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


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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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 35 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxTRS
(7) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
(8) BOUNDS(1, n^1)


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(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:

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

S is empty.
Rewrite Strategy: FULL
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(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(a(x_1)) -> a(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))


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(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:

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

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

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

Rewrite Strategy: FULL
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(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
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(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:

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

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

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

Rewrite Strategy: FULL
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(5) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(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:

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

S is empty.
Rewrite Strategy: FULL
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(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.

"[27, 28, 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]
{(27,28,[b_1|0, encArg_1|0, encode_b_1|0, encode_a_1|0, encode_c_1|0]), (27,34,[a_1|1, c_1|1, b_1|1]), (27,35,[a_1|2]), (28,28,[a_1|0, c_1|0, cons_b_1|0]), (34,28,[encArg_1|1]), (34,34,[a_1|1, c_1|1, b_1|1]), (34,35,[a_1|2]), (35,36,[b_1|2]), (36,37,[a_1|2]), (37,38,[c_1|2]), (38,39,[b_1|2]), (38,44,[a_1|3]), (39,40,[b_1|2]), (39,35,[a_1|2]), (39,53,[a_1|3]), (40,41,[a_1|2]), (41,42,[b_1|2]), (42,43,[a_1|2]), (43,34,[c_1|2]), (43,35,[c_1|2]), (43,53,[c_1|2]), (44,45,[b_1|3]), (45,46,[a_1|3]), (46,47,[c_1|3]), (47,48,[b_1|3]), (48,49,[b_1|3]), (48,53,[a_1|3]), (48,62,[a_1|4]), (49,50,[a_1|3]), (50,51,[b_1|3]), (51,52,[a_1|3]), (52,35,[c_1|3]), (52,53,[c_1|3]), (53,54,[b_1|3]), (54,55,[a_1|3]), (55,56,[c_1|3]), (56,57,[b_1|3]), (57,58,[b_1|3]), (58,59,[a_1|3]), (59,60,[b_1|3]), (60,61,[a_1|3]), (61,37,[c_1|3]), (61,55,[c_1|3]), (62,63,[b_1|4]), (63,64,[a_1|4]), (64,65,[c_1|4]), (65,66,[b_1|4]), (66,67,[b_1|4]), (67,68,[a_1|4]), (68,69,[b_1|4]), (69,70,[a_1|4]), (70,55,[c_1|4])}"
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(8)
BOUNDS(1, n^1)