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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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), 45.8 s]
(2) CpxRelTRS
(3) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 4 ms]
(4) TRS for Loop Detection
(5) DecreasingLoopProof [LOWER BOUND(ID), 5 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__PRIMES -> c(A__SIEVE(a__from(s(s(0)))), A__FROM(s(s(0))))
   A__PRIMES -> c1
   A__FROM(z0) -> c2(MARK(z0))
   A__FROM(z0) -> c3
   A__HEAD(cons(z0, z1)) -> c4(MARK(z0))
   A__HEAD(z0) -> c5
   A__TAIL(cons(z0, z1)) -> c6(MARK(z1))
   A__TAIL(z0) -> c7
   A__IF(true, z0, z1) -> c8(MARK(z0))
   A__IF(false, z0, z1) -> c9(MARK(z1))
   A__IF(z0, z1, z2) -> c10
   A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(A__IF(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))), MARK(z0))
   A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(A__IF(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))), MARK(z1))
   A__FILTER(z0, z1) -> c13
   A__SIEVE(cons(z0, z1)) -> c14(MARK(z0))
   A__SIEVE(z0) -> c15
   MARK(primes) -> c16(A__PRIMES)
   MARK(sieve(z0)) -> c17(A__SIEVE(mark(z0)), MARK(z0))
   MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0))
   MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0))
   MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0))
   MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c24(MARK(z0))
   MARK(0) -> c25
   MARK(cons(z0, z1)) -> c26(MARK(z0))
   MARK(true) -> c27
   MARK(false) -> c28
   MARK(divides(z0, z1)) -> c29(MARK(z0))
   MARK(divides(z0, z1)) -> c30(MARK(z1))

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

   a__primes -> a__sieve(a__from(s(s(0))))
   a__primes -> primes
   a__from(z0) -> cons(mark(z0), from(s(z0)))
   a__from(z0) -> from(z0)
   a__head(cons(z0, z1)) -> mark(z0)
   a__head(z0) -> head(z0)
   a__tail(cons(z0, z1)) -> mark(z1)
   a__tail(z0) -> tail(z0)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1))))
   a__filter(z0, z1) -> filter(z0, z1)
   a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1)))
   a__sieve(z0) -> sieve(z0)
   mark(primes) -> a__primes
   mark(sieve(z0)) -> a__sieve(mark(z0))
   mark(from(z0)) -> a__from(mark(z0))
   mark(head(z0)) -> a__head(mark(z0))
   mark(tail(z0)) -> a__tail(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(cons(z0, z1)) -> cons(mark(z0), z1)
   mark(true) -> true
   mark(false) -> false
   mark(divides(z0, z1)) -> divides(mark(z0), mark(z1))

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__PRIMES -> c(A__SIEVE(a__from(s(s(0)))), A__FROM(s(s(0))))
   A__PRIMES -> c1
   A__FROM(z0) -> c2(MARK(z0))
   A__FROM(z0) -> c3
   A__HEAD(cons(z0, z1)) -> c4(MARK(z0))
   A__HEAD(z0) -> c5
   A__TAIL(cons(z0, z1)) -> c6(MARK(z1))
   A__TAIL(z0) -> c7
   A__IF(true, z0, z1) -> c8(MARK(z0))
   A__IF(false, z0, z1) -> c9(MARK(z1))
   A__IF(z0, z1, z2) -> c10
   A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(A__IF(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))), MARK(z0))
   A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(A__IF(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))), MARK(z1))
   A__FILTER(z0, z1) -> c13
   A__SIEVE(cons(z0, z1)) -> c14(MARK(z0))
   A__SIEVE(z0) -> c15
   MARK(primes) -> c16(A__PRIMES)
   MARK(sieve(z0)) -> c17(A__SIEVE(mark(z0)), MARK(z0))
   MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0))
   MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0))
   MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0))
   MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c24(MARK(z0))
   MARK(0) -> c25
   MARK(cons(z0, z1)) -> c26(MARK(z0))
   MARK(true) -> c27
   MARK(false) -> c28
   MARK(divides(z0, z1)) -> c29(MARK(z0))
   MARK(divides(z0, z1)) -> c30(MARK(z1))

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

   a__primes -> a__sieve(a__from(s(s(0))))
   a__primes -> primes
   a__from(z0) -> cons(mark(z0), from(s(z0)))
   a__from(z0) -> from(z0)
   a__head(cons(z0, z1)) -> mark(z0)
   a__head(z0) -> head(z0)
   a__tail(cons(z0, z1)) -> mark(z1)
   a__tail(z0) -> tail(z0)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1))))
   a__filter(z0, z1) -> filter(z0, z1)
   a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1)))
   a__sieve(z0) -> sieve(z0)
   mark(primes) -> a__primes
   mark(sieve(z0)) -> a__sieve(mark(z0))
   mark(from(z0)) -> a__from(mark(z0))
   mark(head(z0)) -> a__head(mark(z0))
   mark(tail(z0)) -> a__tail(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(cons(z0, z1)) -> cons(mark(z0), z1)
   mark(true) -> true
   mark(false) -> false
   mark(divides(z0, z1)) -> divides(mark(z0), mark(z1))

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__PRIMES -> c(A__SIEVE(a__from(s(s(0)))), A__FROM(s(s(0))))
   A__PRIMES -> c1
   A__FROM(z0) -> c2(MARK(z0))
   A__FROM(z0) -> c3
   A__HEAD(cons(z0, z1)) -> c4(MARK(z0))
   A__HEAD(z0) -> c5
   A__TAIL(cons(z0, z1)) -> c6(MARK(z1))
   A__TAIL(z0) -> c7
   A__IF(true, z0, z1) -> c8(MARK(z0))
   A__IF(false, z0, z1) -> c9(MARK(z1))
   A__IF(z0, z1, z2) -> c10
   A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(A__IF(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))), MARK(z0))
   A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(A__IF(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))), MARK(z1))
   A__FILTER(z0, z1) -> c13
   A__SIEVE(cons(z0, z1)) -> c14(MARK(z0))
   A__SIEVE(z0) -> c15
   MARK(primes) -> c16(A__PRIMES)
   MARK(sieve(z0)) -> c17(A__SIEVE(mark(z0)), MARK(z0))
   MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0))
   MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0))
   MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0))
   MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c24(MARK(z0))
   MARK(0) -> c25
   MARK(cons(z0, z1)) -> c26(MARK(z0))
   MARK(true) -> c27
   MARK(false) -> c28
   MARK(divides(z0, z1)) -> c29(MARK(z0))
   MARK(divides(z0, z1)) -> c30(MARK(z1))

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

   a__primes -> a__sieve(a__from(s(s(0))))
   a__primes -> primes
   a__from(z0) -> cons(mark(z0), from(s(z0)))
   a__from(z0) -> from(z0)
   a__head(cons(z0, z1)) -> mark(z0)
   a__head(z0) -> head(z0)
   a__tail(cons(z0, z1)) -> mark(z1)
   a__tail(z0) -> tail(z0)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1))))
   a__filter(z0, z1) -> filter(z0, z1)
   a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1)))
   a__sieve(z0) -> sieve(z0)
   mark(primes) -> a__primes
   mark(sieve(z0)) -> a__sieve(mark(z0))
   mark(from(z0)) -> a__from(mark(z0))
   mark(head(z0)) -> a__head(mark(z0))
   mark(tail(z0)) -> a__tail(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(cons(z0, z1)) -> cons(mark(z0), z1)
   mark(true) -> true
   mark(false) -> false
   mark(divides(z0, z1)) -> divides(mark(z0), mark(z1))

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(divides(z0, z1)) ->^+ c29(MARK(z0))

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

The pumping substitution is [z0 / divides(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__PRIMES -> c(A__SIEVE(a__from(s(s(0)))), A__FROM(s(s(0))))
   A__PRIMES -> c1
   A__FROM(z0) -> c2(MARK(z0))
   A__FROM(z0) -> c3
   A__HEAD(cons(z0, z1)) -> c4(MARK(z0))
   A__HEAD(z0) -> c5
   A__TAIL(cons(z0, z1)) -> c6(MARK(z1))
   A__TAIL(z0) -> c7
   A__IF(true, z0, z1) -> c8(MARK(z0))
   A__IF(false, z0, z1) -> c9(MARK(z1))
   A__IF(z0, z1, z2) -> c10
   A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(A__IF(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))), MARK(z0))
   A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(A__IF(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))), MARK(z1))
   A__FILTER(z0, z1) -> c13
   A__SIEVE(cons(z0, z1)) -> c14(MARK(z0))
   A__SIEVE(z0) -> c15
   MARK(primes) -> c16(A__PRIMES)
   MARK(sieve(z0)) -> c17(A__SIEVE(mark(z0)), MARK(z0))
   MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0))
   MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0))
   MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0))
   MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c24(MARK(z0))
   MARK(0) -> c25
   MARK(cons(z0, z1)) -> c26(MARK(z0))
   MARK(true) -> c27
   MARK(false) -> c28
   MARK(divides(z0, z1)) -> c29(MARK(z0))
   MARK(divides(z0, z1)) -> c30(MARK(z1))

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

   a__primes -> a__sieve(a__from(s(s(0))))
   a__primes -> primes
   a__from(z0) -> cons(mark(z0), from(s(z0)))
   a__from(z0) -> from(z0)
   a__head(cons(z0, z1)) -> mark(z0)
   a__head(z0) -> head(z0)
   a__tail(cons(z0, z1)) -> mark(z1)
   a__tail(z0) -> tail(z0)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1))))
   a__filter(z0, z1) -> filter(z0, z1)
   a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1)))
   a__sieve(z0) -> sieve(z0)
   mark(primes) -> a__primes
   mark(sieve(z0)) -> a__sieve(mark(z0))
   mark(from(z0)) -> a__from(mark(z0))
   mark(head(z0)) -> a__head(mark(z0))
   mark(tail(z0)) -> a__tail(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(cons(z0, z1)) -> cons(mark(z0), z1)
   mark(true) -> true
   mark(false) -> false
   mark(divides(z0, z1)) -> divides(mark(z0), mark(z1))

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__PRIMES -> c(A__SIEVE(a__from(s(s(0)))), A__FROM(s(s(0))))
   A__PRIMES -> c1
   A__FROM(z0) -> c2(MARK(z0))
   A__FROM(z0) -> c3
   A__HEAD(cons(z0, z1)) -> c4(MARK(z0))
   A__HEAD(z0) -> c5
   A__TAIL(cons(z0, z1)) -> c6(MARK(z1))
   A__TAIL(z0) -> c7
   A__IF(true, z0, z1) -> c8(MARK(z0))
   A__IF(false, z0, z1) -> c9(MARK(z1))
   A__IF(z0, z1, z2) -> c10
   A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(A__IF(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))), MARK(z0))
   A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(A__IF(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))), MARK(z1))
   A__FILTER(z0, z1) -> c13
   A__SIEVE(cons(z0, z1)) -> c14(MARK(z0))
   A__SIEVE(z0) -> c15
   MARK(primes) -> c16(A__PRIMES)
   MARK(sieve(z0)) -> c17(A__SIEVE(mark(z0)), MARK(z0))
   MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0))
   MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0))
   MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0))
   MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c24(MARK(z0))
   MARK(0) -> c25
   MARK(cons(z0, z1)) -> c26(MARK(z0))
   MARK(true) -> c27
   MARK(false) -> c28
   MARK(divides(z0, z1)) -> c29(MARK(z0))
   MARK(divides(z0, z1)) -> c30(MARK(z1))

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

   a__primes -> a__sieve(a__from(s(s(0))))
   a__primes -> primes
   a__from(z0) -> cons(mark(z0), from(s(z0)))
   a__from(z0) -> from(z0)
   a__head(cons(z0, z1)) -> mark(z0)
   a__head(z0) -> head(z0)
   a__tail(cons(z0, z1)) -> mark(z1)
   a__tail(z0) -> tail(z0)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1))))
   a__filter(z0, z1) -> filter(z0, z1)
   a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1)))
   a__sieve(z0) -> sieve(z0)
   mark(primes) -> a__primes
   mark(sieve(z0)) -> a__sieve(mark(z0))
   mark(from(z0)) -> a__from(mark(z0))
   mark(head(z0)) -> a__head(mark(z0))
   mark(tail(z0)) -> a__tail(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(cons(z0, z1)) -> cons(mark(z0), z1)
   mark(true) -> true
   mark(false) -> false
   mark(divides(z0, z1)) -> divides(mark(z0), mark(z1))

Rewrite Strategy: INNERMOST