (73,393,[4_1|2]), (73,399,[0_1|2]), (73,405,[2_1|2]), (73,411,[5_1|2]), (73,418,[1_1|2]), (73,425,[3_1|2]), (73,435,[5_1|2]), (73,443,[1_1|2]), (73,448,[5_1|2]), (73,454,[1_1|2]), (73,461,[3_1|2]), (73,469,[2_1|2]), (73,475,[0_1|2]), (73,485,[0_1|2]), (73,495,[2_1|2]), (73,504,[0_1|2]), (73,514,[3_1|2]), (73,519,[0_1|2]), (73,526,[1_1|2]), (73,531,[4_1|2]), (73,541,[0_1|2]), (73,547,[5_1|2]), (73,557,[3_1|2]), (73,565,[1_1|2]), (73,574,[5_1|2]), (73,582,[3_1|2]), (73,588,[1_1|2]), (73,595,[3_1|2]), (73,603,[5_1|2]), (73,611,[3_1|2]), (73,620,[1_1|2]), (73,629,[5_1|2]), (73,637,[1_1|2]), (73,647,[3_1|2]), (73,655,[1_1|2]), (73,664,[5_1|2]), (73,672,[0_1|2]), (73,680,[5_1|2]), (73,690,[5_1|2]), (73,697,[1_1|2]), (73,707,[4_1|2]), (73,713,[3_1|2]), (73,722,[4_1|2]), (73,732,[4_1|2]), (73,738,[4_1|2]), (73,744,[5_1|2]), (73,751,[3_1|2]), (73,758,[5_1|2]), (74,75,[1_1|2]), (75,76,[0_1|2]), (76,77,[1_1|2]), (77,78,[0_1|2]), (77,304,[3_1|2]), (78,73,[2_1|2]), (78,350,[2_1|2]), (78,367,[2_1|2]), (78,372,[2_1|2]), (78,405,[2_1|2]), (78,469,[2_1|2]), (78,495,[2_1|2]), (78,260,[2_1|2]), (78,455,[2_1|2]), (79,80,[2_1|2]), (80,81,[0_1|2]), (81,82,[4_1|2]), (82,83,[4_1|2]), (83,84,[1_1|2]), (84,85,[0_1|2]), (84,233,[0_1|2]), (84,240,[3_1|2]), (85,86,[1_1|2]), (85,690,[5_1|2]), (85,697,[1_1|2]), (86,73,[5_1|2]), (86,158,[5_1|2]), (86,393,[5_1|2, 4_1|2]), (86,531,[5_1|2]), (86,707,[5_1|2]), (86,722,[5_1|2]), (86,732,[5_1|2]), (86,738,[5_1|2]), (86,102,[5_1|2]), (86,270,[5_1|2]), (86,759,[5_1|2]), (86,313,[3_1|2]), (86,317,[3_1|2]), (86,325,[3_1|2]), (86,333,[5_1|2]), (86,343,[1_1|2]), (86,350,[2_1|2]), (86,358,[1_1|2]), (86,367,[2_1|2]), (86,372,[2_1|2]), (86,381,[1_1|2]), (86,388,[5_1|2]), (86,399,[0_1|2]), (86,405,[2_1|2]), (86,411,[5_1|2]), (86,418,[1_1|2]), (86,425,[3_1|2]), (86,435,[5_1|2]), (87,88,[4_1|2]), (88,89,[4_1|2]), (89,90,[2_1|2]), (90,91,[1_1|2]), (91,92,[0_1|2]), (92,93,[1_1|2]), (93,94,[0_1|2]), (94,95,[5_1|2]), (94,435,[5_1|2]), (95,73,[5_1|2]), (95,101,[5_1|2]), (95,179,[5_1|2]), (95,196,[5_1|2]), (95,269,[5_1|2]), (95,333,[5_1|2]), (95,388,[5_1|2]), (95,411,[5_1|2]), (95,435,[5_1|2]), (95,448,[5_1|2]), (95,547,[5_1|2]), (95,574,[5_1|2]), (95,603,[5_1|2]), (95,629,[5_1|2]), (95,664,[5_1|2]), (95,680,[5_1|2]), 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(737,343,[1_1|2]), (737,350,[2_1|2]), (737,358,[1_1|2]), (737,367,[2_1|2]), (737,372,[2_1|2]), (737,381,[1_1|2]), (737,388,[5_1|2]), (737,399,[0_1|2]), (737,405,[2_1|2]), (737,411,[5_1|2]), (737,418,[1_1|2]), (737,425,[3_1|2]), (737,435,[5_1|2]), (738,739,[3_1|2]), (739,740,[5_1|2]), (740,741,[1_1|2]), (741,742,[0_1|2]), (742,743,[2_1|2]), (743,73,[5_1|2]), (743,101,[5_1|2]), (743,179,[5_1|2]), (743,196,[5_1|2]), (743,269,[5_1|2]), (743,333,[5_1|2]), (743,388,[5_1|2]), (743,411,[5_1|2]), (743,435,[5_1|2]), (743,448,[5_1|2]), (743,547,[5_1|2]), (743,574,[5_1|2]), (743,603,[5_1|2]), (743,629,[5_1|2]), (743,664,[5_1|2]), (743,680,[5_1|2]), (743,690,[5_1|2]), (743,744,[5_1|2]), (743,758,[5_1|2]), (743,190,[5_1|2]), (743,724,[5_1|2]), (743,313,[3_1|2]), (743,317,[3_1|2]), (743,325,[3_1|2]), (743,343,[1_1|2]), (743,350,[2_1|2]), (743,358,[1_1|2]), (743,367,[2_1|2]), (743,372,[2_1|2]), (743,381,[1_1|2]), (743,393,[4_1|2]), (743,399,[0_1|2]), (743,405,[2_1|2]), (743,418,[1_1|2]), (743,425,[3_1|2]), (744,745,[2_1|2]), (745,746,[3_1|2]), (746,747,[1_1|2]), (747,748,[1_1|2]), (748,749,[5_1|2]), (749,750,[4_1|2]), (750,73,[0_1|2]), (750,87,[0_1|2]), (750,96,[0_1|2]), (750,108,[0_1|2]), (750,154,[0_1|2]), (750,163,[0_1|2]), (750,189,[0_1|2]), (750,204,[0_1|2]), (750,217,[0_1|2]), (750,226,[0_1|2]), (750,233,[0_1|2]), (750,276,[0_1|2]), (750,294,[0_1|2]), (750,399,[0_1|2]), (750,475,[0_1|2]), (750,485,[0_1|2]), (750,504,[0_1|2]), (750,519,[0_1|2]), (750,541,[0_1|2]), (750,672,[0_1|2]), (750,412,[0_1|2]), (750,548,[0_1|2]), (750,74,[3_1|2]), (750,79,[3_1|2]), (750,101,[5_1|2]), (750,116,[3_1|2]), (750,124,[1_1|2]), (750,134,[1_1|2]), (750,144,[3_1|2]), (750,158,[4_1|2]), (750,168,[3_1|2]), (750,174,[1_1|2]), (750,179,[5_1|2]), (750,196,[5_1|2]), (750,210,[1_1|2]), (750,240,[3_1|2]), (750,250,[1_1|2]), (750,259,[1_1|2]), (750,269,[5_1|2]), (750,286,[3_1|2]), (750,304,[3_1|2]), (751,752,[3_1|2]), (752,753,[2_1|2]), (753,754,[2_1|2]), (754,755,[3_1|2]), (755,756,[1_1|2]), (756,757,[0_1|2]), (756,304,[3_1|2]), (757,73,[2_1|2]), (757,74,[2_1|2]), (757,79,[2_1|2]), (757,116,[2_1|2]), (757,144,[2_1|2]), (757,168,[2_1|2]), (757,240,[2_1|2]), (757,286,[2_1|2]), (757,304,[2_1|2]), (757,313,[2_1|2]), (757,317,[2_1|2]), (757,325,[2_1|2]), (757,425,[2_1|2]), (757,461,[2_1|2]), (757,514,[2_1|2]), (757,557,[2_1|2]), (757,582,[2_1|2]), (757,595,[2_1|2]), (757,611,[2_1|2]), (757,647,[2_1|2]), (757,713,[2_1|2]), (757,751,[2_1|2]), (758,759,[4_1|2]), (759,760,[3_1|2]), (760,761,[2_1|2]), (761,762,[1_1|2]), (762,763,[1_1|2]), (763,764,[0_1|2]), (764,765,[2_1|2]), (765,766,[0_1|2]), (765,304,[3_1|2]), (766,73,[2_1|2]), (766,350,[2_1|2]), (766,367,[2_1|2]), (766,372,[2_1|2]), (766,405,[2_1|2]), (766,469,[2_1|2]), (766,495,[2_1|2]), (767,768,[4_1|3]), (768,769,[4_1|3]), (769,770,[3_1|3]), (770,532,[1_1|3]), (771,772,[3_1|3]), (772,773,[5_1|3]), (773,774,[1_1|3]), (774,775,[0_1|3]), (774,304,[3_1|2]), (775,73,[2_1|3]), (775,350,[2_1|3]), (775,367,[2_1|3]), (775,372,[2_1|3]), (775,405,[2_1|3]), (775,469,[2_1|3]), (775,495,[2_1|3]), (775,681,[2_1|3]), (775,745,[2_1|3]), (776,777,[5_1|3]), (777,778,[4_1|3]), (778,779,[1_1|3]), (779,780,[1_1|3]), (780,576,[5_1|3])}"
----------------------------------------

(8)
BOUNDS(1, n^1)