'Weak Dependency Graph [60.0]'
------------------------------
Answer:           YES(?,O(n^1))
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  f(f(a())) -> f(g(n__f(n__a())))
     , f(X) -> n__f(X)
     , a() -> n__a()
     , activate(n__f(X)) -> f(activate(X))
     , activate(n__a()) -> a()
     , activate(X) -> X}

Details:         
  We have computed the following set of weak (innermost) dependency pairs:
   {  f^#(f(a())) -> c_0(f^#(g(n__f(n__a()))))
    , f^#(X) -> c_1()
    , a^#() -> c_2()
    , activate^#(n__f(X)) -> c_3(f^#(activate(X)))
    , activate^#(n__a()) -> c_4(a^#())
    , activate^#(X) -> c_5()}
  
  The usable rules are:
   {  activate(n__f(X)) -> f(activate(X))
    , activate(n__a()) -> a()
    , activate(X) -> X
    , f(f(a())) -> f(g(n__f(n__a())))
    , f(X) -> n__f(X)
    , a() -> n__a()}
  
  The estimated dependency graph contains the following edges:
   {f^#(f(a())) -> c_0(f^#(g(n__f(n__a()))))}
     ==> {f^#(X) -> c_1()}
   {activate^#(n__f(X)) -> c_3(f^#(activate(X)))}
     ==> {f^#(X) -> c_1()}
   {activate^#(n__f(X)) -> c_3(f^#(activate(X)))}
     ==> {f^#(f(a())) -> c_0(f^#(g(n__f(n__a()))))}
   {activate^#(n__a()) -> c_4(a^#())}
     ==> {a^#() -> c_2()}
  
  We consider the following path(s):
   1) {  activate^#(n__f(X)) -> c_3(f^#(activate(X)))
       , f^#(f(a())) -> c_0(f^#(g(n__f(n__a()))))}
      
      The usable rules for this path are the following:
      {  activate(n__f(X)) -> f(activate(X))
       , activate(n__a()) -> a()
       , activate(X) -> X
       , f(f(a())) -> f(g(n__f(n__a())))
       , f(X) -> n__f(X)
       , a() -> n__a()}
      
        We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
        
          'Weight Gap Principle'
          ----------------------
          Answer:           YES(?,O(n^1))
          Input Problem:    innermost runtime-complexity with respect to
            Rules:
              {  activate(n__f(X)) -> f(activate(X))
               , activate(n__a()) -> a()
               , activate(X) -> X
               , f(f(a())) -> f(g(n__f(n__a())))
               , f(X) -> n__f(X)
               , a() -> n__a()
               , activate^#(n__f(X)) -> c_3(f^#(activate(X)))
               , f^#(f(a())) -> c_0(f^#(g(n__f(n__a()))))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {  activate(n__a()) -> a()
             , activate(X) -> X}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  activate(n__a()) -> a()
               , activate(X) -> X}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  a() = [0]
                  g(x1) = [1] x1 + [0]
                  n__f(x1) = [1] x1 + [0]
                  n__a() = [0]
                  activate(x1) = [1] x1 + [1]
                  f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [1]
                  c_1() = [0]
                  a^#() = [0]
                  c_2() = [0]
                  activate^#(x1) = [1] x1 + [1]
                  c_3(x1) = [1] x1 + [7]
                  c_4(x1) = [0] x1 + [0]
                  c_5() = [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {activate^#(n__f(X)) -> c_3(f^#(activate(X)))}
            and weakly orienting the rules
            {  activate(n__a()) -> a()
             , activate(X) -> X}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {activate^#(n__f(X)) -> c_3(f^#(activate(X)))}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  a() = [0]
                  g(x1) = [1] x1 + [0]
                  n__f(x1) = [1] x1 + [0]
                  n__a() = [0]
                  activate(x1) = [1] x1 + [1]
                  f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [8]
                  c_1() = [0]
                  a^#() = [0]
                  c_2() = [0]
                  activate^#(x1) = [1] x1 + [9]
                  c_3(x1) = [1] x1 + [0]
                  c_4(x1) = [0] x1 + [0]
                  c_5() = [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {  f(f(a())) -> f(g(n__f(n__a())))
             , a() -> n__a()}
            and weakly orienting the rules
            {  activate^#(n__f(X)) -> c_3(f^#(activate(X)))
             , activate(n__a()) -> a()
             , activate(X) -> X}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  f(f(a())) -> f(g(n__f(n__a())))
               , a() -> n__a()}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  a() = [8]
                  g(x1) = [1] x1 + [0]
                  n__f(x1) = [1] x1 + [0]
                  n__a() = [7]
                  activate(x1) = [1] x1 + [1]
                  f^#(x1) = [1] x1 + [1]
                  c_0(x1) = [1] x1 + [8]
                  c_1() = [0]
                  a^#() = [0]
                  c_2() = [0]
                  activate^#(x1) = [1] x1 + [5]
                  c_3(x1) = [1] x1 + [1]
                  c_4(x1) = [0] x1 + [0]
                  c_5() = [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {f^#(f(a())) -> c_0(f^#(g(n__f(n__a()))))}
            and weakly orienting the rules
            {  f(f(a())) -> f(g(n__f(n__a())))
             , a() -> n__a()
             , activate^#(n__f(X)) -> c_3(f^#(activate(X)))
             , activate(n__a()) -> a()
             , activate(X) -> X}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {f^#(f(a())) -> c_0(f^#(g(n__f(n__a()))))}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  a() = [8]
                  g(x1) = [1] x1 + [0]
                  n__f(x1) = [1] x1 + [0]
                  n__a() = [7]
                  activate(x1) = [1] x1 + [1]
                  f^#(x1) = [1] x1 + [10]
                  c_0(x1) = [1] x1 + [0]
                  c_1() = [0]
                  a^#() = [0]
                  c_2() = [0]
                  activate^#(x1) = [1] x1 + [15]
                  c_3(x1) = [1] x1 + [1]
                  c_4(x1) = [0] x1 + [0]
                  c_5() = [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {f(X) -> n__f(X)}
            and weakly orienting the rules
            {  f^#(f(a())) -> c_0(f^#(g(n__f(n__a()))))
             , f(f(a())) -> f(g(n__f(n__a())))
             , a() -> n__a()
             , activate^#(n__f(X)) -> c_3(f^#(activate(X)))
             , activate(n__a()) -> a()
             , activate(X) -> X}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {f(X) -> n__f(X)}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [8]
                  a() = [0]
                  g(x1) = [1] x1 + [0]
                  n__f(x1) = [1] x1 + [0]
                  n__a() = [0]
                  activate(x1) = [1] x1 + [1]
                  f^#(x1) = [1] x1 + [1]
                  c_0(x1) = [1] x1 + [0]
                  c_1() = [0]
                  a^#() = [0]
                  c_2() = [0]
                  activate^#(x1) = [1] x1 + [9]
                  c_3(x1) = [1] x1 + [1]
                  c_4(x1) = [0] x1 + [0]
                  c_5() = [0]
              
            Finally we apply the subprocessor
            'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
            ------------------------------------------------------------------------------------------
            Answer:           YES(?,O(n^1))
            Input Problem:    innermost relative runtime-complexity with respect to
              Strict Rules: {activate(n__f(X)) -> f(activate(X))}
              Weak Rules:
                {  f(X) -> n__f(X)
                 , f^#(f(a())) -> c_0(f^#(g(n__f(n__a()))))
                 , f(f(a())) -> f(g(n__f(n__a())))
                 , a() -> n__a()
                 , activate^#(n__f(X)) -> c_3(f^#(activate(X)))
                 , activate(n__a()) -> a()
                 , activate(X) -> X}
            
            Details:         
              The problem was solved by processor 'Bounds with default enrichment':
              'Bounds with default enrichment'
              --------------------------------
              Answer:           YES(?,O(n^1))
              Input Problem:    innermost relative runtime-complexity with respect to
                Strict Rules: {activate(n__f(X)) -> f(activate(X))}
                Weak Rules:
                  {  f(X) -> n__f(X)
                   , f^#(f(a())) -> c_0(f^#(g(n__f(n__a()))))
                   , f(f(a())) -> f(g(n__f(n__a())))
                   , a() -> n__a()
                   , activate^#(n__f(X)) -> c_3(f^#(activate(X)))
                   , activate(n__a()) -> a()
                   , activate(X) -> X}
              
              Details:         
                The problem is Match-bounded by 1.
                The enriched problem is compatible with the following automaton:
                {  f_1(5) -> 4
                 , f_1(5) -> 5
                 , f_1(8) -> 4
                 , f_1(8) -> 5
                 , a_0() -> 4
                 , a_1() -> 5
                 , g_0(2) -> 2
                 , g_0(2) -> 4
                 , g_0(2) -> 5
                 , g_1(9) -> 8
                 , n__f_0(2) -> 2
                 , n__f_0(2) -> 4
                 , n__f_0(2) -> 5
                 , n__f_1(5) -> 4
                 , n__f_1(5) -> 5
                 , n__f_1(8) -> 4
                 , n__f_1(8) -> 5
                 , n__f_1(10) -> 9
                 , n__a_0() -> 2
                 , n__a_0() -> 4
                 , n__a_0() -> 5
                 , n__a_1() -> 5
                 , n__a_1() -> 10
                 , activate_0(2) -> 4
                 , activate_1(2) -> 5
                 , f^#_0(2) -> 1
                 , f^#_0(4) -> 3
                 , f^#_1(5) -> 6
                 , f^#_1(8) -> 7
                 , c_0_1(7) -> 3
                 , c_0_1(7) -> 6
                 , activate^#_0(2) -> 1
                 , c_3_0(3) -> 1
                 , c_3_1(6) -> 1}
      
   2) {  activate^#(n__f(X)) -> c_3(f^#(activate(X)))
       , f^#(f(a())) -> c_0(f^#(g(n__f(n__a()))))
       , f^#(X) -> c_1()}
      
      The usable rules for this path are the following:
      {  activate(n__f(X)) -> f(activate(X))
       , activate(n__a()) -> a()
       , activate(X) -> X
       , f(f(a())) -> f(g(n__f(n__a())))
       , f(X) -> n__f(X)
       , a() -> n__a()}
      
        We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
        
          'Weight Gap Principle'
          ----------------------
          Answer:           YES(?,O(n^1))
          Input Problem:    innermost runtime-complexity with respect to
            Rules:
              {  activate(n__f(X)) -> f(activate(X))
               , activate(n__a()) -> a()
               , activate(X) -> X
               , f(f(a())) -> f(g(n__f(n__a())))
               , f(X) -> n__f(X)
               , a() -> n__a()
               , f^#(f(a())) -> c_0(f^#(g(n__f(n__a()))))
               , activate^#(n__f(X)) -> c_3(f^#(activate(X)))
               , f^#(X) -> c_1()}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {  activate(n__a()) -> a()
             , activate(X) -> X
             , f^#(X) -> c_1()}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  activate(n__a()) -> a()
               , activate(X) -> X
               , f^#(X) -> c_1()}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  a() = [0]
                  g(x1) = [1] x1 + [0]
                  n__f(x1) = [1] x1 + [0]
                  n__a() = [0]
                  activate(x1) = [1] x1 + [1]
                  f^#(x1) = [1] x1 + [1]
                  c_0(x1) = [1] x1 + [0]
                  c_1() = [0]
                  a^#() = [0]
                  c_2() = [0]
                  activate^#(x1) = [1] x1 + [1]
                  c_3(x1) = [1] x1 + [1]
                  c_4(x1) = [0] x1 + [0]
                  c_5() = [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {activate^#(n__f(X)) -> c_3(f^#(activate(X)))}
            and weakly orienting the rules
            {  activate(n__a()) -> a()
             , activate(X) -> X
             , f^#(X) -> c_1()}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {activate^#(n__f(X)) -> c_3(f^#(activate(X)))}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  a() = [0]
                  g(x1) = [1] x1 + [0]
                  n__f(x1) = [1] x1 + [0]
                  n__a() = [0]
                  activate(x1) = [1] x1 + [1]
                  f^#(x1) = [1] x1 + [1]
                  c_0(x1) = [1] x1 + [0]
                  c_1() = [0]
                  a^#() = [0]
                  c_2() = [0]
                  activate^#(x1) = [1] x1 + [9]
                  c_3(x1) = [1] x1 + [1]
                  c_4(x1) = [0] x1 + [0]
                  c_5() = [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {a() -> n__a()}
            and weakly orienting the rules
            {  activate^#(n__f(X)) -> c_3(f^#(activate(X)))
             , activate(n__a()) -> a()
             , activate(X) -> X
             , f^#(X) -> c_1()}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {a() -> n__a()}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  a() = [8]
                  g(x1) = [1] x1 + [2]
                  n__f(x1) = [1] x1 + [0]
                  n__a() = [7]
                  activate(x1) = [1] x1 + [1]
                  f^#(x1) = [1] x1 + [1]
                  c_0(x1) = [1] x1 + [1]
                  c_1() = [0]
                  a^#() = [0]
                  c_2() = [0]
                  activate^#(x1) = [1] x1 + [8]
                  c_3(x1) = [1] x1 + [1]
                  c_4(x1) = [0] x1 + [0]
                  c_5() = [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {  f(f(a())) -> f(g(n__f(n__a())))
             , f^#(f(a())) -> c_0(f^#(g(n__f(n__a()))))}
            and weakly orienting the rules
            {  a() -> n__a()
             , activate^#(n__f(X)) -> c_3(f^#(activate(X)))
             , activate(n__a()) -> a()
             , activate(X) -> X
             , f^#(X) -> c_1()}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  f(f(a())) -> f(g(n__f(n__a())))
               , f^#(f(a())) -> c_0(f^#(g(n__f(n__a()))))}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  a() = [8]
                  g(x1) = [1] x1 + [0]
                  n__f(x1) = [1] x1 + [0]
                  n__a() = [7]
                  activate(x1) = [1] x1 + [1]
                  f^#(x1) = [1] x1 + [4]
                  c_0(x1) = [1] x1 + [0]
                  c_1() = [0]
                  a^#() = [0]
                  c_2() = [0]
                  activate^#(x1) = [1] x1 + [13]
                  c_3(x1) = [1] x1 + [8]
                  c_4(x1) = [0] x1 + [0]
                  c_5() = [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {f(X) -> n__f(X)}
            and weakly orienting the rules
            {  f(f(a())) -> f(g(n__f(n__a())))
             , f^#(f(a())) -> c_0(f^#(g(n__f(n__a()))))
             , a() -> n__a()
             , activate^#(n__f(X)) -> c_3(f^#(activate(X)))
             , activate(n__a()) -> a()
             , activate(X) -> X
             , f^#(X) -> c_1()}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {f(X) -> n__f(X)}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [8]
                  a() = [0]
                  g(x1) = [1] x1 + [8]
                  n__f(x1) = [1] x1 + [0]
                  n__a() = [0]
                  activate(x1) = [1] x1 + [1]
                  f^#(x1) = [1] x1 + [1]
                  c_0(x1) = [1] x1 + [0]
                  c_1() = [0]
                  a^#() = [0]
                  c_2() = [0]
                  activate^#(x1) = [1] x1 + [5]
                  c_3(x1) = [1] x1 + [1]
                  c_4(x1) = [0] x1 + [0]
                  c_5() = [0]
              
            Finally we apply the subprocessor
            'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
            ------------------------------------------------------------------------------------------
            Answer:           YES(?,O(n^1))
            Input Problem:    innermost relative runtime-complexity with respect to
              Strict Rules: {activate(n__f(X)) -> f(activate(X))}
              Weak Rules:
                {  f(X) -> n__f(X)
                 , f(f(a())) -> f(g(n__f(n__a())))
                 , f^#(f(a())) -> c_0(f^#(g(n__f(n__a()))))
                 , a() -> n__a()
                 , activate^#(n__f(X)) -> c_3(f^#(activate(X)))
                 , activate(n__a()) -> a()
                 , activate(X) -> X
                 , f^#(X) -> c_1()}
            
            Details:         
              The problem was solved by processor 'Bounds with default enrichment':
              'Bounds with default enrichment'
              --------------------------------
              Answer:           YES(?,O(n^1))
              Input Problem:    innermost relative runtime-complexity with respect to
                Strict Rules: {activate(n__f(X)) -> f(activate(X))}
                Weak Rules:
                  {  f(X) -> n__f(X)
                   , f(f(a())) -> f(g(n__f(n__a())))
                   , f^#(f(a())) -> c_0(f^#(g(n__f(n__a()))))
                   , a() -> n__a()
                   , activate^#(n__f(X)) -> c_3(f^#(activate(X)))
                   , activate(n__a()) -> a()
                   , activate(X) -> X
                   , f^#(X) -> c_1()}
              
              Details:         
                The problem is Match-bounded by 1.
                The enriched problem is compatible with the following automaton:
                {  f_1(5) -> 4
                 , f_1(5) -> 5
                 , a_0() -> 4
                 , a_1() -> 5
                 , g_0(2) -> 2
                 , g_0(2) -> 4
                 , g_0(2) -> 5
                 , g_1(7) -> 5
                 , n__f_0(2) -> 2
                 , n__f_0(2) -> 4
                 , n__f_0(2) -> 5
                 , n__f_1(5) -> 4
                 , n__f_1(5) -> 5
                 , n__f_1(8) -> 7
                 , n__a_0() -> 2
                 , n__a_0() -> 4
                 , n__a_0() -> 5
                 , n__a_1() -> 5
                 , n__a_1() -> 8
                 , activate_0(2) -> 4
                 , activate_1(2) -> 5
                 , f^#_0(2) -> 1
                 , f^#_0(4) -> 3
                 , f^#_1(5) -> 6
                 , c_0_1(6) -> 3
                 , c_0_1(6) -> 6
                 , c_1_0() -> 1
                 , c_1_0() -> 3
                 , c_1_1() -> 6
                 , activate^#_0(2) -> 1
                 , c_3_0(3) -> 1
                 , c_3_1(6) -> 1}
      
   3) {  activate^#(n__f(X)) -> c_3(f^#(activate(X)))
       , f^#(X) -> c_1()}
      
      The usable rules for this path are the following:
      {  activate(n__f(X)) -> f(activate(X))
       , activate(n__a()) -> a()
       , activate(X) -> X
       , f(f(a())) -> f(g(n__f(n__a())))
       , f(X) -> n__f(X)
       , a() -> n__a()}
      
        We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
        
          'Weight Gap Principle'
          ----------------------
          Answer:           YES(?,O(n^1))
          Input Problem:    innermost runtime-complexity with respect to
            Rules:
              {  activate(n__f(X)) -> f(activate(X))
               , activate(n__a()) -> a()
               , activate(X) -> X
               , f(f(a())) -> f(g(n__f(n__a())))
               , f(X) -> n__f(X)
               , a() -> n__a()
               , activate^#(n__f(X)) -> c_3(f^#(activate(X)))
               , f^#(X) -> c_1()}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {  activate(n__a()) -> a()
             , activate(X) -> X}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  activate(n__a()) -> a()
               , activate(X) -> X}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  a() = [0]
                  g(x1) = [1] x1 + [0]
                  n__f(x1) = [1] x1 + [0]
                  n__a() = [0]
                  activate(x1) = [1] x1 + [1]
                  f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  c_1() = [0]
                  a^#() = [0]
                  c_2() = [0]
                  activate^#(x1) = [1] x1 + [1]
                  c_3(x1) = [1] x1 + [3]
                  c_4(x1) = [0] x1 + [0]
                  c_5() = [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {f^#(X) -> c_1()}
            and weakly orienting the rules
            {  activate(n__a()) -> a()
             , activate(X) -> X}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {f^#(X) -> c_1()}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  a() = [0]
                  g(x1) = [1] x1 + [0]
                  n__f(x1) = [1] x1 + [0]
                  n__a() = [0]
                  activate(x1) = [1] x1 + [1]
                  f^#(x1) = [1] x1 + [8]
                  c_0(x1) = [0] x1 + [0]
                  c_1() = [0]
                  a^#() = [0]
                  c_2() = [0]
                  activate^#(x1) = [1] x1 + [1]
                  c_3(x1) = [1] x1 + [0]
                  c_4(x1) = [0] x1 + [0]
                  c_5() = [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {activate^#(n__f(X)) -> c_3(f^#(activate(X)))}
            and weakly orienting the rules
            {  f^#(X) -> c_1()
             , activate(n__a()) -> a()
             , activate(X) -> X}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {activate^#(n__f(X)) -> c_3(f^#(activate(X)))}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  a() = [0]
                  g(x1) = [1] x1 + [3]
                  n__f(x1) = [1] x1 + [0]
                  n__a() = [1]
                  activate(x1) = [1] x1 + [1]
                  f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  c_1() = [0]
                  a^#() = [0]
                  c_2() = [0]
                  activate^#(x1) = [1] x1 + [9]
                  c_3(x1) = [1] x1 + [0]
                  c_4(x1) = [0] x1 + [0]
                  c_5() = [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {  f(f(a())) -> f(g(n__f(n__a())))
             , a() -> n__a()}
            and weakly orienting the rules
            {  activate^#(n__f(X)) -> c_3(f^#(activate(X)))
             , f^#(X) -> c_1()
             , activate(n__a()) -> a()
             , activate(X) -> X}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  f(f(a())) -> f(g(n__f(n__a())))
               , a() -> n__a()}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  a() = [8]
                  g(x1) = [1] x1 + [0]
                  n__f(x1) = [1] x1 + [0]
                  n__a() = [7]
                  activate(x1) = [1] x1 + [1]
                  f^#(x1) = [1] x1 + [5]
                  c_0(x1) = [0] x1 + [0]
                  c_1() = [0]
                  a^#() = [0]
                  c_2() = [0]
                  activate^#(x1) = [1] x1 + [9]
                  c_3(x1) = [1] x1 + [1]
                  c_4(x1) = [0] x1 + [0]
                  c_5() = [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {f(X) -> n__f(X)}
            and weakly orienting the rules
            {  f(f(a())) -> f(g(n__f(n__a())))
             , a() -> n__a()
             , activate^#(n__f(X)) -> c_3(f^#(activate(X)))
             , f^#(X) -> c_1()
             , activate(n__a()) -> a()
             , activate(X) -> X}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {f(X) -> n__f(X)}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [8]
                  a() = [0]
                  g(x1) = [1] x1 + [0]
                  n__f(x1) = [1] x1 + [0]
                  n__a() = [0]
                  activate(x1) = [1] x1 + [1]
                  f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  c_1() = [0]
                  a^#() = [0]
                  c_2() = [0]
                  activate^#(x1) = [1] x1 + [1]
                  c_3(x1) = [1] x1 + [0]
                  c_4(x1) = [0] x1 + [0]
                  c_5() = [0]
              
            Finally we apply the subprocessor
            'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
            ------------------------------------------------------------------------------------------
            Answer:           YES(?,O(n^1))
            Input Problem:    innermost relative runtime-complexity with respect to
              Strict Rules: {activate(n__f(X)) -> f(activate(X))}
              Weak Rules:
                {  f(X) -> n__f(X)
                 , f(f(a())) -> f(g(n__f(n__a())))
                 , a() -> n__a()
                 , activate^#(n__f(X)) -> c_3(f^#(activate(X)))
                 , f^#(X) -> c_1()
                 , activate(n__a()) -> a()
                 , activate(X) -> X}
            
            Details:         
              The problem was solved by processor 'Bounds with default enrichment':
              'Bounds with default enrichment'
              --------------------------------
              Answer:           YES(?,O(n^1))
              Input Problem:    innermost relative runtime-complexity with respect to
                Strict Rules: {activate(n__f(X)) -> f(activate(X))}
                Weak Rules:
                  {  f(X) -> n__f(X)
                   , f(f(a())) -> f(g(n__f(n__a())))
                   , a() -> n__a()
                   , activate^#(n__f(X)) -> c_3(f^#(activate(X)))
                   , f^#(X) -> c_1()
                   , activate(n__a()) -> a()
                   , activate(X) -> X}
              
              Details:         
                The problem is Match-bounded by 1.
                The enriched problem is compatible with the following automaton:
                {  f_1(15) -> 14
                 , f_1(15) -> 15
                 , a_0() -> 14
                 , a_1() -> 15
                 , g_0(3) -> 3
                 , g_0(3) -> 14
                 , g_0(3) -> 15
                 , g_0(4) -> 3
                 , g_0(4) -> 14
                 , g_0(4) -> 15
                 , g_0(5) -> 3
                 , g_0(5) -> 14
                 , g_0(5) -> 15
                 , g_1(17) -> 15
                 , n__f_0(3) -> 4
                 , n__f_0(3) -> 14
                 , n__f_0(3) -> 15
                 , n__f_0(4) -> 4
                 , n__f_0(4) -> 14
                 , n__f_0(4) -> 15
                 , n__f_0(5) -> 4
                 , n__f_0(5) -> 14
                 , n__f_0(5) -> 15
                 , n__f_1(15) -> 14
                 , n__f_1(15) -> 15
                 , n__f_1(18) -> 17
                 , n__a_0() -> 5
                 , n__a_0() -> 14
                 , n__a_0() -> 15
                 , n__a_1() -> 15
                 , n__a_1() -> 18
                 , activate_0(3) -> 14
                 , activate_0(4) -> 14
                 , activate_0(5) -> 14
                 , activate_1(3) -> 15
                 , activate_1(4) -> 15
                 , activate_1(5) -> 15
                 , f^#_0(3) -> 7
                 , f^#_0(4) -> 7
                 , f^#_0(5) -> 7
                 , f^#_0(14) -> 13
                 , f^#_1(15) -> 16
                 , c_1_0() -> 7
                 , c_1_0() -> 13
                 , c_1_1() -> 16
                 , activate^#_0(3) -> 12
                 , activate^#_0(4) -> 12
                 , activate^#_0(5) -> 12
                 , c_3_0(13) -> 12
                 , c_3_1(16) -> 12}
      
   4) {activate^#(n__f(X)) -> c_3(f^#(activate(X)))}
      
      The usable rules for this path are the following:
      {  activate(n__f(X)) -> f(activate(X))
       , activate(n__a()) -> a()
       , activate(X) -> X
       , f(f(a())) -> f(g(n__f(n__a())))
       , f(X) -> n__f(X)
       , a() -> n__a()}
      
        We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
        
          'Weight Gap Principle'
          ----------------------
          Answer:           YES(?,O(n^1))
          Input Problem:    innermost runtime-complexity with respect to
            Rules:
              {  activate(n__f(X)) -> f(activate(X))
               , activate(n__a()) -> a()
               , activate(X) -> X
               , f(f(a())) -> f(g(n__f(n__a())))
               , f(X) -> n__f(X)
               , a() -> n__a()
               , activate^#(n__f(X)) -> c_3(f^#(activate(X)))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {  activate(n__a()) -> a()
             , activate(X) -> X}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  activate(n__a()) -> a()
               , activate(X) -> X}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  a() = [0]
                  g(x1) = [1] x1 + [0]
                  n__f(x1) = [1] x1 + [0]
                  n__a() = [0]
                  activate(x1) = [1] x1 + [1]
                  f^#(x1) = [1] x1 + [1]
                  c_0(x1) = [0] x1 + [0]
                  c_1() = [0]
                  a^#() = [0]
                  c_2() = [0]
                  activate^#(x1) = [1] x1 + [1]
                  c_3(x1) = [1] x1 + [1]
                  c_4(x1) = [0] x1 + [0]
                  c_5() = [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {activate^#(n__f(X)) -> c_3(f^#(activate(X)))}
            and weakly orienting the rules
            {  activate(n__a()) -> a()
             , activate(X) -> X}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {activate^#(n__f(X)) -> c_3(f^#(activate(X)))}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  a() = [0]
                  g(x1) = [1] x1 + [3]
                  n__f(x1) = [1] x1 + [0]
                  n__a() = [1]
                  activate(x1) = [1] x1 + [1]
                  f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  c_1() = [0]
                  a^#() = [0]
                  c_2() = [0]
                  activate^#(x1) = [1] x1 + [9]
                  c_3(x1) = [1] x1 + [0]
                  c_4(x1) = [0] x1 + [0]
                  c_5() = [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {  f(f(a())) -> f(g(n__f(n__a())))
             , a() -> n__a()}
            and weakly orienting the rules
            {  activate^#(n__f(X)) -> c_3(f^#(activate(X)))
             , activate(n__a()) -> a()
             , activate(X) -> X}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  f(f(a())) -> f(g(n__f(n__a())))
               , a() -> n__a()}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  a() = [8]
                  g(x1) = [1] x1 + [0]
                  n__f(x1) = [1] x1 + [0]
                  n__a() = [7]
                  activate(x1) = [1] x1 + [1]
                  f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  c_1() = [0]
                  a^#() = [0]
                  c_2() = [0]
                  activate^#(x1) = [1] x1 + [9]
                  c_3(x1) = [1] x1 + [3]
                  c_4(x1) = [0] x1 + [0]
                  c_5() = [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {f(X) -> n__f(X)}
            and weakly orienting the rules
            {  f(f(a())) -> f(g(n__f(n__a())))
             , a() -> n__a()
             , activate^#(n__f(X)) -> c_3(f^#(activate(X)))
             , activate(n__a()) -> a()
             , activate(X) -> X}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {f(X) -> n__f(X)}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [8]
                  a() = [0]
                  g(x1) = [1] x1 + [0]
                  n__f(x1) = [1] x1 + [0]
                  n__a() = [0]
                  activate(x1) = [1] x1 + [1]
                  f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  c_1() = [0]
                  a^#() = [0]
                  c_2() = [0]
                  activate^#(x1) = [1] x1 + [1]
                  c_3(x1) = [1] x1 + [0]
                  c_4(x1) = [0] x1 + [0]
                  c_5() = [0]
              
            Finally we apply the subprocessor
            'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
            ------------------------------------------------------------------------------------------
            Answer:           YES(?,O(n^1))
            Input Problem:    innermost relative runtime-complexity with respect to
              Strict Rules: {activate(n__f(X)) -> f(activate(X))}
              Weak Rules:
                {  f(X) -> n__f(X)
                 , f(f(a())) -> f(g(n__f(n__a())))
                 , a() -> n__a()
                 , activate^#(n__f(X)) -> c_3(f^#(activate(X)))
                 , activate(n__a()) -> a()
                 , activate(X) -> X}
            
            Details:         
              The problem was solved by processor 'Bounds with default enrichment':
              'Bounds with default enrichment'
              --------------------------------
              Answer:           YES(?,O(n^1))
              Input Problem:    innermost relative runtime-complexity with respect to
                Strict Rules: {activate(n__f(X)) -> f(activate(X))}
                Weak Rules:
                  {  f(X) -> n__f(X)
                   , f(f(a())) -> f(g(n__f(n__a())))
                   , a() -> n__a()
                   , activate^#(n__f(X)) -> c_3(f^#(activate(X)))
                   , activate(n__a()) -> a()
                   , activate(X) -> X}
              
              Details:         
                The problem is Match-bounded by 1.
                The enriched problem is compatible with the following automaton:
                {  f_1(15) -> 14
                 , f_1(15) -> 15
                 , a_0() -> 14
                 , a_1() -> 15
                 , g_0(3) -> 3
                 , g_0(3) -> 14
                 , g_0(3) -> 15
                 , g_0(4) -> 3
                 , g_0(4) -> 14
                 , g_0(4) -> 15
                 , g_0(5) -> 3
                 , g_0(5) -> 14
                 , g_0(5) -> 15
                 , g_1(17) -> 15
                 , n__f_0(3) -> 4
                 , n__f_0(3) -> 14
                 , n__f_0(3) -> 15
                 , n__f_0(4) -> 4
                 , n__f_0(4) -> 14
                 , n__f_0(4) -> 15
                 , n__f_0(5) -> 4
                 , n__f_0(5) -> 14
                 , n__f_0(5) -> 15
                 , n__f_1(15) -> 14
                 , n__f_1(15) -> 15
                 , n__f_1(18) -> 17
                 , n__a_0() -> 5
                 , n__a_0() -> 14
                 , n__a_0() -> 15
                 , n__a_1() -> 15
                 , n__a_1() -> 18
                 , activate_0(3) -> 14
                 , activate_0(4) -> 14
                 , activate_0(5) -> 14
                 , activate_1(3) -> 15
                 , activate_1(4) -> 15
                 , activate_1(5) -> 15
                 , f^#_0(3) -> 7
                 , f^#_0(4) -> 7
                 , f^#_0(5) -> 7
                 , f^#_0(14) -> 13
                 , f^#_1(15) -> 16
                 , activate^#_0(3) -> 12
                 , activate^#_0(4) -> 12
                 , activate^#_0(5) -> 12
                 , c_3_0(13) -> 12
                 , c_3_1(16) -> 12}
      
   5) {activate^#(n__a()) -> c_4(a^#())}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           f(x1) = [0] x1 + [0]
           a() = [0]
           g(x1) = [0] x1 + [0]
           n__f(x1) = [0] x1 + [0]
           n__a() = [0]
           activate(x1) = [0] x1 + [0]
           f^#(x1) = [0] x1 + [0]
           c_0(x1) = [0] x1 + [0]
           c_1() = [0]
           a^#() = [0]
           c_2() = [0]
           activate^#(x1) = [0] x1 + [0]
           c_3(x1) = [0] x1 + [0]
           c_4(x1) = [0] x1 + [0]
           c_5() = [0]
        
        We have applied the subprocessor on the resulting DP-problem:
        
          'Weight Gap Principle'
          ----------------------
          Answer:           YES(?,O(n^1))
          Input Problem:    innermost DP runtime-complexity with respect to
            Strict Rules: {activate^#(n__a()) -> c_4(a^#())}
            Weak Rules: {}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {activate^#(n__a()) -> c_4(a^#())}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {activate^#(n__a()) -> c_4(a^#())}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [0] x1 + [0]
                  a() = [0]
                  g(x1) = [0] x1 + [0]
                  n__f(x1) = [0] x1 + [0]
                  n__a() = [0]
                  activate(x1) = [0] x1 + [0]
                  f^#(x1) = [0] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  c_1() = [0]
                  a^#() = [0]
                  c_2() = [0]
                  activate^#(x1) = [1] x1 + [1]
                  c_3(x1) = [0] x1 + [0]
                  c_4(x1) = [1] x1 + [0]
                  c_5() = [0]
              
            Finally we apply the subprocessor
            'Empty TRS'
            -----------
            Answer:           YES(?,O(1))
            Input Problem:    innermost DP runtime-complexity with respect to
              Strict Rules: {}
              Weak Rules: {activate^#(n__a()) -> c_4(a^#())}
            
            Details:         
              The given problem does not contain any strict rules
      
   6) {  activate^#(n__a()) -> c_4(a^#())
       , a^#() -> c_2()}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           f(x1) = [0] x1 + [0]
           a() = [0]
           g(x1) = [0] x1 + [0]
           n__f(x1) = [0] x1 + [0]
           n__a() = [0]
           activate(x1) = [0] x1 + [0]
           f^#(x1) = [0] x1 + [0]
           c_0(x1) = [0] x1 + [0]
           c_1() = [0]
           a^#() = [0]
           c_2() = [0]
           activate^#(x1) = [0] x1 + [0]
           c_3(x1) = [0] x1 + [0]
           c_4(x1) = [0] x1 + [0]
           c_5() = [0]
        
        We have applied the subprocessor on the resulting DP-problem:
        
          'Weight Gap Principle'
          ----------------------
          Answer:           YES(?,O(n^1))
          Input Problem:    innermost DP runtime-complexity with respect to
            Strict Rules: {a^#() -> c_2()}
            Weak Rules: {activate^#(n__a()) -> c_4(a^#())}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {a^#() -> c_2()}
            and weakly orienting the rules
            {activate^#(n__a()) -> c_4(a^#())}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {a^#() -> c_2()}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [0] x1 + [0]
                  a() = [0]
                  g(x1) = [0] x1 + [0]
                  n__f(x1) = [0] x1 + [0]
                  n__a() = [0]
                  activate(x1) = [0] x1 + [0]
                  f^#(x1) = [0] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  c_1() = [0]
                  a^#() = [1]
                  c_2() = [0]
                  activate^#(x1) = [1] x1 + [1]
                  c_3(x1) = [0] x1 + [0]
                  c_4(x1) = [1] x1 + [0]
                  c_5() = [0]
              
            Finally we apply the subprocessor
            'Empty TRS'
            -----------
            Answer:           YES(?,O(1))
            Input Problem:    innermost DP runtime-complexity with respect to
              Strict Rules: {}
              Weak Rules:
                {  a^#() -> c_2()
                 , activate^#(n__a()) -> c_4(a^#())}
            
            Details:         
              The given problem does not contain any strict rules
      
   7) {activate^#(X) -> c_5()}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           f(x1) = [0] x1 + [0]
           a() = [0]
           g(x1) = [0] x1 + [0]
           n__f(x1) = [0] x1 + [0]
           n__a() = [0]
           activate(x1) = [0] x1 + [0]
           f^#(x1) = [0] x1 + [0]
           c_0(x1) = [0] x1 + [0]
           c_1() = [0]
           a^#() = [0]
           c_2() = [0]
           activate^#(x1) = [0] x1 + [0]
           c_3(x1) = [0] x1 + [0]
           c_4(x1) = [0] x1 + [0]
           c_5() = [0]
        
        We have applied the subprocessor on the resulting DP-problem:
        
          'Weight Gap Principle'
          ----------------------
          Answer:           YES(?,O(n^1))
          Input Problem:    innermost DP runtime-complexity with respect to
            Strict Rules: {activate^#(X) -> c_5()}
            Weak Rules: {}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {activate^#(X) -> c_5()}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {activate^#(X) -> c_5()}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [0] x1 + [0]
                  a() = [0]
                  g(x1) = [0] x1 + [0]
                  n__f(x1) = [0] x1 + [0]
                  n__a() = [0]
                  activate(x1) = [0] x1 + [0]
                  f^#(x1) = [0] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  c_1() = [0]
                  a^#() = [0]
                  c_2() = [0]
                  activate^#(x1) = [1] x1 + [4]
                  c_3(x1) = [0] x1 + [0]
                  c_4(x1) = [0] x1 + [0]
                  c_5() = [0]
              
            Finally we apply the subprocessor
            'Empty TRS'
            -----------
            Answer:           YES(?,O(1))
            Input Problem:    innermost DP runtime-complexity with respect to
              Strict Rules: {}
              Weak Rules: {activate^#(X) -> c_5()}
            
            Details:         
              The given problem does not contain any strict rules