* Step 1: WeightGap 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:
             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 nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(after) = {2}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(0) = [0]          
            p(activate) = [1] x1 + [0] 
               p(after) = [1] x2 + [11]
                p(cons) = [1] x2 + [0] 
                p(from) = [1] x1 + [1] 
             p(n__from) = [1] x1 + [0] 
                   p(s) = [1] x1 + [0] 
          
          Following rules are strictly oriented:
          after(0(),XS) = [1] XS + [11]        
                        > [1] XS + [0]         
                        = XS                   
          
                from(X) = [1] X + [1]          
                        > [1] X + [0]          
                        = cons(X,n__from(s(X)))
          
                from(X) = [1] X + [1]          
                        > [1] X + [0]          
                        = n__from(X)           
          
          
          Following rules are (at-least) weakly oriented:
                     activate(X) =  [1] X + [0]          
                                 >= [1] X + [0]          
                                 =  X                    
          
            activate(n__from(X)) =  [1] X + [0]          
                                 >= [1] X + [1]          
                                 =  from(X)              
          
          after(s(N),cons(X,XS)) =  [1] XS + [11]        
                                 >= [1] XS + [11]        
                                 =  after(N,activate(XS))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: MI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(X)
            after(s(N),cons(X,XS)) -> after(N,activate(XS))
        - Weak TRS:
            after(0(),XS) -> 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:
             runtime complexity wrt. defined symbols {activate,after,from} and constructors {0,cons,n__from,s}
    + Applied Processor:
        MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
        
        The following argument positions are considered usable:
          uargs(after) = {2}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
                 p(0) = [0]                     
          p(activate) = [1] x_1 + [4]           
             p(after) = [7] x_1 + [1] x_2 + [12]
              p(cons) = [1] x_2 + [0]           
              p(from) = [0]                     
           p(n__from) = [0]                     
                 p(s) = [1] x_1 + [1]           
        
        Following rules are strictly oriented:
                   activate(X) = [1] X + [4]          
                               > [1] X + [0]          
                               = X                    
        
          activate(n__from(X)) = [4]                  
                               > [0]                  
                               = from(X)              
        
        after(s(N),cons(X,XS)) = [7] N + [1] XS + [19]
                               > [7] N + [1] XS + [16]
                               = after(N,activate(XS))
        
        
        Following rules are (at-least) weakly oriented:
        after(0(),XS) =  [1] XS + [12]        
                      >= [1] XS + [0]         
                      =  XS                   
        
              from(X) =  [0]                  
                      >= [0]                  
                      =  cons(X,n__from(s(X)))
        
              from(X) =  [0]                  
                      >= [0]                  
                      =  n__from(X)           
        
* Step 3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - 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} / {0/0,cons/2,n__from/1,s/1}
        - Obligation:
             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))