MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
            filter(cons(X,Y),s(N),M) -> cons(X,filter(Y,N,M))
            nats(N) -> cons(N,nats(s(N)))
            sieve(cons(0(),Y)) -> cons(0(),sieve(Y))
            sieve(cons(s(N),Y)) -> cons(s(N),sieve(filter(Y,N,N)))
            zprimes() -> sieve(nats(s(s(0()))))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0} / {0/0,cons/2,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,Y),0(),M) -> c_1(filter#(Y,M,M))
          filter#(cons(X,Y),s(N),M) -> c_2(filter#(Y,N,M))
          nats#(N) -> c_3(nats#(s(N)))
          sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
          sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)),filter#(Y,N,N))
          zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            filter#(cons(X,Y),0(),M) -> c_1(filter#(Y,M,M))
            filter#(cons(X,Y),s(N),M) -> c_2(filter#(Y,N,M))
            nats#(N) -> c_3(nats#(s(N)))
            sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
            sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)),filter#(Y,N,N))
            zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
        - Weak TRS:
            filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
            filter(cons(X,Y),s(N),M) -> cons(X,filter(Y,N,M))
            nats(N) -> cons(N,nats(s(N)))
            sieve(cons(0(),Y)) -> cons(0(),sieve(Y))
            sieve(cons(s(N),Y)) -> cons(s(N),sieve(filter(Y,N,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/2,s/1,c_1/1,c_2/1
            ,c_3/1,c_4/1,c_5/2,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter#,nats#,sieve#,zprimes#} and constructors {0,cons
            ,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
          filter(cons(X,Y),s(N),M) -> cons(X,filter(Y,N,M))
          nats(N) -> cons(N,nats(s(N)))
          filter#(cons(X,Y),0(),M) -> c_1(filter#(Y,M,M))
          filter#(cons(X,Y),s(N),M) -> c_2(filter#(Y,N,M))
          nats#(N) -> c_3(nats#(s(N)))
          sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
          sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)),filter#(Y,N,N))
          zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
* Step 3: Decompose MAYBE
    + Considered Problem:
        - Strict DPs:
            filter#(cons(X,Y),0(),M) -> c_1(filter#(Y,M,M))
            filter#(cons(X,Y),s(N),M) -> c_2(filter#(Y,N,M))
            nats#(N) -> c_3(nats#(s(N)))
            sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
            sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)),filter#(Y,N,N))
            zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
        - Weak TRS:
            filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
            filter(cons(X,Y),s(N),M) -> cons(X,filter(Y,N,M))
            nats(N) -> cons(N,nats(s(N)))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0,cons/2,s/1,c_1/1,c_2/1
            ,c_3/1,c_4/1,c_5/2,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter#,nats#,sieve#,zprimes#} and constructors {0,cons
            ,s}
    + Applied Processor:
        Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    + Details:
        We analyse the complexity of following sub-problems (R) and (S).
        Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
        
        Problem (R)
          - Strict DPs:
              filter#(cons(X,Y),0(),M) -> c_1(filter#(Y,M,M))
              filter#(cons(X,Y),s(N),M) -> c_2(filter#(Y,N,M))
          - Weak DPs:
              nats#(N) -> c_3(nats#(s(N)))
              sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
              sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)),filter#(Y,N,N))
              zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
          - Weak TRS:
              filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
              filter(cons(X,Y),s(N),M) -> cons(X,filter(Y,N,M))
              nats(N) -> cons(N,nats(s(N)))
          - Signature:
              {filter/3,nats/1,sieve/1,zprimes/0,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0,cons/2,s/1,c_1/1,c_2/1
              ,c_3/1,c_4/1,c_5/2,c_6/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {filter#,nats#,sieve#,zprimes#} and constructors {0,cons
              ,s}
        
        Problem (S)
          - Strict DPs:
              nats#(N) -> c_3(nats#(s(N)))
              sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
              sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)),filter#(Y,N,N))
              zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
          - Weak DPs:
              filter#(cons(X,Y),0(),M) -> c_1(filter#(Y,M,M))
              filter#(cons(X,Y),s(N),M) -> c_2(filter#(Y,N,M))
          - Weak TRS:
              filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
              filter(cons(X,Y),s(N),M) -> cons(X,filter(Y,N,M))
              nats(N) -> cons(N,nats(s(N)))
          - Signature:
              {filter/3,nats/1,sieve/1,zprimes/0,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0,cons/2,s/1,c_1/1,c_2/1
              ,c_3/1,c_4/1,c_5/2,c_6/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {filter#,nats#,sieve#,zprimes#} and constructors {0,cons
              ,s}
** Step 3.a:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            filter#(cons(X,Y),0(),M) -> c_1(filter#(Y,M,M))
            filter#(cons(X,Y),s(N),M) -> c_2(filter#(Y,N,M))
        - Weak DPs:
            nats#(N) -> c_3(nats#(s(N)))
            sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
            sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)),filter#(Y,N,N))
            zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
        - Weak TRS:
            filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
            filter(cons(X,Y),s(N),M) -> cons(X,filter(Y,N,M))
            nats(N) -> cons(N,nats(s(N)))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0,cons/2,s/1,c_1/1,c_2/1
            ,c_3/1,c_4/1,c_5/2,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:S:filter#(cons(X,Y),0(),M) -> c_1(filter#(Y,M,M))
             -->_1 filter#(cons(X,Y),s(N),M) -> c_2(filter#(Y,N,M)):2
             -->_1 filter#(cons(X,Y),0(),M) -> c_1(filter#(Y,M,M)):1
          
          2:S:filter#(cons(X,Y),s(N),M) -> c_2(filter#(Y,N,M))
             -->_1 filter#(cons(X,Y),0(),M) -> c_1(filter#(Y,M,M)):1
             -->_1 filter#(cons(X,Y),s(N),M) -> c_2(filter#(Y,N,M)):2
          
          3:W:nats#(N) -> c_3(nats#(s(N)))
             -->_1 nats#(N) -> c_3(nats#(s(N))):3
          
          4:W:sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
             -->_1 sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)),filter#(Y,N,N)):5
             -->_1 sieve#(cons(0(),Y)) -> c_4(sieve#(Y)):4
          
          5:W:sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)),filter#(Y,N,N))
             -->_2 filter#(cons(X,Y),0(),M) -> c_1(filter#(Y,M,M)):1
             -->_2 filter#(cons(X,Y),s(N),M) -> c_2(filter#(Y,N,M)):2
             -->_1 sieve#(cons(0(),Y)) -> c_4(sieve#(Y)):4
             -->_1 sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)),filter#(Y,N,N)):5
          
          6:W:zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
             -->_2 nats#(N) -> c_3(nats#(s(N))):3
             -->_1 sieve#(cons(0(),Y)) -> c_4(sieve#(Y)):4
             -->_1 sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)),filter#(Y,N,N)):5
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: nats#(N) -> c_3(nats#(s(N)))
** Step 3.a:2: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            filter#(cons(X,Y),0(),M) -> c_1(filter#(Y,M,M))
            filter#(cons(X,Y),s(N),M) -> c_2(filter#(Y,N,M))
        - Weak DPs:
            sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
            sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)),filter#(Y,N,N))
            zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
        - Weak TRS:
            filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
            filter(cons(X,Y),s(N),M) -> cons(X,filter(Y,N,M))
            nats(N) -> cons(N,nats(s(N)))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0,cons/2,s/1,c_1/1,c_2/1
            ,c_3/1,c_4/1,c_5/2,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter#,nats#,sieve#,zprimes#} and constructors {0,cons
            ,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:filter#(cons(X,Y),0(),M) -> c_1(filter#(Y,M,M))
             -->_1 filter#(cons(X,Y),s(N),M) -> c_2(filter#(Y,N,M)):2
             -->_1 filter#(cons(X,Y),0(),M) -> c_1(filter#(Y,M,M)):1
          
          2:S:filter#(cons(X,Y),s(N),M) -> c_2(filter#(Y,N,M))
             -->_1 filter#(cons(X,Y),0(),M) -> c_1(filter#(Y,M,M)):1
             -->_1 filter#(cons(X,Y),s(N),M) -> c_2(filter#(Y,N,M)):2
          
          4:W:sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
             -->_1 sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)),filter#(Y,N,N)):5
             -->_1 sieve#(cons(0(),Y)) -> c_4(sieve#(Y)):4
          
          5:W:sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)),filter#(Y,N,N))
             -->_2 filter#(cons(X,Y),0(),M) -> c_1(filter#(Y,M,M)):1
             -->_2 filter#(cons(X,Y),s(N),M) -> c_2(filter#(Y,N,M)):2
             -->_1 sieve#(cons(0(),Y)) -> c_4(sieve#(Y)):4
             -->_1 sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)),filter#(Y,N,N)):5
          
          6:W:zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
             -->_1 sieve#(cons(0(),Y)) -> c_4(sieve#(Y)):4
             -->_1 sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)),filter#(Y,N,N)):5
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          zprimes#() -> c_6(sieve#(nats(s(s(0())))))
** Step 3.a:3: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          filter#(cons(X,Y),0(),M) -> c_1(filter#(Y,M,M))
          filter#(cons(X,Y),s(N),M) -> c_2(filter#(Y,N,M))
      - Weak DPs:
          sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
          sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)),filter#(Y,N,N))
          zprimes#() -> c_6(sieve#(nats(s(s(0())))))
      - Weak TRS:
          filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
          filter(cons(X,Y),s(N),M) -> cons(X,filter(Y,N,M))
          nats(N) -> cons(N,nats(s(N)))
      - Signature:
          {filter/3,nats/1,sieve/1,zprimes/0,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0,cons/2,s/1,c_1/1,c_2/1
          ,c_3/1,c_4/1,c_5/2,c_6/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {filter#,nats#,sieve#,zprimes#} and constructors {0,cons
          ,s}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
** Step 3.b:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            nats#(N) -> c_3(nats#(s(N)))
            sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
            sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)),filter#(Y,N,N))
            zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
        - Weak DPs:
            filter#(cons(X,Y),0(),M) -> c_1(filter#(Y,M,M))
            filter#(cons(X,Y),s(N),M) -> c_2(filter#(Y,N,M))
        - Weak TRS:
            filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
            filter(cons(X,Y),s(N),M) -> cons(X,filter(Y,N,M))
            nats(N) -> cons(N,nats(s(N)))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0,cons/2,s/1,c_1/1,c_2/1
            ,c_3/1,c_4/1,c_5/2,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:S:nats#(N) -> c_3(nats#(s(N)))
             -->_1 nats#(N) -> c_3(nats#(s(N))):1
          
          2:S:sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
             -->_1 sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)),filter#(Y,N,N)):3
             -->_1 sieve#(cons(0(),Y)) -> c_4(sieve#(Y)):2
          
          3:S:sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)),filter#(Y,N,N))
             -->_2 filter#(cons(X,Y),s(N),M) -> c_2(filter#(Y,N,M)):6
             -->_2 filter#(cons(X,Y),0(),M) -> c_1(filter#(Y,M,M)):5
             -->_1 sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)),filter#(Y,N,N)):3
             -->_1 sieve#(cons(0(),Y)) -> c_4(sieve#(Y)):2
          
          4:S:zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
             -->_1 sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)),filter#(Y,N,N)):3
             -->_1 sieve#(cons(0(),Y)) -> c_4(sieve#(Y)):2
             -->_2 nats#(N) -> c_3(nats#(s(N))):1
          
          5:W:filter#(cons(X,Y),0(),M) -> c_1(filter#(Y,M,M))
             -->_1 filter#(cons(X,Y),s(N),M) -> c_2(filter#(Y,N,M)):6
             -->_1 filter#(cons(X,Y),0(),M) -> c_1(filter#(Y,M,M)):5
          
          6:W:filter#(cons(X,Y),s(N),M) -> c_2(filter#(Y,N,M))
             -->_1 filter#(cons(X,Y),s(N),M) -> c_2(filter#(Y,N,M)):6
             -->_1 filter#(cons(X,Y),0(),M) -> c_1(filter#(Y,M,M)):5
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: filter#(cons(X,Y),s(N),M) -> c_2(filter#(Y,N,M))
          5: filter#(cons(X,Y),0(),M) -> c_1(filter#(Y,M,M))
** Step 3.b:2: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            nats#(N) -> c_3(nats#(s(N)))
            sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
            sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)),filter#(Y,N,N))
            zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
        - Weak TRS:
            filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
            filter(cons(X,Y),s(N),M) -> cons(X,filter(Y,N,M))
            nats(N) -> cons(N,nats(s(N)))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0,cons/2,s/1,c_1/1,c_2/1
            ,c_3/1,c_4/1,c_5/2,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter#,nats#,sieve#,zprimes#} and constructors {0,cons
            ,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:nats#(N) -> c_3(nats#(s(N)))
             -->_1 nats#(N) -> c_3(nats#(s(N))):1
          
          2:S:sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
             -->_1 sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)),filter#(Y,N,N)):3
             -->_1 sieve#(cons(0(),Y)) -> c_4(sieve#(Y)):2
          
          3:S:sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)),filter#(Y,N,N))
             -->_1 sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)),filter#(Y,N,N)):3
             -->_1 sieve#(cons(0(),Y)) -> c_4(sieve#(Y)):2
          
          4:S:zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
             -->_1 sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)),filter#(Y,N,N)):3
             -->_1 sieve#(cons(0(),Y)) -> c_4(sieve#(Y)):2
             -->_2 nats#(N) -> c_3(nats#(s(N))):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)))
** Step 3.b:3: Decompose MAYBE
    + Considered Problem:
        - Strict DPs:
            nats#(N) -> c_3(nats#(s(N)))
            sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
            sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)))
            zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
        - Weak TRS:
            filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
            filter(cons(X,Y),s(N),M) -> cons(X,filter(Y,N,M))
            nats(N) -> cons(N,nats(s(N)))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0,cons/2,s/1,c_1/1,c_2/1
            ,c_3/1,c_4/1,c_5/1,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter#,nats#,sieve#,zprimes#} and constructors {0,cons
            ,s}
    + Applied Processor:
        Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    + Details:
        We analyse the complexity of following sub-problems (R) and (S).
        Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
        
        Problem (R)
          - Strict DPs:
              nats#(N) -> c_3(nats#(s(N)))
          - Weak DPs:
              sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
              sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)))
              zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
          - Weak TRS:
              filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
              filter(cons(X,Y),s(N),M) -> cons(X,filter(Y,N,M))
              nats(N) -> cons(N,nats(s(N)))
          - Signature:
              {filter/3,nats/1,sieve/1,zprimes/0,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0,cons/2,s/1,c_1/1,c_2/1
              ,c_3/1,c_4/1,c_5/1,c_6/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {filter#,nats#,sieve#,zprimes#} and constructors {0,cons
              ,s}
        
        Problem (S)
          - Strict DPs:
              sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
              sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)))
              zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
          - Weak DPs:
              nats#(N) -> c_3(nats#(s(N)))
          - Weak TRS:
              filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
              filter(cons(X,Y),s(N),M) -> cons(X,filter(Y,N,M))
              nats(N) -> cons(N,nats(s(N)))
          - Signature:
              {filter/3,nats/1,sieve/1,zprimes/0,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0,cons/2,s/1,c_1/1,c_2/1
              ,c_3/1,c_4/1,c_5/1,c_6/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {filter#,nats#,sieve#,zprimes#} and constructors {0,cons
              ,s}
*** Step 3.b:3.a:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            nats#(N) -> c_3(nats#(s(N)))
        - Weak DPs:
            sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
            sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)))
            zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
        - Weak TRS:
            filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
            filter(cons(X,Y),s(N),M) -> cons(X,filter(Y,N,M))
            nats(N) -> cons(N,nats(s(N)))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0,cons/2,s/1,c_1/1,c_2/1
            ,c_3/1,c_4/1,c_5/1,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:S:nats#(N) -> c_3(nats#(s(N)))
             -->_1 nats#(N) -> c_3(nats#(s(N))):1
          
          2:W:sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
             -->_1 sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N))):3
             -->_1 sieve#(cons(0(),Y)) -> c_4(sieve#(Y)):2
          
          3:W:sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)))
             -->_1 sieve#(cons(0(),Y)) -> c_4(sieve#(Y)):2
             -->_1 sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N))):3
          
          4:W:zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
             -->_2 nats#(N) -> c_3(nats#(s(N))):1
             -->_1 sieve#(cons(0(),Y)) -> c_4(sieve#(Y)):2
             -->_1 sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N))):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
          3: sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)))
*** Step 3.b:3.a:2: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            nats#(N) -> c_3(nats#(s(N)))
        - Weak DPs:
            zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
        - Weak TRS:
            filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
            filter(cons(X,Y),s(N),M) -> cons(X,filter(Y,N,M))
            nats(N) -> cons(N,nats(s(N)))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0,cons/2,s/1,c_1/1,c_2/1
            ,c_3/1,c_4/1,c_5/1,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter#,nats#,sieve#,zprimes#} and constructors {0,cons
            ,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:nats#(N) -> c_3(nats#(s(N)))
             -->_1 nats#(N) -> c_3(nats#(s(N))):1
          
          4:W:zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
             -->_2 nats#(N) -> c_3(nats#(s(N))):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          zprimes#() -> c_6(nats#(s(s(0()))))
*** Step 3.b:3.a:3: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            nats#(N) -> c_3(nats#(s(N)))
        - Weak DPs:
            zprimes#() -> c_6(nats#(s(s(0()))))
        - Weak TRS:
            filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
            filter(cons(X,Y),s(N),M) -> cons(X,filter(Y,N,M))
            nats(N) -> cons(N,nats(s(N)))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0,cons/2,s/1,c_1/1,c_2/1
            ,c_3/1,c_4/1,c_5/1,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter#,nats#,sieve#,zprimes#} and constructors {0,cons
            ,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          nats#(N) -> c_3(nats#(s(N)))
          zprimes#() -> c_6(nats#(s(s(0()))))
*** Step 3.b:3.a:4: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          nats#(N) -> c_3(nats#(s(N)))
      - Weak DPs:
          zprimes#() -> c_6(nats#(s(s(0()))))
      - Signature:
          {filter/3,nats/1,sieve/1,zprimes/0,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0,cons/2,s/1,c_1/1,c_2/1
          ,c_3/1,c_4/1,c_5/1,c_6/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {filter#,nats#,sieve#,zprimes#} and constructors {0,cons
          ,s}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
*** Step 3.b:3.b:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
            sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)))
            zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
        - Weak DPs:
            nats#(N) -> c_3(nats#(s(N)))
        - Weak TRS:
            filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
            filter(cons(X,Y),s(N),M) -> cons(X,filter(Y,N,M))
            nats(N) -> cons(N,nats(s(N)))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0,cons/2,s/1,c_1/1,c_2/1
            ,c_3/1,c_4/1,c_5/1,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:S:sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
             -->_1 sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N))):2
             -->_1 sieve#(cons(0(),Y)) -> c_4(sieve#(Y)):1
          
          2:S:sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)))
             -->_1 sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N))):2
             -->_1 sieve#(cons(0(),Y)) -> c_4(sieve#(Y)):1
          
          3:S:zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
             -->_2 nats#(N) -> c_3(nats#(s(N))):4
             -->_1 sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N))):2
             -->_1 sieve#(cons(0(),Y)) -> c_4(sieve#(Y)):1
          
          4:W:nats#(N) -> c_3(nats#(s(N)))
             -->_1 nats#(N) -> c_3(nats#(s(N))):4
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: nats#(N) -> c_3(nats#(s(N)))
*** Step 3.b:3.b:2: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
            sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)))
            zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
        - Weak TRS:
            filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
            filter(cons(X,Y),s(N),M) -> cons(X,filter(Y,N,M))
            nats(N) -> cons(N,nats(s(N)))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0,cons/2,s/1,c_1/1,c_2/1
            ,c_3/1,c_4/1,c_5/1,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter#,nats#,sieve#,zprimes#} and constructors {0,cons
            ,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
             -->_1 sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N))):2
             -->_1 sieve#(cons(0(),Y)) -> c_4(sieve#(Y)):1
          
          2:S:sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)))
             -->_1 sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N))):2
             -->_1 sieve#(cons(0(),Y)) -> c_4(sieve#(Y)):1
          
          3:S:zprimes#() -> c_6(sieve#(nats(s(s(0())))),nats#(s(s(0()))))
             -->_1 sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N))):2
             -->_1 sieve#(cons(0(),Y)) -> c_4(sieve#(Y)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          zprimes#() -> c_6(sieve#(nats(s(s(0())))))
*** Step 3.b:3.b:3: Decompose MAYBE
    + Considered Problem:
        - Strict DPs:
            sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
            sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)))
            zprimes#() -> c_6(sieve#(nats(s(s(0())))))
        - Weak TRS:
            filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
            filter(cons(X,Y),s(N),M) -> cons(X,filter(Y,N,M))
            nats(N) -> cons(N,nats(s(N)))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0,cons/2,s/1,c_1/1,c_2/1
            ,c_3/1,c_4/1,c_5/1,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter#,nats#,sieve#,zprimes#} and constructors {0,cons
            ,s}
    + Applied Processor:
        Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    + Details:
        We analyse the complexity of following sub-problems (R) and (S).
        Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
        
        Problem (R)
          - Strict DPs:
              sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
              sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)))
          - Weak DPs:
              zprimes#() -> c_6(sieve#(nats(s(s(0())))))
          - Weak TRS:
              filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
              filter(cons(X,Y),s(N),M) -> cons(X,filter(Y,N,M))
              nats(N) -> cons(N,nats(s(N)))
          - Signature:
              {filter/3,nats/1,sieve/1,zprimes/0,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0,cons/2,s/1,c_1/1,c_2/1
              ,c_3/1,c_4/1,c_5/1,c_6/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {filter#,nats#,sieve#,zprimes#} and constructors {0,cons
              ,s}
        
        Problem (S)
          - Strict DPs:
              zprimes#() -> c_6(sieve#(nats(s(s(0())))))
          - Weak DPs:
              sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
              sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)))
          - Weak TRS:
              filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
              filter(cons(X,Y),s(N),M) -> cons(X,filter(Y,N,M))
              nats(N) -> cons(N,nats(s(N)))
          - Signature:
              {filter/3,nats/1,sieve/1,zprimes/0,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0,cons/2,s/1,c_1/1,c_2/1
              ,c_3/1,c_4/1,c_5/1,c_6/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {filter#,nats#,sieve#,zprimes#} and constructors {0,cons
              ,s}
**** Step 3.b:3.b:3.a:1: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
          sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)))
      - Weak DPs:
          zprimes#() -> c_6(sieve#(nats(s(s(0())))))
      - Weak TRS:
          filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
          filter(cons(X,Y),s(N),M) -> cons(X,filter(Y,N,M))
          nats(N) -> cons(N,nats(s(N)))
      - Signature:
          {filter/3,nats/1,sieve/1,zprimes/0,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0,cons/2,s/1,c_1/1,c_2/1
          ,c_3/1,c_4/1,c_5/1,c_6/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {filter#,nats#,sieve#,zprimes#} and constructors {0,cons
          ,s}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
**** Step 3.b:3.b:3.b:1: PredecessorEstimation WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            zprimes#() -> c_6(sieve#(nats(s(s(0())))))
        - Weak DPs:
            sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
            sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)))
        - Weak TRS:
            filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
            filter(cons(X,Y),s(N),M) -> cons(X,filter(Y,N,M))
            nats(N) -> cons(N,nats(s(N)))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0,cons/2,s/1,c_1/1,c_2/1
            ,c_3/1,c_4/1,c_5/1,c_6/1}
        - 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())))))
          2: sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
          3: sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)))
**** Step 3.b:3.b:3.b:2: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
            sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)))
            zprimes#() -> c_6(sieve#(nats(s(s(0())))))
        - Weak TRS:
            filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
            filter(cons(X,Y),s(N),M) -> cons(X,filter(Y,N,M))
            nats(N) -> cons(N,nats(s(N)))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0,cons/2,s/1,c_1/1,c_2/1
            ,c_3/1,c_4/1,c_5/1,c_6/1}
        - 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:sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
             -->_1 sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N))):2
             -->_1 sieve#(cons(0(),Y)) -> c_4(sieve#(Y)):1
          
          2:W:sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)))
             -->_1 sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N))):2
             -->_1 sieve#(cons(0(),Y)) -> c_4(sieve#(Y)):1
          
          3:W:zprimes#() -> c_6(sieve#(nats(s(s(0())))))
             -->_1 sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N))):2
             -->_1 sieve#(cons(0(),Y)) -> c_4(sieve#(Y)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: zprimes#() -> c_6(sieve#(nats(s(s(0())))))
          1: sieve#(cons(0(),Y)) -> c_4(sieve#(Y))
          2: sieve#(cons(s(N),Y)) -> c_5(sieve#(filter(Y,N,N)))
**** Step 3.b:3.b:3.b:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
            filter(cons(X,Y),s(N),M) -> cons(X,filter(Y,N,M))
            nats(N) -> cons(N,nats(s(N)))
        - Signature:
            {filter/3,nats/1,sieve/1,zprimes/0,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0,cons/2,s/1,c_1/1,c_2/1
            ,c_3/1,c_4/1,c_5/1,c_6/1}
        - 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).

MAYBE