/export/starexec/sandbox2/solver/bin/starexec_run_complexity /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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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


The TRS R consists of the following rules:

   a__filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M))
   a__filter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M))
   a__sieve(cons(0, Y)) -> cons(0, sieve(Y))
   a__sieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N)))
   a__nats(N) -> cons(mark(N), nats(s(N)))
   a__zprimes -> a__sieve(a__nats(s(s(0))))
   mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3))
   mark(sieve(X)) -> a__sieve(mark(X))
   mark(nats(X)) -> a__nats(mark(X))
   mark(zprimes) -> a__zprimes
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(0) -> 0
   mark(s(X)) -> s(mark(X))
   a__filter(X1, X2, X3) -> filter(X1, X2, X3)
   a__sieve(X) -> sieve(X)
   a__nats(X) -> nats(X)
   a__zprimes -> zprimes

S is empty.
Rewrite Strategy: INNERMOST
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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 (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   a__filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M))
   a__filter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M))
   a__sieve(cons(0, Y)) -> cons(0, sieve(Y))
   a__sieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N)))
   a__nats(N) -> cons(mark(N), nats(s(N)))
   a__zprimes -> a__sieve(a__nats(s(s(0))))
   mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3))
   mark(sieve(X)) -> a__sieve(mark(X))
   mark(nats(X)) -> a__nats(mark(X))
   mark(zprimes) -> a__zprimes
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(0) -> 0
   mark(s(X)) -> s(mark(X))
   a__filter(X1, X2, X3) -> filter(X1, X2, X3)
   a__sieve(X) -> sieve(X)
   a__nats(X) -> nats(X)
   a__zprimes -> zprimes

S is empty.
Rewrite Strategy: INNERMOST
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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(nats(X)) ->^+ a__nats(mark(X))

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

The pumping substitution is [X / nats(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 (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   a__filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M))
   a__filter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M))
   a__sieve(cons(0, Y)) -> cons(0, sieve(Y))
   a__sieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N)))
   a__nats(N) -> cons(mark(N), nats(s(N)))
   a__zprimes -> a__sieve(a__nats(s(s(0))))
   mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3))
   mark(sieve(X)) -> a__sieve(mark(X))
   mark(nats(X)) -> a__nats(mark(X))
   mark(zprimes) -> a__zprimes
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(0) -> 0
   mark(s(X)) -> s(mark(X))
   a__filter(X1, X2, X3) -> filter(X1, X2, X3)
   a__sieve(X) -> sieve(X)
   a__nats(X) -> nats(X)
   a__zprimes -> zprimes

S is empty.
Rewrite Strategy: INNERMOST
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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 (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   a__filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M))
   a__filter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M))
   a__sieve(cons(0, Y)) -> cons(0, sieve(Y))
   a__sieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N)))
   a__nats(N) -> cons(mark(N), nats(s(N)))
   a__zprimes -> a__sieve(a__nats(s(s(0))))
   mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3))
   mark(sieve(X)) -> a__sieve(mark(X))
   mark(nats(X)) -> a__nats(mark(X))
   mark(zprimes) -> a__zprimes
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(0) -> 0
   mark(s(X)) -> s(mark(X))
   a__filter(X1, X2, X3) -> filter(X1, X2, X3)
   a__sieve(X) -> sieve(X)
   a__nats(X) -> nats(X)
   a__zprimes -> zprimes

S is empty.
Rewrite Strategy: INNERMOST