WORST_CASE(?,O(n^1))
* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            2nd(cons(X,X1)) -> 2nd(cons1(X,activate(X1)))
            2nd(cons1(X,cons(Y,Z))) -> Y
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
        - Signature:
            {2nd/1,activate/1,from/1,s/1} / {cons/2,cons1/2,n__from/1,n__s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd,activate,from,s} and constructors {cons,cons1,n__from
            ,n__s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))),activate#(X1))
          2nd#(cons1(X,cons(Y,Z))) -> c_2()
          activate#(X) -> c_3()
          activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X))
          activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X))
          from#(X) -> c_6()
          from#(X) -> c_7()
          s#(X) -> c_8()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))),activate#(X1))
            2nd#(cons1(X,cons(Y,Z))) -> c_2()
            activate#(X) -> c_3()
            activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X))
            activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X))
            from#(X) -> c_6()
            from#(X) -> c_7()
            s#(X) -> c_8()
        - Weak TRS:
            2nd(cons(X,X1)) -> 2nd(cons1(X,activate(X1)))
            2nd(cons1(X,cons(Y,Z))) -> Y
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
        - Signature:
            {2nd/1,activate/1,from/1,s/1,2nd#/1,activate#/1,from#/1,s#/1} / {cons/2,cons1/2,n__from/1,n__s/1,c_1/2,c_2/0
            ,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,s#} and constructors {cons,cons1
            ,n__from,n__s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,3,6,7,8}
        by application of
          Pre({2,3,6,7,8}) = {1,4,5}.
        Here rules are labelled as follows:
          1: 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))),activate#(X1))
          2: 2nd#(cons1(X,cons(Y,Z))) -> c_2()
          3: activate#(X) -> c_3()
          4: activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X))
          5: activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X))
          6: from#(X) -> c_6()
          7: from#(X) -> c_7()
          8: s#(X) -> c_8()
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))),activate#(X1))
            activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X))
            activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X))
        - Weak DPs:
            2nd#(cons1(X,cons(Y,Z))) -> c_2()
            activate#(X) -> c_3()
            from#(X) -> c_6()
            from#(X) -> c_7()
            s#(X) -> c_8()
        - Weak TRS:
            2nd(cons(X,X1)) -> 2nd(cons1(X,activate(X1)))
            2nd(cons1(X,cons(Y,Z))) -> Y
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
        - Signature:
            {2nd/1,activate/1,from/1,s/1,2nd#/1,activate#/1,from#/1,s#/1} / {cons/2,cons1/2,n__from/1,n__s/1,c_1/2,c_2/0
            ,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,s#} and constructors {cons,cons1
            ,n__from,n__s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))),activate#(X1))
             -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3
             -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):2
             -->_2 activate#(X) -> c_3():5
             -->_1 2nd#(cons1(X,cons(Y,Z))) -> c_2():4
          
          2:S:activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X))
             -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3
             -->_1 from#(X) -> c_7():7
             -->_1 from#(X) -> c_6():6
             -->_2 activate#(X) -> c_3():5
             -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):2
          
          3:S:activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X))
             -->_1 s#(X) -> c_8():8
             -->_2 activate#(X) -> c_3():5
             -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3
             -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):2
          
          4:W:2nd#(cons1(X,cons(Y,Z))) -> c_2()
             
          
          5:W:activate#(X) -> c_3()
             
          
          6:W:from#(X) -> c_6()
             
          
          7:W:from#(X) -> c_7()
             
          
          8:W:s#(X) -> c_8()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: 2nd#(cons1(X,cons(Y,Z))) -> c_2()
          6: from#(X) -> c_6()
          7: from#(X) -> c_7()
          5: activate#(X) -> c_3()
          8: s#(X) -> c_8()
* Step 4: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))),activate#(X1))
            activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X))
            activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X))
        - Weak TRS:
            2nd(cons(X,X1)) -> 2nd(cons1(X,activate(X1)))
            2nd(cons1(X,cons(Y,Z))) -> Y
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
        - Signature:
            {2nd/1,activate/1,from/1,s/1,2nd#/1,activate#/1,from#/1,s#/1} / {cons/2,cons1/2,n__from/1,n__s/1,c_1/2,c_2/0
            ,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,s#} and constructors {cons,cons1
            ,n__from,n__s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))),activate#(X1))
             -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3
             -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):2
          
          2:S:activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X))
             -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3
             -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):2
          
          3:S:activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X))
             -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3
             -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          2nd#(cons(X,X1)) -> c_1(activate#(X1))
          activate#(n__from(X)) -> c_4(activate#(X))
          activate#(n__s(X)) -> c_5(activate#(X))
* Step 5: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            2nd#(cons(X,X1)) -> c_1(activate#(X1))
            activate#(n__from(X)) -> c_4(activate#(X))
            activate#(n__s(X)) -> c_5(activate#(X))
        - Weak TRS:
            2nd(cons(X,X1)) -> 2nd(cons1(X,activate(X1)))
            2nd(cons1(X,cons(Y,Z))) -> Y
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
        - Signature:
            {2nd/1,activate/1,from/1,s/1,2nd#/1,activate#/1,from#/1,s#/1} / {cons/2,cons1/2,n__from/1,n__s/1,c_1/1,c_2/0
            ,c_3/0,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,s#} and constructors {cons,cons1
            ,n__from,n__s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          2nd#(cons(X,X1)) -> c_1(activate#(X1))
          activate#(n__from(X)) -> c_4(activate#(X))
          activate#(n__s(X)) -> c_5(activate#(X))
* Step 6: RemoveHeads WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            2nd#(cons(X,X1)) -> c_1(activate#(X1))
            activate#(n__from(X)) -> c_4(activate#(X))
            activate#(n__s(X)) -> c_5(activate#(X))
        - Signature:
            {2nd/1,activate/1,from/1,s/1,2nd#/1,activate#/1,from#/1,s#/1} / {cons/2,cons1/2,n__from/1,n__s/1,c_1/1,c_2/0
            ,c_3/0,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,s#} and constructors {cons,cons1
            ,n__from,n__s}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:2nd#(cons(X,X1)) -> c_1(activate#(X1))
           -->_1 activate#(n__s(X)) -> c_5(activate#(X)):3
           -->_1 activate#(n__from(X)) -> c_4(activate#(X)):2
        
        2:S:activate#(n__from(X)) -> c_4(activate#(X))
           -->_1 activate#(n__s(X)) -> c_5(activate#(X)):3
           -->_1 activate#(n__from(X)) -> c_4(activate#(X)):2
        
        3:S:activate#(n__s(X)) -> c_5(activate#(X))
           -->_1 activate#(n__s(X)) -> c_5(activate#(X)):3
           -->_1 activate#(n__from(X)) -> c_4(activate#(X)):2
        
        
        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).
        
        [(1,2nd#(cons(X,X1)) -> c_1(activate#(X1)))]
* Step 7: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__from(X)) -> c_4(activate#(X))
            activate#(n__s(X)) -> c_5(activate#(X))
        - Signature:
            {2nd/1,activate/1,from/1,s/1,2nd#/1,activate#/1,from#/1,s#/1} / {cons/2,cons1/2,n__from/1,n__s/1,c_1/1,c_2/0
            ,c_3/0,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,s#} and constructors {cons,cons1
            ,n__from,n__s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(c_4) = {1},
            uargs(c_5) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                  p(2nd) = [0]                  
             p(activate) = [0]                  
                 p(cons) = [1] x1 + [1] x2 + [0]
                p(cons1) = [1] x1 + [1] x2 + [0]
                 p(from) = [0]                  
              p(n__from) = [1] x1 + [0]         
                 p(n__s) = [1] x1 + [1]         
                    p(s) = [0]                  
                 p(2nd#) = [0]                  
            p(activate#) = [2] x1 + [7]         
                p(from#) = [0]                  
                   p(s#) = [0]                  
                  p(c_1) = [0]                  
                  p(c_2) = [0]                  
                  p(c_3) = [0]                  
                  p(c_4) = [1] x1 + [0]         
                  p(c_5) = [1] x1 + [1]         
                  p(c_6) = [0]                  
                  p(c_7) = [0]                  
                  p(c_8) = [0]                  
          
          Following rules are strictly oriented:
          activate#(n__s(X)) = [2] X + [9]      
                             > [2] X + [8]      
                             = c_5(activate#(X))
          
          
          Following rules are (at-least) weakly oriented:
          activate#(n__from(X)) =  [2] X + [7]      
                                >= [2] X + [7]      
                                =  c_4(activate#(X))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 8: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__from(X)) -> c_4(activate#(X))
        - Weak DPs:
            activate#(n__s(X)) -> c_5(activate#(X))
        - Signature:
            {2nd/1,activate/1,from/1,s/1,2nd#/1,activate#/1,from#/1,s#/1} / {cons/2,cons1/2,n__from/1,n__s/1,c_1/1,c_2/0
            ,c_3/0,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,s#} and constructors {cons,cons1
            ,n__from,n__s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(c_4) = {1},
            uargs(c_5) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                  p(2nd) = [0]                  
             p(activate) = [0]                  
                 p(cons) = [1] x1 + [1] x2 + [0]
                p(cons1) = [1] x1 + [1] x2 + [0]
                 p(from) = [0]                  
              p(n__from) = [1] x1 + [1]         
                 p(n__s) = [1] x1 + [0]         
                    p(s) = [0]                  
                 p(2nd#) = [0]                  
            p(activate#) = [1] x1 + [0]         
                p(from#) = [0]                  
                   p(s#) = [0]                  
                  p(c_1) = [0]                  
                  p(c_2) = [0]                  
                  p(c_3) = [0]                  
                  p(c_4) = [1] x1 + [0]         
                  p(c_5) = [1] x1 + [0]         
                  p(c_6) = [0]                  
                  p(c_7) = [0]                  
                  p(c_8) = [0]                  
          
          Following rules are strictly oriented:
          activate#(n__from(X)) = [1] X + [1]      
                                > [1] X + [0]      
                                = c_4(activate#(X))
          
          
          Following rules are (at-least) weakly oriented:
          activate#(n__s(X)) =  [1] X + [0]      
                             >= [1] X + [0]      
                             =  c_5(activate#(X))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 9: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            activate#(n__from(X)) -> c_4(activate#(X))
            activate#(n__s(X)) -> c_5(activate#(X))
        - Signature:
            {2nd/1,activate/1,from/1,s/1,2nd#/1,activate#/1,from#/1,s#/1} / {cons/2,cons1/2,n__from/1,n__s/1,c_1/1,c_2/0
            ,c_3/0,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,s#} and constructors {cons,cons1
            ,n__from,n__s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))