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))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
            sel(0(),cons(X,Y)) -> X
            sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
        - Signature:
            {activate/1,from/1,s/1,sel/2} / {0/0,cons/2,n__from/1,n__s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,from,s,sel} and constructors {0,cons,n__from
            ,n__s}
    + Applied Processor:
        InnermostRuleRemoval
    + Details:
        Arguments of following rules are not normal-forms.
          sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
        All above mentioned rules can be savely removed.
* Step 2: DependencyPairs 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))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
            sel(0(),cons(X,Y)) -> X
        - Signature:
            {activate/1,from/1,s/1,sel/2} / {0/0,cons/2,n__from/1,n__s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,from,s,sel} and constructors {0,cons,n__from
            ,n__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#(activate(X)))
          activate#(n__s(X)) -> c_3(s#(activate(X)))
          from#(X) -> c_4()
          from#(X) -> c_5()
          s#(X) -> c_6()
          sel#(0(),cons(X,Y)) -> c_7()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            activate#(n__from(X)) -> c_2(from#(activate(X)))
            activate#(n__s(X)) -> c_3(s#(activate(X)))
            from#(X) -> c_4()
            from#(X) -> c_5()
            s#(X) -> c_6()
            sel#(0(),cons(X,Y)) -> c_7()
        - Strict TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
            sel(0(),cons(X,Y)) -> X
        - Signature:
            {activate/1,from/1,s/1,sel/2,activate#/1,from#/1,s#/1,sel#/2} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/1
            ,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,from#,s#,sel#} and constructors {0,cons,n__from
            ,n__s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          activate(X) -> X
          activate(n__from(X)) -> from(activate(X))
          activate(n__s(X)) -> s(activate(X))
          from(X) -> cons(X,n__from(n__s(X)))
          from(X) -> n__from(X)
          s(X) -> n__s(X)
          activate#(X) -> c_1()
          activate#(n__from(X)) -> c_2(from#(activate(X)))
          activate#(n__s(X)) -> c_3(s#(activate(X)))
          from#(X) -> c_4()
          from#(X) -> c_5()
          s#(X) -> c_6()
          sel#(0(),cons(X,Y)) -> c_7()
* Step 4: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            activate#(n__from(X)) -> c_2(from#(activate(X)))
            activate#(n__s(X)) -> c_3(s#(activate(X)))
            from#(X) -> c_4()
            from#(X) -> c_5()
            s#(X) -> c_6()
            sel#(0(),cons(X,Y)) -> c_7()
        - Strict TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
        - Signature:
            {activate/1,from/1,s/1,sel/2,activate#/1,from#/1,s#/1,sel#/2} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/1
            ,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,from#,s#,sel#} and constructors {0,cons,n__from
            ,n__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(from) = {1},
            uargs(s) = {1},
            uargs(from#) = {1},
            uargs(s#) = {1},
            uargs(c_2) = {1},
            uargs(c_3) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                    p(0) = [0]          
             p(activate) = [4] x1 + [11]
                 p(cons) = [4]          
                 p(from) = [1] x1 + [5] 
              p(n__from) = [1] x1 + [4] 
                 p(n__s) = [1] x1 + [2] 
                    p(s) = [1] x1 + [5] 
                  p(sel) = [2] x2 + [0] 
            p(activate#) = [4] x1 + [0] 
                p(from#) = [1] x1 + [0] 
                   p(s#) = [1] x1 + [0] 
                 p(sel#) = [0]          
                  p(c_1) = [0]          
                  p(c_2) = [1] x1 + [8] 
                  p(c_3) = [1] x1 + [12]
                  p(c_4) = [0]          
                  p(c_5) = [2]          
                  p(c_6) = [0]          
                  p(c_7) = [1]          
          
          Following rules are strictly oriented:
                   activate(X) = [4] X + [11]            
                               > [1] X + [0]             
                               = X                       
          
          activate(n__from(X)) = [4] X + [27]            
                               > [4] X + [16]            
                               = from(activate(X))       
          
             activate(n__s(X)) = [4] X + [19]            
                               > [4] X + [16]            
                               = s(activate(X))          
          
                       from(X) = [1] X + [5]             
                               > [4]                     
                               = cons(X,n__from(n__s(X)))
          
                       from(X) = [1] X + [5]             
                               > [1] X + [4]             
                               = n__from(X)              
          
                          s(X) = [1] X + [5]             
                               > [1] X + [2]             
                               = n__s(X)                 
          
          
          Following rules are (at-least) weakly oriented:
                   activate#(X) =  [4] X + [0]            
                                >= [0]                    
                                =  c_1()                  
          
          activate#(n__from(X)) =  [4] X + [16]           
                                >= [4] X + [19]           
                                =  c_2(from#(activate(X)))
          
             activate#(n__s(X)) =  [4] X + [8]            
                                >= [4] X + [23]           
                                =  c_3(s#(activate(X)))   
          
                       from#(X) =  [1] X + [0]            
                                >= [0]                    
                                =  c_4()                  
          
                       from#(X) =  [1] X + [0]            
                                >= [2]                    
                                =  c_5()                  
          
                          s#(X) =  [1] X + [0]            
                                >= [0]                    
                                =  c_6()                  
          
            sel#(0(),cons(X,Y)) =  [0]                    
                                >= [1]                    
                                =  c_7()                  
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: PredecessorEstimation WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            activate#(n__from(X)) -> c_2(from#(activate(X)))
            activate#(n__s(X)) -> c_3(s#(activate(X)))
            from#(X) -> c_4()
            from#(X) -> c_5()
            s#(X) -> c_6()
            sel#(0(),cons(X,Y)) -> c_7()
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
        - Signature:
            {activate/1,from/1,s/1,sel/2,activate#/1,from#/1,s#/1,sel#/2} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/1
            ,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,from#,s#,sel#} and constructors {0,cons,n__from
            ,n__s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,4,5,6,7}
        by application of
          Pre({1,4,5,6,7}) = {2,3}.
        Here rules are labelled as follows:
          1: activate#(X) -> c_1()
          2: activate#(n__from(X)) -> c_2(from#(activate(X)))
          3: activate#(n__s(X)) -> c_3(s#(activate(X)))
          4: from#(X) -> c_4()
          5: from#(X) -> c_5()
          6: s#(X) -> c_6()
          7: sel#(0(),cons(X,Y)) -> c_7()
* Step 6: PredecessorEstimation WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__from(X)) -> c_2(from#(activate(X)))
            activate#(n__s(X)) -> c_3(s#(activate(X)))
        - Weak DPs:
            activate#(X) -> c_1()
            from#(X) -> c_4()
            from#(X) -> c_5()
            s#(X) -> c_6()
            sel#(0(),cons(X,Y)) -> c_7()
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
        - Signature:
            {activate/1,from/1,s/1,sel/2,activate#/1,from#/1,s#/1,sel#/2} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/1
            ,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,from#,s#,sel#} and constructors {0,cons,n__from
            ,n__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}) = {}.
        Here rules are labelled as follows:
          1: activate#(n__from(X)) -> c_2(from#(activate(X)))
          2: activate#(n__s(X)) -> c_3(s#(activate(X)))
          3: activate#(X) -> c_1()
          4: from#(X) -> c_4()
          5: from#(X) -> c_5()
          6: s#(X) -> c_6()
          7: sel#(0(),cons(X,Y)) -> c_7()
* Step 7: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            activate#(X) -> c_1()
            activate#(n__from(X)) -> c_2(from#(activate(X)))
            activate#(n__s(X)) -> c_3(s#(activate(X)))
            from#(X) -> c_4()
            from#(X) -> c_5()
            s#(X) -> c_6()
            sel#(0(),cons(X,Y)) -> c_7()
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(activate(X))
            activate(n__s(X)) -> s(activate(X))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
        - Signature:
            {activate/1,from/1,s/1,sel/2,activate#/1,from#/1,s#/1,sel#/2} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/1
            ,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,from#,s#,sel#} and constructors {0,cons,n__from
            ,n__s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:activate#(X) -> c_1()
             
          
          2:W:activate#(n__from(X)) -> c_2(from#(activate(X)))
             -->_1 from#(X) -> c_5():5
             -->_1 from#(X) -> c_4():4
          
          3:W:activate#(n__s(X)) -> c_3(s#(activate(X)))
             -->_1 s#(X) -> c_6():6
          
          4:W:from#(X) -> c_4()
             
          
          5:W:from#(X) -> c_5()
             
          
          6:W:s#(X) -> c_6()
             
          
          7:W:sel#(0(),cons(X,Y)) -> c_7()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          7: sel#(0(),cons(X,Y)) -> c_7()
          3: activate#(n__s(X)) -> c_3(s#(activate(X)))
          6: s#(X) -> c_6()
          2: activate#(n__from(X)) -> c_2(from#(activate(X)))
          4: from#(X) -> c_4()
          5: from#(X) -> c_5()
          1: activate#(X) -> c_1()
* 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))
            from(X) -> cons(X,n__from(n__s(X)))
            from(X) -> n__from(X)
            s(X) -> n__s(X)
        - Signature:
            {activate/1,from/1,s/1,sel/2,activate#/1,from#/1,s#/1,sel#/2} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/1
            ,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,from#,s#,sel#} 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))