WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum WORST_CASE(Omega(n^1),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:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(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:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          activate(x){x -> n__from(x)} =
            activate(n__from(x)) ->^+ from(activate(x))
              = C[activate(x) = activate(x){}]

** Step 1.b:1: 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
            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:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          activate#(X) -> c_1()
          activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X))
          activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X))
          from#(X) -> c_4()
          from#(X) -> c_5()
          s#(X) -> c_6()
          sel#(0(),cons(X,Y)) -> c_7()
          sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X))
            activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X))
            from#(X) -> c_4()
            from#(X) -> c_5()
            s#(X) -> c_6()
            sel#(0(),cons(X,Y)) -> c_7()
            sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z))
        - 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)
            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,activate#/1,from#/1,s#/1,sel#/2} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/2
            ,c_3/2,c_4/0,c_5/0,c_6/0,c_7/0,c_8/2}
        - 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
          {1,4,5,6,7,8}
        by application of
          Pre({1,4,5,6,7,8}) = {2,3}.
        Here rules are labelled as follows:
          1: activate#(X) -> c_1()
          2: activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X))
          3: activate#(n__s(X)) -> c_3(s#(activate(X)),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()
          8: sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z))
** Step 1.b:3: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X))
            activate#(n__s(X)) -> c_3(s#(activate(X)),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()
            sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z))
        - 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)
            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,activate#/1,from#/1,s#/1,sel#/2} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/2
            ,c_3/2,c_4/0,c_5/0,c_6/0,c_7/0,c_8/2}
        - 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:S:activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X))
             -->_2 activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)):2
             -->_1 from#(X) -> c_5():5
             -->_1 from#(X) -> c_4():4
             -->_2 activate#(X) -> c_1():3
             -->_2 activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X)):1
          
          2:S:activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X))
             -->_1 s#(X) -> c_6():6
             -->_2 activate#(X) -> c_1():3
             -->_2 activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)):2
             -->_2 activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X)):1
          
          3:W:activate#(X) -> c_1()
             
          
          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()
             
          
          8:W:sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z))
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          8: sel#(s(X),cons(Y,Z)) -> c_8(sel#(X,activate(Z)),activate#(Z))
          7: sel#(0(),cons(X,Y)) -> c_7()
          4: from#(X) -> c_4()
          5: from#(X) -> c_5()
          3: activate#(X) -> c_1()
          6: s#(X) -> c_6()
** Step 1.b:4: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X))
            activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X))
        - 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)
            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,activate#/1,from#/1,s#/1,sel#/2} / {0/0,cons/2,n__from/1,n__s/1,c_1/0,c_2/2
            ,c_3/2,c_4/0,c_5/0,c_6/0,c_7/0,c_8/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,from#,s#,sel#} and constructors {0,cons,n__from
            ,n__s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X))
             -->_2 activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)):2
             -->_2 activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X)):1
          
          2:S:activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X))
             -->_2 activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)):2
             -->_2 activate#(n__from(X)) -> c_2(from#(activate(X)),activate#(X)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          activate#(n__from(X)) -> c_2(activate#(X))
          activate#(n__s(X)) -> c_3(activate#(X))
** Step 1.b:5: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__from(X)) -> c_2(activate#(X))
            activate#(n__s(X)) -> c_3(activate#(X))
        - 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)
            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,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,c_8/2}
        - 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#(n__from(X)) -> c_2(activate#(X))
          activate#(n__s(X)) -> c_3(activate#(X))
** Step 1.b:6: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__from(X)) -> c_2(activate#(X))
            activate#(n__s(X)) -> c_3(activate#(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,c_8/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,from#,s#,sel#} and constructors {0,cons,n__from
            ,n__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(c_2) = {1},
          uargs(c_3) = {1}
        
        Following symbols are considered usable:
          {activate#,from#,s#,sel#}
        TcT has computed the following interpretation:
                  p(0) = [0]                  
           p(activate) = [1] x1 + [0]         
               p(cons) = [1] x2 + [0]         
               p(from) = [2] x1 + [1]         
            p(n__from) = [1] x1 + [4]         
               p(n__s) = [1] x1 + [4]         
                  p(s) = [1] x1 + [1]         
                p(sel) = [2] x1 + [2] x2 + [1]
          p(activate#) = [4] x1 + [0]         
              p(from#) = [0]                  
                 p(s#) = [0]                  
               p(sel#) = [2] x1 + [2] x2 + [1]
                p(c_1) = [1]                  
                p(c_2) = [1] x1 + [8]         
                p(c_3) = [1] x1 + [15]        
                p(c_4) = [1]                  
                p(c_5) = [4]                  
                p(c_6) = [1]                  
                p(c_7) = [1]                  
                p(c_8) = [1]                  
        
        Following rules are strictly oriented:
        activate#(n__from(X)) = [4] X + [16]     
                              > [4] X + [8]      
                              = c_2(activate#(X))
        
           activate#(n__s(X)) = [4] X + [16]     
                              > [4] X + [15]     
                              = c_3(activate#(X))
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 1.b:7: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            activate#(n__from(X)) -> c_2(activate#(X))
            activate#(n__s(X)) -> c_3(activate#(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,c_8/2}
        - 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(Omega(n^1),O(n^1))