/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 289 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:

   a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z)))
   a____(X, nil) -> mark(X)
   a____(nil, X) -> mark(X)
   a__U11(tt) -> a__U12(tt)
   a__U12(tt) -> tt
   a__isNePal(__(I, __(P, I))) -> a__U11(tt)
   mark(__(X1, X2)) -> a____(mark(X1), mark(X2))
   mark(U11(X)) -> a__U11(mark(X))
   mark(U12(X)) -> a__U12(mark(X))
   mark(isNePal(X)) -> a__isNePal(mark(X))
   mark(nil) -> nil
   mark(tt) -> tt
   a____(X1, X2) -> __(X1, X2)
   a__U11(X) -> U11(X)
   a__U12(X) -> U12(X)
   a__isNePal(X) -> isNePal(X)

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(__(x_1, x_2)) -> __(encArg(x_1), encArg(x_2))
   encArg(nil) -> nil
   encArg(tt) -> tt
   encArg(U11(x_1)) -> U11(encArg(x_1))
   encArg(U12(x_1)) -> U12(encArg(x_1))
   encArg(isNePal(x_1)) -> isNePal(encArg(x_1))
   encArg(cons_a____(x_1, x_2)) -> a____(encArg(x_1), encArg(x_2))
   encArg(cons_a__U11(x_1)) -> a__U11(encArg(x_1))
   encArg(cons_a__U12(x_1)) -> a__U12(encArg(x_1))
   encArg(cons_a__isNePal(x_1)) -> a__isNePal(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a____(x_1, x_2) -> a____(encArg(x_1), encArg(x_2))
   encode___(x_1, x_2) -> __(encArg(x_1), encArg(x_2))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_nil -> nil
   encode_a__U11(x_1) -> a__U11(encArg(x_1))
   encode_tt -> tt
   encode_a__U12(x_1) -> a__U12(encArg(x_1))
   encode_a__isNePal(x_1) -> a__isNePal(encArg(x_1))
   encode_U11(x_1) -> U11(encArg(x_1))
   encode_U12(x_1) -> U12(encArg(x_1))
   encode_isNePal(x_1) -> isNePal(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:

   a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z)))
   a____(X, nil) -> mark(X)
   a____(nil, X) -> mark(X)
   a__U11(tt) -> a__U12(tt)
   a__U12(tt) -> tt
   a__isNePal(__(I, __(P, I))) -> a__U11(tt)
   mark(__(X1, X2)) -> a____(mark(X1), mark(X2))
   mark(U11(X)) -> a__U11(mark(X))
   mark(U12(X)) -> a__U12(mark(X))
   mark(isNePal(X)) -> a__isNePal(mark(X))
   mark(nil) -> nil
   mark(tt) -> tt
   a____(X1, X2) -> __(X1, X2)
   a__U11(X) -> U11(X)
   a__U12(X) -> U12(X)
   a__isNePal(X) -> isNePal(X)

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

   encArg(__(x_1, x_2)) -> __(encArg(x_1), encArg(x_2))
   encArg(nil) -> nil
   encArg(tt) -> tt
   encArg(U11(x_1)) -> U11(encArg(x_1))
   encArg(U12(x_1)) -> U12(encArg(x_1))
   encArg(isNePal(x_1)) -> isNePal(encArg(x_1))
   encArg(cons_a____(x_1, x_2)) -> a____(encArg(x_1), encArg(x_2))
   encArg(cons_a__U11(x_1)) -> a__U11(encArg(x_1))
   encArg(cons_a__U12(x_1)) -> a__U12(encArg(x_1))
   encArg(cons_a__isNePal(x_1)) -> a__isNePal(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a____(x_1, x_2) -> a____(encArg(x_1), encArg(x_2))
   encode___(x_1, x_2) -> __(encArg(x_1), encArg(x_2))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_nil -> nil
   encode_a__U11(x_1) -> a__U11(encArg(x_1))
   encode_tt -> tt
   encode_a__U12(x_1) -> a__U12(encArg(x_1))
   encode_a__isNePal(x_1) -> a__isNePal(encArg(x_1))
   encode_U11(x_1) -> U11(encArg(x_1))
   encode_U12(x_1) -> U12(encArg(x_1))
   encode_isNePal(x_1) -> isNePal(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:

   a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z)))
   a____(X, nil) -> mark(X)
   a____(nil, X) -> mark(X)
   a__U11(tt) -> a__U12(tt)
   a__U12(tt) -> tt
   a__isNePal(__(I, __(P, I))) -> a__U11(tt)
   mark(__(X1, X2)) -> a____(mark(X1), mark(X2))
   mark(U11(X)) -> a__U11(mark(X))
   mark(U12(X)) -> a__U12(mark(X))
   mark(isNePal(X)) -> a__isNePal(mark(X))
   mark(nil) -> nil
   mark(tt) -> tt
   a____(X1, X2) -> __(X1, X2)
   a__U11(X) -> U11(X)
   a__U12(X) -> U12(X)
   a__isNePal(X) -> isNePal(X)

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

   encArg(__(x_1, x_2)) -> __(encArg(x_1), encArg(x_2))
   encArg(nil) -> nil
   encArg(tt) -> tt
   encArg(U11(x_1)) -> U11(encArg(x_1))
   encArg(U12(x_1)) -> U12(encArg(x_1))
   encArg(isNePal(x_1)) -> isNePal(encArg(x_1))
   encArg(cons_a____(x_1, x_2)) -> a____(encArg(x_1), encArg(x_2))
   encArg(cons_a__U11(x_1)) -> a__U11(encArg(x_1))
   encArg(cons_a__U12(x_1)) -> a__U12(encArg(x_1))
   encArg(cons_a__isNePal(x_1)) -> a__isNePal(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a____(x_1, x_2) -> a____(encArg(x_1), encArg(x_2))
   encode___(x_1, x_2) -> __(encArg(x_1), encArg(x_2))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_nil -> nil
   encode_a__U11(x_1) -> a__U11(encArg(x_1))
   encode_tt -> tt
   encode_a__U12(x_1) -> a__U12(encArg(x_1))
   encode_a__isNePal(x_1) -> a__isNePal(encArg(x_1))
   encode_U11(x_1) -> U11(encArg(x_1))
   encode_U12(x_1) -> U12(encArg(x_1))
   encode_isNePal(x_1) -> isNePal(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:

   a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z)))
   a____(X, nil) -> mark(X)
   a____(nil, X) -> mark(X)
   a__U11(tt) -> a__U12(tt)
   a__U12(tt) -> tt
   a__isNePal(__(I, __(P, I))) -> a__U11(tt)
   mark(__(X1, X2)) -> a____(mark(X1), mark(X2))
   mark(U11(X)) -> a__U11(mark(X))
   mark(U12(X)) -> a__U12(mark(X))
   mark(isNePal(X)) -> a__isNePal(mark(X))
   mark(nil) -> nil
   mark(tt) -> tt
   a____(X1, X2) -> __(X1, X2)
   a__U11(X) -> U11(X)
   a__U12(X) -> U12(X)
   a__isNePal(X) -> isNePal(X)

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

   encArg(__(x_1, x_2)) -> __(encArg(x_1), encArg(x_2))
   encArg(nil) -> nil
   encArg(tt) -> tt
   encArg(U11(x_1)) -> U11(encArg(x_1))
   encArg(U12(x_1)) -> U12(encArg(x_1))
   encArg(isNePal(x_1)) -> isNePal(encArg(x_1))
   encArg(cons_a____(x_1, x_2)) -> a____(encArg(x_1), encArg(x_2))
   encArg(cons_a__U11(x_1)) -> a__U11(encArg(x_1))
   encArg(cons_a__U12(x_1)) -> a__U12(encArg(x_1))
   encArg(cons_a__isNePal(x_1)) -> a__isNePal(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a____(x_1, x_2) -> a____(encArg(x_1), encArg(x_2))
   encode___(x_1, x_2) -> __(encArg(x_1), encArg(x_2))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_nil -> nil
   encode_a__U11(x_1) -> a__U11(encArg(x_1))
   encode_tt -> tt
   encode_a__U12(x_1) -> a__U12(encArg(x_1))
   encode_a__isNePal(x_1) -> a__isNePal(encArg(x_1))
   encode_U11(x_1) -> U11(encArg(x_1))
   encode_U12(x_1) -> U12(encArg(x_1))
   encode_isNePal(x_1) -> isNePal(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

mark(U12(X)) ->^+ a__U12(mark(X))

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

The pumping substitution is [X / U12(X)].

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:

   a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z)))
   a____(X, nil) -> mark(X)
   a____(nil, X) -> mark(X)
   a__U11(tt) -> a__U12(tt)
   a__U12(tt) -> tt
   a__isNePal(__(I, __(P, I))) -> a__U11(tt)
   mark(__(X1, X2)) -> a____(mark(X1), mark(X2))
   mark(U11(X)) -> a__U11(mark(X))
   mark(U12(X)) -> a__U12(mark(X))
   mark(isNePal(X)) -> a__isNePal(mark(X))
   mark(nil) -> nil
   mark(tt) -> tt
   a____(X1, X2) -> __(X1, X2)
   a__U11(X) -> U11(X)
   a__U12(X) -> U12(X)
   a__isNePal(X) -> isNePal(X)

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

   encArg(__(x_1, x_2)) -> __(encArg(x_1), encArg(x_2))
   encArg(nil) -> nil
   encArg(tt) -> tt
   encArg(U11(x_1)) -> U11(encArg(x_1))
   encArg(U12(x_1)) -> U12(encArg(x_1))
   encArg(isNePal(x_1)) -> isNePal(encArg(x_1))
   encArg(cons_a____(x_1, x_2)) -> a____(encArg(x_1), encArg(x_2))
   encArg(cons_a__U11(x_1)) -> a__U11(encArg(x_1))
   encArg(cons_a__U12(x_1)) -> a__U12(encArg(x_1))
   encArg(cons_a__isNePal(x_1)) -> a__isNePal(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a____(x_1, x_2) -> a____(encArg(x_1), encArg(x_2))
   encode___(x_1, x_2) -> __(encArg(x_1), encArg(x_2))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_nil -> nil
   encode_a__U11(x_1) -> a__U11(encArg(x_1))
   encode_tt -> tt
   encode_a__U12(x_1) -> a__U12(encArg(x_1))
   encode_a__isNePal(x_1) -> a__isNePal(encArg(x_1))
   encode_U11(x_1) -> U11(encArg(x_1))
   encode_U12(x_1) -> U12(encArg(x_1))
   encode_isNePal(x_1) -> isNePal(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:

   a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z)))
   a____(X, nil) -> mark(X)
   a____(nil, X) -> mark(X)
   a__U11(tt) -> a__U12(tt)
   a__U12(tt) -> tt
   a__isNePal(__(I, __(P, I))) -> a__U11(tt)
   mark(__(X1, X2)) -> a____(mark(X1), mark(X2))
   mark(U11(X)) -> a__U11(mark(X))
   mark(U12(X)) -> a__U12(mark(X))
   mark(isNePal(X)) -> a__isNePal(mark(X))
   mark(nil) -> nil
   mark(tt) -> tt
   a____(X1, X2) -> __(X1, X2)
   a__U11(X) -> U11(X)
   a__U12(X) -> U12(X)
   a__isNePal(X) -> isNePal(X)

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

   encArg(__(x_1, x_2)) -> __(encArg(x_1), encArg(x_2))
   encArg(nil) -> nil
   encArg(tt) -> tt
   encArg(U11(x_1)) -> U11(encArg(x_1))
   encArg(U12(x_1)) -> U12(encArg(x_1))
   encArg(isNePal(x_1)) -> isNePal(encArg(x_1))
   encArg(cons_a____(x_1, x_2)) -> a____(encArg(x_1), encArg(x_2))
   encArg(cons_a__U11(x_1)) -> a__U11(encArg(x_1))
   encArg(cons_a__U12(x_1)) -> a__U12(encArg(x_1))
   encArg(cons_a__isNePal(x_1)) -> a__isNePal(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a____(x_1, x_2) -> a____(encArg(x_1), encArg(x_2))
   encode___(x_1, x_2) -> __(encArg(x_1), encArg(x_2))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_nil -> nil
   encode_a__U11(x_1) -> a__U11(encArg(x_1))
   encode_tt -> tt
   encode_a__U12(x_1) -> a__U12(encArg(x_1))
   encode_a__isNePal(x_1) -> a__isNePal(encArg(x_1))
   encode_U11(x_1) -> U11(encArg(x_1))
   encode_U12(x_1) -> U12(encArg(x_1))
   encode_isNePal(x_1) -> isNePal(encArg(x_1))

Rewrite Strategy: FULL