WORST_CASE(?,O(1))
* Step 1: Sum WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            filter(cons(X),0(),M) -> cons(0())
            filter(cons(X),s(N),M) -> cons(X)
            nats(N) -> cons(N)
            sieve(cons(0())) -> cons(0())
            sieve(cons(s(N))) -> cons(s(N))
            zprimes() -> sieve(nats(s(s(0()))))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0} / {0/0,cons/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter,nats,sieve,zprimes} and constructors {0,cons,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: DependencyPairs WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            filter(cons(X),0(),M) -> cons(0())
            filter(cons(X),s(N),M) -> cons(X)
            nats(N) -> cons(N)
            sieve(cons(0())) -> cons(0())
            sieve(cons(s(N))) -> cons(s(N))
            zprimes() -> sieve(nats(s(s(0()))))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0} / {0/0,cons/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter,nats,sieve,zprimes} and constructors {0,cons,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          filter#(cons(X),0(),M) -> c_1()
          filter#(cons(X),s(N),M) -> c_2()
          nats#(N) -> c_3()
          sieve#(cons(0())) -> c_4()
          sieve#(cons(s(N))) -> c_5()
          zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: PredecessorEstimation WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            filter#(cons(X),0(),M) -> c_1()
            filter#(cons(X),s(N),M) -> c_2()
            nats#(N) -> c_3()
            sieve#(cons(0())) -> c_4()
            sieve#(cons(s(N))) -> c_5()
            zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
        - Weak TRS:
            filter(cons(X),0(),M) -> cons(0())
            filter(cons(X),s(N),M) -> cons(X)
            nats(N) -> cons(N)
            sieve(cons(0())) -> cons(0())
            sieve(cons(s(N))) -> cons(s(N))
            zprimes() -> sieve(nats(s(s(0()))))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0,cons/1,s/1,c_1/0,c_2/0
            ,c_3/0,c_4/0,c_5/0,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter#,nats#,sieve#,zprimes#} and constructors {0,cons
            ,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,3,4,5}
        by application of
          Pre({1,2,3,4,5}) = {6}.
        Here rules are labelled as follows:
          1: filter#(cons(X),0(),M) -> c_1()
          2: filter#(cons(X),s(N),M) -> c_2()
          3: nats#(N) -> c_3()
          4: sieve#(cons(0())) -> c_4()
          5: sieve#(cons(s(N))) -> c_5()
          6: zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
* Step 4: PredecessorEstimation WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
        - Weak DPs:
            filter#(cons(X),0(),M) -> c_1()
            filter#(cons(X),s(N),M) -> c_2()
            nats#(N) -> c_3()
            sieve#(cons(0())) -> c_4()
            sieve#(cons(s(N))) -> c_5()
        - Weak TRS:
            filter(cons(X),0(),M) -> cons(0())
            filter(cons(X),s(N),M) -> cons(X)
            nats(N) -> cons(N)
            sieve(cons(0())) -> cons(0())
            sieve(cons(s(N))) -> cons(s(N))
            zprimes() -> sieve(nats(s(s(0()))))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0,cons/1,s/1,c_1/0,c_2/0
            ,c_3/0,c_4/0,c_5/0,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter#,nats#,sieve#,zprimes#} and constructors {0,cons
            ,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1}
        by application of
          Pre({1}) = {}.
        Here rules are labelled as follows:
          1: zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
          2: filter#(cons(X),0(),M) -> c_1()
          3: filter#(cons(X),s(N),M) -> c_2()
          4: nats#(N) -> c_3()
          5: sieve#(cons(0())) -> c_4()
          6: sieve#(cons(s(N))) -> c_5()
* Step 5: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            filter#(cons(X),0(),M) -> c_1()
            filter#(cons(X),s(N),M) -> c_2()
            nats#(N) -> c_3()
            sieve#(cons(0())) -> c_4()
            sieve#(cons(s(N))) -> c_5()
            zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
        - Weak TRS:
            filter(cons(X),0(),M) -> cons(0())
            filter(cons(X),s(N),M) -> cons(X)
            nats(N) -> cons(N)
            sieve(cons(0())) -> cons(0())
            sieve(cons(s(N))) -> cons(s(N))
            zprimes() -> sieve(nats(s(s(0()))))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0,cons/1,s/1,c_1/0,c_2/0
            ,c_3/0,c_4/0,c_5/0,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter#,nats#,sieve#,zprimes#} and constructors {0,cons
            ,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:filter#(cons(X),0(),M) -> c_1()
             
          
          2:W:filter#(cons(X),s(N),M) -> c_2()
             
          
          3:W:nats#(N) -> c_3()
             
          
          4:W:sieve#(cons(0())) -> c_4()
             
          
          5:W:sieve#(cons(s(N))) -> c_5()
             
          
          6:W:zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
             -->_1 sieve#(cons(s(N))) -> c_5():5
             -->_1 sieve#(cons(0())) -> c_4():4
             -->_2 nats#(N) -> c_3():3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
          5: sieve#(cons(s(N))) -> c_5()
          4: sieve#(cons(0())) -> c_4()
          3: nats#(N) -> c_3()
          2: filter#(cons(X),s(N),M) -> c_2()
          1: filter#(cons(X),0(),M) -> c_1()
* Step 6: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            filter(cons(X),0(),M) -> cons(0())
            filter(cons(X),s(N),M) -> cons(X)
            nats(N) -> cons(N)
            sieve(cons(0())) -> cons(0())
            sieve(cons(s(N))) -> cons(s(N))
            zprimes() -> sieve(nats(s(s(0()))))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0,cons/1,s/1,c_1/0,c_2/0
            ,c_3/0,c_4/0,c_5/0,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter#,nats#,sieve#,zprimes#} and constructors {0,cons
            ,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))