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_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        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)))
          from#(X) -> c_5()
          from#(X) -> c_6()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules 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)))
            from#(X) -> c_5()
            from#(X) -> c_6()
        - 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,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)
          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)))
          from#(X) -> c_5()
          from#(X) -> c_6()
* Step 3: WeightGap 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)))
            from#(X) -> c_5()
            from#(X) -> c_6()
        - Strict 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 = 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(after#) = {2},
            uargs(c_2) = {1},
            uargs(c_4) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                    p(0) = [0]                  
             p(activate) = [1] x1 + [14]        
                p(after) = [0]                  
                 p(cons) = [1] x2 + [4]         
                 p(from) = [1] x1 + [13]        
              p(n__from) = [1] x1 + [3]         
                    p(s) = [1] x1 + [5]         
            p(activate#) = [3] x1 + [0]         
               p(after#) = [1] x1 + [1] x2 + [0]
                p(from#) = [2] x1 + [0]         
                  p(c_1) = [0]                  
                  p(c_2) = [1] x1 + [0]         
                  p(c_3) = [0]                  
                  p(c_4) = [1] x1 + [2]         
                  p(c_5) = [0]                  
                  p(c_6) = [0]                  
          
          Following rules are strictly oriented:
          activate#(n__from(X)) = [3] X + [9]          
                                > [2] X + [0]          
                                = c_2(from#(X))        
          
                    activate(X) = [1] X + [14]         
                                > [1] X + [0]          
                                = X                    
          
           activate(n__from(X)) = [1] X + [17]         
                                > [1] X + [13]         
                                = from(X)              
          
                        from(X) = [1] X + [13]         
                                > [1] X + [12]         
                                = cons(X,n__from(s(X)))
          
                        from(X) = [1] X + [13]         
                                > [1] X + [3]          
                                = n__from(X)           
          
          
          Following rules are (at-least) weakly oriented:
                     activate#(X) =  [3] X + [0]                
                                  >= [0]                        
                                  =  c_1()                      
          
                   after#(0(),XS) =  [1] XS + [0]               
                                  >= [0]                        
                                  =  c_3()                      
          
          after#(s(N),cons(X,XS)) =  [1] N + [1] XS + [9]       
                                  >= [1] N + [1] XS + [16]      
                                  =  c_4(after#(N,activate(XS)))
          
                         from#(X) =  [2] X + [0]                
                                  >= [0]                        
                                  =  c_5()                      
          
                         from#(X) =  [2] X + [0]                
                                  >= [0]                        
                                  =  c_6()                      
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            after#(0(),XS) -> c_3()
            after#(s(N),cons(X,XS)) -> c_4(after#(N,activate(XS)))
            from#(X) -> c_5()
            from#(X) -> c_6()
        - Weak DPs:
            activate#(n__from(X)) -> c_2(from#(X))
        - 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:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2}
        by application of
          Pre({1,2}) = {3}.
        Here rules are labelled as follows:
          1: activate#(X) -> c_1()
          2: after#(0(),XS) -> c_3()
          3: after#(s(N),cons(X,XS)) -> c_4(after#(N,activate(XS)))
          4: from#(X) -> c_5()
          5: from#(X) -> c_6()
          6: activate#(n__from(X)) -> c_2(from#(X))
* Step 5: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            after#(s(N),cons(X,XS)) -> c_4(after#(N,activate(XS)))
            from#(X) -> c_5()
            from#(X) -> c_6()
        - Weak DPs:
            activate#(X) -> c_1()
            activate#(n__from(X)) -> c_2(from#(X))
            after#(0(),XS) -> c_3()
        - 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:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:after#(s(N),cons(X,XS)) -> c_4(after#(N,activate(XS)))
             -->_1 after#(0(),XS) -> c_3():6
             -->_1 after#(s(N),cons(X,XS)) -> c_4(after#(N,activate(XS))):1
          
          2:S:from#(X) -> c_5()
             
          
          3:S:from#(X) -> c_6()
             
          
          4:W:activate#(X) -> c_1()
             
          
          5:W:activate#(n__from(X)) -> c_2(from#(X))
             -->_1 from#(X) -> c_6():3
             -->_1 from#(X) -> c_5():2
          
          6:W:after#(0(),XS) -> c_3()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: activate#(X) -> c_1()
          6: after#(0(),XS) -> c_3()
* Step 6: RemoveHeads WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            after#(s(N),cons(X,XS)) -> c_4(after#(N,activate(XS)))
            from#(X) -> c_5()
            from#(X) -> c_6()
        - Weak DPs:
            activate#(n__from(X)) -> c_2(from#(X))
        - 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:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:after#(s(N),cons(X,XS)) -> c_4(after#(N,activate(XS)))
           -->_1 after#(s(N),cons(X,XS)) -> c_4(after#(N,activate(XS))):1
        
        2:S:from#(X) -> c_5()
           
        
        3:S:from#(X) -> c_6()
           
        
        5:W:activate#(n__from(X)) -> c_2(from#(X))
           -->_1 from#(X) -> c_6():3
           -->_1 from#(X) -> c_5():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).
        
        [(5,activate#(n__from(X)) -> c_2(from#(X)))]
* Step 7: RemoveHeads WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            after#(s(N),cons(X,XS)) -> c_4(after#(N,activate(XS)))
            from#(X) -> c_5()
            from#(X) -> c_6()
        - 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:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:after#(s(N),cons(X,XS)) -> c_4(after#(N,activate(XS)))
           -->_1 after#(s(N),cons(X,XS)) -> c_4(after#(N,activate(XS))):1
        
        2:S:from#(X) -> c_5()
           
        
        3:S:from#(X) -> c_6()
           
        
        
        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,from#(X) -> c_5()),(3,from#(X) -> c_6())]
* Step 8: PredecessorEstimationCP 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:
        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: after#(s(N),cons(X,XS)) -> c_4(after#(N,activate(XS)))
          
        The strictly oriented rules are moved into the weak component.
** Step 8.a:1: NaturalMI 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:
        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_4) = {1}
        
        Following symbols are considered usable:
          {activate,from,activate#,after#,from#}
        TcT has computed the following interpretation:
                  p(0) = [0]                   
           p(activate) = [1] x1 + [0]          
              p(after) = [1] x1 + [8] x2 + [0] 
               p(cons) = [1] x2 + [0]          
               p(from) = [0]                   
            p(n__from) = [0]                   
                  p(s) = [1] x1 + [2]          
          p(activate#) = [0]                   
             p(after#) = [8] x1 + [8] x2 + [10]
              p(from#) = [0]                   
                p(c_1) = [1]                   
                p(c_2) = [1]                   
                p(c_3) = [0]                   
                p(c_4) = [1] x1 + [3]          
                p(c_5) = [1]                   
                p(c_6) = [0]                   
        
        Following rules are strictly oriented:
        after#(s(N),cons(X,XS)) = [8] N + [8] XS + [26]      
                                > [8] N + [8] XS + [13]      
                                = c_4(after#(N,activate(XS)))
        
        
        Following rules are (at-least) weakly oriented:
                 activate(X) =  [1] X + [0]          
                             >= [1] X + [0]          
                             =  X                    
        
        activate(n__from(X)) =  [0]                  
                             >= [0]                  
                             =  from(X)              
        
                     from(X) =  [0]                  
                             >= [0]                  
                             =  cons(X,n__from(s(X)))
        
                     from(X) =  [0]                  
                             >= [0]                  
                             =  n__from(X)           
        
** Step 8.a:2: Assumption 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:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

** Step 8.b:1: RemoveWeakSuffixes 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:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:after#(s(N),cons(X,XS)) -> c_4(after#(N,activate(XS)))
             -->_1 after#(s(N),cons(X,XS)) -> c_4(after#(N,activate(XS))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: after#(s(N),cons(X,XS)) -> c_4(after#(N,activate(XS)))
** Step 8.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - 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))