WORST_CASE(?,O(1))
* Step 1: InnermostRuleRemoval WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__cons(X1,X2)) -> cons(X1,X2)
            activate(n__filter(X1,X2)) -> filter(X1,X2)
            activate(n__from(X)) -> from(X)
            cons(X1,X2) -> n__cons(X1,X2)
            filter(X1,X2) -> n__filter(X1,X2)
            filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y)
                                           ,n__filter(s(s(X)),activate(Z))
                                           ,n__cons(Y,n__filter(X,sieve(Y))))
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            head(cons(X,Y)) -> X
            if(false(),X,Y) -> activate(Y)
            if(true(),X,Y) -> activate(X)
            primes() -> sieve(from(s(s(0()))))
            sieve(cons(X,Y)) -> cons(X,n__filter(X,sieve(activate(Y))))
            tail(cons(X,Y)) -> activate(Y)
        - Signature:
            {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,sieve/1,tail/1} / {0/0,divides/2,false/0,n__cons/2
            ,n__filter/2,n__from/1,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,cons,filter,from,head,if,primes,sieve
            ,tail} and constructors {0,divides,false,n__cons,n__filter,n__from,s,true}
    + Applied Processor:
        InnermostRuleRemoval
    + Details:
        Arguments of following rules are not normal-forms.
          filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y)
                                         ,n__filter(s(s(X)),activate(Z))
                                         ,n__cons(Y,n__filter(X,sieve(Y))))
          head(cons(X,Y)) -> X
          sieve(cons(X,Y)) -> cons(X,n__filter(X,sieve(activate(Y))))
          tail(cons(X,Y)) -> activate(Y)
        All above mentioned rules can be savely removed.
* Step 2: DependencyPairs WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__cons(X1,X2)) -> cons(X1,X2)
            activate(n__filter(X1,X2)) -> filter(X1,X2)
            activate(n__from(X)) -> from(X)
            cons(X1,X2) -> n__cons(X1,X2)
            filter(X1,X2) -> n__filter(X1,X2)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            if(false(),X,Y) -> activate(Y)
            if(true(),X,Y) -> activate(X)
            primes() -> sieve(from(s(s(0()))))
        - Signature:
            {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,sieve/1,tail/1} / {0/0,divides/2,false/0,n__cons/2
            ,n__filter/2,n__from/1,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,cons,filter,from,head,if,primes,sieve
            ,tail} and constructors {0,divides,false,n__cons,n__filter,n__from,s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          activate#(X) -> c_1()
          activate#(n__cons(X1,X2)) -> c_2(cons#(X1,X2))
          activate#(n__filter(X1,X2)) -> c_3(filter#(X1,X2))
          activate#(n__from(X)) -> c_4(from#(X))
          cons#(X1,X2) -> c_5()
          filter#(X1,X2) -> c_6()
          from#(X) -> c_7(cons#(X,n__from(s(X))))
          from#(X) -> c_8()
          if#(false(),X,Y) -> c_9(activate#(Y))
          if#(true(),X,Y) -> c_10(activate#(X))
          primes#() -> c_11(sieve#(from(s(s(0())))),from#(s(s(0()))))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: UsableRules WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            activate#(n__cons(X1,X2)) -> c_2(cons#(X1,X2))
            activate#(n__filter(X1,X2)) -> c_3(filter#(X1,X2))
            activate#(n__from(X)) -> c_4(from#(X))
            cons#(X1,X2) -> c_5()
            filter#(X1,X2) -> c_6()
            from#(X) -> c_7(cons#(X,n__from(s(X))))
            from#(X) -> c_8()
            if#(false(),X,Y) -> c_9(activate#(Y))
            if#(true(),X,Y) -> c_10(activate#(X))
            primes#() -> c_11(sieve#(from(s(s(0())))),from#(s(s(0()))))
        - Weak TRS:
            activate(X) -> X
            activate(n__cons(X1,X2)) -> cons(X1,X2)
            activate(n__filter(X1,X2)) -> filter(X1,X2)
            activate(n__from(X)) -> from(X)
            cons(X1,X2) -> n__cons(X1,X2)
            filter(X1,X2) -> n__filter(X1,X2)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            if(false(),X,Y) -> activate(Y)
            if(true(),X,Y) -> activate(X)
            primes() -> sieve(from(s(s(0()))))
        - Signature:
            {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,sieve/1,tail/1,activate#/1,cons#/2,filter#/2,from#/1
            ,head#/1,if#/3,primes#/0,sieve#/1,tail#/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2,n__from/1,s/1
            ,true/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,cons#,filter#,from#,head#,if#,primes#,sieve#
            ,tail#} and constructors {0,divides,false,n__cons,n__filter,n__from,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          cons(X1,X2) -> n__cons(X1,X2)
          from(X) -> cons(X,n__from(s(X)))
          from(X) -> n__from(X)
          activate#(X) -> c_1()
          activate#(n__cons(X1,X2)) -> c_2(cons#(X1,X2))
          activate#(n__filter(X1,X2)) -> c_3(filter#(X1,X2))
          activate#(n__from(X)) -> c_4(from#(X))
          cons#(X1,X2) -> c_5()
          filter#(X1,X2) -> c_6()
          from#(X) -> c_7(cons#(X,n__from(s(X))))
          from#(X) -> c_8()
          if#(false(),X,Y) -> c_9(activate#(Y))
          if#(true(),X,Y) -> c_10(activate#(X))
          primes#() -> c_11(sieve#(from(s(s(0())))),from#(s(s(0()))))
* Step 4: Trivial WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            activate#(n__cons(X1,X2)) -> c_2(cons#(X1,X2))
            activate#(n__filter(X1,X2)) -> c_3(filter#(X1,X2))
            activate#(n__from(X)) -> c_4(from#(X))
            cons#(X1,X2) -> c_5()
            filter#(X1,X2) -> c_6()
            from#(X) -> c_7(cons#(X,n__from(s(X))))
            from#(X) -> c_8()
            if#(false(),X,Y) -> c_9(activate#(Y))
            if#(true(),X,Y) -> c_10(activate#(X))
            primes#() -> c_11(sieve#(from(s(s(0())))),from#(s(s(0()))))
        - Weak TRS:
            cons(X1,X2) -> n__cons(X1,X2)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
        - Signature:
            {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,sieve/1,tail/1,activate#/1,cons#/2,filter#/2,from#/1
            ,head#/1,if#/3,primes#/0,sieve#/1,tail#/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2,n__from/1,s/1
            ,true/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,cons#,filter#,from#,head#,if#,primes#,sieve#
            ,tail#} and constructors {0,divides,false,n__cons,n__filter,n__from,s,true}
    + Applied Processor:
        Trivial
    + Details:
        Consider the dependency graph
          1:S:activate#(X) -> c_1()
             
          
          2:S:activate#(n__cons(X1,X2)) -> c_2(cons#(X1,X2))
             -->_1 cons#(X1,X2) -> c_5():5
          
          3:S:activate#(n__filter(X1,X2)) -> c_3(filter#(X1,X2))
             -->_1 filter#(X1,X2) -> c_6():6
          
          4:S:activate#(n__from(X)) -> c_4(from#(X))
             -->_1 from#(X) -> c_7(cons#(X,n__from(s(X)))):7
             -->_1 from#(X) -> c_8():8
          
          5:S:cons#(X1,X2) -> c_5()
             
          
          6:S:filter#(X1,X2) -> c_6()
             
          
          7:S:from#(X) -> c_7(cons#(X,n__from(s(X))))
             -->_1 cons#(X1,X2) -> c_5():5
          
          8:S:from#(X) -> c_8()
             
          
          9:S:if#(false(),X,Y) -> c_9(activate#(Y))
             -->_1 activate#(n__from(X)) -> c_4(from#(X)):4
             -->_1 activate#(n__filter(X1,X2)) -> c_3(filter#(X1,X2)):3
             -->_1 activate#(n__cons(X1,X2)) -> c_2(cons#(X1,X2)):2
             -->_1 activate#(X) -> c_1():1
          
          10:S:if#(true(),X,Y) -> c_10(activate#(X))
             -->_1 activate#(n__from(X)) -> c_4(from#(X)):4
             -->_1 activate#(n__filter(X1,X2)) -> c_3(filter#(X1,X2)):3
             -->_1 activate#(n__cons(X1,X2)) -> c_2(cons#(X1,X2)):2
             -->_1 activate#(X) -> c_1():1
          
          11:S:primes#() -> c_11(sieve#(from(s(s(0())))),from#(s(s(0()))))
             -->_2 from#(X) -> c_8():8
             -->_2 from#(X) -> c_7(cons#(X,n__from(s(X)))):7
          
        The dependency graph contains no loops, we remove all dependency pairs.
* Step 5: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            cons(X1,X2) -> n__cons(X1,X2)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
        - Signature:
            {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,sieve/1,tail/1,activate#/1,cons#/2,filter#/2,from#/1
            ,head#/1,if#/3,primes#/0,sieve#/1,tail#/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2,n__from/1,s/1
            ,true/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,cons#,filter#,from#,head#,if#,primes#,sieve#
            ,tail#} and constructors {0,divides,false,n__cons,n__filter,n__from,s,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))