WORST_CASE(?,O(n^2))
* Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            2ndsneg(0(),Z) -> rnil()
            2ndsneg(s(N),cons(X,n__cons(Y,Z))) -> rcons(negrecip(activate(Y)),2ndspos(N,activate(Z)))
            2ndspos(0(),Z) -> rnil()
            2ndspos(s(N),cons(X,n__cons(Y,Z))) -> rcons(posrecip(activate(Y)),2ndsneg(N,activate(Z)))
            activate(X) -> X
            activate(n__cons(X1,X2)) -> cons(X1,X2)
            activate(n__from(X)) -> from(X)
            cons(X1,X2) -> n__cons(X1,X2)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            pi(X) -> 2ndspos(X,from(0()))
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
            square(X) -> times(X,X)
            times(0(),Y) -> 0()
            times(s(X),Y) -> plus(Y,times(X,Y))
        - Signature:
            {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2} / {0/0,n__cons/2,n__from/1
            ,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,cons,from,pi,plus,square
            ,times} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
    + Applied Processor:
        InnermostRuleRemoval
    + Details:
        Arguments of following rules are not normal-forms.
          2ndsneg(s(N),cons(X,n__cons(Y,Z))) -> rcons(negrecip(activate(Y)),2ndspos(N,activate(Z)))
          2ndspos(s(N),cons(X,n__cons(Y,Z))) -> rcons(posrecip(activate(Y)),2ndsneg(N,activate(Z)))
        All above mentioned rules can be savely removed.
* Step 2: DependencyPairs WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            2ndsneg(0(),Z) -> rnil()
            2ndspos(0(),Z) -> rnil()
            activate(X) -> X
            activate(n__cons(X1,X2)) -> cons(X1,X2)
            activate(n__from(X)) -> from(X)
            cons(X1,X2) -> n__cons(X1,X2)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            pi(X) -> 2ndspos(X,from(0()))
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
            square(X) -> times(X,X)
            times(0(),Y) -> 0()
            times(s(X),Y) -> plus(Y,times(X,Y))
        - Signature:
            {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2} / {0/0,n__cons/2,n__from/1
            ,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,cons,from,pi,plus,square
            ,times} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          2ndsneg#(0(),Z) -> c_1()
          2ndspos#(0(),Z) -> c_2()
          activate#(X) -> c_3()
          activate#(n__cons(X1,X2)) -> c_4(cons#(X1,X2))
          activate#(n__from(X)) -> c_5(from#(X))
          cons#(X1,X2) -> c_6()
          from#(X) -> c_7(cons#(X,n__from(s(X))))
          from#(X) -> c_8()
          pi#(X) -> c_9(2ndspos#(X,from(0())),from#(0()))
          plus#(0(),Y) -> c_10()
          plus#(s(X),Y) -> c_11(plus#(X,Y))
          square#(X) -> c_12(times#(X,X))
          times#(0(),Y) -> c_13()
          times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: UsableRules WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            2ndsneg#(0(),Z) -> c_1()
            2ndspos#(0(),Z) -> c_2()
            activate#(X) -> c_3()
            activate#(n__cons(X1,X2)) -> c_4(cons#(X1,X2))
            activate#(n__from(X)) -> c_5(from#(X))
            cons#(X1,X2) -> c_6()
            from#(X) -> c_7(cons#(X,n__from(s(X))))
            from#(X) -> c_8()
            pi#(X) -> c_9(2ndspos#(X,from(0())),from#(0()))
            plus#(0(),Y) -> c_10()
            plus#(s(X),Y) -> c_11(plus#(X,Y))
            square#(X) -> c_12(times#(X,X))
            times#(0(),Y) -> c_13()
            times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y))
        - Weak TRS:
            2ndsneg(0(),Z) -> rnil()
            2ndspos(0(),Z) -> rnil()
            activate(X) -> X
            activate(n__cons(X1,X2)) -> cons(X1,X2)
            activate(n__from(X)) -> from(X)
            cons(X1,X2) -> n__cons(X1,X2)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            pi(X) -> 2ndspos(X,from(0()))
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
            square(X) -> times(X,X)
            times(0(),Y) -> 0()
            times(s(X),Y) -> plus(Y,times(X,Y))
        - Signature:
            {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
            ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2
            ,rnil/0,s/1,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#
            ,times#} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
    + 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)
          plus(0(),Y) -> Y
          plus(s(X),Y) -> s(plus(X,Y))
          times(0(),Y) -> 0()
          times(s(X),Y) -> plus(Y,times(X,Y))
          2ndsneg#(0(),Z) -> c_1()
          2ndspos#(0(),Z) -> c_2()
          activate#(X) -> c_3()
          activate#(n__cons(X1,X2)) -> c_4(cons#(X1,X2))
          activate#(n__from(X)) -> c_5(from#(X))
          cons#(X1,X2) -> c_6()
          from#(X) -> c_7(cons#(X,n__from(s(X))))
          from#(X) -> c_8()
          pi#(X) -> c_9(2ndspos#(X,from(0())),from#(0()))
          plus#(0(),Y) -> c_10()
          plus#(s(X),Y) -> c_11(plus#(X,Y))
          square#(X) -> c_12(times#(X,X))
          times#(0(),Y) -> c_13()
          times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y))
* Step 4: PredecessorEstimation WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            2ndsneg#(0(),Z) -> c_1()
            2ndspos#(0(),Z) -> c_2()
            activate#(X) -> c_3()
            activate#(n__cons(X1,X2)) -> c_4(cons#(X1,X2))
            activate#(n__from(X)) -> c_5(from#(X))
            cons#(X1,X2) -> c_6()
            from#(X) -> c_7(cons#(X,n__from(s(X))))
            from#(X) -> c_8()
            pi#(X) -> c_9(2ndspos#(X,from(0())),from#(0()))
            plus#(0(),Y) -> c_10()
            plus#(s(X),Y) -> c_11(plus#(X,Y))
            square#(X) -> c_12(times#(X,X))
            times#(0(),Y) -> c_13()
            times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y))
        - Weak TRS:
            cons(X1,X2) -> n__cons(X1,X2)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
            times(0(),Y) -> 0()
            times(s(X),Y) -> plus(Y,times(X,Y))
        - Signature:
            {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
            ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2
            ,rnil/0,s/1,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#
            ,times#} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,3,6,8,10,13}
        by application of
          Pre({1,2,3,6,8,10,13}) = {4,5,7,9,11,12,14}.
        Here rules are labelled as follows:
          1: 2ndsneg#(0(),Z) -> c_1()
          2: 2ndspos#(0(),Z) -> c_2()
          3: activate#(X) -> c_3()
          4: activate#(n__cons(X1,X2)) -> c_4(cons#(X1,X2))
          5: activate#(n__from(X)) -> c_5(from#(X))
          6: cons#(X1,X2) -> c_6()
          7: from#(X) -> c_7(cons#(X,n__from(s(X))))
          8: from#(X) -> c_8()
          9: pi#(X) -> c_9(2ndspos#(X,from(0())),from#(0()))
          10: plus#(0(),Y) -> c_10()
          11: plus#(s(X),Y) -> c_11(plus#(X,Y))
          12: square#(X) -> c_12(times#(X,X))
          13: times#(0(),Y) -> c_13()
          14: times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y))
* Step 5: PredecessorEstimation WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            activate#(n__cons(X1,X2)) -> c_4(cons#(X1,X2))
            activate#(n__from(X)) -> c_5(from#(X))
            from#(X) -> c_7(cons#(X,n__from(s(X))))
            pi#(X) -> c_9(2ndspos#(X,from(0())),from#(0()))
            plus#(s(X),Y) -> c_11(plus#(X,Y))
            square#(X) -> c_12(times#(X,X))
            times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y))
        - Weak DPs:
            2ndsneg#(0(),Z) -> c_1()
            2ndspos#(0(),Z) -> c_2()
            activate#(X) -> c_3()
            cons#(X1,X2) -> c_6()
            from#(X) -> c_8()
            plus#(0(),Y) -> c_10()
            times#(0(),Y) -> c_13()
        - Weak TRS:
            cons(X1,X2) -> n__cons(X1,X2)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
            times(0(),Y) -> 0()
            times(s(X),Y) -> plus(Y,times(X,Y))
        - Signature:
            {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
            ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2
            ,rnil/0,s/1,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#
            ,times#} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3}
        by application of
          Pre({1,3}) = {2,4}.
        Here rules are labelled as follows:
          1: activate#(n__cons(X1,X2)) -> c_4(cons#(X1,X2))
          2: activate#(n__from(X)) -> c_5(from#(X))
          3: from#(X) -> c_7(cons#(X,n__from(s(X))))
          4: pi#(X) -> c_9(2ndspos#(X,from(0())),from#(0()))
          5: plus#(s(X),Y) -> c_11(plus#(X,Y))
          6: square#(X) -> c_12(times#(X,X))
          7: times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y))
          8: 2ndsneg#(0(),Z) -> c_1()
          9: 2ndspos#(0(),Z) -> c_2()
          10: activate#(X) -> c_3()
          11: cons#(X1,X2) -> c_6()
          12: from#(X) -> c_8()
          13: plus#(0(),Y) -> c_10()
          14: times#(0(),Y) -> c_13()
* Step 6: PredecessorEstimation WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            activate#(n__from(X)) -> c_5(from#(X))
            pi#(X) -> c_9(2ndspos#(X,from(0())),from#(0()))
            plus#(s(X),Y) -> c_11(plus#(X,Y))
            square#(X) -> c_12(times#(X,X))
            times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y))
        - Weak DPs:
            2ndsneg#(0(),Z) -> c_1()
            2ndspos#(0(),Z) -> c_2()
            activate#(X) -> c_3()
            activate#(n__cons(X1,X2)) -> c_4(cons#(X1,X2))
            cons#(X1,X2) -> c_6()
            from#(X) -> c_7(cons#(X,n__from(s(X))))
            from#(X) -> c_8()
            plus#(0(),Y) -> c_10()
            times#(0(),Y) -> c_13()
        - Weak TRS:
            cons(X1,X2) -> n__cons(X1,X2)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
            times(0(),Y) -> 0()
            times(s(X),Y) -> plus(Y,times(X,Y))
        - Signature:
            {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
            ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2
            ,rnil/0,s/1,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#
            ,times#} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2}
        by application of
          Pre({1,2}) = {}.
        Here rules are labelled as follows:
          1: activate#(n__from(X)) -> c_5(from#(X))
          2: pi#(X) -> c_9(2ndspos#(X,from(0())),from#(0()))
          3: plus#(s(X),Y) -> c_11(plus#(X,Y))
          4: square#(X) -> c_12(times#(X,X))
          5: times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y))
          6: 2ndsneg#(0(),Z) -> c_1()
          7: 2ndspos#(0(),Z) -> c_2()
          8: activate#(X) -> c_3()
          9: activate#(n__cons(X1,X2)) -> c_4(cons#(X1,X2))
          10: cons#(X1,X2) -> c_6()
          11: from#(X) -> c_7(cons#(X,n__from(s(X))))
          12: from#(X) -> c_8()
          13: plus#(0(),Y) -> c_10()
          14: times#(0(),Y) -> c_13()
* Step 7: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            plus#(s(X),Y) -> c_11(plus#(X,Y))
            square#(X) -> c_12(times#(X,X))
            times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y))
        - Weak DPs:
            2ndsneg#(0(),Z) -> c_1()
            2ndspos#(0(),Z) -> c_2()
            activate#(X) -> c_3()
            activate#(n__cons(X1,X2)) -> c_4(cons#(X1,X2))
            activate#(n__from(X)) -> c_5(from#(X))
            cons#(X1,X2) -> c_6()
            from#(X) -> c_7(cons#(X,n__from(s(X))))
            from#(X) -> c_8()
            pi#(X) -> c_9(2ndspos#(X,from(0())),from#(0()))
            plus#(0(),Y) -> c_10()
            times#(0(),Y) -> c_13()
        - Weak TRS:
            cons(X1,X2) -> n__cons(X1,X2)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
            times(0(),Y) -> 0()
            times(s(X),Y) -> plus(Y,times(X,Y))
        - Signature:
            {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
            ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2
            ,rnil/0,s/1,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#
            ,times#} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:plus#(s(X),Y) -> c_11(plus#(X,Y))
             -->_1 plus#(0(),Y) -> c_10():13
             -->_1 plus#(s(X),Y) -> c_11(plus#(X,Y)):1
          
          2:S:square#(X) -> c_12(times#(X,X))
             -->_1 times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y)):3
             -->_1 times#(0(),Y) -> c_13():14
          
          3:S:times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y))
             -->_2 times#(0(),Y) -> c_13():14
             -->_1 plus#(0(),Y) -> c_10():13
             -->_2 times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y)):3
             -->_1 plus#(s(X),Y) -> c_11(plus#(X,Y)):1
          
          4:W:2ndsneg#(0(),Z) -> c_1()
             
          
          5:W:2ndspos#(0(),Z) -> c_2()
             
          
          6:W:activate#(X) -> c_3()
             
          
          7:W:activate#(n__cons(X1,X2)) -> c_4(cons#(X1,X2))
             -->_1 cons#(X1,X2) -> c_6():9
          
          8:W:activate#(n__from(X)) -> c_5(from#(X))
             -->_1 from#(X) -> c_7(cons#(X,n__from(s(X)))):10
             -->_1 from#(X) -> c_8():11
          
          9:W:cons#(X1,X2) -> c_6()
             
          
          10:W:from#(X) -> c_7(cons#(X,n__from(s(X))))
             -->_1 cons#(X1,X2) -> c_6():9
          
          11:W:from#(X) -> c_8()
             
          
          12:W:pi#(X) -> c_9(2ndspos#(X,from(0())),from#(0()))
             -->_2 from#(X) -> c_8():11
             -->_2 from#(X) -> c_7(cons#(X,n__from(s(X)))):10
             -->_1 2ndspos#(0(),Z) -> c_2():5
          
          13:W:plus#(0(),Y) -> c_10()
             
          
          14:W:times#(0(),Y) -> c_13()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          12: pi#(X) -> c_9(2ndspos#(X,from(0())),from#(0()))
          8: activate#(n__from(X)) -> c_5(from#(X))
          11: from#(X) -> c_8()
          10: from#(X) -> c_7(cons#(X,n__from(s(X))))
          7: activate#(n__cons(X1,X2)) -> c_4(cons#(X1,X2))
          9: cons#(X1,X2) -> c_6()
          6: activate#(X) -> c_3()
          5: 2ndspos#(0(),Z) -> c_2()
          4: 2ndsneg#(0(),Z) -> c_1()
          14: times#(0(),Y) -> c_13()
          13: plus#(0(),Y) -> c_10()
* Step 8: RemoveHeads WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            plus#(s(X),Y) -> c_11(plus#(X,Y))
            square#(X) -> c_12(times#(X,X))
            times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y))
        - Weak TRS:
            cons(X1,X2) -> n__cons(X1,X2)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
            times(0(),Y) -> 0()
            times(s(X),Y) -> plus(Y,times(X,Y))
        - Signature:
            {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
            ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2
            ,rnil/0,s/1,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#
            ,times#} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:plus#(s(X),Y) -> c_11(plus#(X,Y))
           -->_1 plus#(s(X),Y) -> c_11(plus#(X,Y)):1
        
        2:S:square#(X) -> c_12(times#(X,X))
           -->_1 times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y)):3
        
        3:S:times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y))
           -->_2 times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y)):3
           -->_1 plus#(s(X),Y) -> c_11(plus#(X,Y)):1
        
        
        Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
        
        [(2,square#(X) -> c_12(times#(X,X)))]
* Step 9: Decompose WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            plus#(s(X),Y) -> c_11(plus#(X,Y))
            times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y))
        - Weak TRS:
            cons(X1,X2) -> n__cons(X1,X2)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
            times(0(),Y) -> 0()
            times(s(X),Y) -> plus(Y,times(X,Y))
        - Signature:
            {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
            ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2
            ,rnil/0,s/1,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#
            ,times#} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
    + Applied Processor:
        Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    + Details:
        We analyse the complexity of following sub-problems (R) and (S).
        Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
        
        Problem (R)
          - Strict DPs:
              plus#(s(X),Y) -> c_11(plus#(X,Y))
          - Weak DPs:
              times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y))
          - Weak TRS:
              cons(X1,X2) -> n__cons(X1,X2)
              from(X) -> cons(X,n__from(s(X)))
              from(X) -> n__from(X)
              plus(0(),Y) -> Y
              plus(s(X),Y) -> s(plus(X,Y))
              times(0(),Y) -> 0()
              times(s(X),Y) -> plus(Y,times(X,Y))
          - Signature:
              {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2
              ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1
              ,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/1,c_12/1
              ,c_13/0,c_14/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#
              ,square#,times#} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
        
        Problem (S)
          - Strict DPs:
              times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y))
          - Weak DPs:
              plus#(s(X),Y) -> c_11(plus#(X,Y))
          - Weak TRS:
              cons(X1,X2) -> n__cons(X1,X2)
              from(X) -> cons(X,n__from(s(X)))
              from(X) -> n__from(X)
              plus(0(),Y) -> Y
              plus(s(X),Y) -> s(plus(X,Y))
              times(0(),Y) -> 0()
              times(s(X),Y) -> plus(Y,times(X,Y))
          - Signature:
              {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2
              ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1
              ,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/1,c_12/1
              ,c_13/0,c_14/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#
              ,square#,times#} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
** Step 9.a:1: UsableRules WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            plus#(s(X),Y) -> c_11(plus#(X,Y))
        - Weak DPs:
            times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y))
        - Weak TRS:
            cons(X1,X2) -> n__cons(X1,X2)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
            times(0(),Y) -> 0()
            times(s(X),Y) -> plus(Y,times(X,Y))
        - Signature:
            {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
            ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2
            ,rnil/0,s/1,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#
            ,times#} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          plus(0(),Y) -> Y
          plus(s(X),Y) -> s(plus(X,Y))
          times(0(),Y) -> 0()
          times(s(X),Y) -> plus(Y,times(X,Y))
          plus#(s(X),Y) -> c_11(plus#(X,Y))
          times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y))
** Step 9.a:2: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            plus#(s(X),Y) -> c_11(plus#(X,Y))
        - Weak DPs:
            times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y))
        - Weak TRS:
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
            times(0(),Y) -> 0()
            times(s(X),Y) -> plus(Y,times(X,Y))
        - Signature:
            {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
            ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2
            ,rnil/0,s/1,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#
            ,times#} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,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: plus#(s(X),Y) -> c_11(plus#(X,Y))
          
        The strictly oriented rules are moved into the weak component.
*** Step 9.a:2.a:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            plus#(s(X),Y) -> c_11(plus#(X,Y))
        - Weak DPs:
            times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y))
        - Weak TRS:
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
            times(0(),Y) -> 0()
            times(s(X),Y) -> plus(Y,times(X,Y))
        - Signature:
            {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
            ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2
            ,rnil/0,s/1,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#
            ,times#} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,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_11) = {1},
          uargs(c_14) = {1,2}
        
        Following symbols are considered usable:
          {plus,times,2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}
        TcT has computed the following interpretation:
                  p(0) = 0                        
            p(2ndsneg) = 0                        
            p(2ndspos) = 0                        
           p(activate) = 0                        
               p(cons) = 0                        
               p(from) = 0                        
            p(n__cons) = x1 + x2                  
            p(n__from) = x1                       
           p(negrecip) = x1                       
                 p(pi) = x1^2                     
               p(plus) = 6 + 2*x1 + x2            
           p(posrecip) = x1                       
              p(rcons) = x2                       
               p(rnil) = 0                        
                  p(s) = 2 + x1                   
             p(square) = 0                        
              p(times) = 2*x1 + 2*x1*x2 + 2*x1^2  
           p(2ndsneg#) = 0                        
           p(2ndspos#) = x1                       
          p(activate#) = x1 + 4*x1^2              
              p(cons#) = 1 + x2 + x2^2            
              p(from#) = 1                        
                p(pi#) = 4*x1 + 2*x1^2            
              p(plus#) = 4*x1                     
            p(square#) = 2 + x1                   
             p(times#) = 2 + 2*x1*x2 + 5*x2 + x2^2
                p(c_1) = 0                        
                p(c_2) = 0                        
                p(c_3) = 4                        
                p(c_4) = 2 + x1                   
                p(c_5) = 0                        
                p(c_6) = 2                        
                p(c_7) = 1                        
                p(c_8) = 0                        
                p(c_9) = 4                        
               p(c_10) = 0                        
               p(c_11) = 6 + x1                   
               p(c_12) = 2 + x1                   
               p(c_13) = 0                        
               p(c_14) = x1 + x2                  
        
        Following rules are strictly oriented:
        plus#(s(X),Y) = 8 + 4*X         
                      > 6 + 4*X         
                      = c_11(plus#(X,Y))
        
        
        Following rules are (at-least) weakly oriented:
        times#(s(X),Y) =  2 + 2*X*Y + 9*Y + Y^2                
                       >= 2 + 2*X*Y + 9*Y + Y^2                
                       =  c_14(plus#(Y,times(X,Y)),times#(X,Y))
        
           plus(0(),Y) =  6 + Y                                
                       >= Y                                    
                       =  Y                                    
        
          plus(s(X),Y) =  10 + 2*X + Y                         
                       >= 8 + 2*X + Y                          
                       =  s(plus(X,Y))                         
        
          times(0(),Y) =  0                                    
                       >= 0                                    
                       =  0()                                  
        
         times(s(X),Y) =  12 + 10*X + 2*X*Y + 2*X^2 + 4*Y      
                       >= 6 + 2*X + 2*X*Y + 2*X^2 + 2*Y        
                       =  plus(Y,times(X,Y))                   
        
*** Step 9.a:2.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            plus#(s(X),Y) -> c_11(plus#(X,Y))
            times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y))
        - Weak TRS:
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
            times(0(),Y) -> 0()
            times(s(X),Y) -> plus(Y,times(X,Y))
        - Signature:
            {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
            ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2
            ,rnil/0,s/1,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#
            ,times#} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 9.a:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            plus#(s(X),Y) -> c_11(plus#(X,Y))
            times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y))
        - Weak TRS:
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
            times(0(),Y) -> 0()
            times(s(X),Y) -> plus(Y,times(X,Y))
        - Signature:
            {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
            ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2
            ,rnil/0,s/1,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#
            ,times#} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:plus#(s(X),Y) -> c_11(plus#(X,Y))
             -->_1 plus#(s(X),Y) -> c_11(plus#(X,Y)):1
          
          2:W:times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y))
             -->_2 times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y)):2
             -->_1 plus#(s(X),Y) -> c_11(plus#(X,Y)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y))
          1: plus#(s(X),Y) -> c_11(plus#(X,Y))
*** Step 9.a:2.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
            times(0(),Y) -> 0()
            times(s(X),Y) -> plus(Y,times(X,Y))
        - Signature:
            {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
            ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2
            ,rnil/0,s/1,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#
            ,times#} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

** Step 9.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y))
        - Weak DPs:
            plus#(s(X),Y) -> c_11(plus#(X,Y))
        - Weak TRS:
            cons(X1,X2) -> n__cons(X1,X2)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
            times(0(),Y) -> 0()
            times(s(X),Y) -> plus(Y,times(X,Y))
        - Signature:
            {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
            ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2
            ,rnil/0,s/1,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#
            ,times#} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y))
             -->_1 plus#(s(X),Y) -> c_11(plus#(X,Y)):2
             -->_2 times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y)):1
          
          2:W:plus#(s(X),Y) -> c_11(plus#(X,Y))
             -->_1 plus#(s(X),Y) -> c_11(plus#(X,Y)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: plus#(s(X),Y) -> c_11(plus#(X,Y))
** Step 9.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y))
        - Weak TRS:
            cons(X1,X2) -> n__cons(X1,X2)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
            times(0(),Y) -> 0()
            times(s(X),Y) -> plus(Y,times(X,Y))
        - Signature:
            {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
            ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2
            ,rnil/0,s/1,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#
            ,times#} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y))
             -->_2 times#(s(X),Y) -> c_14(plus#(Y,times(X,Y)),times#(X,Y)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          times#(s(X),Y) -> c_14(times#(X,Y))
** Step 9.b:3: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            times#(s(X),Y) -> c_14(times#(X,Y))
        - Weak TRS:
            cons(X1,X2) -> n__cons(X1,X2)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
            times(0(),Y) -> 0()
            times(s(X),Y) -> plus(Y,times(X,Y))
        - Signature:
            {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
            ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2
            ,rnil/0,s/1,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/1,c_12/1,c_13/0,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#
            ,times#} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          times#(s(X),Y) -> c_14(times#(X,Y))
** Step 9.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            times#(s(X),Y) -> c_14(times#(X,Y))
        - Signature:
            {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
            ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2
            ,rnil/0,s/1,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/1,c_12/1,c_13/0,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#
            ,times#} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,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: times#(s(X),Y) -> c_14(times#(X,Y))
          
        The strictly oriented rules are moved into the weak component.
*** Step 9.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            times#(s(X),Y) -> c_14(times#(X,Y))
        - Signature:
            {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
            ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2
            ,rnil/0,s/1,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/1,c_12/1,c_13/0,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#
            ,times#} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,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_14) = {1}
        
        Following symbols are considered usable:
          {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}
        TcT has computed the following interpretation:
                  p(0) = [0]                  
            p(2ndsneg) = [0]                  
            p(2ndspos) = [0]                  
           p(activate) = [2] x1 + [1]         
               p(cons) = [1] x2 + [0]         
               p(from) = [1]                  
            p(n__cons) = [1] x1 + [1] x2 + [0]
            p(n__from) = [1] x1 + [0]         
           p(negrecip) = [1] x1 + [0]         
                 p(pi) = [0]                  
               p(plus) = [0]                  
           p(posrecip) = [1] x1 + [0]         
              p(rcons) = [1] x1 + [1] x2 + [0]
               p(rnil) = [0]                  
                  p(s) = [1] x1 + [4]         
             p(square) = [0]                  
              p(times) = [0]                  
           p(2ndsneg#) = [0]                  
           p(2ndspos#) = [0]                  
          p(activate#) = [0]                  
              p(cons#) = [0]                  
              p(from#) = [0]                  
                p(pi#) = [0]                  
              p(plus#) = [0]                  
            p(square#) = [0]                  
             p(times#) = [4] x1 + [0]         
                p(c_1) = [0]                  
                p(c_2) = [0]                  
                p(c_3) = [0]                  
                p(c_4) = [0]                  
                p(c_5) = [0]                  
                p(c_6) = [0]                  
                p(c_7) = [0]                  
                p(c_8) = [0]                  
                p(c_9) = [4] x2 + [1]         
               p(c_10) = [8]                  
               p(c_11) = [2] x1 + [4]         
               p(c_12) = [1] x1 + [2]         
               p(c_13) = [0]                  
               p(c_14) = [1] x1 + [6]         
        
        Following rules are strictly oriented:
        times#(s(X),Y) = [4] X + [16]     
                       > [4] X + [6]      
                       = c_14(times#(X,Y))
        
        
        Following rules are (at-least) weakly oriented:
        
*** Step 9.b:4.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            times#(s(X),Y) -> c_14(times#(X,Y))
        - Signature:
            {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
            ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2
            ,rnil/0,s/1,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/1,c_12/1,c_13/0,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#
            ,times#} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 9.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            times#(s(X),Y) -> c_14(times#(X,Y))
        - Signature:
            {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
            ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2
            ,rnil/0,s/1,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/1,c_12/1,c_13/0,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#
            ,times#} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:times#(s(X),Y) -> c_14(times#(X,Y))
             -->_1 times#(s(X),Y) -> c_14(times#(X,Y)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: times#(s(X),Y) -> c_14(times#(X,Y))
*** Step 9.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
            ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2
            ,rnil/0,s/1,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/1,c_12/1,c_13/0,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#
            ,times#} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))