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

WORST_CASE(?,O(n^1))