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


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WORST_CASE(?, O(n^1))
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 180 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxTRS
(7) CpxTrsMatchBoundsProof [FINISHED, 4 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(1(2(3(4(x1))))) -> 0(2(1(3(4(x1)))))
   0(5(1(2(4(3(x1)))))) -> 0(5(2(1(4(3(x1))))))
   0(5(2(4(1(3(x1)))))) -> 0(1(5(2(4(3(x1))))))
   0(5(3(1(2(4(x1)))))) -> 0(1(5(3(2(4(x1))))))
   0(5(4(1(3(2(x1)))))) -> 0(5(4(3(1(2(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(1(x_1)) -> 1(encArg(x_1))
   encArg(2(x_1)) -> 2(encArg(x_1))
   encArg(3(x_1)) -> 3(encArg(x_1))
   encArg(4(x_1)) -> 4(encArg(x_1))
   encArg(5(x_1)) -> 5(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(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(1(2(3(4(x1))))) -> 0(2(1(3(4(x1)))))
   0(5(1(2(4(3(x1)))))) -> 0(5(2(1(4(3(x1))))))
   0(5(2(4(1(3(x1)))))) -> 0(1(5(2(4(3(x1))))))
   0(5(3(1(2(4(x1)))))) -> 0(1(5(3(2(4(x1))))))
   0(5(4(1(3(2(x1)))))) -> 0(5(4(3(1(2(x1))))))

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

   encArg(1(x_1)) -> 1(encArg(x_1))
   encArg(2(x_1)) -> 2(encArg(x_1))
   encArg(3(x_1)) -> 3(encArg(x_1))
   encArg(4(x_1)) -> 4(encArg(x_1))
   encArg(5(x_1)) -> 5(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(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(1(2(3(4(x1))))) -> 0(2(1(3(4(x1)))))
   0(5(1(2(4(3(x1)))))) -> 0(5(2(1(4(3(x1))))))
   0(5(2(4(1(3(x1)))))) -> 0(1(5(2(4(3(x1))))))
   0(5(3(1(2(4(x1)))))) -> 0(1(5(3(2(4(x1))))))
   0(5(4(1(3(2(x1)))))) -> 0(5(4(3(1(2(x1))))))

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

   encArg(1(x_1)) -> 1(encArg(x_1))
   encArg(2(x_1)) -> 2(encArg(x_1))
   encArg(3(x_1)) -> 3(encArg(x_1))
   encArg(4(x_1)) -> 4(encArg(x_1))
   encArg(5(x_1)) -> 5(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(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(1(2(3(4(x1))))) -> 0(2(1(3(4(x1)))))
   0(5(1(2(4(3(x1)))))) -> 0(5(2(1(4(3(x1))))))
   0(5(2(4(1(3(x1)))))) -> 0(1(5(2(4(3(x1))))))
   0(5(3(1(2(4(x1)))))) -> 0(1(5(3(2(4(x1))))))
   0(5(4(1(3(2(x1)))))) -> 0(5(4(3(1(2(x1))))))
   encArg(1(x_1)) -> 1(encArg(x_1))
   encArg(2(x_1)) -> 2(encArg(x_1))
   encArg(3(x_1)) -> 3(encArg(x_1))
   encArg(4(x_1)) -> 4(encArg(x_1))
   encArg(5(x_1)) -> 5(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(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 2. 
The certificate found is represented by the following graph.

"[17, 18, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75]
{(17,18,[0_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]), (17,27,[0_1|1]), (17,31,[0_1|1]), (17,36,[0_1|1]), (17,41,[0_1|1]), (17,46,[0_1|1]), (17,51,[1_1|1, 2_1|1, 3_1|1, 4_1|1, 5_1|1, 0_1|1]), (17,52,[0_1|2]), (17,56,[0_1|2]), (17,61,[0_1|2]), (17,66,[0_1|2]), (17,71,[0_1|2]), (18,18,[1_1|0, 2_1|0, 3_1|0, 4_1|0, 5_1|0, cons_0_1|0]), (27,28,[2_1|1]), (28,29,[1_1|1]), (29,30,[3_1|1]), (30,18,[4_1|1]), (31,32,[5_1|1]), (32,33,[2_1|1]), (33,34,[1_1|1]), (34,35,[4_1|1]), (35,18,[3_1|1]), (36,37,[1_1|1]), (37,38,[5_1|1]), (38,39,[2_1|1]), (39,40,[4_1|1]), (40,18,[3_1|1]), (41,42,[1_1|1]), (42,43,[5_1|1]), (43,44,[3_1|1]), (44,45,[2_1|1]), (45,18,[4_1|1]), (46,47,[5_1|1]), (47,48,[4_1|1]), (48,49,[3_1|1]), (49,50,[1_1|1]), (50,18,[2_1|1]), (51,18,[encArg_1|1]), (51,51,[1_1|1, 2_1|1, 3_1|1, 4_1|1, 5_1|1, 0_1|1]), (51,52,[0_1|2]), (51,56,[0_1|2]), (51,61,[0_1|2]), (51,66,[0_1|2]), (51,71,[0_1|2]), (52,53,[2_1|2]), (53,54,[1_1|2]), (54,55,[3_1|2]), (55,51,[4_1|2]), (56,57,[5_1|2]), (57,58,[2_1|2]), (58,59,[1_1|2]), (59,60,[4_1|2]), (60,51,[3_1|2]), (61,62,[1_1|2]), (62,63,[5_1|2]), (63,64,[2_1|2]), (64,65,[4_1|2]), (65,51,[3_1|2]), (66,67,[1_1|2]), (67,68,[5_1|2]), (68,69,[3_1|2]), (69,70,[2_1|2]), (70,51,[4_1|2]), (71,72,[5_1|2]), (72,73,[4_1|2]), (73,74,[3_1|2]), (74,75,[1_1|2]), (75,51,[2_1|2])}"
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(8)
BOUNDS(1, n^1)