/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(NON_POLY, ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(EXP, INF).

(0) CpxTRS
(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(2) TRS for Loop Detection
(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
(4) BEST
    (5) proven lower bound
        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
        (7) BOUNDS(n^1, INF)
    (8) TRS for Loop Detection
        (9) DecreasingLoopProof [FINISHED, 522 ms]
        (10) BOUNDS(EXP, INF)


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


The TRS R consists of the following rules:

   a__pairNs -> cons(0, incr(oddNs))
   a__oddNs -> a__incr(a__pairNs)
   a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS))
   a__take(0, XS) -> nil
   a__take(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS))
   a__zip(nil, XS) -> nil
   a__zip(X, nil) -> nil
   a__zip(cons(X, XS), cons(Y, YS)) -> cons(pair(mark(X), mark(Y)), zip(XS, YS))
   a__tail(cons(X, XS)) -> mark(XS)
   a__repItems(nil) -> nil
   a__repItems(cons(X, XS)) -> cons(mark(X), cons(X, repItems(XS)))
   mark(pairNs) -> a__pairNs
   mark(incr(X)) -> a__incr(mark(X))
   mark(oddNs) -> a__oddNs
   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
   mark(zip(X1, X2)) -> a__zip(mark(X1), mark(X2))
   mark(tail(X)) -> a__tail(mark(X))
   mark(repItems(X)) -> a__repItems(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(0) -> 0
   mark(s(X)) -> s(mark(X))
   mark(nil) -> nil
   mark(pair(X1, X2)) -> pair(mark(X1), mark(X2))
   a__pairNs -> pairNs
   a__incr(X) -> incr(X)
   a__oddNs -> oddNs
   a__take(X1, X2) -> take(X1, X2)
   a__zip(X1, X2) -> zip(X1, X2)
   a__tail(X) -> tail(X)
   a__repItems(X) -> repItems(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(EXP, INF).


The TRS R consists of the following rules:

   a__pairNs -> cons(0, incr(oddNs))
   a__oddNs -> a__incr(a__pairNs)
   a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS))
   a__take(0, XS) -> nil
   a__take(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS))
   a__zip(nil, XS) -> nil
   a__zip(X, nil) -> nil
   a__zip(cons(X, XS), cons(Y, YS)) -> cons(pair(mark(X), mark(Y)), zip(XS, YS))
   a__tail(cons(X, XS)) -> mark(XS)
   a__repItems(nil) -> nil
   a__repItems(cons(X, XS)) -> cons(mark(X), cons(X, repItems(XS)))
   mark(pairNs) -> a__pairNs
   mark(incr(X)) -> a__incr(mark(X))
   mark(oddNs) -> a__oddNs
   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
   mark(zip(X1, X2)) -> a__zip(mark(X1), mark(X2))
   mark(tail(X)) -> a__tail(mark(X))
   mark(repItems(X)) -> a__repItems(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(0) -> 0
   mark(s(X)) -> s(mark(X))
   mark(nil) -> nil
   mark(pair(X1, X2)) -> pair(mark(X1), mark(X2))
   a__pairNs -> pairNs
   a__incr(X) -> incr(X)
   a__oddNs -> oddNs
   a__take(X1, X2) -> take(X1, X2)
   a__zip(X1, X2) -> zip(X1, X2)
   a__tail(X) -> tail(X)
   a__repItems(X) -> repItems(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

mark(repItems(X)) ->^+ a__repItems(mark(X))

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

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

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(EXP, INF).


The TRS R consists of the following rules:

   a__pairNs -> cons(0, incr(oddNs))
   a__oddNs -> a__incr(a__pairNs)
   a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS))
   a__take(0, XS) -> nil
   a__take(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS))
   a__zip(nil, XS) -> nil
   a__zip(X, nil) -> nil
   a__zip(cons(X, XS), cons(Y, YS)) -> cons(pair(mark(X), mark(Y)), zip(XS, YS))
   a__tail(cons(X, XS)) -> mark(XS)
   a__repItems(nil) -> nil
   a__repItems(cons(X, XS)) -> cons(mark(X), cons(X, repItems(XS)))
   mark(pairNs) -> a__pairNs
   mark(incr(X)) -> a__incr(mark(X))
   mark(oddNs) -> a__oddNs
   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
   mark(zip(X1, X2)) -> a__zip(mark(X1), mark(X2))
   mark(tail(X)) -> a__tail(mark(X))
   mark(repItems(X)) -> a__repItems(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(0) -> 0
   mark(s(X)) -> s(mark(X))
   mark(nil) -> nil
   mark(pair(X1, X2)) -> pair(mark(X1), mark(X2))
   a__pairNs -> pairNs
   a__incr(X) -> incr(X)
   a__oddNs -> oddNs
   a__take(X1, X2) -> take(X1, X2)
   a__zip(X1, X2) -> zip(X1, X2)
   a__tail(X) -> tail(X)
   a__repItems(X) -> repItems(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(EXP, INF).


The TRS R consists of the following rules:

   a__pairNs -> cons(0, incr(oddNs))
   a__oddNs -> a__incr(a__pairNs)
   a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS))
   a__take(0, XS) -> nil
   a__take(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS))
   a__zip(nil, XS) -> nil
   a__zip(X, nil) -> nil
   a__zip(cons(X, XS), cons(Y, YS)) -> cons(pair(mark(X), mark(Y)), zip(XS, YS))
   a__tail(cons(X, XS)) -> mark(XS)
   a__repItems(nil) -> nil
   a__repItems(cons(X, XS)) -> cons(mark(X), cons(X, repItems(XS)))
   mark(pairNs) -> a__pairNs
   mark(incr(X)) -> a__incr(mark(X))
   mark(oddNs) -> a__oddNs
   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
   mark(zip(X1, X2)) -> a__zip(mark(X1), mark(X2))
   mark(tail(X)) -> a__tail(mark(X))
   mark(repItems(X)) -> a__repItems(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(0) -> 0
   mark(s(X)) -> s(mark(X))
   mark(nil) -> nil
   mark(pair(X1, X2)) -> pair(mark(X1), mark(X2))
   a__pairNs -> pairNs
   a__incr(X) -> incr(X)
   a__oddNs -> oddNs
   a__take(X1, X2) -> take(X1, X2)
   a__zip(X1, X2) -> zip(X1, X2)
   a__tail(X) -> tail(X)
   a__repItems(X) -> repItems(X)

S is empty.
Rewrite Strategy: FULL
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(9) DecreasingLoopProof (FINISHED)
The following loop(s) give(s) rise to the lower bound EXP:

The rewrite sequence

mark(repItems(cons(X11_0, X22_0))) ->^+ cons(mark(mark(X11_0)), cons(mark(X11_0), repItems(X22_0)))

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

The pumping substitution is [X11_0 / repItems(cons(X11_0, X22_0))].

The result substitution is [ ].



The rewrite sequence

mark(repItems(cons(X11_0, X22_0))) ->^+ cons(mark(mark(X11_0)), cons(mark(X11_0), repItems(X22_0)))

gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0].

The pumping substitution is [X11_0 / repItems(cons(X11_0, X22_0))].

The result substitution is [ ].




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(10)
BOUNDS(EXP, INF)