WORST_CASE(?,O(n^1))
* Step 1: 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
            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:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          activate#(X) -> c_1()
          activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X))
          activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X))
          after#(0(),XS) -> c_4()
          after#(s(N),cons(X,XS)) -> c_5(after#(N,activate(XS)),activate#(XS))
          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:
            activate#(X) -> c_1()
            activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X))
            activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X))
            after#(0(),XS) -> c_4()
            after#(s(N),cons(X,XS)) -> c_5(after#(N,activate(XS)),activate#(XS))
            from#(X) -> c_6()
            from#(X) -> c_7()
            s#(X) -> c_8()
        - Weak 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,activate#/1,after#/2,from#/1,s#/1} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/2
            ,c_3/2,c_4/0,c_5/2,c_6/0,c_7/0,c_8/0}
        - 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
          {1,4,5,6,7,8}
        by application of
          Pre({1,4,5,6,7,8}) = {2,3}.
        Here rules are labelled as follows:
          1: activate#(X) -> c_1()
          2: activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X))
          3: activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X))
          4: after#(0(),XS) -> c_4()
          5: after#(s(N),cons(X,XS)) -> c_5(after#(N,activate(XS)),activate#(XS))
          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:
            activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X))
            activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X))
        - Weak DPs:
            activate#(X) -> c_1()
            after#(0(),XS) -> c_4()
            after#(s(N),cons(X,XS)) -> c_5(after#(N,activate(XS)),activate#(XS))
            from#(X) -> c_6()
            from#(X) -> c_7()
            s#(X) -> c_8()
        - Weak 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,activate#/1,after#/2,from#/1,s#/1} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/2
            ,c_3/2,c_4/0,c_5/2,c_6/0,c_7/0,c_8/0}
        - 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:activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X))
             -->_2 activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)):2
             -->_1 from#(X) -> c_7():7
             -->_1 from#(X) -> c_6():6
             -->_2 activate#(X) -> c_1():3
             -->_2 activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X)):1
          
          2:S:activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X))
             -->_1 s#(X) -> c_8():8
             -->_2 activate#(X) -> c_1():3
             -->_2 activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)):2
             -->_2 activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X)):1
          
          3:W:activate#(X) -> c_1()
             
          
          4:W:after#(0(),XS) -> c_4()
             
          
          5:W:after#(s(N),cons(X,XS)) -> c_5(after#(N,activate(XS)),activate#(XS))
             
          
          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.
          5: after#(s(N),cons(X,XS)) -> c_5(after#(N,activate(XS)),activate#(XS))
          4: after#(0(),XS) -> c_4()
          6: from#(X) -> c_6()
          7: from#(X) -> c_7()
          3: activate#(X) -> c_1()
          8: s#(X) -> c_8()
* Step 4: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X))
            activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X))
        - Weak 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,activate#/1,after#/2,from#/1,s#/1} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/2
            ,c_3/2,c_4/0,c_5/2,c_6/0,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,after#,from#,s#} and constructors {0,cons
            ,n__from,n__s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X))
             -->_2 activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)):2
             -->_2 activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X)):1
          
          2:S:activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X))
             -->_2 activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)):2
             -->_2 activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          activate#(n__from(X)) -> c_2(activate#(X))
          activate#(n__s(X)) -> c_3(activate#(X))
* Step 5: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__from(X)) -> c_2(activate#(X))
            activate#(n__s(X)) -> c_3(activate#(X))
        - Weak 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,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_5/2,c_6/0,c_7/0,c_8/0}
        - 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#(n__from(X)) -> c_2(activate#(X))
          activate#(n__s(X)) -> c_3(activate#(X))
* Step 6: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__from(X)) -> c_2(activate#(X))
            activate#(n__s(X)) -> c_3(activate#(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_5/2,c_6/0,c_7/0,c_8/0}
        - 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 = 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_2) = {1},
            uargs(c_3) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                    p(0) = [0]                  
             p(activate) = [0]                  
                p(after) = [0]                  
                 p(cons) = [1] x1 + [1] x2 + [0]
                 p(from) = [0]                  
              p(n__from) = [1] x1 + [8]         
                 p(n__s) = [1] x1 + [10]        
                    p(s) = [0]                  
            p(activate#) = [2] x1 + [10]        
               p(after#) = [8] x2 + [2]         
                p(from#) = [1]                  
                   p(s#) = [2] x1 + [4]         
                  p(c_1) = [4]                  
                  p(c_2) = [1] x1 + [10]        
                  p(c_3) = [1] x1 + [0]         
                  p(c_4) = [0]                  
                  p(c_5) = [0]                  
                  p(c_6) = [0]                  
                  p(c_7) = [0]                  
                  p(c_8) = [0]                  
          
          Following rules are strictly oriented:
          activate#(n__from(X)) = [2] X + [26]     
                                > [2] X + [20]     
                                = c_2(activate#(X))
          
             activate#(n__s(X)) = [2] X + [30]     
                                > [2] X + [10]     
                                = c_3(activate#(X))
          
          
          Following rules are (at-least) weakly oriented:
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 7: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            activate#(n__from(X)) -> c_2(activate#(X))
            activate#(n__s(X)) -> c_3(activate#(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_5/2,c_6/0,c_7/0,c_8/0}
        - 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))