WORST_CASE(?,O(n^1))
* Step 1: InnermostRuleRemoval 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:
        InnermostRuleRemoval
    + Details:
        Arguments of following rules are not normal-forms.
          after(s(N),cons(X,XS)) -> after(N,activate(XS))
        All above mentioned rules can be savely removed.
* Step 2: WeightGap 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
            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:
        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(from) = {1},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(0) = [0]         
            p(activate) = [2] x1 + [0]
               p(after) = [2] x2 + [0]
                p(cons) = [1] x2 + [0]
                p(from) = [1] x1 + [0]
             p(n__from) = [1] x1 + [0]
                p(n__s) = [1] x1 + [0]
                   p(s) = [1] x1 + [1]
          
          Following rules are strictly oriented:
          s(X) = [1] X + [1]
               > [1] X + [0]
               = n__s(X)    
          
          
          Following rules are (at-least) weakly oriented:
                   activate(X) =  [2] X + [0]             
                               >= [1] X + [0]             
                               =  X                       
          
          activate(n__from(X)) =  [2] X + [0]             
                               >= [2] X + [0]             
                               =  from(activate(X))       
          
             activate(n__s(X)) =  [2] X + [0]             
                               >= [2] X + [1]             
                               =  s(activate(X))          
          
                 after(0(),XS) =  [2] XS + [0]            
                               >= [1] XS + [0]            
                               =  XS                      
          
                       from(X) =  [1] X + [0]             
                               >= [1] X + [0]             
                               =  cons(X,n__from(n__s(X)))
          
                       from(X) =  [1] X + [0]             
                               >= [1] X + [0]             
                               =  n__from(X)              
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap 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
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
        - Weak TRS:
            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:
        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(from) = {1},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(0) = [0]         
            p(activate) = [2] x1 + [0]
               p(after) = [2] x2 + [0]
                p(cons) = [1] x2 + [0]
                p(from) = [1] x1 + [1]
             p(n__from) = [1] x1 + [0]
                p(n__s) = [1] x1 + [0]
                   p(s) = [1] x1 + [0]
          
          Following rules are strictly oriented:
          from(X) = [1] X + [1]             
                  > [1] X + [0]             
                  = cons(X,n__from(n__s(X)))
          
          from(X) = [1] X + [1]             
                  > [1] X + [0]             
                  = n__from(X)              
          
          
          Following rules are (at-least) weakly oriented:
                   activate(X) =  [2] X + [0]      
                               >= [1] X + [0]      
                               =  X                
          
          activate(n__from(X)) =  [2] X + [0]      
                               >= [2] X + [1]      
                               =  from(activate(X))
          
             activate(n__s(X)) =  [2] X + [0]      
                               >= [2] X + [0]      
                               =  s(activate(X))   
          
                 after(0(),XS) =  [2] XS + [0]     
                               >= [1] XS + [0]     
                               =  XS               
          
                          s(X) =  [1] X + [0]      
                               >= [1] X + [0]      
                               =  n__s(X)          
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: WeightGap 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
        - Weak TRS:
            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:
        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(from) = {1},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(0) = [0]         
            p(activate) = [2] x1 + [0]
               p(after) = [2] x2 + [1]
                p(cons) = [1] x2 + [0]
                p(from) = [1] x1 + [0]
             p(n__from) = [1] x1 + [0]
                p(n__s) = [1] x1 + [0]
                   p(s) = [1] x1 + [0]
          
          Following rules are strictly oriented:
          after(0(),XS) = [2] XS + [1]
                        > [1] XS + [0]
                        = XS          
          
          
          Following rules are (at-least) weakly oriented:
                   activate(X) =  [2] X + [0]             
                               >= [1] X + [0]             
                               =  X                       
          
          activate(n__from(X)) =  [2] X + [0]             
                               >= [2] X + [0]             
                               =  from(activate(X))       
          
             activate(n__s(X)) =  [2] X + [0]             
                               >= [2] X + [0]             
                               =  s(activate(X))          
          
                       from(X) =  [1] X + [0]             
                               >= [1] X + [0]             
                               =  cons(X,n__from(n__s(X)))
          
                       from(X) =  [1] X + [0]             
                               >= [1] X + [0]             
                               =  n__from(X)              
          
                          s(X) =  [1] X + [0]             
                               >= [1] X + [0]             
                               =  n__s(X)                 
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: WeightGap 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))
        - Weak TRS:
            after(0(),XS) -> 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:
        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(from) = {1},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(0) = [8]                  
            p(activate) = [1] x1 + [9]         
               p(after) = [1] x1 + [2] x2 + [9]
                p(cons) = [7]                  
                p(from) = [1] x1 + [7]         
             p(n__from) = [1] x1 + [2]         
                p(n__s) = [1] x1 + [0]         
                   p(s) = [1] x1 + [3]         
          
          Following rules are strictly oriented:
          activate(X) = [1] X + [9]
                      > [1] X + [0]
                      = X          
          
          
          Following rules are (at-least) weakly oriented:
          activate(n__from(X)) =  [1] X + [11]            
                               >= [1] X + [16]            
                               =  from(activate(X))       
          
             activate(n__s(X)) =  [1] X + [9]             
                               >= [1] X + [12]            
                               =  s(activate(X))          
          
                 after(0(),XS) =  [2] XS + [17]           
                               >= [1] XS + [0]            
                               =  XS                      
          
                       from(X) =  [1] X + [7]             
                               >= [7]                     
                               =  cons(X,n__from(n__s(X)))
          
                       from(X) =  [1] X + [7]             
                               >= [1] X + [2]             
                               =  n__from(X)              
          
                          s(X) =  [1] X + [3]             
                               >= [1] X + [0]             
                               =  n__s(X)                 
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 6: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
        - Weak TRS:
            activate(X) -> X
            after(0(),XS) -> 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:
        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(from) = {1},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(0) = [6]                  
            p(activate) = [2] x1 + [0]         
               p(after) = [5] x1 + [2] x2 + [1]
                p(cons) = [0]                  
                p(from) = [1] x1 + [0]         
             p(n__from) = [1] x1 + [0]         
                p(n__s) = [1] x1 + [3]         
                   p(s) = [1] x1 + [3]         
          
          Following rules are strictly oriented:
          activate(n__s(X)) = [2] X + [6]   
                            > [2] X + [3]   
                            = s(activate(X))
          
          
          Following rules are (at-least) weakly oriented:
                   activate(X) =  [2] X + [0]             
                               >= [1] X + [0]             
                               =  X                       
          
          activate(n__from(X)) =  [2] X + [0]             
                               >= [2] X + [0]             
                               =  from(activate(X))       
          
                 after(0(),XS) =  [2] XS + [31]           
                               >= [1] XS + [0]            
                               =  XS                      
          
                       from(X) =  [1] X + [0]             
                               >= [0]                     
                               =  cons(X,n__from(n__s(X)))
          
                       from(X) =  [1] X + [0]             
                               >= [1] X + [0]             
                               =  n__from(X)              
          
                          s(X) =  [1] X + [3]             
                               >= [1] X + [3]             
                               =  n__s(X)                 
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 7: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            activate(n__from(X)) -> from(activate(X))
        - Weak TRS:
            activate(X) -> X
            activate(n__s(X)) -> s(activate(X))
            after(0(),XS) -> 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:
        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(from) = {1},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(0) = [0]         
            p(activate) = [3] x1 + [0]
               p(after) = [2] x2 + [0]
                p(cons) = [1] x2 + [1]
                p(from) = [1] x1 + [2]
             p(n__from) = [1] x1 + [1]
                p(n__s) = [1] x1 + [0]
                   p(s) = [1] x1 + [0]
          
          Following rules are strictly oriented:
          activate(n__from(X)) = [3] X + [3]      
                               > [3] X + [2]      
                               = from(activate(X))
          
          
          Following rules are (at-least) weakly oriented:
                activate(X) =  [3] X + [0]             
                            >= [1] X + [0]             
                            =  X                       
          
          activate(n__s(X)) =  [3] X + [0]             
                            >= [3] X + [0]             
                            =  s(activate(X))          
          
              after(0(),XS) =  [2] XS + [0]            
                            >= [1] XS + [0]            
                            =  XS                      
          
                    from(X) =  [1] X + [2]             
                            >= [1] X + [2]             
                            =  cons(X,n__from(n__s(X)))
          
                    from(X) =  [1] X + [2]             
                            >= [1] X + [1]             
                            =  n__from(X)              
          
                       s(X) =  [1] X + [0]             
                            >= [1] X + [0]             
                            =  n__s(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 TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            after(0(),XS) -> 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:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))