WORST_CASE(Omega(n^1),?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 1113 ms]
(2) CpxRelTRS
(3) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(4) TRS for Loop Detection
(5) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
(6) BEST
    (7) proven lower bound
        (8) LowerBoundPropagationProof [FINISHED, 0 ms]
        (9) BOUNDS(n^1, INF)
    (10) TRS for Loop Detection


----------------------------------------

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


The TRS R consists of the following rules:

   A__FILTER(cons(z0, z1), 0, z2) -> c
   A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0))
   A__FILTER(z0, z1, z2) -> c2
   A__SIEVE(cons(0, z0)) -> c3
   A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0))
   A__SIEVE(z0) -> c5
   A__NATS(z0) -> c6(MARK(z0))
   A__NATS(z0) -> c7
   A__ZPRIMES -> c8(A__SIEVE(a__nats(s(s(0)))), A__NATS(s(s(0))))
   A__ZPRIMES -> c9
   MARK(filter(z0, z1, z2)) -> c10(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z0))
   MARK(filter(z0, z1, z2)) -> c11(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z1))
   MARK(filter(z0, z1, z2)) -> c12(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z2))
   MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0))
   MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0))
   MARK(zprimes) -> c15(A__ZPRIMES)
   MARK(cons(z0, z1)) -> c16(MARK(z0))
   MARK(0) -> c17
   MARK(s(z0)) -> c18(MARK(z0))

The (relative) TRS S consists of the following rules:

   a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2))
   a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3))
   a__filter(z0, z1, z2) -> filter(z0, z1, z2)
   a__sieve(cons(0, z0)) -> cons(0, sieve(z0))
   a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0)))
   a__sieve(z0) -> sieve(z0)
   a__nats(z0) -> cons(mark(z0), nats(s(z0)))
   a__nats(z0) -> nats(z0)
   a__zprimes -> a__sieve(a__nats(s(s(0))))
   a__zprimes -> zprimes
   mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2))
   mark(sieve(z0)) -> a__sieve(mark(z0))
   mark(nats(z0)) -> a__nats(mark(z0))
   mark(zprimes) -> a__zprimes
   mark(cons(z0, z1)) -> cons(mark(z0), z1)
   mark(0) -> 0
   mark(s(z0)) -> s(mark(z0))

Rewrite Strategy: INNERMOST
----------------------------------------

(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   A__FILTER(cons(z0, z1), 0, z2) -> c
   A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0))
   A__FILTER(z0, z1, z2) -> c2
   A__SIEVE(cons(0, z0)) -> c3
   A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0))
   A__SIEVE(z0) -> c5
   A__NATS(z0) -> c6(MARK(z0))
   A__NATS(z0) -> c7
   A__ZPRIMES -> c8(A__SIEVE(a__nats(s(s(0)))), A__NATS(s(s(0))))
   A__ZPRIMES -> c9
   MARK(filter(z0, z1, z2)) -> c10(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z0))
   MARK(filter(z0, z1, z2)) -> c11(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z1))
   MARK(filter(z0, z1, z2)) -> c12(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z2))
   MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0))
   MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0))
   MARK(zprimes) -> c15(A__ZPRIMES)
   MARK(cons(z0, z1)) -> c16(MARK(z0))
   MARK(0) -> c17
   MARK(s(z0)) -> c18(MARK(z0))

The (relative) TRS S consists of the following rules:

   a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2))
   a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3))
   a__filter(z0, z1, z2) -> filter(z0, z1, z2)
   a__sieve(cons(0, z0)) -> cons(0, sieve(z0))
   a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0)))
   a__sieve(z0) -> sieve(z0)
   a__nats(z0) -> cons(mark(z0), nats(s(z0)))
   a__nats(z0) -> nats(z0)
   a__zprimes -> a__sieve(a__nats(s(s(0))))
   a__zprimes -> zprimes
   mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2))
   mark(sieve(z0)) -> a__sieve(mark(z0))
   mark(nats(z0)) -> a__nats(mark(z0))
   mark(zprimes) -> a__zprimes
   mark(cons(z0, z1)) -> cons(mark(z0), z1)
   mark(0) -> 0
   mark(s(z0)) -> s(mark(z0))

Rewrite Strategy: INNERMOST
----------------------------------------

(3) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(4)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   A__FILTER(cons(z0, z1), 0, z2) -> c
   A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0))
   A__FILTER(z0, z1, z2) -> c2
   A__SIEVE(cons(0, z0)) -> c3
   A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0))
   A__SIEVE(z0) -> c5
   A__NATS(z0) -> c6(MARK(z0))
   A__NATS(z0) -> c7
   A__ZPRIMES -> c8(A__SIEVE(a__nats(s(s(0)))), A__NATS(s(s(0))))
   A__ZPRIMES -> c9
   MARK(filter(z0, z1, z2)) -> c10(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z0))
   MARK(filter(z0, z1, z2)) -> c11(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z1))
   MARK(filter(z0, z1, z2)) -> c12(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z2))
   MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0))
   MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0))
   MARK(zprimes) -> c15(A__ZPRIMES)
   MARK(cons(z0, z1)) -> c16(MARK(z0))
   MARK(0) -> c17
   MARK(s(z0)) -> c18(MARK(z0))

The (relative) TRS S consists of the following rules:

   a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2))
   a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3))
   a__filter(z0, z1, z2) -> filter(z0, z1, z2)
   a__sieve(cons(0, z0)) -> cons(0, sieve(z0))
   a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0)))
   a__sieve(z0) -> sieve(z0)
   a__nats(z0) -> cons(mark(z0), nats(s(z0)))
   a__nats(z0) -> nats(z0)
   a__zprimes -> a__sieve(a__nats(s(s(0))))
   a__zprimes -> zprimes
   mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2))
   mark(sieve(z0)) -> a__sieve(mark(z0))
   mark(nats(z0)) -> a__nats(mark(z0))
   mark(zprimes) -> a__zprimes
   mark(cons(z0, z1)) -> cons(mark(z0), z1)
   mark(0) -> 0
   mark(s(z0)) -> s(mark(z0))

Rewrite Strategy: INNERMOST
----------------------------------------

(5) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

MARK(cons(z0, z1)) ->^+ c16(MARK(z0))

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

The pumping substitution is [z0 / cons(z0, z1)].

The result substitution is [ ].




----------------------------------------

(6)
Complex Obligation (BEST)

----------------------------------------

(7)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   A__FILTER(cons(z0, z1), 0, z2) -> c
   A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0))
   A__FILTER(z0, z1, z2) -> c2
   A__SIEVE(cons(0, z0)) -> c3
   A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0))
   A__SIEVE(z0) -> c5
   A__NATS(z0) -> c6(MARK(z0))
   A__NATS(z0) -> c7
   A__ZPRIMES -> c8(A__SIEVE(a__nats(s(s(0)))), A__NATS(s(s(0))))
   A__ZPRIMES -> c9
   MARK(filter(z0, z1, z2)) -> c10(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z0))
   MARK(filter(z0, z1, z2)) -> c11(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z1))
   MARK(filter(z0, z1, z2)) -> c12(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z2))
   MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0))
   MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0))
   MARK(zprimes) -> c15(A__ZPRIMES)
   MARK(cons(z0, z1)) -> c16(MARK(z0))
   MARK(0) -> c17
   MARK(s(z0)) -> c18(MARK(z0))

The (relative) TRS S consists of the following rules:

   a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2))
   a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3))
   a__filter(z0, z1, z2) -> filter(z0, z1, z2)
   a__sieve(cons(0, z0)) -> cons(0, sieve(z0))
   a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0)))
   a__sieve(z0) -> sieve(z0)
   a__nats(z0) -> cons(mark(z0), nats(s(z0)))
   a__nats(z0) -> nats(z0)
   a__zprimes -> a__sieve(a__nats(s(s(0))))
   a__zprimes -> zprimes
   mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2))
   mark(sieve(z0)) -> a__sieve(mark(z0))
   mark(nats(z0)) -> a__nats(mark(z0))
   mark(zprimes) -> a__zprimes
   mark(cons(z0, z1)) -> cons(mark(z0), z1)
   mark(0) -> 0
   mark(s(z0)) -> s(mark(z0))

Rewrite Strategy: INNERMOST
----------------------------------------

(8) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(9)
BOUNDS(n^1, INF)

----------------------------------------

(10)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   A__FILTER(cons(z0, z1), 0, z2) -> c
   A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0))
   A__FILTER(z0, z1, z2) -> c2
   A__SIEVE(cons(0, z0)) -> c3
   A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0))
   A__SIEVE(z0) -> c5
   A__NATS(z0) -> c6(MARK(z0))
   A__NATS(z0) -> c7
   A__ZPRIMES -> c8(A__SIEVE(a__nats(s(s(0)))), A__NATS(s(s(0))))
   A__ZPRIMES -> c9
   MARK(filter(z0, z1, z2)) -> c10(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z0))
   MARK(filter(z0, z1, z2)) -> c11(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z1))
   MARK(filter(z0, z1, z2)) -> c12(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z2))
   MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0))
   MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0))
   MARK(zprimes) -> c15(A__ZPRIMES)
   MARK(cons(z0, z1)) -> c16(MARK(z0))
   MARK(0) -> c17
   MARK(s(z0)) -> c18(MARK(z0))

The (relative) TRS S consists of the following rules:

   a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2))
   a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3))
   a__filter(z0, z1, z2) -> filter(z0, z1, z2)
   a__sieve(cons(0, z0)) -> cons(0, sieve(z0))
   a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0)))
   a__sieve(z0) -> sieve(z0)
   a__nats(z0) -> cons(mark(z0), nats(s(z0)))
   a__nats(z0) -> nats(z0)
   a__zprimes -> a__sieve(a__nats(s(s(0))))
   a__zprimes -> zprimes
   mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2))
   mark(sieve(z0)) -> a__sieve(mark(z0))
   mark(nats(z0)) -> a__nats(mark(z0))
   mark(zprimes) -> a__zprimes
   mark(cons(z0, z1)) -> cons(mark(z0), z1)
   mark(0) -> 0
   mark(s(z0)) -> s(mark(z0))

Rewrite Strategy: INNERMOST