WORST_CASE(Omega(n^1),?)
* Step 1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            FILTER(s(s(z0)),cons(z1,z2)) -> c6(IF(divides(s(s(z0)),z1)
                                                 ,filter(s(s(z0)),z2)
                                                 ,cons(z1,filter(z0,sieve(z1))))
                                              ,FILTER(s(s(z0)),z2))
            FILTER(s(s(z0)),cons(z1,z2)) -> c7(IF(divides(s(s(z0)),z1)
                                                 ,filter(s(s(z0)),z2)
                                                 ,cons(z1,filter(z0,sieve(z1))))
                                              ,FILTER(z0,sieve(z1))
                                              ,SIEVE(z1))
            FROM(z0) -> c1(FROM(s(z0)))
            HEAD(cons(z0,z1)) -> c2()
            IF(false(),z0,z1) -> c5()
            IF(true(),z0,z1) -> c4()
            PRIMES() -> c(SIEVE(from(s(s(0())))),FROM(s(s(0()))))
            SIEVE(cons(z0,z1)) -> c8(FILTER(z0,sieve(z1)),SIEVE(z1))
            TAIL(cons(z0,z1)) -> c3()
        - Weak TRS:
            filter(s(s(z0)),cons(z1,z2)) -> if(divides(s(s(z0)),z1),filter(s(s(z0)),z2),cons(z1,filter(z0,sieve(z1))))
            from(z0) -> cons(z0,from(s(z0)))
            head(cons(z0,z1)) -> z0
            if(false(),z0,z1) -> z1
            if(true(),z0,z1) -> z0
            primes() -> sieve(from(s(s(0()))))
            sieve(cons(z0,z1)) -> cons(z0,filter(z0,sieve(z1)))
            tail(cons(z0,z1)) -> z1
        - Signature:
            {FILTER/2,FROM/1,HEAD/1,IF/3,PRIMES/0,SIEVE/1,TAIL/1,filter/2,from/1,head/1,if/3,primes/0,sieve/1
            ,tail/1} / {0/0,c/2,c1/1,c2/0,c3/0,c4/0,c5/0,c6/2,c7/3,c8/2,cons/2,divides/2,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {FILTER,FROM,HEAD,IF,PRIMES,SIEVE,TAIL,filter,from,head,if
            ,primes,sieve,tail} and constructors {0,c,c1,c2,c3,c4,c5,c6,c7,c8,cons,divides,false,s,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            FILTER(s(s(z0)),cons(z1,z2)) -> c6(IF(divides(s(s(z0)),z1)
                                                 ,filter(s(s(z0)),z2)
                                                 ,cons(z1,filter(z0,sieve(z1))))
                                              ,FILTER(s(s(z0)),z2))
            FILTER(s(s(z0)),cons(z1,z2)) -> c7(IF(divides(s(s(z0)),z1)
                                                 ,filter(s(s(z0)),z2)
                                                 ,cons(z1,filter(z0,sieve(z1))))
                                              ,FILTER(z0,sieve(z1))
                                              ,SIEVE(z1))
            FROM(z0) -> c1(FROM(s(z0)))
            HEAD(cons(z0,z1)) -> c2()
            IF(false(),z0,z1) -> c5()
            IF(true(),z0,z1) -> c4()
            PRIMES() -> c(SIEVE(from(s(s(0())))),FROM(s(s(0()))))
            SIEVE(cons(z0,z1)) -> c8(FILTER(z0,sieve(z1)),SIEVE(z1))
            TAIL(cons(z0,z1)) -> c3()
        - Weak TRS:
            filter(s(s(z0)),cons(z1,z2)) -> if(divides(s(s(z0)),z1),filter(s(s(z0)),z2),cons(z1,filter(z0,sieve(z1))))
            from(z0) -> cons(z0,from(s(z0)))
            head(cons(z0,z1)) -> z0
            if(false(),z0,z1) -> z1
            if(true(),z0,z1) -> z0
            primes() -> sieve(from(s(s(0()))))
            sieve(cons(z0,z1)) -> cons(z0,filter(z0,sieve(z1)))
            tail(cons(z0,z1)) -> z1
        - Signature:
            {FILTER/2,FROM/1,HEAD/1,IF/3,PRIMES/0,SIEVE/1,TAIL/1,filter/2,from/1,head/1,if/3,primes/0,sieve/1
            ,tail/1} / {0/0,c/2,c1/1,c2/0,c3/0,c4/0,c5/0,c6/2,c7/3,c8/2,cons/2,divides/2,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {FILTER,FROM,HEAD,IF,PRIMES,SIEVE,TAIL,filter,from,head,if
            ,primes,sieve,tail} and constructors {0,c,c1,c2,c3,c4,c5,c6,c7,c8,cons,divides,false,s,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 3: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            FILTER(s(s(z0)),cons(z1,z2)) -> c6(IF(divides(s(s(z0)),z1)
                                                 ,filter(s(s(z0)),z2)
                                                 ,cons(z1,filter(z0,sieve(z1))))
                                              ,FILTER(s(s(z0)),z2))
            FILTER(s(s(z0)),cons(z1,z2)) -> c7(IF(divides(s(s(z0)),z1)
                                                 ,filter(s(s(z0)),z2)
                                                 ,cons(z1,filter(z0,sieve(z1))))
                                              ,FILTER(z0,sieve(z1))
                                              ,SIEVE(z1))
            FROM(z0) -> c1(FROM(s(z0)))
            HEAD(cons(z0,z1)) -> c2()
            IF(false(),z0,z1) -> c5()
            IF(true(),z0,z1) -> c4()
            PRIMES() -> c(SIEVE(from(s(s(0())))),FROM(s(s(0()))))
            SIEVE(cons(z0,z1)) -> c8(FILTER(z0,sieve(z1)),SIEVE(z1))
            TAIL(cons(z0,z1)) -> c3()
        - Weak TRS:
            filter(s(s(z0)),cons(z1,z2)) -> if(divides(s(s(z0)),z1),filter(s(s(z0)),z2),cons(z1,filter(z0,sieve(z1))))
            from(z0) -> cons(z0,from(s(z0)))
            head(cons(z0,z1)) -> z0
            if(false(),z0,z1) -> z1
            if(true(),z0,z1) -> z0
            primes() -> sieve(from(s(s(0()))))
            sieve(cons(z0,z1)) -> cons(z0,filter(z0,sieve(z1)))
            tail(cons(z0,z1)) -> z1
        - Signature:
            {FILTER/2,FROM/1,HEAD/1,IF/3,PRIMES/0,SIEVE/1,TAIL/1,filter/2,from/1,head/1,if/3,primes/0,sieve/1
            ,tail/1} / {0/0,c/2,c1/1,c2/0,c3/0,c4/0,c5/0,c6/2,c7/3,c8/2,cons/2,divides/2,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {FILTER,FROM,HEAD,IF,PRIMES,SIEVE,TAIL,filter,from,head,if
            ,primes,sieve,tail} and constructors {0,c,c1,c2,c3,c4,c5,c6,c7,c8,cons,divides,false,s,true}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          FILTER(s(s(x)),z){z -> cons(y,z)} =
            FILTER(s(s(x)),cons(y,z)) ->^+ c6(IF(divides(s(s(x)),y),filter(s(s(x)),z),cons(y,filter(x,sieve(y))))
                                             ,FILTER(s(s(x)),z))
              = C[FILTER(s(s(x)),z) = FILTER(s(s(x)),z){}]

WORST_CASE(Omega(n^1),?)