*** 1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        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)
      Weak DP Rules:
        
      Weak TRS Rules:
        
      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
        basic terms: {activate,add,dbl,first,s,sqr,terms}/{0,cons,n__add,n__dbl,n__first,n__s,n__terms,nil,recip}
    Applied Processor:
      InnermostRuleRemoval
    Proof:
      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.
*** 1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        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)
      Weak DP Rules:
        
      Weak TRS Rules:
        
      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
        basic terms: {activate,add,dbl,first,s,sqr,terms}/{0,cons,n__add,n__dbl,n__first,n__s,n__terms,nil,recip}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      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.
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP 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()
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        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
        basic terms: {activate#,add#,dbl#,first#,s#,sqr#,terms#}/{0,cons,n__add,n__dbl,n__first,n__s,n__terms,nil,recip}
    Applied Processor:
      UsableRules
    Proof:
      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()
*** 1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP 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()
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      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
        basic terms: {activate#,add#,dbl#,first#,s#,sqr#,terms#}/{0,cons,n__add,n__dbl,n__first,n__s,n__terms,nil,recip}
    Applied Processor:
      Trivial
    Proof:
      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.
*** 1.1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      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
        basic terms: {activate#,add#,dbl#,first#,s#,sqr#,terms#}/{0,cons,n__add,n__dbl,n__first,n__s,n__terms,nil,recip}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).