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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 34 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(a(a(x1)))))) -> a(a(a(a(b(a(b(a(b(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(cons_b(x_1)) -> b(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_a(x_1) -> a(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:

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

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

   encArg(a(x_1)) -> a(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))

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:

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

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

   encArg(a(x_1)) -> a(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))

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:

   b(a(b(a(a(a(x1)))))) -> a(a(a(a(b(a(b(a(b(x1)))))))))
   encArg(a(x_1)) -> a(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))

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 5. 
The certificate found is represented by the following graph.

"[24, 25, 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, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101]
{(24,25,[b_1|0, encArg_1|0, encode_b_1|0, encode_a_1|0]), (24,29,[a_1|1, b_1|1]), (24,30,[a_1|2]), (25,25,[a_1|0, cons_b_1|0]), (29,25,[encArg_1|1]), (29,29,[a_1|1, b_1|1]), (29,30,[a_1|2]), (30,31,[a_1|2]), (31,32,[a_1|2]), (32,33,[a_1|2]), (33,34,[b_1|2]), (33,46,[a_1|3]), (34,35,[a_1|2]), (35,36,[b_1|2]), (35,30,[a_1|2]), (35,38,[a_1|3]), (36,37,[a_1|2]), (37,29,[b_1|2]), (37,30,[b_1|2, a_1|2]), (37,31,[b_1|2]), (37,32,[b_1|2]), (37,46,[a_1|3]), (38,39,[a_1|3]), (39,40,[a_1|3]), (40,41,[a_1|3]), (41,42,[b_1|3]), (41,54,[a_1|4]), (42,43,[a_1|3]), (43,44,[b_1|3]), (43,46,[a_1|3]), (43,62,[a_1|4]), (44,45,[a_1|3]), (45,32,[b_1|3]), (45,33,[b_1|3]), (45,46,[a_1|3, b_1|3]), (45,47,[b_1|3]), (46,47,[a_1|3]), (47,48,[a_1|3]), (48,49,[a_1|3]), (49,50,[b_1|3]), (50,51,[a_1|3]), (51,52,[b_1|3]), (51,38,[a_1|3]), (51,70,[a_1|4]), (52,53,[a_1|3]), (53,31,[b_1|3]), (53,39,[b_1|3]), (53,47,[b_1|3]), (54,55,[a_1|4]), (55,56,[a_1|4]), (56,57,[a_1|4]), (57,58,[b_1|4]), (58,59,[a_1|4]), (59,60,[b_1|4]), (60,61,[a_1|4]), (61,47,[b_1|4]), (62,63,[a_1|4]), (63,64,[a_1|4]), (64,65,[a_1|4]), (65,66,[b_1|4]), (65,86,[a_1|5]), (66,67,[a_1|4]), (67,68,[b_1|4]), (68,69,[a_1|4]), (69,48,[b_1|4]), (69,49,[b_1|4]), (69,78,[a_1|4]), (70,71,[a_1|4]), (71,72,[a_1|4]), (72,73,[a_1|4]), (73,74,[b_1|4]), (74,75,[a_1|4]), (75,76,[b_1|4]), (75,94,[a_1|5]), (76,77,[a_1|4]), (77,54,[b_1|4]), (78,79,[a_1|4]), (79,80,[a_1|4]), (80,81,[a_1|4]), (81,82,[b_1|4]), (82,83,[a_1|4]), (83,84,[b_1|4]), (83,70,[a_1|4]), (84,85,[a_1|4]), (85,39,[b_1|4]), (85,71,[b_1|4]), (86,87,[a_1|5]), (87,88,[a_1|5]), (88,89,[a_1|5]), (89,90,[b_1|5]), (90,91,[a_1|5]), (91,92,[b_1|5]), (92,93,[a_1|5]), (93,79,[b_1|5]), (94,95,[a_1|5]), (95,96,[a_1|5]), (96,97,[a_1|5]), (97,98,[b_1|5]), (98,99,[a_1|5]), (99,100,[b_1|5]), (100,101,[a_1|5]), (101,57,[b_1|5])}"
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(8)
BOUNDS(1, n^1)