/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 229 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:

   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   from(X) -> cons(X, n__from(n__s(X)))
   first(X1, X2) -> n__first(X1, X2)
   from(X) -> n__from(X)
   s(X) -> n__s(X)
   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(X) -> X

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(0) -> 0
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(n__first(x_1, x_2)) -> n__first(encArg(x_1), encArg(x_2))
   encArg(n__from(x_1)) -> n__from(encArg(x_1))
   encArg(n__s(x_1)) -> n__s(encArg(x_1))
   encArg(cons_first(x_1, x_2)) -> first(encArg(x_1), encArg(x_2))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_first(x_1, x_2) -> first(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_nil -> nil
   encode_s(x_1) -> s(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_n__first(x_1, x_2) -> n__first(encArg(x_1), encArg(x_2))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))
   encode_n__from(x_1) -> n__from(encArg(x_1))
   encode_n__s(x_1) -> n__s(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:

   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   from(X) -> cons(X, n__from(n__s(X)))
   first(X1, X2) -> n__first(X1, X2)
   from(X) -> n__from(X)
   s(X) -> n__s(X)
   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(X) -> X

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

   encArg(0) -> 0
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(n__first(x_1, x_2)) -> n__first(encArg(x_1), encArg(x_2))
   encArg(n__from(x_1)) -> n__from(encArg(x_1))
   encArg(n__s(x_1)) -> n__s(encArg(x_1))
   encArg(cons_first(x_1, x_2)) -> first(encArg(x_1), encArg(x_2))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_first(x_1, x_2) -> first(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_nil -> nil
   encode_s(x_1) -> s(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_n__first(x_1, x_2) -> n__first(encArg(x_1), encArg(x_2))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))
   encode_n__from(x_1) -> n__from(encArg(x_1))
   encode_n__s(x_1) -> n__s(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:

   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   from(X) -> cons(X, n__from(n__s(X)))
   first(X1, X2) -> n__first(X1, X2)
   from(X) -> n__from(X)
   s(X) -> n__s(X)
   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(X) -> X

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

   encArg(0) -> 0
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(n__first(x_1, x_2)) -> n__first(encArg(x_1), encArg(x_2))
   encArg(n__from(x_1)) -> n__from(encArg(x_1))
   encArg(n__s(x_1)) -> n__s(encArg(x_1))
   encArg(cons_first(x_1, x_2)) -> first(encArg(x_1), encArg(x_2))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_first(x_1, x_2) -> first(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_nil -> nil
   encode_s(x_1) -> s(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_n__first(x_1, x_2) -> n__first(encArg(x_1), encArg(x_2))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))
   encode_n__from(x_1) -> n__from(encArg(x_1))
   encode_n__s(x_1) -> n__s(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:

   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   from(X) -> cons(X, n__from(n__s(X)))
   first(X1, X2) -> n__first(X1, X2)
   from(X) -> n__from(X)
   s(X) -> n__s(X)
   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(X) -> X

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

   encArg(0) -> 0
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(n__first(x_1, x_2)) -> n__first(encArg(x_1), encArg(x_2))
   encArg(n__from(x_1)) -> n__from(encArg(x_1))
   encArg(n__s(x_1)) -> n__s(encArg(x_1))
   encArg(cons_first(x_1, x_2)) -> first(encArg(x_1), encArg(x_2))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_first(x_1, x_2) -> first(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_nil -> nil
   encode_s(x_1) -> s(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_n__first(x_1, x_2) -> n__first(encArg(x_1), encArg(x_2))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))
   encode_n__from(x_1) -> n__from(encArg(x_1))
   encode_n__s(x_1) -> n__s(encArg(x_1))

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

The rewrite sequence

activate(n__first(X1, X2)) ->^+ first(activate(X1), activate(X2))

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

The pumping substitution is [X1 / n__first(X1, X2)].

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:

   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   from(X) -> cons(X, n__from(n__s(X)))
   first(X1, X2) -> n__first(X1, X2)
   from(X) -> n__from(X)
   s(X) -> n__s(X)
   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(X) -> X

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

   encArg(0) -> 0
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(n__first(x_1, x_2)) -> n__first(encArg(x_1), encArg(x_2))
   encArg(n__from(x_1)) -> n__from(encArg(x_1))
   encArg(n__s(x_1)) -> n__s(encArg(x_1))
   encArg(cons_first(x_1, x_2)) -> first(encArg(x_1), encArg(x_2))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_first(x_1, x_2) -> first(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_nil -> nil
   encode_s(x_1) -> s(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_n__first(x_1, x_2) -> n__first(encArg(x_1), encArg(x_2))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))
   encode_n__from(x_1) -> n__from(encArg(x_1))
   encode_n__s(x_1) -> n__s(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:

   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   from(X) -> cons(X, n__from(n__s(X)))
   first(X1, X2) -> n__first(X1, X2)
   from(X) -> n__from(X)
   s(X) -> n__s(X)
   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(X) -> X

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

   encArg(0) -> 0
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(n__first(x_1, x_2)) -> n__first(encArg(x_1), encArg(x_2))
   encArg(n__from(x_1)) -> n__from(encArg(x_1))
   encArg(n__s(x_1)) -> n__s(encArg(x_1))
   encArg(cons_first(x_1, x_2)) -> first(encArg(x_1), encArg(x_2))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_first(x_1, x_2) -> first(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_nil -> nil
   encode_s(x_1) -> s(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_n__first(x_1, x_2) -> n__first(encArg(x_1), encArg(x_2))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))
   encode_n__from(x_1) -> n__from(encArg(x_1))
   encode_n__s(x_1) -> n__s(encArg(x_1))

Rewrite Strategy: INNERMOST