/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (full) 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), 160 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 (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   r0(0(x1)) -> 0(r0(x1))
   r0(1(x1)) -> 1(r0(x1))
   r0(m(x1)) -> m(r0(x1))
   r1(0(x1)) -> 0(r1(x1))
   r1(1(x1)) -> 1(r1(x1))
   r1(m(x1)) -> m(r1(x1))
   r0(b(x1)) -> qr(0(b(x1)))
   r1(b(x1)) -> qr(1(b(x1)))
   0(qr(x1)) -> qr(0(x1))
   1(qr(x1)) -> qr(1(x1))
   m(qr(x1)) -> ql(m(x1))
   0(ql(x1)) -> ql(0(x1))
   1(ql(x1)) -> ql(1(x1))
   b(ql(0(x1))) -> 0(b(r0(x1)))
   b(ql(1(x1))) -> 1(b(r1(x1)))

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

(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(qr(x_1)) -> qr(encArg(x_1))
   encArg(ql(x_1)) -> ql(encArg(x_1))
   encArg(cons_r0(x_1)) -> r0(encArg(x_1))
   encArg(cons_r1(x_1)) -> r1(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(encArg(x_1))
   encArg(cons_1(x_1)) -> 1(encArg(x_1))
   encArg(cons_m(x_1)) -> m(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_r0(x_1) -> r0(encArg(x_1))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_1(x_1) -> 1(encArg(x_1))
   encode_m(x_1) -> m(encArg(x_1))
   encode_r1(x_1) -> r1(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_qr(x_1) -> qr(encArg(x_1))
   encode_ql(x_1) -> ql(encArg(x_1))


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


The TRS R consists of the following rules:

   r0(0(x1)) -> 0(r0(x1))
   r0(1(x1)) -> 1(r0(x1))
   r0(m(x1)) -> m(r0(x1))
   r1(0(x1)) -> 0(r1(x1))
   r1(1(x1)) -> 1(r1(x1))
   r1(m(x1)) -> m(r1(x1))
   r0(b(x1)) -> qr(0(b(x1)))
   r1(b(x1)) -> qr(1(b(x1)))
   0(qr(x1)) -> qr(0(x1))
   1(qr(x1)) -> qr(1(x1))
   m(qr(x1)) -> ql(m(x1))
   0(ql(x1)) -> ql(0(x1))
   1(ql(x1)) -> ql(1(x1))
   b(ql(0(x1))) -> 0(b(r0(x1)))
   b(ql(1(x1))) -> 1(b(r1(x1)))

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

   encArg(qr(x_1)) -> qr(encArg(x_1))
   encArg(ql(x_1)) -> ql(encArg(x_1))
   encArg(cons_r0(x_1)) -> r0(encArg(x_1))
   encArg(cons_r1(x_1)) -> r1(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(encArg(x_1))
   encArg(cons_1(x_1)) -> 1(encArg(x_1))
   encArg(cons_m(x_1)) -> m(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_r0(x_1) -> r0(encArg(x_1))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_1(x_1) -> 1(encArg(x_1))
   encode_m(x_1) -> m(encArg(x_1))
   encode_r1(x_1) -> r1(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_qr(x_1) -> qr(encArg(x_1))
   encode_ql(x_1) -> ql(encArg(x_1))

Rewrite Strategy: FULL
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(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(n^1, INF).


The TRS R consists of the following rules:

   r0(0(x1)) -> 0(r0(x1))
   r0(1(x1)) -> 1(r0(x1))
   r0(m(x1)) -> m(r0(x1))
   r1(0(x1)) -> 0(r1(x1))
   r1(1(x1)) -> 1(r1(x1))
   r1(m(x1)) -> m(r1(x1))
   r0(b(x1)) -> qr(0(b(x1)))
   r1(b(x1)) -> qr(1(b(x1)))
   0(qr(x1)) -> qr(0(x1))
   1(qr(x1)) -> qr(1(x1))
   m(qr(x1)) -> ql(m(x1))
   0(ql(x1)) -> ql(0(x1))
   1(ql(x1)) -> ql(1(x1))
   b(ql(0(x1))) -> 0(b(r0(x1)))
   b(ql(1(x1))) -> 1(b(r1(x1)))

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

   encArg(qr(x_1)) -> qr(encArg(x_1))
   encArg(ql(x_1)) -> ql(encArg(x_1))
   encArg(cons_r0(x_1)) -> r0(encArg(x_1))
   encArg(cons_r1(x_1)) -> r1(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(encArg(x_1))
   encArg(cons_1(x_1)) -> 1(encArg(x_1))
   encArg(cons_m(x_1)) -> m(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_r0(x_1) -> r0(encArg(x_1))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_1(x_1) -> 1(encArg(x_1))
   encode_m(x_1) -> m(encArg(x_1))
   encode_r1(x_1) -> r1(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_qr(x_1) -> qr(encArg(x_1))
   encode_ql(x_1) -> ql(encArg(x_1))

Rewrite Strategy: FULL
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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 (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   r0(0(x1)) -> 0(r0(x1))
   r0(1(x1)) -> 1(r0(x1))
   r0(m(x1)) -> m(r0(x1))
   r1(0(x1)) -> 0(r1(x1))
   r1(1(x1)) -> 1(r1(x1))
   r1(m(x1)) -> m(r1(x1))
   r0(b(x1)) -> qr(0(b(x1)))
   r1(b(x1)) -> qr(1(b(x1)))
   0(qr(x1)) -> qr(0(x1))
   1(qr(x1)) -> qr(1(x1))
   m(qr(x1)) -> ql(m(x1))
   0(ql(x1)) -> ql(0(x1))
   1(ql(x1)) -> ql(1(x1))
   b(ql(0(x1))) -> 0(b(r0(x1)))
   b(ql(1(x1))) -> 1(b(r1(x1)))

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

   encArg(qr(x_1)) -> qr(encArg(x_1))
   encArg(ql(x_1)) -> ql(encArg(x_1))
   encArg(cons_r0(x_1)) -> r0(encArg(x_1))
   encArg(cons_r1(x_1)) -> r1(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(encArg(x_1))
   encArg(cons_1(x_1)) -> 1(encArg(x_1))
   encArg(cons_m(x_1)) -> m(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_r0(x_1) -> r0(encArg(x_1))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_1(x_1) -> 1(encArg(x_1))
   encode_m(x_1) -> m(encArg(x_1))
   encode_r1(x_1) -> r1(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_qr(x_1) -> qr(encArg(x_1))
   encode_ql(x_1) -> ql(encArg(x_1))

Rewrite Strategy: FULL
----------------------------------------

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

The rewrite sequence

1(qr(x1)) ->^+ qr(1(x1))

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

The pumping substitution is [x1 / qr(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 (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   r0(0(x1)) -> 0(r0(x1))
   r0(1(x1)) -> 1(r0(x1))
   r0(m(x1)) -> m(r0(x1))
   r1(0(x1)) -> 0(r1(x1))
   r1(1(x1)) -> 1(r1(x1))
   r1(m(x1)) -> m(r1(x1))
   r0(b(x1)) -> qr(0(b(x1)))
   r1(b(x1)) -> qr(1(b(x1)))
   0(qr(x1)) -> qr(0(x1))
   1(qr(x1)) -> qr(1(x1))
   m(qr(x1)) -> ql(m(x1))
   0(ql(x1)) -> ql(0(x1))
   1(ql(x1)) -> ql(1(x1))
   b(ql(0(x1))) -> 0(b(r0(x1)))
   b(ql(1(x1))) -> 1(b(r1(x1)))

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

   encArg(qr(x_1)) -> qr(encArg(x_1))
   encArg(ql(x_1)) -> ql(encArg(x_1))
   encArg(cons_r0(x_1)) -> r0(encArg(x_1))
   encArg(cons_r1(x_1)) -> r1(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(encArg(x_1))
   encArg(cons_1(x_1)) -> 1(encArg(x_1))
   encArg(cons_m(x_1)) -> m(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_r0(x_1) -> r0(encArg(x_1))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_1(x_1) -> 1(encArg(x_1))
   encode_m(x_1) -> m(encArg(x_1))
   encode_r1(x_1) -> r1(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_qr(x_1) -> qr(encArg(x_1))
   encode_ql(x_1) -> ql(encArg(x_1))

Rewrite Strategy: FULL
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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 (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   r0(0(x1)) -> 0(r0(x1))
   r0(1(x1)) -> 1(r0(x1))
   r0(m(x1)) -> m(r0(x1))
   r1(0(x1)) -> 0(r1(x1))
   r1(1(x1)) -> 1(r1(x1))
   r1(m(x1)) -> m(r1(x1))
   r0(b(x1)) -> qr(0(b(x1)))
   r1(b(x1)) -> qr(1(b(x1)))
   0(qr(x1)) -> qr(0(x1))
   1(qr(x1)) -> qr(1(x1))
   m(qr(x1)) -> ql(m(x1))
   0(ql(x1)) -> ql(0(x1))
   1(ql(x1)) -> ql(1(x1))
   b(ql(0(x1))) -> 0(b(r0(x1)))
   b(ql(1(x1))) -> 1(b(r1(x1)))

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

   encArg(qr(x_1)) -> qr(encArg(x_1))
   encArg(ql(x_1)) -> ql(encArg(x_1))
   encArg(cons_r0(x_1)) -> r0(encArg(x_1))
   encArg(cons_r1(x_1)) -> r1(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(encArg(x_1))
   encArg(cons_1(x_1)) -> 1(encArg(x_1))
   encArg(cons_m(x_1)) -> m(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_r0(x_1) -> r0(encArg(x_1))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_1(x_1) -> 1(encArg(x_1))
   encode_m(x_1) -> m(encArg(x_1))
   encode_r1(x_1) -> r1(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_qr(x_1) -> qr(encArg(x_1))
   encode_ql(x_1) -> ql(encArg(x_1))

Rewrite Strategy: FULL