WORST_CASE(?,O(n^1))
* Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            after(0(),XS) -> XS
            after(s(N),cons(X,XS)) -> after(N,activate(XS))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
        - Signature:
            {activate/1,after/2,from/1,s/1} / {0/0,cons/2,n__from/1,n__s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,after,from,s} and constructors {0,cons,n__from
            ,n__s}
    + Applied Processor:
        InnermostRuleRemoval
    + Details:
        Arguments of following rules are not normal-forms.
          after(s(N),cons(X,XS)) -> after(N,activate(XS))
        All above mentioned rules can be savely removed.
* Step 2: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            after(0(),XS) -> XS
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
        - Signature:
            {activate/1,after/2,from/1,s/1} / {0/0,cons/2,n__from/1,n__s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,after,from,s} and constructors {0,cons,n__from
            ,n__s}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          activate#(X) -> c_1()
          activate#(n__from(X)) -> c_2(from#(activate(X)))
          activate#(n__s(X)) -> c_3(s#(activate(X)))
          after#(0(),XS) -> c_4()
          from#(X) -> c_5()
          from#(X) -> c_6()
          s#(X) -> c_7()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            activate#(n__from(X)) -> c_2(from#(activate(X)))
            activate#(n__s(X)) -> c_3(s#(activate(X)))
            after#(0(),XS) -> c_4()
            from#(X) -> c_5()
            from#(X) -> c_6()
            s#(X) -> c_7()
        - Strict TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            after(0(),XS) -> XS
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
        - Signature:
            {activate/1,after/2,from/1,s/1,activate#/1,after#/2,from#/1,s#/1} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/1
            ,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,after#,from#,s#} and constructors {0,cons
            ,n__from,n__s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          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)
          activate#(X) -> c_1()
          activate#(n__from(X)) -> c_2(from#(activate(X)))
          activate#(n__s(X)) -> c_3(s#(activate(X)))
          after#(0(),XS) -> c_4()
          from#(X) -> c_5()
          from#(X) -> c_6()
          s#(X) -> c_7()
* Step 4: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            activate#(n__from(X)) -> c_2(from#(activate(X)))
            activate#(n__s(X)) -> c_3(s#(activate(X)))
            after#(0(),XS) -> c_4()
            from#(X) -> c_5()
            from#(X) -> c_6()
            s#(X) -> c_7()
        - Strict TRS:
            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:
            {activate/1,after/2,from/1,s/1,activate#/1,after#/2,from#/1,s#/1} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/1
            ,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,after#,from#,s#} and constructors {0,cons
            ,n__from,n__s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs}
    + 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(from) = {1},
            uargs(s) = {1},
            uargs(from#) = {1},
            uargs(s#) = {1},
            uargs(c_2) = {1},
            uargs(c_3) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                    p(0) = [0]         
             p(activate) = [2] x1 + [5]
                p(after) = [2] x2 + [2]
                 p(cons) = [1] x1 + [4]
                 p(from) = [1] x1 + [7]
              p(n__from) = [1] x1 + [4]
                 p(n__s) = [1] x1 + [3]
                    p(s) = [1] x1 + [5]
            p(activate#) = [6] x1 + [0]
               p(after#) = [2] x2 + [0]
                p(from#) = [1] x1 + [0]
                   p(s#) = [1] x1 + [0]
                  p(c_1) = [0]         
                  p(c_2) = [1] x1 + [0]
                  p(c_3) = [1] x1 + [0]
                  p(c_4) = [0]         
                  p(c_5) = [0]         
                  p(c_6) = [0]         
                  p(c_7) = [0]         
          
          Following rules are strictly oriented:
          activate#(n__from(X)) = [6] X + [24]            
                                > [2] X + [5]             
                                = c_2(from#(activate(X))) 
          
             activate#(n__s(X)) = [6] X + [18]            
                                > [2] X + [5]             
                                = c_3(s#(activate(X)))    
          
                    activate(X) = [2] X + [5]             
                                > [1] X + [0]             
                                = X                       
          
           activate(n__from(X)) = [2] X + [13]            
                                > [2] X + [12]            
                                = from(activate(X))       
          
              activate(n__s(X)) = [2] X + [11]            
                                > [2] X + [10]            
                                = s(activate(X))          
          
                        from(X) = [1] X + [7]             
                                > [1] X + [4]             
                                = cons(X,n__from(n__s(X)))
          
                        from(X) = [1] X + [7]             
                                > [1] X + [4]             
                                = n__from(X)              
          
                           s(X) = [1] X + [5]             
                                > [1] X + [3]             
                                = n__s(X)                 
          
          
          Following rules are (at-least) weakly oriented:
            activate#(X) =  [6] X + [0] 
                         >= [0]         
                         =  c_1()       
          
          after#(0(),XS) =  [2] XS + [0]
                         >= [0]         
                         =  c_4()       
          
                from#(X) =  [1] X + [0] 
                         >= [0]         
                         =  c_5()       
          
                from#(X) =  [1] X + [0] 
                         >= [0]         
                         =  c_6()       
          
                   s#(X) =  [1] X + [0] 
                         >= [0]         
                         =  c_7()       
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: PredecessorEstimation WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            after#(0(),XS) -> c_4()
            from#(X) -> c_5()
            from#(X) -> c_6()
            s#(X) -> c_7()
        - Weak DPs:
            activate#(n__from(X)) -> c_2(from#(activate(X)))
            activate#(n__s(X)) -> c_3(s#(activate(X)))
        - Weak TRS:
            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:
            {activate/1,after/2,from/1,s/1,activate#/1,after#/2,from#/1,s#/1} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/1
            ,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,after#,from#,s#} and constructors {0,cons
            ,n__from,n__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#(X) -> c_1()
          2: after#(0(),XS) -> c_4()
          3: from#(X) -> c_5()
          4: from#(X) -> c_6()
          5: s#(X) -> c_7()
          6: activate#(n__from(X)) -> c_2(from#(activate(X)))
          7: activate#(n__s(X)) -> c_3(s#(activate(X)))
* Step 6: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            from#(X) -> c_5()
            from#(X) -> c_6()
            s#(X) -> c_7()
        - Weak DPs:
            activate#(X) -> c_1()
            activate#(n__from(X)) -> c_2(from#(activate(X)))
            activate#(n__s(X)) -> c_3(s#(activate(X)))
            after#(0(),XS) -> c_4()
        - Weak TRS:
            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:
            {activate/1,after/2,from/1,s/1,activate#/1,after#/2,from#/1,s#/1} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/1
            ,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,after#,from#,s#} and constructors {0,cons
            ,n__from,n__s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:from#(X) -> c_5()
             
          
          2:S:from#(X) -> c_6()
             
          
          3:S:s#(X) -> c_7()
             
          
          4:W:activate#(X) -> c_1()
             
          
          5:W:activate#(n__from(X)) -> c_2(from#(activate(X)))
             -->_1 from#(X) -> c_6():2
             -->_1 from#(X) -> c_5():1
          
          6:W:activate#(n__s(X)) -> c_3(s#(activate(X)))
             -->_1 s#(X) -> c_7():3
          
          7:W:after#(0(),XS) -> c_4()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          7: after#(0(),XS) -> c_4()
          4: activate#(X) -> c_1()
* Step 7: Decompose WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            from#(X) -> c_5()
            from#(X) -> c_6()
            s#(X) -> c_7()
        - Weak DPs:
            activate#(n__from(X)) -> c_2(from#(activate(X)))
            activate#(n__s(X)) -> c_3(s#(activate(X)))
        - Weak TRS:
            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:
            {activate/1,after/2,from/1,s/1,activate#/1,after#/2,from#/1,s#/1} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/1
            ,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,after#,from#,s#} and constructors {0,cons
            ,n__from,n__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:
              from#(X) -> c_5()
          - Weak DPs:
              activate#(n__from(X)) -> c_2(from#(activate(X)))
              activate#(n__s(X)) -> c_3(s#(activate(X)))
              from#(X) -> c_6()
              s#(X) -> c_7()
          - Weak TRS:
              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:
              {activate/1,after/2,from/1,s/1,activate#/1,after#/2,from#/1,s#/1} / {0/0,cons/2,n__from/1,n__s/1,c_1/0
              ,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {activate#,after#,from#,s#} and constructors {0,cons
              ,n__from,n__s}
        
        Problem (S)
          - Strict DPs:
              from#(X) -> c_6()
              s#(X) -> c_7()
          - Weak DPs:
              activate#(n__from(X)) -> c_2(from#(activate(X)))
              activate#(n__s(X)) -> c_3(s#(activate(X)))
              from#(X) -> c_5()
          - Weak TRS:
              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:
              {activate/1,after/2,from/1,s/1,activate#/1,after#/2,from#/1,s#/1} / {0/0,cons/2,n__from/1,n__s/1,c_1/0
              ,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {activate#,after#,from#,s#} and constructors {0,cons
              ,n__from,n__s}
** Step 7.a:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            from#(X) -> c_5()
        - Weak DPs:
            activate#(n__from(X)) -> c_2(from#(activate(X)))
            activate#(n__s(X)) -> c_3(s#(activate(X)))
            from#(X) -> c_6()
            s#(X) -> c_7()
        - Weak TRS:
            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:
            {activate/1,after/2,from/1,s/1,activate#/1,after#/2,from#/1,s#/1} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/1
            ,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,after#,from#,s#} and constructors {0,cons
            ,n__from,n__s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:from#(X) -> c_5()
             
          
          2:W:from#(X) -> c_6()
             
          
          3:W:s#(X) -> c_7()
             
          
          5:W:activate#(n__from(X)) -> c_2(from#(activate(X)))
             -->_1 from#(X) -> c_5():1
             -->_1 from#(X) -> c_6():2
          
          6:W:activate#(n__s(X)) -> c_3(s#(activate(X)))
             -->_1 s#(X) -> c_7():3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: activate#(n__s(X)) -> c_3(s#(activate(X)))
          3: s#(X) -> c_7()
          2: from#(X) -> c_6()
** Step 7.a:2: Trivial WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            from#(X) -> c_5()
        - Weak DPs:
            activate#(n__from(X)) -> c_2(from#(activate(X)))
        - Weak TRS:
            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:
            {activate/1,after/2,from/1,s/1,activate#/1,after#/2,from#/1,s#/1} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/1
            ,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,after#,from#,s#} and constructors {0,cons
            ,n__from,n__s}
    + Applied Processor:
        Trivial
    + Details:
        Consider the dependency graph
          1:S:from#(X) -> c_5()
             
          
          5:W:activate#(n__from(X)) -> c_2(from#(activate(X)))
             -->_1 from#(X) -> c_5():1
          
        The dependency graph contains no loops, we remove all dependency pairs.
** Step 7.a:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            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:
            {activate/1,after/2,from/1,s/1,activate#/1,after#/2,from#/1,s#/1} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/1
            ,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,after#,from#,s#} and constructors {0,cons
            ,n__from,n__s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

** Step 7.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            from#(X) -> c_6()
            s#(X) -> c_7()
        - Weak DPs:
            activate#(n__from(X)) -> c_2(from#(activate(X)))
            activate#(n__s(X)) -> c_3(s#(activate(X)))
            from#(X) -> c_5()
        - Weak TRS:
            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:
            {activate/1,after/2,from/1,s/1,activate#/1,after#/2,from#/1,s#/1} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/1
            ,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,after#,from#,s#} and constructors {0,cons
            ,n__from,n__s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:from#(X) -> c_6()
             
          
          2:S:s#(X) -> c_7()
             
          
          3:W:activate#(n__from(X)) -> c_2(from#(activate(X)))
             -->_1 from#(X) -> c_5():5
             -->_1 from#(X) -> c_6():1
          
          4:W:activate#(n__s(X)) -> c_3(s#(activate(X)))
             -->_1 s#(X) -> c_7():2
          
          5:W:from#(X) -> c_5()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: from#(X) -> c_5()
** Step 7.b:2: Decompose WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            from#(X) -> c_6()
            s#(X) -> c_7()
        - Weak DPs:
            activate#(n__from(X)) -> c_2(from#(activate(X)))
            activate#(n__s(X)) -> c_3(s#(activate(X)))
        - Weak TRS:
            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:
            {activate/1,after/2,from/1,s/1,activate#/1,after#/2,from#/1,s#/1} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/1
            ,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,after#,from#,s#} and constructors {0,cons
            ,n__from,n__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:
              from#(X) -> c_6()
          - Weak DPs:
              activate#(n__from(X)) -> c_2(from#(activate(X)))
              activate#(n__s(X)) -> c_3(s#(activate(X)))
              s#(X) -> c_7()
          - Weak TRS:
              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:
              {activate/1,after/2,from/1,s/1,activate#/1,after#/2,from#/1,s#/1} / {0/0,cons/2,n__from/1,n__s/1,c_1/0
              ,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {activate#,after#,from#,s#} and constructors {0,cons
              ,n__from,n__s}
        
        Problem (S)
          - Strict DPs:
              s#(X) -> c_7()
          - Weak DPs:
              activate#(n__from(X)) -> c_2(from#(activate(X)))
              activate#(n__s(X)) -> c_3(s#(activate(X)))
              from#(X) -> c_6()
          - Weak TRS:
              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:
              {activate/1,after/2,from/1,s/1,activate#/1,after#/2,from#/1,s#/1} / {0/0,cons/2,n__from/1,n__s/1,c_1/0
              ,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {activate#,after#,from#,s#} and constructors {0,cons
              ,n__from,n__s}
*** Step 7.b:2.a:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            from#(X) -> c_6()
        - Weak DPs:
            activate#(n__from(X)) -> c_2(from#(activate(X)))
            activate#(n__s(X)) -> c_3(s#(activate(X)))
            s#(X) -> c_7()
        - Weak TRS:
            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:
            {activate/1,after/2,from/1,s/1,activate#/1,after#/2,from#/1,s#/1} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/1
            ,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,after#,from#,s#} and constructors {0,cons
            ,n__from,n__s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:from#(X) -> c_6()
             
          
          2:W:s#(X) -> c_7()
             
          
          3:W:activate#(n__from(X)) -> c_2(from#(activate(X)))
             -->_1 from#(X) -> c_6():1
          
          4:W:activate#(n__s(X)) -> c_3(s#(activate(X)))
             -->_1 s#(X) -> c_7():2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: activate#(n__s(X)) -> c_3(s#(activate(X)))
          2: s#(X) -> c_7()
*** Step 7.b:2.a:2: Trivial WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            from#(X) -> c_6()
        - Weak DPs:
            activate#(n__from(X)) -> c_2(from#(activate(X)))
        - Weak TRS:
            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:
            {activate/1,after/2,from/1,s/1,activate#/1,after#/2,from#/1,s#/1} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/1
            ,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,after#,from#,s#} and constructors {0,cons
            ,n__from,n__s}
    + Applied Processor:
        Trivial
    + Details:
        Consider the dependency graph
          1:S:from#(X) -> c_6()
             
          
          3:W:activate#(n__from(X)) -> c_2(from#(activate(X)))
             -->_1 from#(X) -> c_6():1
          
        The dependency graph contains no loops, we remove all dependency pairs.
*** Step 7.b:2.a:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            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:
            {activate/1,after/2,from/1,s/1,activate#/1,after#/2,from#/1,s#/1} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/1
            ,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,after#,from#,s#} and constructors {0,cons
            ,n__from,n__s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 7.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            s#(X) -> c_7()
        - Weak DPs:
            activate#(n__from(X)) -> c_2(from#(activate(X)))
            activate#(n__s(X)) -> c_3(s#(activate(X)))
            from#(X) -> c_6()
        - Weak TRS:
            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:
            {activate/1,after/2,from/1,s/1,activate#/1,after#/2,from#/1,s#/1} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/1
            ,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,after#,from#,s#} and constructors {0,cons
            ,n__from,n__s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:s#(X) -> c_7()
             
          
          2:W:activate#(n__from(X)) -> c_2(from#(activate(X)))
             -->_1 from#(X) -> c_6():4
          
          3:W:activate#(n__s(X)) -> c_3(s#(activate(X)))
             -->_1 s#(X) -> c_7():1
          
          4:W:from#(X) -> c_6()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: activate#(n__from(X)) -> c_2(from#(activate(X)))
          4: from#(X) -> c_6()
*** Step 7.b:2.b:2: Trivial WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            s#(X) -> c_7()
        - Weak DPs:
            activate#(n__s(X)) -> c_3(s#(activate(X)))
        - Weak TRS:
            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:
            {activate/1,after/2,from/1,s/1,activate#/1,after#/2,from#/1,s#/1} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/1
            ,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,after#,from#,s#} and constructors {0,cons
            ,n__from,n__s}
    + Applied Processor:
        Trivial
    + Details:
        Consider the dependency graph
          1:S:s#(X) -> c_7()
             
          
          3:W:activate#(n__s(X)) -> c_3(s#(activate(X)))
             -->_1 s#(X) -> c_7():1
          
        The dependency graph contains no loops, we remove all dependency pairs.
*** Step 7.b:2.b:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            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:
            {activate/1,after/2,from/1,s/1,activate#/1,after#/2,from#/1,s#/1} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/1
            ,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,after#,from#,s#} and constructors {0,cons
            ,n__from,n__s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))