/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 42 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(c(b(x1))) -> b(a(b(a(x1))))
   b(x1) -> c(a(c(x1)))
   a(a(x1)) -> a(b(c(a(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(c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_c(x_1) -> c(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(c(b(x1))) -> b(a(b(a(x1))))
   b(x1) -> c(a(c(x1)))
   a(a(x1)) -> a(b(c(a(x1))))

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

   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_c(x_1) -> c(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(c(b(x1))) -> b(a(b(a(x1))))
   b(x1) -> c(a(c(x1)))
   a(a(x1)) -> a(b(c(a(x1))))

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

   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_c(x_1) -> c(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(c(b(x1))) -> b(a(b(a(x1))))
   b(x1) -> c(a(c(x1)))
   a(a(x1)) -> a(b(c(a(x1))))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_c(x_1) -> c(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 6. 
The certificate found is represented by the following graph.

"[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, 80, 81, 82, 83, 84, 85, 86]
{(39,40,[a_1|0, b_1|0, encArg_1|0, encode_a_1|0, encode_c_1|0, encode_b_1|0]), (39,41,[c_1|1]), (39,43,[c_1|1, a_1|1, b_1|1]), (39,44,[c_1|2]), (39,46,[b_1|2]), (39,49,[a_1|2]), (39,52,[c_1|3]), (40,40,[c_1|0, cons_a_1|0, cons_b_1|0]), (41,42,[a_1|1]), (42,40,[c_1|1]), (43,40,[encArg_1|1]), (43,43,[c_1|1, a_1|1, b_1|1]), (43,46,[b_1|2]), (43,49,[a_1|2]), (43,44,[c_1|2]), (43,52,[c_1|3]), (44,45,[a_1|2]), (44,46,[b_1|2]), (44,54,[b_1|3]), (44,52,[c_1|3]), (44,64,[c_1|4]), (45,43,[c_1|2]), (46,47,[a_1|2]), (46,54,[b_1|3]), (46,64,[c_1|4]), (47,48,[b_1|2]), (47,57,[c_1|3]), (48,43,[a_1|2]), (48,46,[a_1|2, b_1|2]), (48,49,[a_1|2]), (48,59,[a_1|3]), (48,54,[b_1|3, a_1|2]), (48,52,[c_1|3]), (48,64,[c_1|4]), (48,68,[a_1|2]), (49,50,[b_1|2]), (49,62,[c_1|3]), (50,51,[c_1|2]), (51,43,[a_1|2]), (51,49,[a_1|2]), (51,46,[b_1|2]), (51,59,[a_1|3]), (51,54,[b_1|3]), (51,52,[c_1|3]), (51,64,[c_1|4]), (52,53,[a_1|3]), (52,68,[b_1|4]), (52,76,[c_1|5]), (53,46,[c_1|3]), (54,55,[a_1|3]), (55,56,[b_1|3]), (55,66,[c_1|4]), (56,46,[a_1|3]), (56,54,[a_1|3]), (56,59,[a_1|3]), (56,68,[a_1|3]), (56,73,[a_1|4]), (57,58,[a_1|3]), (57,54,[b_1|3]), (57,68,[b_1|4]), (57,64,[c_1|4]), (57,76,[c_1|5]), (58,48,[c_1|3]), (59,60,[b_1|3]), (59,71,[c_1|4]), (60,61,[c_1|3]), (61,49,[a_1|3]), (61,47,[a_1|3]), (61,55,[a_1|3]), (61,68,[b_1|4]), (61,76,[c_1|5]), (61,69,[a_1|3]), (62,63,[a_1|3]), (63,50,[c_1|3]), (64,65,[a_1|4]), (65,54,[c_1|4]), (66,67,[a_1|4]), (67,56,[c_1|4]), (68,69,[a_1|4]), (69,70,[b_1|4]), (69,78,[c_1|5]), (70,54,[a_1|4]), (70,68,[a_1|4]), (70,73,[a_1|4]), (70,80,[a_1|5]), (71,72,[a_1|4]), (72,60,[c_1|4]), (73,74,[b_1|4]), (73,83,[c_1|5]), (74,75,[c_1|4]), (75,55,[a_1|4]), (75,69,[a_1|4]), (76,77,[a_1|5]), (77,68,[c_1|5]), (78,79,[a_1|5]), (79,70,[c_1|5]), (80,81,[b_1|5]), (80,85,[c_1|6]), (81,82,[c_1|5]), (82,69,[a_1|5]), (83,84,[a_1|5]), (84,74,[c_1|5]), (85,86,[a_1|6]), (86,81,[c_1|6])}"
----------------------------------------

(8)
BOUNDS(1, n^1)