/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 35 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:

   natsFrom(N) -> cons(N, n__natsFrom(s(N)))
   fst(pair(XS, YS)) -> XS
   snd(pair(XS, YS)) -> YS
   splitAt(0, XS) -> pair(nil, XS)
   splitAt(s(N), cons(X, XS)) -> u(splitAt(N, activate(XS)), N, X, activate(XS))
   u(pair(YS, ZS), N, X, XS) -> pair(cons(activate(X), YS), ZS)
   head(cons(N, XS)) -> N
   tail(cons(N, XS)) -> activate(XS)
   sel(N, XS) -> head(afterNth(N, XS))
   take(N, XS) -> fst(splitAt(N, XS))
   afterNth(N, XS) -> snd(splitAt(N, XS))
   natsFrom(X) -> n__natsFrom(X)
   activate(n__natsFrom(X)) -> natsFrom(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:

   natsFrom(N) -> cons(N, n__natsFrom(s(N)))
   fst(pair(XS, YS)) -> XS
   snd(pair(XS, YS)) -> YS
   splitAt(0, XS) -> pair(nil, XS)
   splitAt(s(N), cons(X, XS)) -> u(splitAt(N, activate(XS)), N, X, activate(XS))
   u(pair(YS, ZS), N, X, XS) -> pair(cons(activate(X), YS), ZS)
   head(cons(N, XS)) -> N
   tail(cons(N, XS)) -> activate(XS)
   sel(N, XS) -> head(afterNth(N, XS))
   take(N, XS) -> fst(splitAt(N, XS))
   afterNth(N, XS) -> snd(splitAt(N, XS))
   natsFrom(X) -> n__natsFrom(X)
   activate(n__natsFrom(X)) -> natsFrom(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

splitAt(s(N), cons(X, XS)) ->^+ u(splitAt(N, XS), N, X, activate(XS))

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

The pumping substitution is [N / s(N), XS / cons(X, XS)].

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:

   natsFrom(N) -> cons(N, n__natsFrom(s(N)))
   fst(pair(XS, YS)) -> XS
   snd(pair(XS, YS)) -> YS
   splitAt(0, XS) -> pair(nil, XS)
   splitAt(s(N), cons(X, XS)) -> u(splitAt(N, activate(XS)), N, X, activate(XS))
   u(pair(YS, ZS), N, X, XS) -> pair(cons(activate(X), YS), ZS)
   head(cons(N, XS)) -> N
   tail(cons(N, XS)) -> activate(XS)
   sel(N, XS) -> head(afterNth(N, XS))
   take(N, XS) -> fst(splitAt(N, XS))
   afterNth(N, XS) -> snd(splitAt(N, XS))
   natsFrom(X) -> n__natsFrom(X)
   activate(n__natsFrom(X)) -> natsFrom(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:

   natsFrom(N) -> cons(N, n__natsFrom(s(N)))
   fst(pair(XS, YS)) -> XS
   snd(pair(XS, YS)) -> YS
   splitAt(0, XS) -> pair(nil, XS)
   splitAt(s(N), cons(X, XS)) -> u(splitAt(N, activate(XS)), N, X, activate(XS))
   u(pair(YS, ZS), N, X, XS) -> pair(cons(activate(X), YS), ZS)
   head(cons(N, XS)) -> N
   tail(cons(N, XS)) -> activate(XS)
   sel(N, XS) -> head(afterNth(N, XS))
   take(N, XS) -> fst(splitAt(N, XS))
   afterNth(N, XS) -> snd(splitAt(N, XS))
   natsFrom(X) -> n__natsFrom(X)
   activate(n__natsFrom(X)) -> natsFrom(X)
   activate(X) -> X

S is empty.
Rewrite Strategy: FULL