/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 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:

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

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

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

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

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

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

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

   c(a(a(b(c(a(x1)))))) -> a(a(b(c(c(a(a(b(c(x1)))))))))
   encArg(a(x_1)) -> a(encArg(x_1))
   encArg(b(x_1)) -> b(encArg(x_1))
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encode_c(x_1) -> c(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
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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 5. 
The certificate found is represented by the following graph.

"[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]
{(30,31,[c_1|0, encArg_1|0, encode_c_1|0, encode_a_1|0, encode_b_1|0]), (30,32,[a_1|1, b_1|1, c_1|1]), (30,33,[a_1|2]), (31,31,[a_1|0, b_1|0, cons_c_1|0]), (32,31,[encArg_1|1]), (32,32,[a_1|1, b_1|1, c_1|1]), (32,33,[a_1|2]), (33,34,[a_1|2]), (34,35,[b_1|2]), (35,36,[c_1|2]), (35,41,[a_1|3]), (36,37,[c_1|2]), (36,33,[a_1|2]), (36,41,[a_1|3]), (37,38,[a_1|2]), (38,39,[a_1|2]), (39,40,[b_1|2]), (40,32,[c_1|2]), (40,33,[c_1|2, a_1|2]), (40,41,[c_1|2, a_1|3]), (41,42,[a_1|3]), (42,43,[b_1|3]), (43,44,[c_1|3]), (43,49,[a_1|4]), (44,45,[c_1|3]), (44,41,[a_1|3]), (44,57,[a_1|4]), (45,46,[a_1|3]), (46,47,[a_1|3]), (47,48,[b_1|3]), (48,33,[c_1|3]), (48,41,[c_1|3]), (48,34,[c_1|3]), (48,42,[c_1|3]), (48,57,[c_1|3]), (49,50,[a_1|4]), (50,51,[b_1|4]), (51,52,[c_1|4]), (52,53,[c_1|4]), (52,57,[a_1|4]), (52,65,[a_1|5]), (53,54,[a_1|4]), (54,55,[a_1|4]), (55,56,[b_1|4]), (56,41,[c_1|4]), (56,57,[c_1|4]), (57,58,[a_1|4]), (58,59,[b_1|4]), (59,60,[c_1|4]), (60,61,[c_1|4]), (61,62,[a_1|4]), (62,63,[a_1|4]), (63,64,[b_1|4]), (64,42,[c_1|4]), (64,58,[c_1|4]), (65,66,[a_1|5]), (66,67,[b_1|5]), (67,68,[c_1|5]), (68,69,[c_1|5]), (69,70,[a_1|5]), (70,71,[a_1|5]), (71,72,[b_1|5]), (72,58,[c_1|5])}"
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(8)
BOUNDS(1, n^1)