WORST_CASE(?,O(n^1))
* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__first(X1,X2)) -> first(X1,X2)
            activate(n__from(X)) -> from(X)
            first(X1,X2) -> n__first(X1,X2)
            first(0(),X) -> nil()
            first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
        - Signature:
            {activate/1,first/2,from/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,first,from} and constructors {0,cons,n__first
            ,n__from,nil,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          activate#(X) -> c_1()
          activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
          activate#(n__from(X)) -> c_3(from#(X))
          first#(X1,X2) -> c_4()
          first#(0(),X) -> c_5()
          first#(s(X),cons(Y,Z)) -> c_6(activate#(Z))
          from#(X) -> c_7()
          from#(X) -> c_8()
        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__first(X1,X2)) -> c_2(first#(X1,X2))
            activate#(n__from(X)) -> c_3(from#(X))
            first#(X1,X2) -> c_4()
            first#(0(),X) -> c_5()
            first#(s(X),cons(Y,Z)) -> c_6(activate#(Z))
            from#(X) -> c_7()
            from#(X) -> c_8()
        - Strict TRS:
            activate(X) -> X
            activate(n__first(X1,X2)) -> first(X1,X2)
            activate(n__from(X)) -> from(X)
            first(X1,X2) -> n__first(X1,X2)
            first(0(),X) -> nil()
            first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
        - Signature:
            {activate/1,first/2,from/1,activate#/1,first#/2,from#/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1,c_1/0
            ,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,first#,from#} and constructors {0,cons,n__first
            ,n__from,nil,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          activate#(X) -> c_1()
          activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
          activate#(n__from(X)) -> c_3(from#(X))
          first#(X1,X2) -> c_4()
          first#(0(),X) -> c_5()
          first#(s(X),cons(Y,Z)) -> c_6(activate#(Z))
          from#(X) -> c_7()
          from#(X) -> c_8()
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
            activate#(n__from(X)) -> c_3(from#(X))
            first#(X1,X2) -> c_4()
            first#(0(),X) -> c_5()
            first#(s(X),cons(Y,Z)) -> c_6(activate#(Z))
            from#(X) -> c_7()
            from#(X) -> c_8()
        - Signature:
            {activate/1,first/2,from/1,activate#/1,first#/2,from#/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1,c_1/0
            ,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,first#,from#} and constructors {0,cons,n__first
            ,n__from,nil,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,4,5,7,8}
        by application of
          Pre({1,4,5,7,8}) = {2,3,6}.
        Here rules are labelled as follows:
          1: activate#(X) -> c_1()
          2: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
          3: activate#(n__from(X)) -> c_3(from#(X))
          4: first#(X1,X2) -> c_4()
          5: first#(0(),X) -> c_5()
          6: first#(s(X),cons(Y,Z)) -> c_6(activate#(Z))
          7: from#(X) -> c_7()
          8: from#(X) -> c_8()
* Step 4: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
            activate#(n__from(X)) -> c_3(from#(X))
            first#(s(X),cons(Y,Z)) -> c_6(activate#(Z))
        - Weak DPs:
            activate#(X) -> c_1()
            first#(X1,X2) -> c_4()
            first#(0(),X) -> c_5()
            from#(X) -> c_7()
            from#(X) -> c_8()
        - Signature:
            {activate/1,first/2,from/1,activate#/1,first#/2,from#/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1,c_1/0
            ,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,first#,from#} and constructors {0,cons,n__first
            ,n__from,nil,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2}
        by application of
          Pre({2}) = {3}.
        Here rules are labelled as follows:
          1: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
          2: activate#(n__from(X)) -> c_3(from#(X))
          3: first#(s(X),cons(Y,Z)) -> c_6(activate#(Z))
          4: activate#(X) -> c_1()
          5: first#(X1,X2) -> c_4()
          6: first#(0(),X) -> c_5()
          7: from#(X) -> c_7()
          8: from#(X) -> c_8()
* Step 5: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
            first#(s(X),cons(Y,Z)) -> c_6(activate#(Z))
        - Weak DPs:
            activate#(X) -> c_1()
            activate#(n__from(X)) -> c_3(from#(X))
            first#(X1,X2) -> c_4()
            first#(0(),X) -> c_5()
            from#(X) -> c_7()
            from#(X) -> c_8()
        - Signature:
            {activate/1,first/2,from/1,activate#/1,first#/2,from#/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1,c_1/0
            ,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,first#,from#} and constructors {0,cons,n__first
            ,n__from,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
             -->_1 first#(s(X),cons(Y,Z)) -> c_6(activate#(Z)):2
             -->_1 first#(0(),X) -> c_5():6
             -->_1 first#(X1,X2) -> c_4():5
          
          2:S:first#(s(X),cons(Y,Z)) -> c_6(activate#(Z))
             -->_1 activate#(n__from(X)) -> c_3(from#(X)):4
             -->_1 activate#(X) -> c_1():3
             -->_1 activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)):1
          
          3:W:activate#(X) -> c_1()
             
          
          4:W:activate#(n__from(X)) -> c_3(from#(X))
             -->_1 from#(X) -> c_8():8
             -->_1 from#(X) -> c_7():7
          
          5:W:first#(X1,X2) -> c_4()
             
          
          6:W:first#(0(),X) -> c_5()
             
          
          7:W:from#(X) -> c_7()
             
          
          8:W:from#(X) -> c_8()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: first#(X1,X2) -> c_4()
          6: first#(0(),X) -> c_5()
          3: activate#(X) -> c_1()
          4: activate#(n__from(X)) -> c_3(from#(X))
          7: from#(X) -> c_7()
          8: from#(X) -> c_8()
* Step 6: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
            first#(s(X),cons(Y,Z)) -> c_6(activate#(Z))
        - Signature:
            {activate/1,first/2,from/1,activate#/1,first#/2,from#/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1,c_1/0
            ,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,first#,from#} and constructors {0,cons,n__first
            ,n__from,nil,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:
          2: first#(s(X),cons(Y,Z)) -> c_6(activate#(Z))
          
        Consider the set of all dependency pairs
          1: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
          2: first#(s(X),cons(Y,Z)) -> c_6(activate#(Z))
        Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1))
        SPACE(?,?)on application of the dependency pairs
          {2}
        These cover all (indirect) predecessors of dependency pairs
          {1,2}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
** Step 6.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
            first#(s(X),cons(Y,Z)) -> c_6(activate#(Z))
        - Signature:
            {activate/1,first/2,from/1,activate#/1,first#/2,from#/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1,c_1/0
            ,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,first#,from#} and constructors {0,cons,n__first
            ,n__from,nil,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_2) = {1},
          uargs(c_6) = {1}
        
        Following symbols are considered usable:
          {activate#,first#,from#}
        TcT has computed the following interpretation:
                  p(0) = [1]          
           p(activate) = [0]          
               p(cons) = [1] x2 + [2] 
              p(first) = [8]          
               p(from) = [1] x1 + [1] 
           p(n__first) = [1] x2 + [0] 
            p(n__from) = [0]          
                p(nil) = [0]          
                  p(s) = [1] x1 + [1] 
          p(activate#) = [8] x1 + [0] 
             p(first#) = [8] x2 + [0] 
              p(from#) = [1] x1 + [2] 
                p(c_1) = [1]          
                p(c_2) = [1] x1 + [0] 
                p(c_3) = [2] x1 + [1] 
                p(c_4) = [0]          
                p(c_5) = [2]          
                p(c_6) = [1] x1 + [14]
                p(c_7) = [1]          
                p(c_8) = [8]          
        
        Following rules are strictly oriented:
        first#(s(X),cons(Y,Z)) = [8] Z + [16]     
                               > [8] Z + [14]     
                               = c_6(activate#(Z))
        
        
        Following rules are (at-least) weakly oriented:
        activate#(n__first(X1,X2)) =  [8] X2 + [0]      
                                   >= [8] X2 + [0]      
                                   =  c_2(first#(X1,X2))
        
** Step 6.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
        - Weak DPs:
            first#(s(X),cons(Y,Z)) -> c_6(activate#(Z))
        - Signature:
            {activate/1,first/2,from/1,activate#/1,first#/2,from#/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1,c_1/0
            ,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,first#,from#} and constructors {0,cons,n__first
            ,n__from,nil,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
            first#(s(X),cons(Y,Z)) -> c_6(activate#(Z))
        - Signature:
            {activate/1,first/2,from/1,activate#/1,first#/2,from#/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1,c_1/0
            ,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,first#,from#} and constructors {0,cons,n__first
            ,n__from,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
             -->_1 first#(s(X),cons(Y,Z)) -> c_6(activate#(Z)):2
          
          2:W:first#(s(X),cons(Y,Z)) -> c_6(activate#(Z))
             -->_1 activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
          2: first#(s(X),cons(Y,Z)) -> c_6(activate#(Z))
** Step 6.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {activate/1,first/2,from/1,activate#/1,first#/2,from#/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1,c_1/0
            ,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,first#,from#} and constructors {0,cons,n__first
            ,n__from,nil,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))