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)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            sel(0(),cons(X,Y)) -> X
            sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
        - Signature:
            {activate/1,from/1,sel/2} / {0/0,cons/2,n__from/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,from,sel} 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))
          from#(X) -> c_3()
          from#(X) -> c_4()
          sel#(0(),cons(X,Y)) -> c_5()
          sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
        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))
            from#(X) -> c_3()
            from#(X) -> c_4()
            sel#(0(),cons(X,Y)) -> c_5()
            sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
        - Strict TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(X)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            sel(0(),cons(X,Y)) -> X
            sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
        - Signature:
            {activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0
            ,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,from#,sel#} 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))
          from#(X) -> c_3()
          from#(X) -> c_4()
          sel#(0(),cons(X,Y)) -> c_5()
          sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
* Step 3: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            activate#(n__from(X)) -> c_2(from#(X))
            from#(X) -> c_3()
            from#(X) -> c_4()
            sel#(0(),cons(X,Y)) -> c_5()
            sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
        - 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,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0
            ,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,from#,sel#} 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(sel#) = {2},
            uargs(c_2) = {1},
            uargs(c_6) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                    p(0) = [0]         
             p(activate) = [1] x1 + [7]
                 p(cons) = [1] x2 + [0]
                 p(from) = [1] x1 + [2]
              p(n__from) = [1] x1 + [0]
                    p(s) = [1] x1 + [0]
                  p(sel) = [0]         
            p(activate#) = [0]         
                p(from#) = [0]         
                 p(sel#) = [1] x2 + [0]
                  p(c_1) = [0]         
                  p(c_2) = [1] x1 + [0]
                  p(c_3) = [0]         
                  p(c_4) = [0]         
                  p(c_5) = [0]         
                  p(c_6) = [1] x1 + [0]
          
          Following rules are strictly oriented:
                   activate(X) = [1] X + [7]          
                               > [1] X + [0]          
                               = X                    
          
          activate(n__from(X)) = [1] X + [7]          
                               > [1] X + [2]          
                               = from(X)              
          
                       from(X) = [1] X + [2]          
                               > [1] X + [0]          
                               = cons(X,n__from(s(X)))
          
                       from(X) = [1] X + [2]          
                               > [1] X + [0]          
                               = n__from(X)           
          
          
          Following rules are (at-least) weakly oriented:
                   activate#(X) =  [0]                     
                                >= [0]                     
                                =  c_1()                   
          
          activate#(n__from(X)) =  [0]                     
                                >= [0]                     
                                =  c_2(from#(X))           
          
                       from#(X) =  [0]                     
                                >= [0]                     
                                =  c_3()                   
          
                       from#(X) =  [0]                     
                                >= [0]                     
                                =  c_4()                   
          
            sel#(0(),cons(X,Y)) =  [1] Y + [0]             
                                >= [0]                     
                                =  c_5()                   
          
           sel#(s(X),cons(Y,Z)) =  [1] Z + [0]             
                                >= [1] Z + [7]             
                                =  c_6(sel#(X,activate(Z)))
          
        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()
            activate#(n__from(X)) -> c_2(from#(X))
            from#(X) -> c_3()
            from#(X) -> c_4()
            sel#(0(),cons(X,Y)) -> c_5()
            sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
        - 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,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0
            ,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,from#,sel#} and constructors {0,cons,n__from,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3,4,5}
        by application of
          Pre({1,3,4,5}) = {2,6}.
        Here rules are labelled as follows:
          1: activate#(X) -> c_1()
          2: activate#(n__from(X)) -> c_2(from#(X))
          3: from#(X) -> c_3()
          4: from#(X) -> c_4()
          5: sel#(0(),cons(X,Y)) -> c_5()
          6: sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
* Step 5: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__from(X)) -> c_2(from#(X))
            sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
        - Weak DPs:
            activate#(X) -> c_1()
            from#(X) -> c_3()
            from#(X) -> c_4()
            sel#(0(),cons(X,Y)) -> c_5()
        - 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,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0
            ,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,from#,sel#} 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}
        by application of
          Pre({1}) = {}.
        Here rules are labelled as follows:
          1: activate#(n__from(X)) -> c_2(from#(X))
          2: sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
          3: activate#(X) -> c_1()
          4: from#(X) -> c_3()
          5: from#(X) -> c_4()
          6: sel#(0(),cons(X,Y)) -> c_5()
* Step 6: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
        - Weak DPs:
            activate#(X) -> c_1()
            activate#(n__from(X)) -> c_2(from#(X))
            from#(X) -> c_3()
            from#(X) -> c_4()
            sel#(0(),cons(X,Y)) -> c_5()
        - 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,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0
            ,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,from#,sel#} and constructors {0,cons,n__from,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
             -->_1 sel#(0(),cons(X,Y)) -> c_5():6
             -->_1 sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z))):1
          
          2:W:activate#(X) -> c_1()
             
          
          3:W:activate#(n__from(X)) -> c_2(from#(X))
             -->_1 from#(X) -> c_4():5
             -->_1 from#(X) -> c_3():4
          
          4:W:from#(X) -> c_3()
             
          
          5:W:from#(X) -> c_4()
             
          
          6:W:sel#(0(),cons(X,Y)) -> c_5()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: activate#(n__from(X)) -> c_2(from#(X))
          4: from#(X) -> c_3()
          5: from#(X) -> c_4()
          2: activate#(X) -> c_1()
          6: sel#(0(),cons(X,Y)) -> c_5()
* Step 7: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
        - 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,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0
            ,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,from#,sel#} 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: sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
          
        The strictly oriented rules are moved into the weak component.
** Step 7.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
        - 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,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0
            ,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,from#,sel#} 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_6) = {1}
        
        Following symbols are considered usable:
          {activate#,from#,sel#}
        TcT has computed the following interpretation:
                  p(0) = [0]         
           p(activate) = [5]         
               p(cons) = [3]         
               p(from) = [1]         
            p(n__from) = [1] x1 + [9]
                  p(s) = [1] x1 + [9]
                p(sel) = [0]         
          p(activate#) = [0]         
              p(from#) = [0]         
               p(sel#) = [1] x1 + [6]
                p(c_1) = [0]         
                p(c_2) = [0]         
                p(c_3) = [0]         
                p(c_4) = [2]         
                p(c_5) = [2]         
                p(c_6) = [1] x1 + [0]
        
        Following rules are strictly oriented:
        sel#(s(X),cons(Y,Z)) = [1] X + [15]            
                             > [1] X + [6]             
                             = c_6(sel#(X,activate(Z)))
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 7.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
        - 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,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0
            ,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,from#,sel#} and constructors {0,cons,n__from,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

** Step 7.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
        - 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,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0
            ,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,from#,sel#} and constructors {0,cons,n__from,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
             -->_1 sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
** Step 7.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,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0
            ,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,from#,sel#} 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))