*** 1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        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)
      Weak DP Rules:
        
      Weak TRS Rules:
        
      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
        basic terms: {activate,cons,filter,from,head,if,primes,sieve,tail}/{0,divides,false,n__cons,n__filter,n__from,s,true}
    Applied Processor:
      InnermostRuleRemoval
    Proof:
      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.
*** 1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        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()))))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      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
        basic terms: {activate,cons,filter,from,head,if,primes,sieve,tail}/{0,divides,false,n__cons,n__filter,n__from,s,true}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      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.
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        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()))))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        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
        basic terms: {activate#,cons#,filter#,from#,head#,if#,primes#,sieve#,tail#}/{0,divides,false,n__cons,n__filter,n__from,s,true}
    Applied Processor:
      UsableRules
    Proof:
      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()))))
*** 1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        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()))))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        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
        basic terms: {activate#,cons#,filter#,from#,head#,if#,primes#,sieve#,tail#}/{0,divides,false,n__cons,n__filter,n__from,s,true}
    Applied Processor:
      Trivial
    Proof:
      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.
*** 1.1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        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
        basic terms: {activate#,cons#,filter#,from#,head#,if#,primes#,sieve#,tail#}/{0,divides,false,n__cons,n__filter,n__from,s,true}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).