/export/starexec/sandbox/solver/bin/starexec_run_complexity /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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) CpxTRS
(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(2) TRS for Loop Detection
(3) DecreasingLoopProof [LOWER BOUND(ID), 39 ms]
(4) BEST
    (5) proven lower bound
        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
        (7) BOUNDS(n^1, INF)
    (8) TRS for Loop Detection


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(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   eq(n__0, n__0) -> true
   eq(n__s(X), n__s(Y)) -> eq(activate(X), activate(Y))
   eq(X, Y) -> false
   inf(X) -> cons(X, n__inf(s(X)))
   take(0, X) -> nil
   take(s(X), cons(Y, L)) -> cons(activate(Y), n__take(activate(X), activate(L)))
   length(nil) -> 0
   length(cons(X, L)) -> s(n__length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   inf(X) -> n__inf(X)
   take(X1, X2) -> n__take(X1, X2)
   length(X) -> n__length(X)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(X)
   activate(n__inf(X)) -> inf(X)
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__length(X)) -> length(X)
   activate(X) -> X

S is empty.
Rewrite Strategy: FULL
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(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(2)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   eq(n__0, n__0) -> true
   eq(n__s(X), n__s(Y)) -> eq(activate(X), activate(Y))
   eq(X, Y) -> false
   inf(X) -> cons(X, n__inf(s(X)))
   take(0, X) -> nil
   take(s(X), cons(Y, L)) -> cons(activate(Y), n__take(activate(X), activate(L)))
   length(nil) -> 0
   length(cons(X, L)) -> s(n__length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   inf(X) -> n__inf(X)
   take(X1, X2) -> n__take(X1, X2)
   length(X) -> n__length(X)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(X)
   activate(n__inf(X)) -> inf(X)
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__length(X)) -> length(X)
   activate(X) -> X

S is empty.
Rewrite Strategy: FULL
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(3) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

activate(n__length(cons(X1_0, L2_0))) ->^+ s(n__length(activate(L2_0)))

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

The pumping substitution is [L2_0 / n__length(cons(X1_0, L2_0))].

The result substitution is [ ].




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(4)
Complex Obligation (BEST)

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(5)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   eq(n__0, n__0) -> true
   eq(n__s(X), n__s(Y)) -> eq(activate(X), activate(Y))
   eq(X, Y) -> false
   inf(X) -> cons(X, n__inf(s(X)))
   take(0, X) -> nil
   take(s(X), cons(Y, L)) -> cons(activate(Y), n__take(activate(X), activate(L)))
   length(nil) -> 0
   length(cons(X, L)) -> s(n__length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   inf(X) -> n__inf(X)
   take(X1, X2) -> n__take(X1, X2)
   length(X) -> n__length(X)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(X)
   activate(n__inf(X)) -> inf(X)
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__length(X)) -> length(X)
   activate(X) -> X

S is empty.
Rewrite Strategy: FULL
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(6) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(7)
BOUNDS(n^1, INF)

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(8)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   eq(n__0, n__0) -> true
   eq(n__s(X), n__s(Y)) -> eq(activate(X), activate(Y))
   eq(X, Y) -> false
   inf(X) -> cons(X, n__inf(s(X)))
   take(0, X) -> nil
   take(s(X), cons(Y, L)) -> cons(activate(Y), n__take(activate(X), activate(L)))
   length(nil) -> 0
   length(cons(X, L)) -> s(n__length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   inf(X) -> n__inf(X)
   take(X1, X2) -> n__take(X1, X2)
   length(X) -> n__length(X)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(X)
   activate(n__inf(X)) -> inf(X)
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__length(X)) -> length(X)
   activate(X) -> X

S is empty.
Rewrite Strategy: FULL