/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(Omega(n^1), ?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 62 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
(7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) TRS for Loop Detection


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(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   2(7(x1)) -> 1(8(x1))
   2(8(1(x1))) -> 8(x1)
   2(8(x1)) -> 4(x1)
   5(9(x1)) -> 0(x1)
   4(x1) -> 5(2(3(x1)))
   5(3(x1)) -> 6(0(x1))
   2(8(x1)) -> 7(x1)
   4(7(x1)) -> 1(3(x1))
   5(2(6(x1))) -> 6(2(4(x1)))
   9(7(x1)) -> 7(5(x1))
   7(2(x1)) -> 4(x1)
   7(0(x1)) -> 9(3(x1))
   6(9(x1)) -> 9(x1)
   9(5(9(x1))) -> 5(7(x1))
   4(x1) -> 9(6(6(x1)))
   9(x1) -> 6(7(x1))
   6(2(x1)) -> 7(7(x1))
   2(4(x1)) -> 0(7(x1))
   6(6(x1)) -> 3(x1)
   0(3(x1)) -> 5(3(x1))

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(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(1(x_1)) -> 1(encArg(x_1))
   encArg(8(x_1)) -> 8(encArg(x_1))
   encArg(3(x_1)) -> 3(encArg(x_1))
   encArg(cons_2(x_1)) -> 2(encArg(x_1))
   encArg(cons_5(x_1)) -> 5(encArg(x_1))
   encArg(cons_4(x_1)) -> 4(encArg(x_1))
   encArg(cons_9(x_1)) -> 9(encArg(x_1))
   encArg(cons_7(x_1)) -> 7(encArg(x_1))
   encArg(cons_6(x_1)) -> 6(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(encArg(x_1))
   encode_2(x_1) -> 2(encArg(x_1))
   encode_7(x_1) -> 7(encArg(x_1))
   encode_1(x_1) -> 1(encArg(x_1))
   encode_8(x_1) -> 8(encArg(x_1))
   encode_4(x_1) -> 4(encArg(x_1))
   encode_5(x_1) -> 5(encArg(x_1))
   encode_9(x_1) -> 9(encArg(x_1))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_3(x_1) -> 3(encArg(x_1))
   encode_6(x_1) -> 6(encArg(x_1))


----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   2(7(x1)) -> 1(8(x1))
   2(8(1(x1))) -> 8(x1)
   2(8(x1)) -> 4(x1)
   5(9(x1)) -> 0(x1)
   4(x1) -> 5(2(3(x1)))
   5(3(x1)) -> 6(0(x1))
   2(8(x1)) -> 7(x1)
   4(7(x1)) -> 1(3(x1))
   5(2(6(x1))) -> 6(2(4(x1)))
   9(7(x1)) -> 7(5(x1))
   7(2(x1)) -> 4(x1)
   7(0(x1)) -> 9(3(x1))
   6(9(x1)) -> 9(x1)
   9(5(9(x1))) -> 5(7(x1))
   4(x1) -> 9(6(6(x1)))
   9(x1) -> 6(7(x1))
   6(2(x1)) -> 7(7(x1))
   2(4(x1)) -> 0(7(x1))
   6(6(x1)) -> 3(x1)
   0(3(x1)) -> 5(3(x1))

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

   encArg(1(x_1)) -> 1(encArg(x_1))
   encArg(8(x_1)) -> 8(encArg(x_1))
   encArg(3(x_1)) -> 3(encArg(x_1))
   encArg(cons_2(x_1)) -> 2(encArg(x_1))
   encArg(cons_5(x_1)) -> 5(encArg(x_1))
   encArg(cons_4(x_1)) -> 4(encArg(x_1))
   encArg(cons_9(x_1)) -> 9(encArg(x_1))
   encArg(cons_7(x_1)) -> 7(encArg(x_1))
   encArg(cons_6(x_1)) -> 6(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(encArg(x_1))
   encode_2(x_1) -> 2(encArg(x_1))
   encode_7(x_1) -> 7(encArg(x_1))
   encode_1(x_1) -> 1(encArg(x_1))
   encode_8(x_1) -> 8(encArg(x_1))
   encode_4(x_1) -> 4(encArg(x_1))
   encode_5(x_1) -> 5(encArg(x_1))
   encode_9(x_1) -> 9(encArg(x_1))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_3(x_1) -> 3(encArg(x_1))
   encode_6(x_1) -> 6(encArg(x_1))

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

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   2(7(x1)) -> 1(8(x1))
   2(8(1(x1))) -> 8(x1)
   2(8(x1)) -> 4(x1)
   5(9(x1)) -> 0(x1)
   4(x1) -> 5(2(3(x1)))
   5(3(x1)) -> 6(0(x1))
   2(8(x1)) -> 7(x1)
   4(7(x1)) -> 1(3(x1))
   5(2(6(x1))) -> 6(2(4(x1)))
   9(7(x1)) -> 7(5(x1))
   7(2(x1)) -> 4(x1)
   7(0(x1)) -> 9(3(x1))
   6(9(x1)) -> 9(x1)
   9(5(9(x1))) -> 5(7(x1))
   4(x1) -> 9(6(6(x1)))
   9(x1) -> 6(7(x1))
   6(2(x1)) -> 7(7(x1))
   2(4(x1)) -> 0(7(x1))
   6(6(x1)) -> 3(x1)
   0(3(x1)) -> 5(3(x1))

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

   encArg(1(x_1)) -> 1(encArg(x_1))
   encArg(8(x_1)) -> 8(encArg(x_1))
   encArg(3(x_1)) -> 3(encArg(x_1))
   encArg(cons_2(x_1)) -> 2(encArg(x_1))
   encArg(cons_5(x_1)) -> 5(encArg(x_1))
   encArg(cons_4(x_1)) -> 4(encArg(x_1))
   encArg(cons_9(x_1)) -> 9(encArg(x_1))
   encArg(cons_7(x_1)) -> 7(encArg(x_1))
   encArg(cons_6(x_1)) -> 6(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(encArg(x_1))
   encode_2(x_1) -> 2(encArg(x_1))
   encode_7(x_1) -> 7(encArg(x_1))
   encode_1(x_1) -> 1(encArg(x_1))
   encode_8(x_1) -> 8(encArg(x_1))
   encode_4(x_1) -> 4(encArg(x_1))
   encode_5(x_1) -> 5(encArg(x_1))
   encode_9(x_1) -> 9(encArg(x_1))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_3(x_1) -> 3(encArg(x_1))
   encode_6(x_1) -> 6(encArg(x_1))

Rewrite Strategy: INNERMOST
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(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(6)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   2(7(x1)) -> 1(8(x1))
   2(8(1(x1))) -> 8(x1)
   2(8(x1)) -> 4(x1)
   5(9(x1)) -> 0(x1)
   4(x1) -> 5(2(3(x1)))
   5(3(x1)) -> 6(0(x1))
   2(8(x1)) -> 7(x1)
   4(7(x1)) -> 1(3(x1))
   5(2(6(x1))) -> 6(2(4(x1)))
   9(7(x1)) -> 7(5(x1))
   7(2(x1)) -> 4(x1)
   7(0(x1)) -> 9(3(x1))
   6(9(x1)) -> 9(x1)
   9(5(9(x1))) -> 5(7(x1))
   4(x1) -> 9(6(6(x1)))
   9(x1) -> 6(7(x1))
   6(2(x1)) -> 7(7(x1))
   2(4(x1)) -> 0(7(x1))
   6(6(x1)) -> 3(x1)
   0(3(x1)) -> 5(3(x1))

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

   encArg(1(x_1)) -> 1(encArg(x_1))
   encArg(8(x_1)) -> 8(encArg(x_1))
   encArg(3(x_1)) -> 3(encArg(x_1))
   encArg(cons_2(x_1)) -> 2(encArg(x_1))
   encArg(cons_5(x_1)) -> 5(encArg(x_1))
   encArg(cons_4(x_1)) -> 4(encArg(x_1))
   encArg(cons_9(x_1)) -> 9(encArg(x_1))
   encArg(cons_7(x_1)) -> 7(encArg(x_1))
   encArg(cons_6(x_1)) -> 6(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(encArg(x_1))
   encode_2(x_1) -> 2(encArg(x_1))
   encode_7(x_1) -> 7(encArg(x_1))
   encode_1(x_1) -> 1(encArg(x_1))
   encode_8(x_1) -> 8(encArg(x_1))
   encode_4(x_1) -> 4(encArg(x_1))
   encode_5(x_1) -> 5(encArg(x_1))
   encode_9(x_1) -> 9(encArg(x_1))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_3(x_1) -> 3(encArg(x_1))
   encode_6(x_1) -> 6(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(7) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

0(3(x1)) ->^+ 6(0(x1))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [x1 / 3(x1)].

The result substitution is [ ].




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(8)
Complex Obligation (BEST)

----------------------------------------

(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   2(7(x1)) -> 1(8(x1))
   2(8(1(x1))) -> 8(x1)
   2(8(x1)) -> 4(x1)
   5(9(x1)) -> 0(x1)
   4(x1) -> 5(2(3(x1)))
   5(3(x1)) -> 6(0(x1))
   2(8(x1)) -> 7(x1)
   4(7(x1)) -> 1(3(x1))
   5(2(6(x1))) -> 6(2(4(x1)))
   9(7(x1)) -> 7(5(x1))
   7(2(x1)) -> 4(x1)
   7(0(x1)) -> 9(3(x1))
   6(9(x1)) -> 9(x1)
   9(5(9(x1))) -> 5(7(x1))
   4(x1) -> 9(6(6(x1)))
   9(x1) -> 6(7(x1))
   6(2(x1)) -> 7(7(x1))
   2(4(x1)) -> 0(7(x1))
   6(6(x1)) -> 3(x1)
   0(3(x1)) -> 5(3(x1))

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

   encArg(1(x_1)) -> 1(encArg(x_1))
   encArg(8(x_1)) -> 8(encArg(x_1))
   encArg(3(x_1)) -> 3(encArg(x_1))
   encArg(cons_2(x_1)) -> 2(encArg(x_1))
   encArg(cons_5(x_1)) -> 5(encArg(x_1))
   encArg(cons_4(x_1)) -> 4(encArg(x_1))
   encArg(cons_9(x_1)) -> 9(encArg(x_1))
   encArg(cons_7(x_1)) -> 7(encArg(x_1))
   encArg(cons_6(x_1)) -> 6(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(encArg(x_1))
   encode_2(x_1) -> 2(encArg(x_1))
   encode_7(x_1) -> 7(encArg(x_1))
   encode_1(x_1) -> 1(encArg(x_1))
   encode_8(x_1) -> 8(encArg(x_1))
   encode_4(x_1) -> 4(encArg(x_1))
   encode_5(x_1) -> 5(encArg(x_1))
   encode_9(x_1) -> 9(encArg(x_1))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_3(x_1) -> 3(encArg(x_1))
   encode_6(x_1) -> 6(encArg(x_1))

Rewrite Strategy: INNERMOST
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(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(11)
BOUNDS(n^1, INF)

----------------------------------------

(12)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   2(7(x1)) -> 1(8(x1))
   2(8(1(x1))) -> 8(x1)
   2(8(x1)) -> 4(x1)
   5(9(x1)) -> 0(x1)
   4(x1) -> 5(2(3(x1)))
   5(3(x1)) -> 6(0(x1))
   2(8(x1)) -> 7(x1)
   4(7(x1)) -> 1(3(x1))
   5(2(6(x1))) -> 6(2(4(x1)))
   9(7(x1)) -> 7(5(x1))
   7(2(x1)) -> 4(x1)
   7(0(x1)) -> 9(3(x1))
   6(9(x1)) -> 9(x1)
   9(5(9(x1))) -> 5(7(x1))
   4(x1) -> 9(6(6(x1)))
   9(x1) -> 6(7(x1))
   6(2(x1)) -> 7(7(x1))
   2(4(x1)) -> 0(7(x1))
   6(6(x1)) -> 3(x1)
   0(3(x1)) -> 5(3(x1))

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

   encArg(1(x_1)) -> 1(encArg(x_1))
   encArg(8(x_1)) -> 8(encArg(x_1))
   encArg(3(x_1)) -> 3(encArg(x_1))
   encArg(cons_2(x_1)) -> 2(encArg(x_1))
   encArg(cons_5(x_1)) -> 5(encArg(x_1))
   encArg(cons_4(x_1)) -> 4(encArg(x_1))
   encArg(cons_9(x_1)) -> 9(encArg(x_1))
   encArg(cons_7(x_1)) -> 7(encArg(x_1))
   encArg(cons_6(x_1)) -> 6(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(encArg(x_1))
   encode_2(x_1) -> 2(encArg(x_1))
   encode_7(x_1) -> 7(encArg(x_1))
   encode_1(x_1) -> 1(encArg(x_1))
   encode_8(x_1) -> 8(encArg(x_1))
   encode_4(x_1) -> 4(encArg(x_1))
   encode_5(x_1) -> 5(encArg(x_1))
   encode_9(x_1) -> 9(encArg(x_1))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_3(x_1) -> 3(encArg(x_1))
   encode_6(x_1) -> 6(encArg(x_1))

Rewrite Strategy: INNERMOST