/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 78 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxTRS
(7) CpxTrsMatchBoundsProof [FINISHED, 0 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:

   a(a(b(a(a(b(a(x1))))))) -> a(b(a(a(b(a(a(a(b(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(b(x_1)) -> b(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(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:

   a(a(b(a(a(b(a(x1))))))) -> a(b(a(a(b(a(a(a(b(x1)))))))))

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

   encArg(b(x_1)) -> b(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(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:

   a(a(b(a(a(b(a(x1))))))) -> a(b(a(a(b(a(a(a(b(x1)))))))))

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

   encArg(b(x_1)) -> b(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(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:

   a(a(b(a(a(b(a(x1))))))) -> a(b(a(a(b(a(a(a(b(x1)))))))))
   encArg(b(x_1)) -> b(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(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 4. 
The certificate found is represented by the following graph.

"[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, 76, 77, 78, 79]
{(29,30,[a_1|0, encArg_1|0, encode_a_1|0, encode_b_1|0]), (29,31,[b_1|1, a_1|1]), (29,32,[a_1|2]), (30,30,[b_1|0, cons_a_1|0]), (31,30,[encArg_1|1]), (31,31,[b_1|1, a_1|1]), (31,32,[a_1|2]), (32,33,[b_1|2]), (33,34,[a_1|2]), (33,40,[a_1|3]), (34,35,[a_1|2]), (35,36,[b_1|2]), (36,37,[a_1|2]), (36,48,[a_1|3]), (37,38,[a_1|2]), (37,32,[a_1|2]), (37,56,[a_1|3]), (38,39,[a_1|2]), (39,31,[b_1|2]), (39,32,[b_1|2]), (39,34,[b_1|2]), (39,37,[b_1|2]), (39,40,[b_1|2]), (39,48,[b_1|2]), (40,41,[b_1|3]), (41,42,[a_1|3]), (42,43,[a_1|3]), (43,44,[b_1|3]), (44,45,[a_1|3]), (45,46,[a_1|3]), (46,47,[a_1|3]), (47,34,[b_1|3]), (47,40,[b_1|3]), (47,58,[b_1|3]), (48,49,[b_1|3]), (49,50,[a_1|3]), (49,40,[a_1|3]), (49,64,[a_1|4]), (50,51,[a_1|3]), (51,52,[b_1|3]), (52,53,[a_1|3]), (52,48,[a_1|3]), (52,72,[a_1|4]), (53,54,[a_1|3]), (53,32,[a_1|2]), (53,56,[a_1|3]), (54,55,[a_1|3]), (55,37,[b_1|3]), (55,48,[b_1|3]), (55,61,[b_1|3]), (56,57,[b_1|3]), (57,58,[a_1|3]), (58,59,[a_1|3]), (59,60,[b_1|3]), (60,61,[a_1|3]), (61,62,[a_1|3]), (62,63,[a_1|3]), (63,32,[b_1|3]), (63,35,[b_1|3]), (63,38,[b_1|3]), (63,56,[b_1|3]), (63,39,[b_1|3]), (64,65,[b_1|4]), (65,66,[a_1|4]), (66,67,[a_1|4]), (67,68,[b_1|4]), (68,69,[a_1|4]), (69,70,[a_1|4]), (70,71,[a_1|4]), (71,58,[b_1|4]), (72,73,[b_1|4]), (73,74,[a_1|4]), (73,64,[a_1|4]), (74,75,[a_1|4]), (75,76,[b_1|4]), (76,77,[a_1|4]), (76,72,[a_1|4]), (77,78,[a_1|4]), (77,56,[a_1|3]), (78,79,[a_1|4]), (79,61,[b_1|4])}"
----------------------------------------

(8)
BOUNDS(1, n^1)