/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 60 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


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

(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:

   0(q0(0(x1))) -> 0(0(q0(x1)))
   0(q0(h(x1))) -> 0(0(q0(h(x1))))
   0(q0(1(x1))) -> 0(1(q0(x1)))
   1(q0(0(x1))) -> 0(0(q1(x1)))
   1(q0(h(x1))) -> 0(0(q1(h(x1))))
   1(q0(1(x1))) -> 0(1(q1(x1)))
   1(q1(0(x1))) -> 1(0(q1(x1)))
   1(q1(h(x1))) -> 1(0(q1(h(x1))))
   1(q1(1(x1))) -> 1(1(q1(x1)))
   0(q1(0(x1))) -> 0(0(q2(x1)))
   0(q1(h(x1))) -> 0(0(q2(h(x1))))
   0(q1(1(x1))) -> 0(1(q2(x1)))
   1(q2(0(x1))) -> 1(0(q2(x1)))
   1(q2(h(x1))) -> 1(0(q2(h(x1))))
   1(q2(1(x1))) -> 1(1(q2(x1)))
   0(q2(x1)) -> q3(1(x1))
   1(q3(x1)) -> q3(1(x1))
   0(q3(x1)) -> q4(0(x1))
   1(q4(x1)) -> q4(1(x1))
   0(q4(0(x1))) -> 1(0(q5(x1)))
   0(q4(h(x1))) -> 1(0(q5(h(x1))))
   0(q4(1(x1))) -> 1(1(q5(x1)))
   1(q5(0(x1))) -> 0(0(q1(x1)))
   1(q5(h(x1))) -> 0(0(q1(h(x1))))
   1(q5(1(x1))) -> 0(1(q1(x1)))
   h(q0(x1)) -> h(0(q0(x1)))
   h(q1(x1)) -> h(0(q1(x1)))
   h(q2(x1)) -> h(0(q2(x1)))
   h(q3(x1)) -> h(0(q3(x1)))
   h(q4(x1)) -> h(0(q4(x1)))
   h(q5(x1)) -> h(0(q5(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(q0(x_1)) -> q0(encArg(x_1))
   encArg(q1(x_1)) -> q1(encArg(x_1))
   encArg(q2(x_1)) -> q2(encArg(x_1))
   encArg(q3(x_1)) -> q3(encArg(x_1))
   encArg(q4(x_1)) -> q4(encArg(x_1))
   encArg(q5(x_1)) -> q5(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(encArg(x_1))
   encArg(cons_1(x_1)) -> 1(encArg(x_1))
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_q0(x_1) -> q0(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_1(x_1) -> 1(encArg(x_1))
   encode_q1(x_1) -> q1(encArg(x_1))
   encode_q2(x_1) -> q2(encArg(x_1))
   encode_q3(x_1) -> q3(encArg(x_1))
   encode_q4(x_1) -> q4(encArg(x_1))
   encode_q5(x_1) -> q5(encArg(x_1))


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

(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:

   0(q0(0(x1))) -> 0(0(q0(x1)))
   0(q0(h(x1))) -> 0(0(q0(h(x1))))
   0(q0(1(x1))) -> 0(1(q0(x1)))
   1(q0(0(x1))) -> 0(0(q1(x1)))
   1(q0(h(x1))) -> 0(0(q1(h(x1))))
   1(q0(1(x1))) -> 0(1(q1(x1)))
   1(q1(0(x1))) -> 1(0(q1(x1)))
   1(q1(h(x1))) -> 1(0(q1(h(x1))))
   1(q1(1(x1))) -> 1(1(q1(x1)))
   0(q1(0(x1))) -> 0(0(q2(x1)))
   0(q1(h(x1))) -> 0(0(q2(h(x1))))
   0(q1(1(x1))) -> 0(1(q2(x1)))
   1(q2(0(x1))) -> 1(0(q2(x1)))
   1(q2(h(x1))) -> 1(0(q2(h(x1))))
   1(q2(1(x1))) -> 1(1(q2(x1)))
   0(q2(x1)) -> q3(1(x1))
   1(q3(x1)) -> q3(1(x1))
   0(q3(x1)) -> q4(0(x1))
   1(q4(x1)) -> q4(1(x1))
   0(q4(0(x1))) -> 1(0(q5(x1)))
   0(q4(h(x1))) -> 1(0(q5(h(x1))))
   0(q4(1(x1))) -> 1(1(q5(x1)))
   1(q5(0(x1))) -> 0(0(q1(x1)))
   1(q5(h(x1))) -> 0(0(q1(h(x1))))
   1(q5(1(x1))) -> 0(1(q1(x1)))
   h(q0(x1)) -> h(0(q0(x1)))
   h(q1(x1)) -> h(0(q1(x1)))
   h(q2(x1)) -> h(0(q2(x1)))
   h(q3(x1)) -> h(0(q3(x1)))
   h(q4(x1)) -> h(0(q4(x1)))
   h(q5(x1)) -> h(0(q5(x1)))

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

   encArg(q0(x_1)) -> q0(encArg(x_1))
   encArg(q1(x_1)) -> q1(encArg(x_1))
   encArg(q2(x_1)) -> q2(encArg(x_1))
   encArg(q3(x_1)) -> q3(encArg(x_1))
   encArg(q4(x_1)) -> q4(encArg(x_1))
   encArg(q5(x_1)) -> q5(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(encArg(x_1))
   encArg(cons_1(x_1)) -> 1(encArg(x_1))
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_q0(x_1) -> q0(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_1(x_1) -> 1(encArg(x_1))
   encode_q1(x_1) -> q1(encArg(x_1))
   encode_q2(x_1) -> q2(encArg(x_1))
   encode_q3(x_1) -> q3(encArg(x_1))
   encode_q4(x_1) -> q4(encArg(x_1))
   encode_q5(x_1) -> q5(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(n^1, INF).


The TRS R consists of the following rules:

   0(q0(0(x1))) -> 0(0(q0(x1)))
   0(q0(h(x1))) -> 0(0(q0(h(x1))))
   0(q0(1(x1))) -> 0(1(q0(x1)))
   1(q0(0(x1))) -> 0(0(q1(x1)))
   1(q0(h(x1))) -> 0(0(q1(h(x1))))
   1(q0(1(x1))) -> 0(1(q1(x1)))
   1(q1(0(x1))) -> 1(0(q1(x1)))
   1(q1(h(x1))) -> 1(0(q1(h(x1))))
   1(q1(1(x1))) -> 1(1(q1(x1)))
   0(q1(0(x1))) -> 0(0(q2(x1)))
   0(q1(h(x1))) -> 0(0(q2(h(x1))))
   0(q1(1(x1))) -> 0(1(q2(x1)))
   1(q2(0(x1))) -> 1(0(q2(x1)))
   1(q2(h(x1))) -> 1(0(q2(h(x1))))
   1(q2(1(x1))) -> 1(1(q2(x1)))
   0(q2(x1)) -> q3(1(x1))
   1(q3(x1)) -> q3(1(x1))
   0(q3(x1)) -> q4(0(x1))
   1(q4(x1)) -> q4(1(x1))
   0(q4(0(x1))) -> 1(0(q5(x1)))
   0(q4(h(x1))) -> 1(0(q5(h(x1))))
   0(q4(1(x1))) -> 1(1(q5(x1)))
   1(q5(0(x1))) -> 0(0(q1(x1)))
   1(q5(h(x1))) -> 0(0(q1(h(x1))))
   1(q5(1(x1))) -> 0(1(q1(x1)))
   h(q0(x1)) -> h(0(q0(x1)))
   h(q1(x1)) -> h(0(q1(x1)))
   h(q2(x1)) -> h(0(q2(x1)))
   h(q3(x1)) -> h(0(q3(x1)))
   h(q4(x1)) -> h(0(q4(x1)))
   h(q5(x1)) -> h(0(q5(x1)))

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

   encArg(q0(x_1)) -> q0(encArg(x_1))
   encArg(q1(x_1)) -> q1(encArg(x_1))
   encArg(q2(x_1)) -> q2(encArg(x_1))
   encArg(q3(x_1)) -> q3(encArg(x_1))
   encArg(q4(x_1)) -> q4(encArg(x_1))
   encArg(q5(x_1)) -> q5(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(encArg(x_1))
   encArg(cons_1(x_1)) -> 1(encArg(x_1))
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_q0(x_1) -> q0(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_1(x_1) -> 1(encArg(x_1))
   encode_q1(x_1) -> q1(encArg(x_1))
   encode_q2(x_1) -> q2(encArg(x_1))
   encode_q3(x_1) -> q3(encArg(x_1))
   encode_q4(x_1) -> q4(encArg(x_1))
   encode_q5(x_1) -> q5(encArg(x_1))

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

(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:

   0(q0(0(x1))) -> 0(0(q0(x1)))
   0(q0(h(x1))) -> 0(0(q0(h(x1))))
   0(q0(1(x1))) -> 0(1(q0(x1)))
   1(q0(0(x1))) -> 0(0(q1(x1)))
   1(q0(h(x1))) -> 0(0(q1(h(x1))))
   1(q0(1(x1))) -> 0(1(q1(x1)))
   1(q1(0(x1))) -> 1(0(q1(x1)))
   1(q1(h(x1))) -> 1(0(q1(h(x1))))
   1(q1(1(x1))) -> 1(1(q1(x1)))
   0(q1(0(x1))) -> 0(0(q2(x1)))
   0(q1(h(x1))) -> 0(0(q2(h(x1))))
   0(q1(1(x1))) -> 0(1(q2(x1)))
   1(q2(0(x1))) -> 1(0(q2(x1)))
   1(q2(h(x1))) -> 1(0(q2(h(x1))))
   1(q2(1(x1))) -> 1(1(q2(x1)))
   0(q2(x1)) -> q3(1(x1))
   1(q3(x1)) -> q3(1(x1))
   0(q3(x1)) -> q4(0(x1))
   1(q4(x1)) -> q4(1(x1))
   0(q4(0(x1))) -> 1(0(q5(x1)))
   0(q4(h(x1))) -> 1(0(q5(h(x1))))
   0(q4(1(x1))) -> 1(1(q5(x1)))
   1(q5(0(x1))) -> 0(0(q1(x1)))
   1(q5(h(x1))) -> 0(0(q1(h(x1))))
   1(q5(1(x1))) -> 0(1(q1(x1)))
   h(q0(x1)) -> h(0(q0(x1)))
   h(q1(x1)) -> h(0(q1(x1)))
   h(q2(x1)) -> h(0(q2(x1)))
   h(q3(x1)) -> h(0(q3(x1)))
   h(q4(x1)) -> h(0(q4(x1)))
   h(q5(x1)) -> h(0(q5(x1)))

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

   encArg(q0(x_1)) -> q0(encArg(x_1))
   encArg(q1(x_1)) -> q1(encArg(x_1))
   encArg(q2(x_1)) -> q2(encArg(x_1))
   encArg(q3(x_1)) -> q3(encArg(x_1))
   encArg(q4(x_1)) -> q4(encArg(x_1))
   encArg(q5(x_1)) -> q5(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(encArg(x_1))
   encArg(cons_1(x_1)) -> 1(encArg(x_1))
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_q0(x_1) -> q0(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_1(x_1) -> 1(encArg(x_1))
   encode_q1(x_1) -> q1(encArg(x_1))
   encode_q2(x_1) -> q2(encArg(x_1))
   encode_q3(x_1) -> q3(encArg(x_1))
   encode_q4(x_1) -> q4(encArg(x_1))
   encode_q5(x_1) -> q5(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(q3(x1)) ->^+ q3(1(x1))

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

The pumping substitution is [x1 / q3(x1)].

The result substitution is [ ].




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

(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:

   0(q0(0(x1))) -> 0(0(q0(x1)))
   0(q0(h(x1))) -> 0(0(q0(h(x1))))
   0(q0(1(x1))) -> 0(1(q0(x1)))
   1(q0(0(x1))) -> 0(0(q1(x1)))
   1(q0(h(x1))) -> 0(0(q1(h(x1))))
   1(q0(1(x1))) -> 0(1(q1(x1)))
   1(q1(0(x1))) -> 1(0(q1(x1)))
   1(q1(h(x1))) -> 1(0(q1(h(x1))))
   1(q1(1(x1))) -> 1(1(q1(x1)))
   0(q1(0(x1))) -> 0(0(q2(x1)))
   0(q1(h(x1))) -> 0(0(q2(h(x1))))
   0(q1(1(x1))) -> 0(1(q2(x1)))
   1(q2(0(x1))) -> 1(0(q2(x1)))
   1(q2(h(x1))) -> 1(0(q2(h(x1))))
   1(q2(1(x1))) -> 1(1(q2(x1)))
   0(q2(x1)) -> q3(1(x1))
   1(q3(x1)) -> q3(1(x1))
   0(q3(x1)) -> q4(0(x1))
   1(q4(x1)) -> q4(1(x1))
   0(q4(0(x1))) -> 1(0(q5(x1)))
   0(q4(h(x1))) -> 1(0(q5(h(x1))))
   0(q4(1(x1))) -> 1(1(q5(x1)))
   1(q5(0(x1))) -> 0(0(q1(x1)))
   1(q5(h(x1))) -> 0(0(q1(h(x1))))
   1(q5(1(x1))) -> 0(1(q1(x1)))
   h(q0(x1)) -> h(0(q0(x1)))
   h(q1(x1)) -> h(0(q1(x1)))
   h(q2(x1)) -> h(0(q2(x1)))
   h(q3(x1)) -> h(0(q3(x1)))
   h(q4(x1)) -> h(0(q4(x1)))
   h(q5(x1)) -> h(0(q5(x1)))

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

   encArg(q0(x_1)) -> q0(encArg(x_1))
   encArg(q1(x_1)) -> q1(encArg(x_1))
   encArg(q2(x_1)) -> q2(encArg(x_1))
   encArg(q3(x_1)) -> q3(encArg(x_1))
   encArg(q4(x_1)) -> q4(encArg(x_1))
   encArg(q5(x_1)) -> q5(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(encArg(x_1))
   encArg(cons_1(x_1)) -> 1(encArg(x_1))
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_q0(x_1) -> q0(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_1(x_1) -> 1(encArg(x_1))
   encode_q1(x_1) -> q1(encArg(x_1))
   encode_q2(x_1) -> q2(encArg(x_1))
   encode_q3(x_1) -> q3(encArg(x_1))
   encode_q4(x_1) -> q4(encArg(x_1))
   encode_q5(x_1) -> q5(encArg(x_1))

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

(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:

   0(q0(0(x1))) -> 0(0(q0(x1)))
   0(q0(h(x1))) -> 0(0(q0(h(x1))))
   0(q0(1(x1))) -> 0(1(q0(x1)))
   1(q0(0(x1))) -> 0(0(q1(x1)))
   1(q0(h(x1))) -> 0(0(q1(h(x1))))
   1(q0(1(x1))) -> 0(1(q1(x1)))
   1(q1(0(x1))) -> 1(0(q1(x1)))
   1(q1(h(x1))) -> 1(0(q1(h(x1))))
   1(q1(1(x1))) -> 1(1(q1(x1)))
   0(q1(0(x1))) -> 0(0(q2(x1)))
   0(q1(h(x1))) -> 0(0(q2(h(x1))))
   0(q1(1(x1))) -> 0(1(q2(x1)))
   1(q2(0(x1))) -> 1(0(q2(x1)))
   1(q2(h(x1))) -> 1(0(q2(h(x1))))
   1(q2(1(x1))) -> 1(1(q2(x1)))
   0(q2(x1)) -> q3(1(x1))
   1(q3(x1)) -> q3(1(x1))
   0(q3(x1)) -> q4(0(x1))
   1(q4(x1)) -> q4(1(x1))
   0(q4(0(x1))) -> 1(0(q5(x1)))
   0(q4(h(x1))) -> 1(0(q5(h(x1))))
   0(q4(1(x1))) -> 1(1(q5(x1)))
   1(q5(0(x1))) -> 0(0(q1(x1)))
   1(q5(h(x1))) -> 0(0(q1(h(x1))))
   1(q5(1(x1))) -> 0(1(q1(x1)))
   h(q0(x1)) -> h(0(q0(x1)))
   h(q1(x1)) -> h(0(q1(x1)))
   h(q2(x1)) -> h(0(q2(x1)))
   h(q3(x1)) -> h(0(q3(x1)))
   h(q4(x1)) -> h(0(q4(x1)))
   h(q5(x1)) -> h(0(q5(x1)))

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

   encArg(q0(x_1)) -> q0(encArg(x_1))
   encArg(q1(x_1)) -> q1(encArg(x_1))
   encArg(q2(x_1)) -> q2(encArg(x_1))
   encArg(q3(x_1)) -> q3(encArg(x_1))
   encArg(q4(x_1)) -> q4(encArg(x_1))
   encArg(q5(x_1)) -> q5(encArg(x_1))
   encArg(cons_0(x_1)) -> 0(encArg(x_1))
   encArg(cons_1(x_1)) -> 1(encArg(x_1))
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_q0(x_1) -> q0(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_1(x_1) -> 1(encArg(x_1))
   encode_q1(x_1) -> q1(encArg(x_1))
   encode_q2(x_1) -> q2(encArg(x_1))
   encode_q3(x_1) -> q3(encArg(x_1))
   encode_q4(x_1) -> q4(encArg(x_1))
   encode_q5(x_1) -> q5(encArg(x_1))

Rewrite Strategy: FULL