* Step 1: DependencyPairs WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__first(X1,X2)) -> first(X1,X2)
            activate(n__terms(X)) -> terms(X)
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            first(X1,X2) -> n__first(X1,X2)
            first(0(),X) -> nil()
            first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(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,sqr/1,terms/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,add,dbl,first,sqr,terms} and constructors {0
            ,cons,n__first,n__terms,nil,recip,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          activate#(X) -> c_1()
          activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
          activate#(n__terms(X)) -> c_3(terms#(X))
          add#(0(),X) -> c_4()
          add#(s(X),Y) -> c_5(add#(X,Y))
          dbl#(0()) -> c_6()
          dbl#(s(X)) -> c_7(dbl#(X))
          first#(X1,X2) -> c_8()
          first#(0(),X) -> c_9()
          first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
          sqr#(0()) -> c_11()
          sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
          terms#(N) -> c_13(sqr#(N))
          terms#(X) -> c_14()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
            activate#(n__terms(X)) -> c_3(terms#(X))
            add#(0(),X) -> c_4()
            add#(s(X),Y) -> c_5(add#(X,Y))
            dbl#(0()) -> c_6()
            dbl#(s(X)) -> c_7(dbl#(X))
            first#(X1,X2) -> c_8()
            first#(0(),X) -> c_9()
            first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
            sqr#(0()) -> c_11()
            sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
            terms#(N) -> c_13(sqr#(N))
            terms#(X) -> c_14()
        - Weak TRS:
            activate(X) -> X
            activate(n__first(X1,X2)) -> first(X1,X2)
            activate(n__terms(X)) -> terms(X)
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            first(X1,X2) -> n__first(X1,X2)
            first(0(),X) -> nil()
            first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(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,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0
            ,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1
            ,c_11/0,c_12/3,c_13/1,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,sqr#
            ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          add(0(),X) -> X
          add(s(X),Y) -> s(add(X,Y))
          dbl(0()) -> 0()
          dbl(s(X)) -> s(s(dbl(X)))
          sqr(0()) -> 0()
          sqr(s(X)) -> s(add(sqr(X),dbl(X)))
          activate#(X) -> c_1()
          activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
          activate#(n__terms(X)) -> c_3(terms#(X))
          add#(0(),X) -> c_4()
          add#(s(X),Y) -> c_5(add#(X,Y))
          dbl#(0()) -> c_6()
          dbl#(s(X)) -> c_7(dbl#(X))
          first#(X1,X2) -> c_8()
          first#(0(),X) -> c_9()
          first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
          sqr#(0()) -> c_11()
          sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
          terms#(N) -> c_13(sqr#(N))
          terms#(X) -> c_14()
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
            activate#(n__terms(X)) -> c_3(terms#(X))
            add#(0(),X) -> c_4()
            add#(s(X),Y) -> c_5(add#(X,Y))
            dbl#(0()) -> c_6()
            dbl#(s(X)) -> c_7(dbl#(X))
            first#(X1,X2) -> c_8()
            first#(0(),X) -> c_9()
            first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
            sqr#(0()) -> c_11()
            sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
            terms#(N) -> c_13(sqr#(N))
            terms#(X) -> c_14()
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {activate/1,add/2,dbl/1,first/2,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0
            ,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1
            ,c_11/0,c_12/3,c_13/1,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,sqr#
            ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,4,6,8,9,11,14}
        by application of
          Pre({1,4,6,8,9,11,14}) = {2,3,5,7,10,12,13}.
        Here rules are labelled as follows:
          1: activate#(X) -> c_1()
          2: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
          3: activate#(n__terms(X)) -> c_3(terms#(X))
          4: add#(0(),X) -> c_4()
          5: add#(s(X),Y) -> c_5(add#(X,Y))
          6: dbl#(0()) -> c_6()
          7: dbl#(s(X)) -> c_7(dbl#(X))
          8: first#(X1,X2) -> c_8()
          9: first#(0(),X) -> c_9()
          10: first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
          11: sqr#(0()) -> c_11()
          12: sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
          13: terms#(N) -> c_13(sqr#(N))
          14: terms#(X) -> c_14()
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
            activate#(n__terms(X)) -> c_3(terms#(X))
            add#(s(X),Y) -> c_5(add#(X,Y))
            dbl#(s(X)) -> c_7(dbl#(X))
            first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
            sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
            terms#(N) -> c_13(sqr#(N))
        - Weak DPs:
            activate#(X) -> c_1()
            add#(0(),X) -> c_4()
            dbl#(0()) -> c_6()
            first#(X1,X2) -> c_8()
            first#(0(),X) -> c_9()
            sqr#(0()) -> c_11()
            terms#(X) -> c_14()
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {activate/1,add/2,dbl/1,first/2,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0
            ,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1
            ,c_11/0,c_12/3,c_13/1,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,sqr#
            ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
             -->_1 first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)):5
             -->_1 first#(0(),X) -> c_9():12
             -->_1 first#(X1,X2) -> c_8():11
          
          2:S:activate#(n__terms(X)) -> c_3(terms#(X))
             -->_1 terms#(N) -> c_13(sqr#(N)):7
             -->_1 terms#(X) -> c_14():14
          
          3:S:add#(s(X),Y) -> c_5(add#(X,Y))
             -->_1 add#(0(),X) -> c_4():9
             -->_1 add#(s(X),Y) -> c_5(add#(X,Y)):3
          
          4:S:dbl#(s(X)) -> c_7(dbl#(X))
             -->_1 dbl#(0()) -> c_6():10
             -->_1 dbl#(s(X)) -> c_7(dbl#(X)):4
          
          5:S:first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
             -->_1 activate#(X) -> c_1():8
             -->_1 activate#(n__terms(X)) -> c_3(terms#(X)):2
             -->_1 activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)):1
          
          6:S:sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
             -->_2 sqr#(0()) -> c_11():13
             -->_3 dbl#(0()) -> c_6():10
             -->_1 add#(0(),X) -> c_4():9
             -->_2 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):6
             -->_3 dbl#(s(X)) -> c_7(dbl#(X)):4
             -->_1 add#(s(X),Y) -> c_5(add#(X,Y)):3
          
          7:S:terms#(N) -> c_13(sqr#(N))
             -->_1 sqr#(0()) -> c_11():13
             -->_1 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):6
          
          8:W:activate#(X) -> c_1()
             
          
          9:W:add#(0(),X) -> c_4()
             
          
          10:W:dbl#(0()) -> c_6()
             
          
          11:W:first#(X1,X2) -> c_8()
             
          
          12:W:first#(0(),X) -> c_9()
             
          
          13:W:sqr#(0()) -> c_11()
             
          
          14:W:terms#(X) -> c_14()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          11: first#(X1,X2) -> c_8()
          12: first#(0(),X) -> c_9()
          14: terms#(X) -> c_14()
          9: add#(0(),X) -> c_4()
          10: dbl#(0()) -> c_6()
          13: sqr#(0()) -> c_11()
          8: activate#(X) -> c_1()
* Step 5: PredecessorEstimationCP WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
            activate#(n__terms(X)) -> c_3(terms#(X))
            add#(s(X),Y) -> c_5(add#(X,Y))
            dbl#(s(X)) -> c_7(dbl#(X))
            first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
            sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
            terms#(N) -> c_13(sqr#(N))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {activate/1,add/2,dbl/1,first/2,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0
            ,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1
            ,c_11/0,c_12/3,c_13/1,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,sqr#
            ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
          4: dbl#(s(X)) -> c_7(dbl#(X))
          7: terms#(N) -> c_13(sqr#(N))
          
        Consider the set of all dependency pairs
          1: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
          2: activate#(n__terms(X)) -> c_3(terms#(X))
          3: add#(s(X),Y) -> c_5(add#(X,Y))
          4: dbl#(s(X)) -> c_7(dbl#(X))
          5: first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
          6: sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
          7: terms#(N) -> c_13(sqr#(N))
        Processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^2))
        SPACE(?,?)on application of the dependency pairs
          {1,4,7}
        These cover all (indirect) predecessors of dependency pairs
          {1,2,4,5,7}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
** Step 5.a:1: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
            activate#(n__terms(X)) -> c_3(terms#(X))
            add#(s(X),Y) -> c_5(add#(X,Y))
            dbl#(s(X)) -> c_7(dbl#(X))
            first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
            sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
            terms#(N) -> c_13(sqr#(N))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {activate/1,add/2,dbl/1,first/2,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0
            ,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1
            ,c_11/0,c_12/3,c_13/1,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,sqr#
            ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
    + Applied Processor:
        NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        The following argument positions are considered usable:
          uargs(c_2) = {1},
          uargs(c_3) = {1},
          uargs(c_5) = {1},
          uargs(c_7) = {1},
          uargs(c_10) = {1},
          uargs(c_12) = {1,2,3},
          uargs(c_13) = {1}
        
        Following symbols are considered usable:
          {activate#,add#,dbl#,first#,sqr#,terms#}
        TcT has computed the following interpretation:
                  p(0) = [0]                                       
                         [0]                                       
                         [1]                                       
           p(activate) = [0]                                       
                         [0]                                       
                         [0]                                       
                p(add) = [1 1 0]      [0]                          
                         [0 0 0] x1 + [0]                          
                         [0 0 1]      [1]                          
               p(cons) = [0 0 0]      [1]                          
                         [0 1 1] x2 + [1]                          
                         [0 0 0]      [0]                          
                p(dbl) = [0 0 1]      [1]                          
                         [0 0 0] x1 + [0]                          
                         [0 0 0]      [1]                          
              p(first) = [0]                                       
                         [0]                                       
                         [0]                                       
           p(n__first) = [0 1 0]      [0 0 0]      [0]             
                         [0 0 1] x1 + [0 1 0] x2 + [1]             
                         [0 0 0]      [0 0 0]      [1]             
           p(n__terms) = [0 0 0]      [0]                          
                         [0 1 0] x1 + [1]                          
                         [0 0 0]      [1]                          
                p(nil) = [0]                                       
                         [0]                                       
                         [0]                                       
              p(recip) = [0]                                       
                         [0]                                       
                         [0]                                       
                  p(s) = [0 0 0]      [0]                          
                         [0 1 1] x1 + [0]                          
                         [0 0 1]      [1]                          
                p(sqr) = [1 0 0]      [0]                          
                         [1 0 0] x1 + [0]                          
                         [0 0 1]      [1]                          
              p(terms) = [0]                                       
                         [0]                                       
                         [0]                                       
          p(activate#) = [0 1 0]      [1]                          
                         [0 0 1] x1 + [0]                          
                         [1 0 1]      [1]                          
               p(add#) = [0]                                       
                         [1]                                       
                         [0]                                       
               p(dbl#) = [0 0 1]      [0]                          
                         [0 1 1] x1 + [0]                          
                         [0 0 1]      [1]                          
             p(first#) = [0 1 0]      [0]                          
                         [0 0 0] x2 + [1]                          
                         [1 0 0]      [0]                          
               p(sqr#) = [0 1 0]      [0]                          
                         [0 0 1] x1 + [1]                          
                         [0 0 0]      [1]                          
             p(terms#) = [0 1 0]      [1]                          
                         [0 0 0] x1 + [1]                          
                         [0 0 0]      [1]                          
                p(c_1) = [0]                                       
                         [0]                                       
                         [0]                                       
                p(c_2) = [1 1 0]      [0]                          
                         [0 0 0] x1 + [0]                          
                         [0 0 0]      [0]                          
                p(c_3) = [1 1 0]      [0]                          
                         [0 0 1] x1 + [0]                          
                         [0 1 0]      [1]                          
                p(c_4) = [0]                                       
                         [0]                                       
                         [0]                                       
                p(c_5) = [1 0 0]      [0]                          
                         [0 1 0] x1 + [0]                          
                         [0 0 0]      [0]                          
                p(c_6) = [0]                                       
                         [0]                                       
                         [0]                                       
                p(c_7) = [1 0 0]      [0]                          
                         [0 1 0] x1 + [1]                          
                         [0 0 0]      [1]                          
                p(c_8) = [0]                                       
                         [0]                                       
                         [0]                                       
                p(c_9) = [0]                                       
                         [0]                                       
                         [0]                                       
               p(c_10) = [1 1 0]      [0]                          
                         [0 0 0] x1 + [0]                          
                         [0 0 0]      [1]                          
               p(c_11) = [0]                                       
                         [0]                                       
                         [0]                                       
               p(c_12) = [1 0 0]      [1 0 0]      [1 0 0]      [0]
                         [0 0 0] x1 + [0 0 0] x2 + [0 0 1] x3 + [1]
                         [0 0 0]      [0 0 1]      [0 0 0]      [0]
               p(c_13) = [1 0 0]      [0]                          
                         [0 0 0] x1 + [1]                          
                         [0 0 0]      [0]                          
               p(c_14) = [0]                                       
                         [0]                                       
                         [0]                                       
        
        Following rules are strictly oriented:
        activate#(n__first(X1,X2)) = [0 0 1]      [0 1 0]      [2]
                                     [0 0 0] X1 + [0 0 0] X2 + [1]
                                     [0 1 0]      [0 0 0]      [2]
                                   > [0 1 0]      [1]             
                                     [0 0 0] X2 + [0]             
                                     [0 0 0]      [0]             
                                   = c_2(first#(X1,X2))           
        
                        dbl#(s(X)) = [0 0 1]     [1]              
                                     [0 1 2] X + [1]              
                                     [0 0 1]     [2]              
                                   > [0 0 1]     [0]              
                                     [0 1 1] X + [1]              
                                     [0 0 0]     [1]              
                                   = c_7(dbl#(X))                 
        
                         terms#(N) = [0 1 0]     [1]              
                                     [0 0 0] N + [1]              
                                     [0 0 0]     [1]              
                                   > [0 1 0]     [0]              
                                     [0 0 0] N + [1]              
                                     [0 0 0]     [0]              
                                   = c_13(sqr#(N))                
        
        
        Following rules are (at-least) weakly oriented:
        activate#(n__terms(X)) =  [0 1 0]     [2]                          
                                  [0 0 0] X + [1]                          
                                  [0 0 0]     [2]                          
                               >= [0 1 0]     [2]                          
                                  [0 0 0] X + [1]                          
                                  [0 0 0]     [2]                          
                               =  c_3(terms#(X))                           
        
                  add#(s(X),Y) =  [0]                                      
                                  [1]                                      
                                  [0]                                      
                               >= [0]                                      
                                  [1]                                      
                                  [0]                                      
                               =  c_5(add#(X,Y))                           
        
        first#(s(X),cons(Y,Z)) =  [0 1 1]     [1]                          
                                  [0 0 0] Z + [1]                          
                                  [0 0 0]     [1]                          
                               >= [0 1 1]     [1]                          
                                  [0 0 0] Z + [0]                          
                                  [0 0 0]     [1]                          
                               =  c_10(activate#(Z))                       
        
                    sqr#(s(X)) =  [0 1 1]     [0]                          
                                  [0 0 1] X + [2]                          
                                  [0 0 0]     [1]                          
                               >= [0 1 1]     [0]                          
                                  [0 0 1] X + [2]                          
                                  [0 0 0]     [1]                          
                               =  c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
        
** Step 5.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__terms(X)) -> c_3(terms#(X))
            add#(s(X),Y) -> c_5(add#(X,Y))
            first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
            sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
        - Weak DPs:
            activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
            dbl#(s(X)) -> c_7(dbl#(X))
            terms#(N) -> c_13(sqr#(N))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {activate/1,add/2,dbl/1,first/2,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0
            ,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1
            ,c_11/0,c_12/3,c_13/1,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,sqr#
            ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            add#(s(X),Y) -> c_5(add#(X,Y))
            sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
        - Weak DPs:
            activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
            activate#(n__terms(X)) -> c_3(terms#(X))
            dbl#(s(X)) -> c_7(dbl#(X))
            first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
            terms#(N) -> c_13(sqr#(N))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {activate/1,add/2,dbl/1,first/2,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0
            ,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1
            ,c_11/0,c_12/3,c_13/1,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,sqr#
            ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:add#(s(X),Y) -> c_5(add#(X,Y))
             -->_1 add#(s(X),Y) -> c_5(add#(X,Y)):1
          
          2:S:sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
             -->_3 dbl#(s(X)) -> c_7(dbl#(X)):5
             -->_2 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):2
             -->_1 add#(s(X),Y) -> c_5(add#(X,Y)):1
          
          3:W:activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
             -->_1 first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)):6
          
          4:W:activate#(n__terms(X)) -> c_3(terms#(X))
             -->_1 terms#(N) -> c_13(sqr#(N)):7
          
          5:W:dbl#(s(X)) -> c_7(dbl#(X))
             -->_1 dbl#(s(X)) -> c_7(dbl#(X)):5
          
          6:W:first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
             -->_1 activate#(n__terms(X)) -> c_3(terms#(X)):4
             -->_1 activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)):3
          
          7:W:terms#(N) -> c_13(sqr#(N))
             -->_1 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: dbl#(s(X)) -> c_7(dbl#(X))
** Step 5.b:2: SimplifyRHS WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            add#(s(X),Y) -> c_5(add#(X,Y))
            sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
        - Weak DPs:
            activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
            activate#(n__terms(X)) -> c_3(terms#(X))
            first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
            terms#(N) -> c_13(sqr#(N))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {activate/1,add/2,dbl/1,first/2,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0
            ,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1
            ,c_11/0,c_12/3,c_13/1,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,sqr#
            ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:add#(s(X),Y) -> c_5(add#(X,Y))
             -->_1 add#(s(X),Y) -> c_5(add#(X,Y)):1
          
          2:S:sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
             -->_2 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):2
             -->_1 add#(s(X),Y) -> c_5(add#(X,Y)):1
          
          3:W:activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
             -->_1 first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)):6
          
          4:W:activate#(n__terms(X)) -> c_3(terms#(X))
             -->_1 terms#(N) -> c_13(sqr#(N)):7
          
          6:W:first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
             -->_1 activate#(n__terms(X)) -> c_3(terms#(X)):4
             -->_1 activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)):3
          
          7:W:terms#(N) -> c_13(sqr#(N))
             -->_1 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X))
** Step 5.b:3: DecomposeDG WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            add#(s(X),Y) -> c_5(add#(X,Y))
            sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X))
        - Weak DPs:
            activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
            activate#(n__terms(X)) -> c_3(terms#(X))
            first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
            terms#(N) -> c_13(sqr#(N))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {activate/1,add/2,dbl/1,first/2,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0
            ,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1
            ,c_11/0,c_12/2,c_13/1,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,sqr#
            ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
          activate#(n__terms(X)) -> c_3(terms#(X))
          first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
          sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X))
          terms#(N) -> c_13(sqr#(N))
        and a lower component
          add#(s(X),Y) -> c_5(add#(X,Y))
        Further, following extension rules are added to the lower component.
          activate#(n__first(X1,X2)) -> first#(X1,X2)
          activate#(n__terms(X)) -> terms#(X)
          first#(s(X),cons(Y,Z)) -> activate#(Z)
          sqr#(s(X)) -> add#(sqr(X),dbl(X))
          sqr#(s(X)) -> sqr#(X)
          terms#(N) -> sqr#(N)
*** Step 5.b:3.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X))
        - Weak DPs:
            activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
            activate#(n__terms(X)) -> c_3(terms#(X))
            first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
            terms#(N) -> c_13(sqr#(N))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {activate/1,add/2,dbl/1,first/2,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0
            ,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1
            ,c_11/0,c_12/2,c_13/1,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,sqr#
            ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X))
          
        The strictly oriented rules are moved into the weak component.
**** Step 5.b:3.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X))
        - Weak DPs:
            activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
            activate#(n__terms(X)) -> c_3(terms#(X))
            first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
            terms#(N) -> c_13(sqr#(N))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {activate/1,add/2,dbl/1,first/2,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0
            ,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1
            ,c_11/0,c_12/2,c_13/1,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,sqr#
            ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_2) = {1},
          uargs(c_3) = {1},
          uargs(c_10) = {1},
          uargs(c_12) = {1,2},
          uargs(c_13) = {1}
        
        Following symbols are considered usable:
          {activate#,add#,dbl#,first#,sqr#,terms#}
        TcT has computed the following interpretation:
                  p(0) = [4]                  
           p(activate) = [2] x1 + [0]         
                p(add) = [1] x1 + [12]        
               p(cons) = [1] x1 + [1] x2 + [0]
                p(dbl) = [2]                  
              p(first) = [0]                  
           p(n__first) = [1] x2 + [1]         
           p(n__terms) = [1] x1 + [0]         
                p(nil) = [0]                  
              p(recip) = [1]                  
                  p(s) = [1] x1 + [8]         
                p(sqr) = [2] x1 + [10]        
              p(terms) = [0]                  
          p(activate#) = [8] x1 + [9]         
               p(add#) = [0]                  
               p(dbl#) = [0]                  
             p(first#) = [8] x2 + [9]         
               p(sqr#) = [1] x1 + [9]         
             p(terms#) = [1] x1 + [9]         
                p(c_1) = [0]                  
                p(c_2) = [1] x1 + [8]         
                p(c_3) = [1] x1 + [0]         
                p(c_4) = [0]                  
                p(c_5) = [0]                  
                p(c_6) = [0]                  
                p(c_7) = [2] x1 + [0]         
                p(c_8) = [0]                  
                p(c_9) = [0]                  
               p(c_10) = [1] x1 + [0]         
               p(c_11) = [2]                  
               p(c_12) = [1] x1 + [1] x2 + [0]
               p(c_13) = [1] x1 + [0]         
               p(c_14) = [0]                  
        
        Following rules are strictly oriented:
        sqr#(s(X)) = [1] X + [17]                     
                   > [1] X + [9]                      
                   = c_12(add#(sqr(X),dbl(X)),sqr#(X))
        
        
        Following rules are (at-least) weakly oriented:
        activate#(n__first(X1,X2)) =  [8] X2 + [17]      
                                   >= [8] X2 + [17]      
                                   =  c_2(first#(X1,X2)) 
        
            activate#(n__terms(X)) =  [8] X + [9]        
                                   >= [1] X + [9]        
                                   =  c_3(terms#(X))     
        
            first#(s(X),cons(Y,Z)) =  [8] Y + [8] Z + [9]
                                   >= [8] Z + [9]        
                                   =  c_10(activate#(Z)) 
        
                         terms#(N) =  [1] N + [9]        
                                   >= [1] N + [9]        
                                   =  c_13(sqr#(N))      
        
**** Step 5.b:3.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
            activate#(n__terms(X)) -> c_3(terms#(X))
            first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
            sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X))
            terms#(N) -> c_13(sqr#(N))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {activate/1,add/2,dbl/1,first/2,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0
            ,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1
            ,c_11/0,c_12/2,c_13/1,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,sqr#
            ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 5.b:3.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
            activate#(n__terms(X)) -> c_3(terms#(X))
            first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
            sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X))
            terms#(N) -> c_13(sqr#(N))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {activate/1,add/2,dbl/1,first/2,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0
            ,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1
            ,c_11/0,c_12/2,c_13/1,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,sqr#
            ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
             -->_1 first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)):3
          
          2:W:activate#(n__terms(X)) -> c_3(terms#(X))
             -->_1 terms#(N) -> c_13(sqr#(N)):5
          
          3:W:first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
             -->_1 activate#(n__terms(X)) -> c_3(terms#(X)):2
             -->_1 activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)):1
          
          4:W:sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X))
             -->_2 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X)):4
          
          5:W:terms#(N) -> c_13(sqr#(N))
             -->_1 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X)):4
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
          3: first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
          2: activate#(n__terms(X)) -> c_3(terms#(X))
          5: terms#(N) -> c_13(sqr#(N))
          4: sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X))
**** Step 5.b:3.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {activate/1,add/2,dbl/1,first/2,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0
            ,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1
            ,c_11/0,c_12/2,c_13/1,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,sqr#
            ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 5.b:3.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            add#(s(X),Y) -> c_5(add#(X,Y))
        - Weak DPs:
            activate#(n__first(X1,X2)) -> first#(X1,X2)
            activate#(n__terms(X)) -> terms#(X)
            first#(s(X),cons(Y,Z)) -> activate#(Z)
            sqr#(s(X)) -> add#(sqr(X),dbl(X))
            sqr#(s(X)) -> sqr#(X)
            terms#(N) -> sqr#(N)
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {activate/1,add/2,dbl/1,first/2,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0
            ,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1
            ,c_11/0,c_12/2,c_13/1,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,sqr#
            ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: add#(s(X),Y) -> c_5(add#(X,Y))
          
        The strictly oriented rules are moved into the weak component.
**** Step 5.b:3.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            add#(s(X),Y) -> c_5(add#(X,Y))
        - Weak DPs:
            activate#(n__first(X1,X2)) -> first#(X1,X2)
            activate#(n__terms(X)) -> terms#(X)
            first#(s(X),cons(Y,Z)) -> activate#(Z)
            sqr#(s(X)) -> add#(sqr(X),dbl(X))
            sqr#(s(X)) -> sqr#(X)
            terms#(N) -> sqr#(N)
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {activate/1,add/2,dbl/1,first/2,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0
            ,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1
            ,c_11/0,c_12/2,c_13/1,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,sqr#
            ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(c_5) = {1}
        
        Following symbols are considered usable:
          {add,dbl,sqr,activate#,add#,dbl#,first#,sqr#,terms#}
        TcT has computed the following interpretation:
                  p(0) = 0                    
           p(activate) = 4 + 2*x1 + x1^2      
                p(add) = x1 + x2              
               p(cons) = 1 + x1 + x2          
                p(dbl) = 2*x1                 
              p(first) = 4 + x2^2             
           p(n__first) = x1 + x2              
           p(n__terms) = x1                   
                p(nil) = 0                    
              p(recip) = 0                    
                  p(s) = 1 + x1               
                p(sqr) = 2 + x1^2             
              p(terms) = x1^2                 
          p(activate#) = 3 + 5*x1 + 4*x1^2    
               p(add#) = x1 + x2              
               p(dbl#) = 2 + x1^2             
             p(first#) = x1*x2 + x1^2 + 4*x2^2
               p(sqr#) = 2*x1^2               
             p(terms#) = 1 + 4*x1^2           
                p(c_1) = 0                    
                p(c_2) = x1                   
                p(c_3) = 1 + x1               
                p(c_4) = 0                    
                p(c_5) = x1                   
                p(c_6) = 0                    
                p(c_7) = 1                    
                p(c_8) = 0                    
                p(c_9) = 0                    
               p(c_10) = 1 + x1               
               p(c_11) = 1                    
               p(c_12) = 1                    
               p(c_13) = 1                    
               p(c_14) = 0                    
        
        Following rules are strictly oriented:
        add#(s(X),Y) = 1 + X + Y     
                     > X + Y         
                     = c_5(add#(X,Y))
        
        
        Following rules are (at-least) weakly oriented:
        activate#(n__first(X1,X2)) =  3 + 5*X1 + 8*X1*X2 + 4*X1^2 + 5*X2 + 4*X2^2                  
                                   >= X1*X2 + X1^2 + 4*X2^2                                        
                                   =  first#(X1,X2)                                                
        
            activate#(n__terms(X)) =  3 + 5*X + 4*X^2                                              
                                   >= 1 + 4*X^2                                                    
                                   =  terms#(X)                                                    
        
            first#(s(X),cons(Y,Z)) =  6 + 3*X + X*Y + X*Z + X^2 + 9*Y + 8*Y*Z + 4*Y^2 + 9*Z + 4*Z^2
                                   >= 3 + 5*Z + 4*Z^2                                              
                                   =  activate#(Z)                                                 
        
                        sqr#(s(X)) =  2 + 4*X + 2*X^2                                              
                                   >= 2 + 2*X + X^2                                                
                                   =  add#(sqr(X),dbl(X))                                          
        
                        sqr#(s(X)) =  2 + 4*X + 2*X^2                                              
                                   >= 2*X^2                                                        
                                   =  sqr#(X)                                                      
        
                         terms#(N) =  1 + 4*N^2                                                    
                                   >= 2*N^2                                                        
                                   =  sqr#(N)                                                      
        
                        add(0(),X) =  X                                                            
                                   >= X                                                            
                                   =  X                                                            
        
                       add(s(X),Y) =  1 + X + Y                                                    
                                   >= 1 + X + Y                                                    
                                   =  s(add(X,Y))                                                  
        
                          dbl(0()) =  0                                                            
                                   >= 0                                                            
                                   =  0()                                                          
        
                         dbl(s(X)) =  2 + 2*X                                                      
                                   >= 2 + 2*X                                                      
                                   =  s(s(dbl(X)))                                                 
        
                          sqr(0()) =  2                                                            
                                   >= 0                                                            
                                   =  0()                                                          
        
                         sqr(s(X)) =  3 + 2*X + X^2                                                
                                   >= 3 + 2*X + X^2                                                
                                   =  s(add(sqr(X),dbl(X)))                                        
        
**** Step 5.b:3.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            activate#(n__first(X1,X2)) -> first#(X1,X2)
            activate#(n__terms(X)) -> terms#(X)
            add#(s(X),Y) -> c_5(add#(X,Y))
            first#(s(X),cons(Y,Z)) -> activate#(Z)
            sqr#(s(X)) -> add#(sqr(X),dbl(X))
            sqr#(s(X)) -> sqr#(X)
            terms#(N) -> sqr#(N)
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {activate/1,add/2,dbl/1,first/2,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0
            ,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1
            ,c_11/0,c_12/2,c_13/1,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,sqr#
            ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 5.b:3.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            activate#(n__first(X1,X2)) -> first#(X1,X2)
            activate#(n__terms(X)) -> terms#(X)
            add#(s(X),Y) -> c_5(add#(X,Y))
            first#(s(X),cons(Y,Z)) -> activate#(Z)
            sqr#(s(X)) -> add#(sqr(X),dbl(X))
            sqr#(s(X)) -> sqr#(X)
            terms#(N) -> sqr#(N)
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {activate/1,add/2,dbl/1,first/2,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0
            ,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1
            ,c_11/0,c_12/2,c_13/1,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,sqr#
            ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:activate#(n__first(X1,X2)) -> first#(X1,X2)
             -->_1 first#(s(X),cons(Y,Z)) -> activate#(Z):4
          
          2:W:activate#(n__terms(X)) -> terms#(X)
             -->_1 terms#(N) -> sqr#(N):7
          
          3:W:add#(s(X),Y) -> c_5(add#(X,Y))
             -->_1 add#(s(X),Y) -> c_5(add#(X,Y)):3
          
          4:W:first#(s(X),cons(Y,Z)) -> activate#(Z)
             -->_1 activate#(n__terms(X)) -> terms#(X):2
             -->_1 activate#(n__first(X1,X2)) -> first#(X1,X2):1
          
          5:W:sqr#(s(X)) -> add#(sqr(X),dbl(X))
             -->_1 add#(s(X),Y) -> c_5(add#(X,Y)):3
          
          6:W:sqr#(s(X)) -> sqr#(X)
             -->_1 sqr#(s(X)) -> sqr#(X):6
             -->_1 sqr#(s(X)) -> add#(sqr(X),dbl(X)):5
          
          7:W:terms#(N) -> sqr#(N)
             -->_1 sqr#(s(X)) -> sqr#(X):6
             -->_1 sqr#(s(X)) -> add#(sqr(X),dbl(X)):5
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: activate#(n__first(X1,X2)) -> first#(X1,X2)
          4: first#(s(X),cons(Y,Z)) -> activate#(Z)
          2: activate#(n__terms(X)) -> terms#(X)
          7: terms#(N) -> sqr#(N)
          6: sqr#(s(X)) -> sqr#(X)
          5: sqr#(s(X)) -> add#(sqr(X),dbl(X))
          3: add#(s(X),Y) -> c_5(add#(X,Y))
**** Step 5.b:3.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {activate/1,add/2,dbl/1,first/2,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0
            ,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1
            ,c_11/0,c_12/2,c_13/1,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,sqr#
            ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^3))