/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 51 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:

   filter(cons(X, Y), 0, M) -> cons(0, n__filter(activate(Y), M, M))
   filter(cons(X, Y), s(N), M) -> cons(X, n__filter(activate(Y), N, M))
   sieve(cons(0, Y)) -> cons(0, n__sieve(activate(Y)))
   sieve(cons(s(N), Y)) -> cons(s(N), n__sieve(filter(activate(Y), N, N)))
   nats(N) -> cons(N, n__nats(s(N)))
   zprimes -> sieve(nats(s(s(0))))
   filter(X1, X2, X3) -> n__filter(X1, X2, X3)
   sieve(X) -> n__sieve(X)
   nats(X) -> n__nats(X)
   activate(n__filter(X1, X2, X3)) -> filter(X1, X2, X3)
   activate(n__sieve(X)) -> sieve(X)
   activate(n__nats(X)) -> nats(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:

   filter(cons(X, Y), 0, M) -> cons(0, n__filter(activate(Y), M, M))
   filter(cons(X, Y), s(N), M) -> cons(X, n__filter(activate(Y), N, M))
   sieve(cons(0, Y)) -> cons(0, n__sieve(activate(Y)))
   sieve(cons(s(N), Y)) -> cons(s(N), n__sieve(filter(activate(Y), N, N)))
   nats(N) -> cons(N, n__nats(s(N)))
   zprimes -> sieve(nats(s(s(0))))
   filter(X1, X2, X3) -> n__filter(X1, X2, X3)
   sieve(X) -> n__sieve(X)
   nats(X) -> n__nats(X)
   activate(n__filter(X1, X2, X3)) -> filter(X1, X2, X3)
   activate(n__sieve(X)) -> sieve(X)
   activate(n__nats(X)) -> nats(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

sieve(cons(s(N), n__sieve(X1_0))) ->^+ cons(s(N), n__sieve(filter(sieve(X1_0), N, N)))

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

The pumping substitution is [X1_0 / cons(s(N), n__sieve(X1_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:

   filter(cons(X, Y), 0, M) -> cons(0, n__filter(activate(Y), M, M))
   filter(cons(X, Y), s(N), M) -> cons(X, n__filter(activate(Y), N, M))
   sieve(cons(0, Y)) -> cons(0, n__sieve(activate(Y)))
   sieve(cons(s(N), Y)) -> cons(s(N), n__sieve(filter(activate(Y), N, N)))
   nats(N) -> cons(N, n__nats(s(N)))
   zprimes -> sieve(nats(s(s(0))))
   filter(X1, X2, X3) -> n__filter(X1, X2, X3)
   sieve(X) -> n__sieve(X)
   nats(X) -> n__nats(X)
   activate(n__filter(X1, X2, X3)) -> filter(X1, X2, X3)
   activate(n__sieve(X)) -> sieve(X)
   activate(n__nats(X)) -> nats(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:

   filter(cons(X, Y), 0, M) -> cons(0, n__filter(activate(Y), M, M))
   filter(cons(X, Y), s(N), M) -> cons(X, n__filter(activate(Y), N, M))
   sieve(cons(0, Y)) -> cons(0, n__sieve(activate(Y)))
   sieve(cons(s(N), Y)) -> cons(s(N), n__sieve(filter(activate(Y), N, N)))
   nats(N) -> cons(N, n__nats(s(N)))
   zprimes -> sieve(nats(s(s(0))))
   filter(X1, X2, X3) -> n__filter(X1, X2, X3)
   sieve(X) -> n__sieve(X)
   nats(X) -> n__nats(X)
   activate(n__filter(X1, X2, X3)) -> filter(X1, X2, X3)
   activate(n__sieve(X)) -> sieve(X)
   activate(n__nats(X)) -> nats(X)
   activate(X) -> X

S is empty.
Rewrite Strategy: FULL