WORST_CASE(?,O(n^1))
* Step 1: NaturalMI 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:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        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:
          {activate,after,from}
        TcT has computed the following interpretation:
                 p(0) = [0]         
          p(activate) = [1] x1 + [2]
             p(after) = [1] x2 + [0]
              p(cons) = [1] x2 + [2]
              p(from) = [1] x1 + [2]
           p(n__from) = [1] x1 + [0]
                 p(s) = [0]         
        
        Following rules are strictly oriented:
        activate(X) = [1] X + [2]
                    > [1] X + [0]
                    = X          
        
            from(X) = [1] X + [2]
                    > [1] X + [0]
                    = n__from(X) 
        
        
        Following rules are (at-least) weakly oriented:
          activate(n__from(X)) =  [1] X + [2]          
                               >= [1] X + [2]          
                               =  from(X)              
        
                 after(0(),XS) =  [1] XS + [0]         
                               >= [1] XS + [0]         
                               =  XS                   
        
        after(s(N),cons(X,XS)) =  [1] XS + [2]         
                               >= [1] XS + [2]         
                               =  after(N,activate(XS))
        
                       from(X) =  [1] X + [2]          
                               >= [2]                  
                               =  cons(X,n__from(s(X)))
        
* Step 2: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            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)))
        - Weak TRS:
            activate(X) -> 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:
        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) = [2] x1 + [1] x2 + [2]
                p(cons) = [1] x2 + [0]         
                p(from) = [1]                  
             p(n__from) = [0]                  
                   p(s) = [1] x1 + [4]         
          
          Following rules are strictly oriented:
                   after(0(),XS) = [1] XS + [2]         
                                 > [1] XS + [0]         
                                 = XS                   
          
          after(s(N),cons(X,XS)) = [2] N + [1] XS + [10]
                                 > [2] N + [1] XS + [2] 
                                 = after(N,activate(XS))
          
                         from(X) = [1]                  
                                 > [0]                  
                                 = cons(X,n__from(s(X)))
          
          
          Following rules are (at-least) weakly oriented:
                   activate(X) =  [1] X + [0]
                               >= [1] X + [0]
                               =  X          
          
          activate(n__from(X)) =  [0]        
                               >= [1]        
                               =  from(X)    
          
                       from(X) =  [1]        
                               >= [0]        
                               =  n__from(X) 
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            activate(n__from(X)) -> from(X)
        - Weak TRS:
            activate(X) -> 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:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        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:
          {activate,after,from}
        TcT has computed the following interpretation:
                 p(0) = [1]                  
          p(activate) = [1] x1 + [2]         
             p(after) = [2] x1 + [8] x2 + [0]
              p(cons) = [1] x2 + [0]         
              p(from) = [8]                  
           p(n__from) = [8]                  
                 p(s) = [1] x1 + [8]         
        
        Following rules are strictly oriented:
        activate(n__from(X)) = [10]   
                             > [8]    
                             = from(X)
        
        
        Following rules are (at-least) weakly oriented:
                   activate(X) =  [1] X + [2]          
                               >= [1] X + [0]          
                               =  X                    
        
                 after(0(),XS) =  [8] XS + [2]         
                               >= [1] XS + [0]         
                               =  XS                   
        
        after(s(N),cons(X,XS)) =  [2] N + [8] XS + [16]
                               >= [2] N + [8] XS + [16]
                               =  after(N,activate(XS))
        
                       from(X) =  [8]                  
                               >= [8]                  
                               =  cons(X,n__from(s(X)))
        
                       from(X) =  [8]                  
                               >= [8]                  
                               =  n__from(X)           
        
* Step 4: 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:
            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))