* Step 1: Sum WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__add(X1,X2)) -> add(X1,X2)
            activate(n__dbl(X)) -> dbl(X)
            activate(n__first(X1,X2)) -> first(X1,X2)
            activate(n__s(X)) -> s(X)
            activate(n__terms(X)) -> terms(X)
            add(X1,X2) -> n__add(X1,X2)
            add(0(),X) -> X
            add(s(X),Y) -> s(n__add(activate(X),Y))
            dbl(X) -> n__dbl(X)
            dbl(0()) -> 0()
            dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
            first(X1,X2) -> n__first(X1,X2)
            first(0(),X) -> nil()
            first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
            s(X) -> n__s(X)
            sqr(0()) -> 0()
            sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X))))
            terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
            terms(X) -> n__terms(X)
        - Signature:
            {activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1
            ,n__terms/1,nil/0,recip/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,add,dbl,first,s,sqr,terms} and constructors {0
            ,cons,n__add,n__dbl,n__first,n__s,n__terms,nil,recip}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: InnermostRuleRemoval WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__add(X1,X2)) -> add(X1,X2)
            activate(n__dbl(X)) -> dbl(X)
            activate(n__first(X1,X2)) -> first(X1,X2)
            activate(n__s(X)) -> s(X)
            activate(n__terms(X)) -> terms(X)
            add(X1,X2) -> n__add(X1,X2)
            add(0(),X) -> X
            add(s(X),Y) -> s(n__add(activate(X),Y))
            dbl(X) -> n__dbl(X)
            dbl(0()) -> 0()
            dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
            first(X1,X2) -> n__first(X1,X2)
            first(0(),X) -> nil()
            first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
            s(X) -> n__s(X)
            sqr(0()) -> 0()
            sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X))))
            terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
            terms(X) -> n__terms(X)
        - Signature:
            {activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1
            ,n__terms/1,nil/0,recip/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,add,dbl,first,s,sqr,terms} and constructors {0
            ,cons,n__add,n__dbl,n__first,n__s,n__terms,nil,recip}
    + Applied Processor:
        InnermostRuleRemoval
    + Details:
        Arguments of following rules are not normal-forms.
          add(s(X),Y) -> s(n__add(activate(X),Y))
          dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
          first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
          sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X))))
        All above mentioned rules can be savely removed.
* Step 3: DependencyPairs WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__add(X1,X2)) -> add(X1,X2)
            activate(n__dbl(X)) -> dbl(X)
            activate(n__first(X1,X2)) -> first(X1,X2)
            activate(n__s(X)) -> s(X)
            activate(n__terms(X)) -> terms(X)
            add(X1,X2) -> n__add(X1,X2)
            add(0(),X) -> X
            dbl(X) -> n__dbl(X)
            dbl(0()) -> 0()
            first(X1,X2) -> n__first(X1,X2)
            first(0(),X) -> nil()
            s(X) -> n__s(X)
            sqr(0()) -> 0()
            terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
            terms(X) -> n__terms(X)
        - Signature:
            {activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1
            ,n__terms/1,nil/0,recip/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,add,dbl,first,s,sqr,terms} and constructors {0
            ,cons,n__add,n__dbl,n__first,n__s,n__terms,nil,recip}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          activate#(X) -> c_1()
          activate#(n__add(X1,X2)) -> c_2(add#(X1,X2))
          activate#(n__dbl(X)) -> c_3(dbl#(X))
          activate#(n__first(X1,X2)) -> c_4(first#(X1,X2))
          activate#(n__s(X)) -> c_5(s#(X))
          activate#(n__terms(X)) -> c_6(terms#(X))
          add#(X1,X2) -> c_7()
          add#(0(),X) -> c_8()
          dbl#(X) -> c_9()
          dbl#(0()) -> c_10()
          first#(X1,X2) -> c_11()
          first#(0(),X) -> c_12()
          s#(X) -> c_13()
          sqr#(0()) -> c_14()
          terms#(N) -> c_15(sqr#(N),s#(N))
          terms#(X) -> c_16()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 4: UsableRules WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            activate#(n__add(X1,X2)) -> c_2(add#(X1,X2))
            activate#(n__dbl(X)) -> c_3(dbl#(X))
            activate#(n__first(X1,X2)) -> c_4(first#(X1,X2))
            activate#(n__s(X)) -> c_5(s#(X))
            activate#(n__terms(X)) -> c_6(terms#(X))
            add#(X1,X2) -> c_7()
            add#(0(),X) -> c_8()
            dbl#(X) -> c_9()
            dbl#(0()) -> c_10()
            first#(X1,X2) -> c_11()
            first#(0(),X) -> c_12()
            s#(X) -> c_13()
            sqr#(0()) -> c_14()
            terms#(N) -> c_15(sqr#(N),s#(N))
            terms#(X) -> c_16()
        - Weak TRS:
            activate(X) -> X
            activate(n__add(X1,X2)) -> add(X1,X2)
            activate(n__dbl(X)) -> dbl(X)
            activate(n__first(X1,X2)) -> first(X1,X2)
            activate(n__s(X)) -> s(X)
            activate(n__terms(X)) -> terms(X)
            add(X1,X2) -> n__add(X1,X2)
            add(0(),X) -> X
            dbl(X) -> n__dbl(X)
            dbl(0()) -> 0()
            first(X1,X2) -> n__first(X1,X2)
            first(0(),X) -> nil()
            s(X) -> n__s(X)
            sqr(0()) -> 0()
            terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
            terms(X) -> n__terms(X)
        - Signature:
            {activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,s#/1,sqr#/1
            ,terms#/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1,n__terms/1,nil/0,recip/1,c_1/0,c_2/1,c_3/1
            ,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/2,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,s#,sqr#
            ,terms#} and constructors {0,cons,n__add,n__dbl,n__first,n__s,n__terms,nil,recip}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          activate#(X) -> c_1()
          activate#(n__add(X1,X2)) -> c_2(add#(X1,X2))
          activate#(n__dbl(X)) -> c_3(dbl#(X))
          activate#(n__first(X1,X2)) -> c_4(first#(X1,X2))
          activate#(n__s(X)) -> c_5(s#(X))
          activate#(n__terms(X)) -> c_6(terms#(X))
          add#(X1,X2) -> c_7()
          add#(0(),X) -> c_8()
          dbl#(X) -> c_9()
          dbl#(0()) -> c_10()
          first#(X1,X2) -> c_11()
          first#(0(),X) -> c_12()
          s#(X) -> c_13()
          sqr#(0()) -> c_14()
          terms#(N) -> c_15(sqr#(N),s#(N))
          terms#(X) -> c_16()
* Step 5: Trivial WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            activate#(n__add(X1,X2)) -> c_2(add#(X1,X2))
            activate#(n__dbl(X)) -> c_3(dbl#(X))
            activate#(n__first(X1,X2)) -> c_4(first#(X1,X2))
            activate#(n__s(X)) -> c_5(s#(X))
            activate#(n__terms(X)) -> c_6(terms#(X))
            add#(X1,X2) -> c_7()
            add#(0(),X) -> c_8()
            dbl#(X) -> c_9()
            dbl#(0()) -> c_10()
            first#(X1,X2) -> c_11()
            first#(0(),X) -> c_12()
            s#(X) -> c_13()
            sqr#(0()) -> c_14()
            terms#(N) -> c_15(sqr#(N),s#(N))
            terms#(X) -> c_16()
        - Signature:
            {activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,s#/1,sqr#/1
            ,terms#/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1,n__terms/1,nil/0,recip/1,c_1/0,c_2/1,c_3/1
            ,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/2,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,s#,sqr#
            ,terms#} and constructors {0,cons,n__add,n__dbl,n__first,n__s,n__terms,nil,recip}
    + Applied Processor:
        Trivial
    + Details:
        Consider the dependency graph
          1:S:activate#(X) -> c_1()
             
          
          2:S:activate#(n__add(X1,X2)) -> c_2(add#(X1,X2))
             -->_1 add#(0(),X) -> c_8():8
             -->_1 add#(X1,X2) -> c_7():7
          
          3:S:activate#(n__dbl(X)) -> c_3(dbl#(X))
             -->_1 dbl#(0()) -> c_10():10
             -->_1 dbl#(X) -> c_9():9
          
          4:S:activate#(n__first(X1,X2)) -> c_4(first#(X1,X2))
             -->_1 first#(0(),X) -> c_12():12
             -->_1 first#(X1,X2) -> c_11():11
          
          5:S:activate#(n__s(X)) -> c_5(s#(X))
             -->_1 s#(X) -> c_13():13
          
          6:S:activate#(n__terms(X)) -> c_6(terms#(X))
             -->_1 terms#(N) -> c_15(sqr#(N),s#(N)):15
             -->_1 terms#(X) -> c_16():16
          
          7:S:add#(X1,X2) -> c_7()
             
          
          8:S:add#(0(),X) -> c_8()
             
          
          9:S:dbl#(X) -> c_9()
             
          
          10:S:dbl#(0()) -> c_10()
             
          
          11:S:first#(X1,X2) -> c_11()
             
          
          12:S:first#(0(),X) -> c_12()
             
          
          13:S:s#(X) -> c_13()
             
          
          14:S:sqr#(0()) -> c_14()
             
          
          15:S:terms#(N) -> c_15(sqr#(N),s#(N))
             -->_1 sqr#(0()) -> c_14():14
             -->_2 s#(X) -> c_13():13
          
          16:S:terms#(X) -> c_16()
             
          
        The dependency graph contains no loops, we remove all dependency pairs.
* Step 6: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,s#/1,sqr#/1
            ,terms#/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1,n__terms/1,nil/0,recip/1,c_1/0,c_2/1,c_3/1
            ,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/2,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,s#,sqr#
            ,terms#} and constructors {0,cons,n__add,n__dbl,n__first,n__s,n__terms,nil,recip}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))