/export/starexec/sandbox2/solver/bin/starexec_run_tct_rci /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            2nd(cons(X,X1)) -> 2nd(cons1(X,activate(X1)))
            2nd(cons1(X,cons(Y,Z))) -> Y
            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:
            {2nd/1,activate/1,from/1,s/1} / {cons/2,cons1/2,n__from/1,n__s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd,activate,from,s} and constructors {cons,cons1,n__from
            ,n__s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            2nd(cons(X,X1)) -> 2nd(cons1(X,activate(X1)))
            2nd(cons1(X,cons(Y,Z))) -> Y
            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:
            {2nd/1,activate/1,from/1,s/1} / {cons/2,cons1/2,n__from/1,n__s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd,activate,from,s} and constructors {cons,cons1,n__from
            ,n__s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            2nd(cons(X,X1)) -> 2nd(cons1(X,activate(X1)))
            2nd(cons1(X,cons(Y,Z))) -> Y
            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:
            {2nd/1,activate/1,from/1,s/1} / {cons/2,cons1/2,n__from/1,n__s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd,activate,from,s} and constructors {cons,cons1,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:
            2nd(cons(X,X1)) -> 2nd(cons1(X,activate(X1)))
            2nd(cons1(X,cons(Y,Z))) -> Y
            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:
            {2nd/1,activate/1,from/1,s/1} / {cons/2,cons1/2,n__from/1,n__s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd,activate,from,s} and constructors {cons,cons1,n__from
            ,n__s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))),activate#(X1))
          2nd#(cons1(X,cons(Y,Z))) -> c_2()
          activate#(X) -> c_3()
          activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X))
          activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X))
          from#(X) -> c_6()
          from#(X) -> c_7()
          s#(X) -> c_8()
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))),activate#(X1))
            2nd#(cons1(X,cons(Y,Z))) -> c_2()
            activate#(X) -> c_3()
            activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X))
            activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X))
            from#(X) -> c_6()
            from#(X) -> c_7()
            s#(X) -> c_8()
        - Weak TRS:
            2nd(cons(X,X1)) -> 2nd(cons1(X,activate(X1)))
            2nd(cons1(X,cons(Y,Z))) -> Y
            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:
            {2nd/1,activate/1,from/1,s/1,2nd#/1,activate#/1,from#/1,s#/1} / {cons/2,cons1/2,n__from/1,n__s/1,c_1/2,c_2/0
            ,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,s#} and constructors {cons,cons1
            ,n__from,n__s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,3,6,7,8}
        by application of
          Pre({2,3,6,7,8}) = {1,4,5}.
        Here rules are labelled as follows:
          1: 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))),activate#(X1))
          2: 2nd#(cons1(X,cons(Y,Z))) -> c_2()
          3: activate#(X) -> c_3()
          4: activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X))
          5: activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X))
          6: from#(X) -> c_6()
          7: from#(X) -> c_7()
          8: s#(X) -> c_8()
** Step 1.b:3: RemoveWeakSuffixes.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))),activate#(X1))
            activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X))
            activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X))
        - Weak DPs:
            2nd#(cons1(X,cons(Y,Z))) -> c_2()
            activate#(X) -> c_3()
            from#(X) -> c_6()
            from#(X) -> c_7()
            s#(X) -> c_8()
        - Weak TRS:
            2nd(cons(X,X1)) -> 2nd(cons1(X,activate(X1)))
            2nd(cons1(X,cons(Y,Z))) -> Y
            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:
            {2nd/1,activate/1,from/1,s/1,2nd#/1,activate#/1,from#/1,s#/1} / {cons/2,cons1/2,n__from/1,n__s/1,c_1/2,c_2/0
            ,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,s#} and constructors {cons,cons1
            ,n__from,n__s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))),activate#(X1))
             -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3
             -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):2
             -->_2 activate#(X) -> c_3():5
             -->_1 2nd#(cons1(X,cons(Y,Z))) -> c_2():4
          
          2:S:activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X))
             -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3
             -->_1 from#(X) -> c_7():7
             -->_1 from#(X) -> c_6():6
             -->_2 activate#(X) -> c_3():5
             -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):2
          
          3:S:activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X))
             -->_1 s#(X) -> c_8():8
             -->_2 activate#(X) -> c_3():5
             -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3
             -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):2
          
          4:W:2nd#(cons1(X,cons(Y,Z))) -> c_2()
             
          
          5:W:activate#(X) -> c_3()
             
          
          6:W:from#(X) -> c_6()
             
          
          7:W:from#(X) -> c_7()
             
          
          8:W:s#(X) -> c_8()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: 2nd#(cons1(X,cons(Y,Z))) -> c_2()
          6: from#(X) -> c_6()
          7: from#(X) -> c_7()
          5: activate#(X) -> c_3()
          8: s#(X) -> c_8()
** Step 1.b:4: SimplifyRHS.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))),activate#(X1))
            activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X))
            activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X))
        - Weak TRS:
            2nd(cons(X,X1)) -> 2nd(cons1(X,activate(X1)))
            2nd(cons1(X,cons(Y,Z))) -> Y
            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:
            {2nd/1,activate/1,from/1,s/1,2nd#/1,activate#/1,from#/1,s#/1} / {cons/2,cons1/2,n__from/1,n__s/1,c_1/2,c_2/0
            ,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,s#} and constructors {cons,cons1
            ,n__from,n__s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))),activate#(X1))
             -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3
             -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):2
          
          2:S:activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X))
             -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3
             -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):2
          
          3:S:activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X))
             -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3
             -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          2nd#(cons(X,X1)) -> c_1(activate#(X1))
          activate#(n__from(X)) -> c_4(activate#(X))
          activate#(n__s(X)) -> c_5(activate#(X))
** Step 1.b:5: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            2nd#(cons(X,X1)) -> c_1(activate#(X1))
            activate#(n__from(X)) -> c_4(activate#(X))
            activate#(n__s(X)) -> c_5(activate#(X))
        - Weak TRS:
            2nd(cons(X,X1)) -> 2nd(cons1(X,activate(X1)))
            2nd(cons1(X,cons(Y,Z))) -> Y
            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:
            {2nd/1,activate/1,from/1,s/1,2nd#/1,activate#/1,from#/1,s#/1} / {cons/2,cons1/2,n__from/1,n__s/1,c_1/1,c_2/0
            ,c_3/0,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,s#} and constructors {cons,cons1
            ,n__from,n__s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          2nd#(cons(X,X1)) -> c_1(activate#(X1))
          activate#(n__from(X)) -> c_4(activate#(X))
          activate#(n__s(X)) -> c_5(activate#(X))
** Step 1.b:6: RemoveHeads.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            2nd#(cons(X,X1)) -> c_1(activate#(X1))
            activate#(n__from(X)) -> c_4(activate#(X))
            activate#(n__s(X)) -> c_5(activate#(X))
        - Signature:
            {2nd/1,activate/1,from/1,s/1,2nd#/1,activate#/1,from#/1,s#/1} / {cons/2,cons1/2,n__from/1,n__s/1,c_1/1,c_2/0
            ,c_3/0,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,s#} and constructors {cons,cons1
            ,n__from,n__s}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:2nd#(cons(X,X1)) -> c_1(activate#(X1))
           -->_1 activate#(n__s(X)) -> c_5(activate#(X)):3
           -->_1 activate#(n__from(X)) -> c_4(activate#(X)):2
        
        2:S:activate#(n__from(X)) -> c_4(activate#(X))
           -->_1 activate#(n__s(X)) -> c_5(activate#(X)):3
           -->_1 activate#(n__from(X)) -> c_4(activate#(X)):2
        
        3:S:activate#(n__s(X)) -> c_5(activate#(X))
           -->_1 activate#(n__s(X)) -> c_5(activate#(X)):3
           -->_1 activate#(n__from(X)) -> c_4(activate#(X)):2
        
        
        Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
        
        [(1,2nd#(cons(X,X1)) -> c_1(activate#(X1)))]
** Step 1.b:7: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__from(X)) -> c_4(activate#(X))
            activate#(n__s(X)) -> c_5(activate#(X))
        - Signature:
            {2nd/1,activate/1,from/1,s/1,2nd#/1,activate#/1,from#/1,s#/1} / {cons/2,cons1/2,n__from/1,n__s/1,c_1/1,c_2/0
            ,c_3/0,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,s#} and constructors {cons,cons1
            ,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_4) = {1},
          uargs(c_5) = {1}
        
        Following symbols are considered usable:
          {2nd#,activate#,from#,s#}
        TcT has computed the following interpretation:
                p(2nd) = [1] x1 + [8]
           p(activate) = [1]         
               p(cons) = [1] x1 + [0]
              p(cons1) = [1]         
               p(from) = [1] x1 + [0]
            p(n__from) = [1] x1 + [0]
               p(n__s) = [1] x1 + [1]
                  p(s) = [1] x1 + [1]
               p(2nd#) = [1]         
          p(activate#) = [1] x1 + [0]
              p(from#) = [2] x1 + [1]
                 p(s#) = [0]         
                p(c_1) = [0]         
                p(c_2) = [1]         
                p(c_3) = [2]         
                p(c_4) = [1] x1 + [0]
                p(c_5) = [1] x1 + [0]
                p(c_6) = [2]         
                p(c_7) = [1]         
                p(c_8) = [8]         
        
        Following rules are strictly oriented:
        activate#(n__s(X)) = [1] X + [1]      
                           > [1] X + [0]      
                           = c_5(activate#(X))
        
        
        Following rules are (at-least) weakly oriented:
        activate#(n__from(X)) =  [1] X + [0]      
                              >= [1] X + [0]      
                              =  c_4(activate#(X))
        
** Step 1.b:8: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__from(X)) -> c_4(activate#(X))
        - Weak DPs:
            activate#(n__s(X)) -> c_5(activate#(X))
        - Signature:
            {2nd/1,activate/1,from/1,s/1,2nd#/1,activate#/1,from#/1,s#/1} / {cons/2,cons1/2,n__from/1,n__s/1,c_1/1,c_2/0
            ,c_3/0,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,s#} and constructors {cons,cons1
            ,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_4) = {1},
          uargs(c_5) = {1}
        
        Following symbols are considered usable:
          {2nd#,activate#,from#,s#}
        TcT has computed the following interpretation:
                p(2nd) = [1] x1 + [0] 
           p(activate) = [1] x1 + [1] 
               p(cons) = [1] x2 + [1] 
              p(cons1) = [2]          
               p(from) = [2] x1 + [2] 
            p(n__from) = [1] x1 + [12]
               p(n__s) = [1] x1 + [9] 
                  p(s) = [1] x1 + [2] 
               p(2nd#) = [1]          
          p(activate#) = [1] x1 + [5] 
              p(from#) = [8] x1 + [1] 
                 p(s#) = [8] x1 + [0] 
                p(c_1) = [1] x1 + [1] 
                p(c_2) = [2]          
                p(c_3) = [0]          
                p(c_4) = [1] x1 + [0] 
                p(c_5) = [1] x1 + [3] 
                p(c_6) = [2]          
                p(c_7) = [1]          
                p(c_8) = [1]          
        
        Following rules are strictly oriented:
        activate#(n__from(X)) = [1] X + [17]     
                              > [1] X + [5]      
                              = c_4(activate#(X))
        
        
        Following rules are (at-least) weakly oriented:
        activate#(n__s(X)) =  [1] X + [14]     
                           >= [1] X + [8]      
                           =  c_5(activate#(X))
        
** Step 1.b:9: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            activate#(n__from(X)) -> c_4(activate#(X))
            activate#(n__s(X)) -> c_5(activate#(X))
        - Signature:
            {2nd/1,activate/1,from/1,s/1,2nd#/1,activate#/1,from#/1,s#/1} / {cons/2,cons1/2,n__from/1,n__s/1,c_1/1,c_2/0
            ,c_3/0,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,s#} and constructors {cons,cons1
            ,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))