Derivational Complexity: TRS pair #487102998

details

property	value
status	complete
benchmark	86499.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n137.star.cs.uiowa.edu
space	ICFP_2010

run statistics

property	value
solver	AProVE
configuration	rcdcRelativeAlsoLower
runtime (wallclock)	293.488 seconds
cpu usage	787.37
user time	779.607
system time	7.76288
max virtual memory	1.8965728E7
max residence set size	1.4926684E7

stage attributes

key	value
starexec-result	WORST_CASE(?, O(n^1))

output

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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 56 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxTRS
(7) CpxTrsMatchBoundsProof [FINISHED, 108 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:

   0(0(1(1(x1)))) -> 2(3(1(x1)))
   1(0(1(3(x1)))) -> 4(3(3(x1)))
   2(1(3(4(2(x1))))) -> 5(1(1(3(2(x1)))))
   1(3(0(1(0(2(x1)))))) -> 1(1(3(3(2(x1)))))
   3(3(3(4(2(0(x1)))))) -> 3(3(0(1(1(3(x1))))))
   0(2(5(1(5(5(2(2(3(x1))))))))) -> 5(3(0(1(1(3(0(0(3(x1)))))))))
   0(4(0(2(1(2(5(1(0(x1))))))))) -> 0(4(5(1(4(3(4(4(x1))))))))
   5(0(4(2(4(4(1(5(2(x1))))))))) -> 5(1(5(1(3(2(2(5(2(x1)))))))))
   1(2(5(4(1(4(0(1(0(5(x1)))))))))) -> 4(2(1(1(2(1(3(4(5(x1)))))))))
   1(1(2(3(1(0(5(5(4(1(4(x1))))))))))) -> 1(3(3(2(1(2(3(0(1(4(x1))))))))))
   2(2(0(2(0(3(1(0(2(4(5(x1))))))))))) -> 2(3(2(5(3(3(2(1(3(5(x1))))))))))
   1(5(3(2(5(0(3(0(4(1(1(2(x1)))))))))))) -> 1(5(3(4(4(0(5(1(4(2(2(x1)))))))))))
   4(1(0(0(1(3(4(1(0(5(4(3(x1)))))))))))) -> 4(4(3(0(3(1(0(3(0(3(3(x1)))))))))))
   0(1(2(5(2(1(3(3(0(1(3(1(3(x1))))))))))))) -> 2(3(5(2(4(0(0(3(0(3(1(3(x1))))))))))))
   0(3(5(4(4(2(0(3(3(3(5(1(2(x1))))))))))))) -> 3(4(4(0(5(1(1(2(1(0(1(5(0(2(x1))))))))))))))
   2(1(2(3(3(0(0(0(0(0(5(1(2(x1))))))))))))) -> 0(2(5(2(4(5(0(3(2(1(3(5(x1))))))))))))
   4(5(2(0(5(1(5(5(0(2(0(5(5(x1))))))))))))) -> 4(4(1(3(4(1(2(5(1(3(1(x1)))))))))))
   0(5(2(5(0(4(1(5(1(0(5(5(3(1(x1)))))))))))))) -> 5(1(2(5(1(3(3(0(4(4(5(5(x1))))))))))))
   2(2(1(4(3(4(2(1(5(3(1(1(5(5(x1)))))))))))))) -> 5(3(0(5(4(3(2(0(4(5(3(3(5(x1)))))))))))))
   1(1(2(2(1(3(4(2(0(3(0(0(5(2(5(x1))))))))))))))) -> 1(1(0(1(1(5(1(3(5(2(3(4(5(1(5(x1)))))))))))))))
   2(0(0(5(3(4(0(5(5(3(5(1(3(2(1(x1))))))))))))))) -> 5(2(5(2(4(4(2(5(5(0(5(4(3(1(x1))))))))))))))
   0(3(1(2(0(3(1(3(1(2(1(1(3(1(1(2(x1)))))))))))))))) -> 5(1(1(4(2(3(5(2(0(4(0(1(0(5(4(x1)))))))))))))))
   1(4(2(2(0(0(4(3(1(2(4(1(5(5(1(0(x1)))))))))))))))) -> 1(1(2(0(1(4(3(1(3(4(4(3(5(3(3(x1)))))))))))))))
   2(3(5(2(1(4(1(1(4(0(4(0(0(0(0(0(2(x1))))))))))))))))) -> 1(1(4(3(4(0(4(1(1(4(5(5(5(2(2(0(5(x1)))))))))))))))))
   1(3(2(1(0(3(4(3(0(5(1(1(2(0(4(1(5(4(0(x1))))))))))))))))))) -> 5(5(2(1(0(4(5(5(3(0(0(4(3(3(0(2(3(1(0(x1)))))))))))))))))))
   2(0(0(1(4(2(3(4(2(1(3(1(5(4(2(0(5(4(5(5(x1)))))))))))))))))))) -> 0(1(2(5(4(5(1(4(0(2(1(3(4(1(5(1(4(0(4(2(x1))))))))))))))))))))
   3(3(4(0(2(3(5(5(4(5(0(5(4(4(5(2(0(2(0(2(x1)))))))))))))))))))) -> 3(3(4(1(0(3(0(2(3(0(2(4(1(1(5(0(0(2(3(2(x1))))))))))))))))))))
   1(4(2(1(0(0(1(0(2(3(2(1(1(2(4(5(1(5(5(4(5(x1))))))))))))))))))))) -> 1(1(5(2(5(3(1(3(1(5(3(4(4(5(4(2(2(1(4(2(x1))))))))))))))))))))
   5(3(0(4(3(4(1(3(3(0(3(0(5(2(2(2(5(1(3(0(1(x1))))))))))))))))))))) -> 5(2(1(3(1(5(5(3(2(0(3(2(2(0(1(2(3(2(4(2(x1))))))))))))))))))))
   5(5(1(3(0(2(4(1(0(2(5(4(1(5(5(4(4(0(4(2(3(x1))))))))))))))))))))) -> 2(4(0(0(2(4(0(3(4(5(0(5(2(2(3(2(0(3(3(4(3(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(cons_0(x_1)) -> 0(encArg(x_1))
   encArg(cons_1(x_1)) -> 1(encArg(x_1))
   encArg(cons_2(x_1)) -> 2(encArg(x_1))
   encArg(cons_3(x_1)) -> 3(encArg(x_1))
   encArg(cons_5(x_1)) -> 5(encArg(x_1))
   encArg(cons_4(x_1)) -> 4(encArg(x_1))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_1(x_1) -> 1(encArg(x_1))
   encode_2(x_1) -> 2(encArg(x_1))
   encode_3(x_1) -> 3(encArg(x_1))
   encode_4(x_1) -> 4(encArg(x_1))
   encode_5(x_1) -> 5(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:

   0(0(1(1(x1)))) -> 2(3(1(x1)))
   1(0(1(3(x1)))) -> 4(3(3(x1)))
   2(1(3(4(2(x1))))) -> 5(1(1(3(2(x1)))))
   1(3(0(1(0(2(x1)))))) -> 1(1(3(3(2(x1)))))
   3(3(3(4(2(0(x1)))))) -> 3(3(0(1(1(3(x1))))))
   0(2(5(1(5(5(2(2(3(x1))))))))) -> 5(3(0(1(1(3(0(0(3(x1)))))))))
   0(4(0(2(1(2(5(1(0(x1))))))))) -> 0(4(5(1(4(3(4(4(x1))))))))
   5(0(4(2(4(4(1(5(2(x1))))))))) -> 5(1(5(1(3(2(2(5(2(x1)))))))))
   1(2(5(4(1(4(0(1(0(5(x1)))))))))) -> 4(2(1(1(2(1(3(4(5(x1)))))))))
   1(1(2(3(1(0(5(5(4(1(4(x1))))))))))) -> 1(3(3(2(1(2(3(0(1(4(x1))))))))))
   2(2(0(2(0(3(1(0(2(4(5(x1))))))))))) -> 2(3(2(5(3(3(2(1(3(5(x1))))))))))
   1(5(3(2(5(0(3(0(4(1(1(2(x1)))))))))))) -> 1(5(3(4(4(0(5(1(4(2(2(x1)))))))))))

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actions all output return to Derivational Complexity: TRS