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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 71 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.

"[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]
{(32,33,[c_1|0, encArg_1|0, encode_c_1|0, encode_a_1|0, encode_b_1|0]), (32,34,[a_1|1, b_1|1, c_1|1]), (32,35,[a_1|2]), (33,33,[a_1|0, b_1|0, cons_c_1|0]), (34,33,[encArg_1|1]), (34,34,[a_1|1, b_1|1, c_1|1]), (34,35,[a_1|2]), (35,36,[a_1|2]), (36,37,[b_1|2]), (37,38,[c_1|2]), (37,43,[a_1|3]), (38,39,[c_1|2]), (38,35,[a_1|2]), (38,43,[a_1|3]), (39,40,[a_1|2]), (40,41,[a_1|2]), (41,42,[b_1|2]), (42,34,[c_1|2]), (42,35,[c_1|2, a_1|2]), (42,43,[c_1|2, a_1|3]), (43,44,[a_1|3]), (44,45,[b_1|3]), (45,46,[c_1|3]), (45,51,[a_1|4]), (46,47,[c_1|3]), (46,43,[a_1|3]), (46,59,[a_1|4]), (47,48,[a_1|3]), (48,49,[a_1|3]), (49,50,[b_1|3]), (50,35,[c_1|3]), (50,43,[c_1|3]), (50,36,[c_1|3]), (50,44,[c_1|3]), (50,59,[c_1|3]), (51,52,[a_1|4]), (52,53,[b_1|4]), (53,54,[c_1|4]), (54,55,[c_1|4]), (54,59,[a_1|4]), (54,67,[a_1|5]), (55,56,[a_1|4]), (56,57,[a_1|4]), (57,58,[b_1|4]), (58,43,[c_1|4]), (58,59,[c_1|4]), (59,60,[a_1|4]), (60,61,[b_1|4]), (61,62,[c_1|4]), (62,63,[c_1|4]), (63,64,[a_1|4]), (64,65,[a_1|4]), (65,66,[b_1|4]), (66,44,[c_1|4]), (66,60,[c_1|4]), (67,68,[a_1|5]), (68,69,[b_1|5]), (69,70,[c_1|5]), (70,71,[c_1|5]), (71,72,[a_1|5]), (72,73,[a_1|5]), (73,74,[b_1|5]), (74,60,[c_1|5])}"
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(8)
BOUNDS(1, n^1)