MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y))))
            from(X) -> cons(X,from(s(X)))
            head(cons(X,Y)) -> X
            if(false(),X,Y) -> Y
            if(true(),X,Y) -> X
            primes() -> sieve(from(s(s(0()))))
            sieve(cons(X,Y)) -> cons(X,filter(X,sieve(Y)))
            tail(cons(X,Y)) -> Y
        - Signature:
            {filter/2,from/1,head/1,if/3,primes/0,sieve/1,tail/1} / {0/0,cons/2,divides/2,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter,from,head,if,primes,sieve
            ,tail} and constructors {0,cons,divides,false,s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          filter#(s(s(X)),cons(Y,Z)) -> c_1(if#(divides(s(s(X)),Y),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y))))
                                           ,filter#(s(s(X)),Z)
                                           ,filter#(X,sieve(Y))
                                           ,sieve#(Y))
          from#(X) -> c_2(from#(s(X)))
          head#(cons(X,Y)) -> c_3()
          if#(false(),X,Y) -> c_4()
          if#(true(),X,Y) -> c_5()
          primes#() -> c_6(sieve#(from(s(s(0())))),from#(s(s(0()))))
          sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y))
          tail#(cons(X,Y)) -> c_8()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            filter#(s(s(X)),cons(Y,Z)) -> c_1(if#(divides(s(s(X)),Y),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y))))
                                             ,filter#(s(s(X)),Z)
                                             ,filter#(X,sieve(Y))
                                             ,sieve#(Y))
            from#(X) -> c_2(from#(s(X)))
            head#(cons(X,Y)) -> c_3()
            if#(false(),X,Y) -> c_4()
            if#(true(),X,Y) -> c_5()
            primes#() -> c_6(sieve#(from(s(s(0())))),from#(s(s(0()))))
            sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y))
            tail#(cons(X,Y)) -> c_8()
        - Weak TRS:
            filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y))))
            from(X) -> cons(X,from(s(X)))
            head(cons(X,Y)) -> X
            if(false(),X,Y) -> Y
            if(true(),X,Y) -> X
            primes() -> sieve(from(s(s(0()))))
            sieve(cons(X,Y)) -> cons(X,filter(X,sieve(Y)))
            tail(cons(X,Y)) -> Y
        - 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,cons/2,divides/2,false/0,s/1,true/0,c_1/4,c_2/1,c_3/0,c_4/0,c_5/0,c_6/2,c_7/2,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter#,from#,head#,if#,primes#,sieve#
            ,tail#} and constructors {0,cons,divides,false,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y))))
          from(X) -> cons(X,from(s(X)))
          sieve(cons(X,Y)) -> cons(X,filter(X,sieve(Y)))
          filter#(s(s(X)),cons(Y,Z)) -> c_1(if#(divides(s(s(X)),Y),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y))))
                                           ,filter#(s(s(X)),Z)
                                           ,filter#(X,sieve(Y))
                                           ,sieve#(Y))
          from#(X) -> c_2(from#(s(X)))
          head#(cons(X,Y)) -> c_3()
          if#(false(),X,Y) -> c_4()
          if#(true(),X,Y) -> c_5()
          primes#() -> c_6(sieve#(from(s(s(0())))),from#(s(s(0()))))
          sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y))
          tail#(cons(X,Y)) -> c_8()
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            filter#(s(s(X)),cons(Y,Z)) -> c_1(if#(divides(s(s(X)),Y),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y))))
                                             ,filter#(s(s(X)),Z)
                                             ,filter#(X,sieve(Y))
                                             ,sieve#(Y))
            from#(X) -> c_2(from#(s(X)))
            head#(cons(X,Y)) -> c_3()
            if#(false(),X,Y) -> c_4()
            if#(true(),X,Y) -> c_5()
            primes#() -> c_6(sieve#(from(s(s(0())))),from#(s(s(0()))))
            sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y))
            tail#(cons(X,Y)) -> c_8()
        - Weak TRS:
            filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y))))
            from(X) -> cons(X,from(s(X)))
            sieve(cons(X,Y)) -> cons(X,filter(X,sieve(Y)))
        - 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,cons/2,divides/2,false/0,s/1,true/0,c_1/4,c_2/1,c_3/0,c_4/0,c_5/0,c_6/2,c_7/2,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter#,from#,head#,if#,primes#,sieve#
            ,tail#} and constructors {0,cons,divides,false,s,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {3,4,5,8}
        by application of
          Pre({3,4,5,8}) = {}.
        Here rules are labelled as follows:
          1: filter#(s(s(X)),cons(Y,Z)) -> c_1(if#(divides(s(s(X)),Y),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y))))
                                              ,filter#(s(s(X)),Z)
                                              ,filter#(X,sieve(Y))
                                              ,sieve#(Y))
          2: from#(X) -> c_2(from#(s(X)))
          3: head#(cons(X,Y)) -> c_3()
          4: if#(false(),X,Y) -> c_4()
          5: if#(true(),X,Y) -> c_5()
          6: primes#() -> c_6(sieve#(from(s(s(0())))),from#(s(s(0()))))
          7: sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y))
          8: tail#(cons(X,Y)) -> c_8()
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            filter#(s(s(X)),cons(Y,Z)) -> c_1(if#(divides(s(s(X)),Y),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y))))
                                             ,filter#(s(s(X)),Z)
                                             ,filter#(X,sieve(Y))
                                             ,sieve#(Y))
            from#(X) -> c_2(from#(s(X)))
            primes#() -> c_6(sieve#(from(s(s(0())))),from#(s(s(0()))))
            sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y))
        - Weak DPs:
            head#(cons(X,Y)) -> c_3()
            if#(false(),X,Y) -> c_4()
            if#(true(),X,Y) -> c_5()
            tail#(cons(X,Y)) -> c_8()
        - Weak TRS:
            filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y))))
            from(X) -> cons(X,from(s(X)))
            sieve(cons(X,Y)) -> cons(X,filter(X,sieve(Y)))
        - 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,cons/2,divides/2,false/0,s/1,true/0,c_1/4,c_2/1,c_3/0,c_4/0,c_5/0,c_6/2,c_7/2,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter#,from#,head#,if#,primes#,sieve#
            ,tail#} and constructors {0,cons,divides,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:filter#(s(s(X)),cons(Y,Z)) -> c_1(if#(divides(s(s(X)),Y),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y))))
                                               ,filter#(s(s(X)),Z)
                                               ,filter#(X,sieve(Y))
                                               ,sieve#(Y))
             -->_4 sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y)):4
             -->_3 filter#(s(s(X)),cons(Y,Z)) -> c_1(if#(divides(s(s(X)),Y)
                                                        ,filter(s(s(X)),Z)
                                                        ,cons(Y,filter(X,sieve(Y))))
                                                    ,filter#(s(s(X)),Z)
                                                    ,filter#(X,sieve(Y))
                                                    ,sieve#(Y)):1
             -->_2 filter#(s(s(X)),cons(Y,Z)) -> c_1(if#(divides(s(s(X)),Y)
                                                        ,filter(s(s(X)),Z)
                                                        ,cons(Y,filter(X,sieve(Y))))
                                                    ,filter#(s(s(X)),Z)
                                                    ,filter#(X,sieve(Y))
                                                    ,sieve#(Y)):1
          
          2:S:from#(X) -> c_2(from#(s(X)))
             -->_1 from#(X) -> c_2(from#(s(X))):2
          
          3:S:primes#() -> c_6(sieve#(from(s(s(0())))),from#(s(s(0()))))
             -->_1 sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y)):4
             -->_2 from#(X) -> c_2(from#(s(X))):2
          
          4:S:sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y))
             -->_2 sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y)):4
             -->_1 filter#(s(s(X)),cons(Y,Z)) -> c_1(if#(divides(s(s(X)),Y)
                                                        ,filter(s(s(X)),Z)
                                                        ,cons(Y,filter(X,sieve(Y))))
                                                    ,filter#(s(s(X)),Z)
                                                    ,filter#(X,sieve(Y))
                                                    ,sieve#(Y)):1
          
          5:W:head#(cons(X,Y)) -> c_3()
             
          
          6:W:if#(false(),X,Y) -> c_4()
             
          
          7:W:if#(true(),X,Y) -> c_5()
             
          
          8:W:tail#(cons(X,Y)) -> c_8()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          8: tail#(cons(X,Y)) -> c_8()
          7: if#(true(),X,Y) -> c_5()
          6: if#(false(),X,Y) -> c_4()
          5: head#(cons(X,Y)) -> c_3()
* Step 5: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            filter#(s(s(X)),cons(Y,Z)) -> c_1(if#(divides(s(s(X)),Y),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y))))
                                             ,filter#(s(s(X)),Z)
                                             ,filter#(X,sieve(Y))
                                             ,sieve#(Y))
            from#(X) -> c_2(from#(s(X)))
            primes#() -> c_6(sieve#(from(s(s(0())))),from#(s(s(0()))))
            sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y))
        - Weak TRS:
            filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y))))
            from(X) -> cons(X,from(s(X)))
            sieve(cons(X,Y)) -> cons(X,filter(X,sieve(Y)))
        - 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,cons/2,divides/2,false/0,s/1,true/0,c_1/4,c_2/1,c_3/0,c_4/0,c_5/0,c_6/2,c_7/2,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter#,from#,head#,if#,primes#,sieve#
            ,tail#} and constructors {0,cons,divides,false,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:filter#(s(s(X)),cons(Y,Z)) -> c_1(if#(divides(s(s(X)),Y),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y))))
                                               ,filter#(s(s(X)),Z)
                                               ,filter#(X,sieve(Y))
                                               ,sieve#(Y))
             -->_4 sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y)):4
             -->_3 filter#(s(s(X)),cons(Y,Z)) -> c_1(if#(divides(s(s(X)),Y)
                                                        ,filter(s(s(X)),Z)
                                                        ,cons(Y,filter(X,sieve(Y))))
                                                    ,filter#(s(s(X)),Z)
                                                    ,filter#(X,sieve(Y))
                                                    ,sieve#(Y)):1
             -->_2 filter#(s(s(X)),cons(Y,Z)) -> c_1(if#(divides(s(s(X)),Y)
                                                        ,filter(s(s(X)),Z)
                                                        ,cons(Y,filter(X,sieve(Y))))
                                                    ,filter#(s(s(X)),Z)
                                                    ,filter#(X,sieve(Y))
                                                    ,sieve#(Y)):1
          
          2:S:from#(X) -> c_2(from#(s(X)))
             -->_1 from#(X) -> c_2(from#(s(X))):2
          
          3:S:primes#() -> c_6(sieve#(from(s(s(0())))),from#(s(s(0()))))
             -->_1 sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y)):4
             -->_2 from#(X) -> c_2(from#(s(X))):2
          
          4:S:sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y))
             -->_2 sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y)):4
             -->_1 filter#(s(s(X)),cons(Y,Z)) -> c_1(if#(divides(s(s(X)),Y)
                                                        ,filter(s(s(X)),Z)
                                                        ,cons(Y,filter(X,sieve(Y))))
                                                    ,filter#(s(s(X)),Z)
                                                    ,filter#(X,sieve(Y))
                                                    ,sieve#(Y)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          filter#(s(s(X)),cons(Y,Z)) -> c_1(filter#(s(s(X)),Z),filter#(X,sieve(Y)),sieve#(Y))
* Step 6: Decompose MAYBE
    + Considered Problem:
        - Strict DPs:
            filter#(s(s(X)),cons(Y,Z)) -> c_1(filter#(s(s(X)),Z),filter#(X,sieve(Y)),sieve#(Y))
            from#(X) -> c_2(from#(s(X)))
            primes#() -> c_6(sieve#(from(s(s(0())))),from#(s(s(0()))))
            sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y))
        - Weak TRS:
            filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y))))
            from(X) -> cons(X,from(s(X)))
            sieve(cons(X,Y)) -> cons(X,filter(X,sieve(Y)))
        - 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,cons/2,divides/2,false/0,s/1,true/0,c_1/3,c_2/1,c_3/0,c_4/0,c_5/0,c_6/2,c_7/2,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter#,from#,head#,if#,primes#,sieve#
            ,tail#} and constructors {0,cons,divides,false,s,true}
    + 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#(s(s(X)),cons(Y,Z)) -> c_1(filter#(s(s(X)),Z),filter#(X,sieve(Y)),sieve#(Y))
              sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y))
          - Weak DPs:
              from#(X) -> c_2(from#(s(X)))
              primes#() -> c_6(sieve#(from(s(s(0())))),from#(s(s(0()))))
          - Weak TRS:
              filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y))))
              from(X) -> cons(X,from(s(X)))
              sieve(cons(X,Y)) -> cons(X,filter(X,sieve(Y)))
          - 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,cons/2,divides/2,false/0,s/1,true/0,c_1/3,c_2/1,c_3/0,c_4/0,c_5/0,c_6/2,c_7/2,c_8/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {filter#,from#,head#,if#,primes#,sieve#
              ,tail#} and constructors {0,cons,divides,false,s,true}
        
        Problem (S)
          - Strict DPs:
              from#(X) -> c_2(from#(s(X)))
              primes#() -> c_6(sieve#(from(s(s(0())))),from#(s(s(0()))))
          - Weak DPs:
              filter#(s(s(X)),cons(Y,Z)) -> c_1(filter#(s(s(X)),Z),filter#(X,sieve(Y)),sieve#(Y))
              sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y))
          - Weak TRS:
              filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y))))
              from(X) -> cons(X,from(s(X)))
              sieve(cons(X,Y)) -> cons(X,filter(X,sieve(Y)))
          - 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,cons/2,divides/2,false/0,s/1,true/0,c_1/3,c_2/1,c_3/0,c_4/0,c_5/0,c_6/2,c_7/2,c_8/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {filter#,from#,head#,if#,primes#,sieve#
              ,tail#} and constructors {0,cons,divides,false,s,true}
** Step 6.a:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            filter#(s(s(X)),cons(Y,Z)) -> c_1(filter#(s(s(X)),Z),filter#(X,sieve(Y)),sieve#(Y))
            sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y))
        - Weak DPs:
            from#(X) -> c_2(from#(s(X)))
            primes#() -> c_6(sieve#(from(s(s(0())))),from#(s(s(0()))))
        - Weak TRS:
            filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y))))
            from(X) -> cons(X,from(s(X)))
            sieve(cons(X,Y)) -> cons(X,filter(X,sieve(Y)))
        - 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,cons/2,divides/2,false/0,s/1,true/0,c_1/3,c_2/1,c_3/0,c_4/0,c_5/0,c_6/2,c_7/2,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter#,from#,head#,if#,primes#,sieve#
            ,tail#} and constructors {0,cons,divides,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:filter#(s(s(X)),cons(Y,Z)) -> c_1(filter#(s(s(X)),Z),filter#(X,sieve(Y)),sieve#(Y))
             -->_3 sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y)):4
             -->_2 filter#(s(s(X)),cons(Y,Z)) -> c_1(filter#(s(s(X)),Z),filter#(X,sieve(Y)),sieve#(Y)):1
             -->_1 filter#(s(s(X)),cons(Y,Z)) -> c_1(filter#(s(s(X)),Z),filter#(X,sieve(Y)),sieve#(Y)):1
          
          2:W:from#(X) -> c_2(from#(s(X)))
             -->_1 from#(X) -> c_2(from#(s(X))):2
          
          3:W:primes#() -> c_6(sieve#(from(s(s(0())))),from#(s(s(0()))))
             -->_2 from#(X) -> c_2(from#(s(X))):2
             -->_1 sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y)):4
          
          4:S:sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y))
             -->_1 filter#(s(s(X)),cons(Y,Z)) -> c_1(filter#(s(s(X)),Z),filter#(X,sieve(Y)),sieve#(Y)):1
             -->_2 sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y)):4
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: from#(X) -> c_2(from#(s(X)))
** Step 6.a:2: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            filter#(s(s(X)),cons(Y,Z)) -> c_1(filter#(s(s(X)),Z),filter#(X,sieve(Y)),sieve#(Y))
            sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y))
        - Weak DPs:
            primes#() -> c_6(sieve#(from(s(s(0())))),from#(s(s(0()))))
        - Weak TRS:
            filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y))))
            from(X) -> cons(X,from(s(X)))
            sieve(cons(X,Y)) -> cons(X,filter(X,sieve(Y)))
        - 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,cons/2,divides/2,false/0,s/1,true/0,c_1/3,c_2/1,c_3/0,c_4/0,c_5/0,c_6/2,c_7/2,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter#,from#,head#,if#,primes#,sieve#
            ,tail#} and constructors {0,cons,divides,false,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:filter#(s(s(X)),cons(Y,Z)) -> c_1(filter#(s(s(X)),Z),filter#(X,sieve(Y)),sieve#(Y))
             -->_3 sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y)):4
             -->_2 filter#(s(s(X)),cons(Y,Z)) -> c_1(filter#(s(s(X)),Z),filter#(X,sieve(Y)),sieve#(Y)):1
             -->_1 filter#(s(s(X)),cons(Y,Z)) -> c_1(filter#(s(s(X)),Z),filter#(X,sieve(Y)),sieve#(Y)):1
          
          3:W:primes#() -> c_6(sieve#(from(s(s(0())))),from#(s(s(0()))))
             -->_1 sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y)):4
          
          4:S:sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y))
             -->_1 filter#(s(s(X)),cons(Y,Z)) -> c_1(filter#(s(s(X)),Z),filter#(X,sieve(Y)),sieve#(Y)):1
             -->_2 sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y)):4
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          primes#() -> c_6(sieve#(from(s(s(0())))))
** Step 6.a:3: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          filter#(s(s(X)),cons(Y,Z)) -> c_1(filter#(s(s(X)),Z),filter#(X,sieve(Y)),sieve#(Y))
          sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y))
      - Weak DPs:
          primes#() -> c_6(sieve#(from(s(s(0())))))
      - Weak TRS:
          filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y))))
          from(X) -> cons(X,from(s(X)))
          sieve(cons(X,Y)) -> cons(X,filter(X,sieve(Y)))
      - 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,cons/2,divides/2,false/0,s/1,true/0,c_1/3,c_2/1,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2,c_8/0}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {filter#,from#,head#,if#,primes#,sieve#
          ,tail#} and constructors {0,cons,divides,false,s,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
** Step 6.b:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            from#(X) -> c_2(from#(s(X)))
            primes#() -> c_6(sieve#(from(s(s(0())))),from#(s(s(0()))))
        - Weak DPs:
            filter#(s(s(X)),cons(Y,Z)) -> c_1(filter#(s(s(X)),Z),filter#(X,sieve(Y)),sieve#(Y))
            sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y))
        - Weak TRS:
            filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y))))
            from(X) -> cons(X,from(s(X)))
            sieve(cons(X,Y)) -> cons(X,filter(X,sieve(Y)))
        - 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,cons/2,divides/2,false/0,s/1,true/0,c_1/3,c_2/1,c_3/0,c_4/0,c_5/0,c_6/2,c_7/2,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter#,from#,head#,if#,primes#,sieve#
            ,tail#} and constructors {0,cons,divides,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:from#(X) -> c_2(from#(s(X)))
             -->_1 from#(X) -> c_2(from#(s(X))):1
          
          2:S:primes#() -> c_6(sieve#(from(s(s(0())))),from#(s(s(0()))))
             -->_1 sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y)):4
             -->_2 from#(X) -> c_2(from#(s(X))):1
          
          3:W:filter#(s(s(X)),cons(Y,Z)) -> c_1(filter#(s(s(X)),Z),filter#(X,sieve(Y)),sieve#(Y))
             -->_3 sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y)):4
             -->_2 filter#(s(s(X)),cons(Y,Z)) -> c_1(filter#(s(s(X)),Z),filter#(X,sieve(Y)),sieve#(Y)):3
             -->_1 filter#(s(s(X)),cons(Y,Z)) -> c_1(filter#(s(s(X)),Z),filter#(X,sieve(Y)),sieve#(Y)):3
          
          4:W:sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y))
             -->_2 sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y)):4
             -->_1 filter#(s(s(X)),cons(Y,Z)) -> c_1(filter#(s(s(X)),Z),filter#(X,sieve(Y)),sieve#(Y)):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: sieve#(cons(X,Y)) -> c_7(filter#(X,sieve(Y)),sieve#(Y))
          3: filter#(s(s(X)),cons(Y,Z)) -> c_1(filter#(s(s(X)),Z),filter#(X,sieve(Y)),sieve#(Y))
** Step 6.b:2: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            from#(X) -> c_2(from#(s(X)))
            primes#() -> c_6(sieve#(from(s(s(0())))),from#(s(s(0()))))
        - Weak TRS:
            filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y))))
            from(X) -> cons(X,from(s(X)))
            sieve(cons(X,Y)) -> cons(X,filter(X,sieve(Y)))
        - 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,cons/2,divides/2,false/0,s/1,true/0,c_1/3,c_2/1,c_3/0,c_4/0,c_5/0,c_6/2,c_7/2,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter#,from#,head#,if#,primes#,sieve#
            ,tail#} and constructors {0,cons,divides,false,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:from#(X) -> c_2(from#(s(X)))
             -->_1 from#(X) -> c_2(from#(s(X))):1
          
          2:S:primes#() -> c_6(sieve#(from(s(s(0())))),from#(s(s(0()))))
             -->_2 from#(X) -> c_2(from#(s(X))):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          primes#() -> c_6(from#(s(s(0()))))
** Step 6.b:3: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            from#(X) -> c_2(from#(s(X)))
            primes#() -> c_6(from#(s(s(0()))))
        - Weak TRS:
            filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y))))
            from(X) -> cons(X,from(s(X)))
            sieve(cons(X,Y)) -> cons(X,filter(X,sieve(Y)))
        - 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,cons/2,divides/2,false/0,s/1,true/0,c_1/3,c_2/1,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter#,from#,head#,if#,primes#,sieve#
            ,tail#} and constructors {0,cons,divides,false,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          from#(X) -> c_2(from#(s(X)))
          primes#() -> c_6(from#(s(s(0()))))
** Step 6.b:4: Decompose MAYBE
    + Considered Problem:
        - Strict DPs:
            from#(X) -> c_2(from#(s(X)))
            primes#() -> c_6(from#(s(s(0()))))
        - 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,cons/2,divides/2,false/0,s/1,true/0,c_1/3,c_2/1,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter#,from#,head#,if#,primes#,sieve#
            ,tail#} and constructors {0,cons,divides,false,s,true}
    + 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:
              from#(X) -> c_2(from#(s(X)))
          - Weak DPs:
              primes#() -> c_6(from#(s(s(0()))))
          - 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,cons/2,divides/2,false/0,s/1,true/0,c_1/3,c_2/1,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2,c_8/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {filter#,from#,head#,if#,primes#,sieve#
              ,tail#} and constructors {0,cons,divides,false,s,true}
        
        Problem (S)
          - Strict DPs:
              primes#() -> c_6(from#(s(s(0()))))
          - Weak DPs:
              from#(X) -> c_2(from#(s(X)))
          - 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,cons/2,divides/2,false/0,s/1,true/0,c_1/3,c_2/1,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2,c_8/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {filter#,from#,head#,if#,primes#,sieve#
              ,tail#} and constructors {0,cons,divides,false,s,true}
*** Step 6.b:4.a:1: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          from#(X) -> c_2(from#(s(X)))
      - Weak DPs:
          primes#() -> c_6(from#(s(s(0()))))
      - 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,cons/2,divides/2,false/0,s/1,true/0,c_1/3,c_2/1,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2,c_8/0}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {filter#,from#,head#,if#,primes#,sieve#
          ,tail#} and constructors {0,cons,divides,false,s,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
*** Step 6.b:4.b:1: PredecessorEstimation WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            primes#() -> c_6(from#(s(s(0()))))
        - Weak DPs:
            from#(X) -> c_2(from#(s(X)))
        - 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,cons/2,divides/2,false/0,s/1,true/0,c_1/3,c_2/1,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter#,from#,head#,if#,primes#,sieve#
            ,tail#} and constructors {0,cons,divides,false,s,true}
    + 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: primes#() -> c_6(from#(s(s(0()))))
          2: from#(X) -> c_2(from#(s(X)))
*** Step 6.b:4.b:2: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            from#(X) -> c_2(from#(s(X)))
            primes#() -> c_6(from#(s(s(0()))))
        - 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,cons/2,divides/2,false/0,s/1,true/0,c_1/3,c_2/1,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter#,from#,head#,if#,primes#,sieve#
            ,tail#} and constructors {0,cons,divides,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:from#(X) -> c_2(from#(s(X)))
             -->_1 from#(X) -> c_2(from#(s(X))):1
          
          2:W:primes#() -> c_6(from#(s(s(0()))))
             -->_1 from#(X) -> c_2(from#(s(X))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: primes#() -> c_6(from#(s(s(0()))))
          1: from#(X) -> c_2(from#(s(X)))
*** Step 6.b:4.b:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - 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,cons/2,divides/2,false/0,s/1,true/0,c_1/3,c_2/1,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {filter#,from#,head#,if#,primes#,sieve#
            ,tail#} and constructors {0,cons,divides,false,s,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

MAYBE