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

Details:         
  We have computed the following set of weak (innermost) dependency pairs:
   {  a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))
    , mark^#(f(X)) -> c_1(a__f^#(mark(X)))
    , mark^#(a()) -> c_2()
    , mark^#(g(X)) -> c_3()
    , a__f^#(X) -> c_4()}
  
  The usable rules are:
   {  mark(f(X)) -> a__f(mark(X))
    , mark(a()) -> a()
    , mark(g(X)) -> g(X)
    , a__f(f(a())) -> a__f(g(f(a())))
    , a__f(X) -> f(X)}
  
  The estimated dependency graph contains the following edges:
   {a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))}
     ==> {a__f^#(X) -> c_4()}
   {mark^#(f(X)) -> c_1(a__f^#(mark(X)))}
     ==> {a__f^#(X) -> c_4()}
   {mark^#(f(X)) -> c_1(a__f^#(mark(X)))}
     ==> {a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))}
  
  We consider the following path(s):
   1) {  mark^#(f(X)) -> c_1(a__f^#(mark(X)))
       , a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))
       , a__f^#(X) -> c_4()}
      
      The usable rules for this path are the following:
      {  mark(f(X)) -> a__f(mark(X))
       , mark(a()) -> a()
       , mark(g(X)) -> g(X)
       , a__f(f(a())) -> a__f(g(f(a())))
       , a__f(X) -> f(X)}
      
        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:
              {  mark(f(X)) -> a__f(mark(X))
               , mark(a()) -> a()
               , mark(g(X)) -> g(X)
               , a__f(f(a())) -> a__f(g(f(a())))
               , a__f(X) -> f(X)
               , a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))
               , mark^#(f(X)) -> c_1(a__f^#(mark(X)))
               , a__f^#(X) -> c_4()}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {  mark(a()) -> a()
             , mark(g(X)) -> g(X)
             , a__f^#(X) -> c_4()}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  mark(a()) -> a()
               , mark(g(X)) -> g(X)
               , a__f^#(X) -> c_4()}
              
              Details:
                 Interpretation Functions:
                  a__f(x1) = [1] x1 + [0]
                  f(x1) = [1] x1 + [0]
                  a() = [0]
                  g(x1) = [1] x1 + [0]
                  mark(x1) = [1] x1 + [1]
                  a__f^#(x1) = [1] x1 + [1]
                  c_0(x1) = [1] x1 + [0]
                  mark^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [1]
                  c_2() = [0]
                  c_3() = [0]
                  c_4() = [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {mark^#(f(X)) -> c_1(a__f^#(mark(X)))}
            and weakly orienting the rules
            {  mark(a()) -> a()
             , mark(g(X)) -> g(X)
             , a__f^#(X) -> c_4()}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {mark^#(f(X)) -> c_1(a__f^#(mark(X)))}
              
              Details:
                 Interpretation Functions:
                  a__f(x1) = [1] x1 + [0]
                  f(x1) = [1] x1 + [0]
                  a() = [0]
                  g(x1) = [1] x1 + [0]
                  mark(x1) = [1] x1 + [1]
                  a__f^#(x1) = [1] x1 + [13]
                  c_0(x1) = [1] x1 + [0]
                  mark^#(x1) = [1] x1 + [15]
                  c_1(x1) = [1] x1 + [0]
                  c_2() = [0]
                  c_3() = [0]
                  c_4() = [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {mark(f(X)) -> a__f(mark(X))}
            and weakly orienting the rules
            {  mark^#(f(X)) -> c_1(a__f^#(mark(X)))
             , mark(a()) -> a()
             , mark(g(X)) -> g(X)
             , a__f^#(X) -> c_4()}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {mark(f(X)) -> a__f(mark(X))}
              
              Details:
                 Interpretation Functions:
                  a__f(x1) = [1] x1 + [0]
                  f(x1) = [1] x1 + [1]
                  a() = [1]
                  g(x1) = [1] x1 + [0]
                  mark(x1) = [1] x1 + [0]
                  a__f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [0]
                  mark^#(x1) = [1] x1 + [8]
                  c_1(x1) = [1] x1 + [0]
                  c_2() = [0]
                  c_3() = [0]
                  c_4() = [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:
                {  a__f(f(a())) -> a__f(g(f(a())))
                 , a__f(X) -> f(X)
                 , a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))}
              Weak Rules:
                {  mark(f(X)) -> a__f(mark(X))
                 , mark^#(f(X)) -> c_1(a__f^#(mark(X)))
                 , mark(a()) -> a()
                 , mark(g(X)) -> g(X)
                 , a__f^#(X) -> c_4()}
            
            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:
                  {  a__f(f(a())) -> a__f(g(f(a())))
                   , a__f(X) -> f(X)
                   , a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))}
                Weak Rules:
                  {  mark(f(X)) -> a__f(mark(X))
                   , mark^#(f(X)) -> c_1(a__f^#(mark(X)))
                   , mark(a()) -> a()
                   , mark(g(X)) -> g(X)
                   , a__f^#(X) -> c_4()}
              
              Details:         
                The problem is Match-bounded by 3.
                The enriched problem is compatible with the following automaton:
                {  a__f_0(10) -> 10
                 , a__f_1(12) -> 10
                 , a__f_1(16) -> 16
                 , a__f_2(18) -> 16
                 , f_0(2) -> 2
                 , f_0(3) -> 2
                 , f_0(4) -> 2
                 , f_1(10) -> 10
                 , f_1(14) -> 13
                 , f_2(12) -> 10
                 , f_2(16) -> 16
                 , f_2(20) -> 19
                 , f_3(18) -> 16
                 , a_0() -> 3
                 , a_0() -> 10
                 , a_1() -> 14
                 , a_1() -> 16
                 , a_2() -> 20
                 , g_0(2) -> 4
                 , g_0(2) -> 10
                 , g_0(3) -> 4
                 , g_0(3) -> 10
                 , g_0(4) -> 4
                 , g_0(4) -> 10
                 , g_1(2) -> 16
                 , g_1(3) -> 16
                 , g_1(4) -> 16
                 , g_1(13) -> 12
                 , g_2(19) -> 18
                 , mark_0(2) -> 10
                 , mark_0(3) -> 10
                 , mark_0(4) -> 10
                 , mark_1(2) -> 16
                 , mark_1(3) -> 16
                 , mark_1(4) -> 16
                 , a__f^#_0(2) -> 6
                 , a__f^#_0(3) -> 6
                 , a__f^#_0(4) -> 6
                 , a__f^#_0(10) -> 9
                 , a__f^#_1(12) -> 11
                 , a__f^#_1(16) -> 15
                 , a__f^#_2(18) -> 17
                 , c_0_1(11) -> 6
                 , c_0_1(11) -> 9
                 , c_0_2(17) -> 15
                 , mark^#_0(2) -> 8
                 , mark^#_0(3) -> 8
                 , mark^#_0(4) -> 8
                 , c_1_0(9) -> 8
                 , c_1_1(15) -> 8
                 , c_4_0() -> 6
                 , c_4_0() -> 9
                 , c_4_1() -> 11
                 , c_4_1() -> 15
                 , c_4_2() -> 17}
      
   2) {  mark^#(f(X)) -> c_1(a__f^#(mark(X)))
       , a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))}
      
      The usable rules for this path are the following:
      {  mark(f(X)) -> a__f(mark(X))
       , mark(a()) -> a()
       , mark(g(X)) -> g(X)
       , a__f(f(a())) -> a__f(g(f(a())))
       , a__f(X) -> f(X)}
      
        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:
              {  mark(f(X)) -> a__f(mark(X))
               , mark(a()) -> a()
               , mark(g(X)) -> g(X)
               , a__f(f(a())) -> a__f(g(f(a())))
               , a__f(X) -> f(X)
               , mark^#(f(X)) -> c_1(a__f^#(mark(X)))
               , a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {  mark(a()) -> a()
             , mark(g(X)) -> g(X)}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  mark(a()) -> a()
               , mark(g(X)) -> g(X)}
              
              Details:
                 Interpretation Functions:
                  a__f(x1) = [1] x1 + [0]
                  f(x1) = [1] x1 + [0]
                  a() = [0]
                  g(x1) = [1] x1 + [0]
                  mark(x1) = [1] x1 + [1]
                  a__f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [1]
                  mark^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [0]
                  c_2() = [0]
                  c_3() = [0]
                  c_4() = [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {mark^#(f(X)) -> c_1(a__f^#(mark(X)))}
            and weakly orienting the rules
            {  mark(a()) -> a()
             , mark(g(X)) -> g(X)}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {mark^#(f(X)) -> c_1(a__f^#(mark(X)))}
              
              Details:
                 Interpretation Functions:
                  a__f(x1) = [1] x1 + [0]
                  f(x1) = [1] x1 + [0]
                  a() = [0]
                  g(x1) = [1] x1 + [0]
                  mark(x1) = [1] x1 + [1]
                  a__f^#(x1) = [1] x1 + [1]
                  c_0(x1) = [1] x1 + [3]
                  mark^#(x1) = [1] x1 + [9]
                  c_1(x1) = [1] x1 + [0]
                  c_2() = [0]
                  c_3() = [0]
                  c_4() = [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {mark(f(X)) -> a__f(mark(X))}
            and weakly orienting the rules
            {  mark^#(f(X)) -> c_1(a__f^#(mark(X)))
             , mark(a()) -> a()
             , mark(g(X)) -> g(X)}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {mark(f(X)) -> a__f(mark(X))}
              
              Details:
                 Interpretation Functions:
                  a__f(x1) = [1] x1 + [0]
                  f(x1) = [1] x1 + [1]
                  a() = [0]
                  g(x1) = [1] x1 + [8]
                  mark(x1) = [1] x1 + [7]
                  a__f^#(x1) = [1] x1 + [2]
                  c_0(x1) = [1] x1 + [8]
                  mark^#(x1) = [1] x1 + [13]
                  c_1(x1) = [1] x1 + [1]
                  c_2() = [0]
                  c_3() = [0]
                  c_4() = [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:
                {  a__f(f(a())) -> a__f(g(f(a())))
                 , a__f(X) -> f(X)
                 , a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))}
              Weak Rules:
                {  mark(f(X)) -> a__f(mark(X))
                 , mark^#(f(X)) -> c_1(a__f^#(mark(X)))
                 , mark(a()) -> a()
                 , mark(g(X)) -> g(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:
                  {  a__f(f(a())) -> a__f(g(f(a())))
                   , a__f(X) -> f(X)
                   , a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))}
                Weak Rules:
                  {  mark(f(X)) -> a__f(mark(X))
                   , mark^#(f(X)) -> c_1(a__f^#(mark(X)))
                   , mark(a()) -> a()
                   , mark(g(X)) -> g(X)}
              
              Details:         
                The problem is Match-bounded by 3.
                The enriched problem is compatible with the following automaton:
                {  a__f_0(4) -> 4
                 , a__f_1(6) -> 4
                 , a__f_1(10) -> 10
                 , a__f_2(12) -> 10
                 , f_0(2) -> 2
                 , f_1(4) -> 4
                 , f_1(8) -> 7
                 , f_2(6) -> 4
                 , f_2(10) -> 10
                 , f_2(14) -> 13
                 , f_3(12) -> 10
                 , a_0() -> 2
                 , a_0() -> 4
                 , a_1() -> 8
                 , a_1() -> 10
                 , a_2() -> 14
                 , g_0(2) -> 2
                 , g_0(2) -> 4
                 , g_1(2) -> 10
                 , g_1(7) -> 6
                 , g_2(13) -> 12
                 , mark_0(2) -> 4
                 , mark_1(2) -> 10
                 , a__f^#_0(2) -> 1
                 , a__f^#_0(4) -> 3
                 , a__f^#_1(6) -> 5
                 , a__f^#_1(10) -> 9
                 , a__f^#_2(12) -> 11
                 , c_0_1(5) -> 1
                 , c_0_1(5) -> 3
                 , c_0_2(11) -> 9
                 , mark^#_0(2) -> 1
                 , c_1_0(3) -> 1
                 , c_1_1(9) -> 1}
      
   3) {  mark^#(f(X)) -> c_1(a__f^#(mark(X)))
       , a__f^#(X) -> c_4()}
      
      The usable rules for this path are the following:
      {  mark(f(X)) -> a__f(mark(X))
       , mark(a()) -> a()
       , mark(g(X)) -> g(X)
       , a__f(f(a())) -> a__f(g(f(a())))
       , a__f(X) -> f(X)}
      
        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:
              {  mark(f(X)) -> a__f(mark(X))
               , mark(a()) -> a()
               , mark(g(X)) -> g(X)
               , a__f(f(a())) -> a__f(g(f(a())))
               , a__f(X) -> f(X)
               , mark^#(f(X)) -> c_1(a__f^#(mark(X)))
               , a__f^#(X) -> c_4()}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {  mark(a()) -> a()
             , mark(g(X)) -> g(X)}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  mark(a()) -> a()
               , mark(g(X)) -> g(X)}
              
              Details:
                 Interpretation Functions:
                  a__f(x1) = [1] x1 + [0]
                  f(x1) = [1] x1 + [0]
                  a() = [0]
                  g(x1) = [1] x1 + [0]
                  mark(x1) = [1] x1 + [1]
                  a__f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  mark^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [0]
                  c_2() = [0]
                  c_3() = [0]
                  c_4() = [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {a__f^#(X) -> c_4()}
            and weakly orienting the rules
            {  mark(a()) -> a()
             , mark(g(X)) -> g(X)}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {a__f^#(X) -> c_4()}
              
              Details:
                 Interpretation Functions:
                  a__f(x1) = [1] x1 + [0]
                  f(x1) = [1] x1 + [0]
                  a() = [0]
                  g(x1) = [1] x1 + [0]
                  mark(x1) = [1] x1 + [1]
                  a__f^#(x1) = [1] x1 + [8]
                  c_0(x1) = [0] x1 + [0]
                  mark^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [3]
                  c_2() = [0]
                  c_3() = [0]
                  c_4() = [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {mark^#(f(X)) -> c_1(a__f^#(mark(X)))}
            and weakly orienting the rules
            {  a__f^#(X) -> c_4()
             , mark(a()) -> a()
             , mark(g(X)) -> g(X)}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {mark^#(f(X)) -> c_1(a__f^#(mark(X)))}
              
              Details:
                 Interpretation Functions:
                  a__f(x1) = [1] x1 + [0]
                  f(x1) = [1] x1 + [0]
                  a() = [0]
                  g(x1) = [1] x1 + [0]
                  mark(x1) = [1] x1 + [1]
                  a__f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  mark^#(x1) = [1] x1 + [9]
                  c_1(x1) = [1] x1 + [1]
                  c_2() = [0]
                  c_3() = [0]
                  c_4() = [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {a__f(X) -> f(X)}
            and weakly orienting the rules
            {  mark^#(f(X)) -> c_1(a__f^#(mark(X)))
             , a__f^#(X) -> c_4()
             , mark(a()) -> a()
             , mark(g(X)) -> g(X)}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {a__f(X) -> f(X)}
              
              Details:
                 Interpretation Functions:
                  a__f(x1) = [1] x1 + [8]
                  f(x1) = [1] x1 + [0]
                  a() = [0]
                  g(x1) = [1] x1 + [0]
                  mark(x1) = [1] x1 + [1]
                  a__f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  mark^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [0]
                  c_2() = [0]
                  c_3() = [0]
                  c_4() = [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:
                {  mark(f(X)) -> a__f(mark(X))
                 , a__f(f(a())) -> a__f(g(f(a())))}
              Weak Rules:
                {  a__f(X) -> f(X)
                 , mark^#(f(X)) -> c_1(a__f^#(mark(X)))
                 , a__f^#(X) -> c_4()
                 , mark(a()) -> a()
                 , mark(g(X)) -> g(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:
                  {  mark(f(X)) -> a__f(mark(X))
                   , a__f(f(a())) -> a__f(g(f(a())))}
                Weak Rules:
                  {  a__f(X) -> f(X)
                   , mark^#(f(X)) -> c_1(a__f^#(mark(X)))
                   , a__f^#(X) -> c_4()
                   , mark(a()) -> a()
                   , mark(g(X)) -> g(X)}
              
              Details:         
                The problem is Match-bounded by 2.
                The enriched problem is compatible with the following automaton:
                {  a__f_1(11) -> 10
                 , a__f_1(11) -> 11
                 , a__f_2(13) -> 10
                 , a__f_2(13) -> 11
                 , f_0(2) -> 2
                 , f_0(3) -> 2
                 , f_0(4) -> 2
                 , f_1(11) -> 10
                 , f_1(11) -> 11
                 , f_2(13) -> 10
                 , f_2(13) -> 11
                 , f_2(15) -> 14
                 , a_0() -> 3
                 , a_0() -> 10
                 , a_1() -> 11
                 , a_2() -> 15
                 , g_0(2) -> 4
                 , g_0(2) -> 10
                 , g_0(3) -> 4
                 , g_0(3) -> 10
                 , g_0(4) -> 4
                 , g_0(4) -> 10
                 , g_1(2) -> 11
                 , g_1(3) -> 11
                 , g_1(4) -> 11
                 , g_2(14) -> 13
                 , mark_0(2) -> 10
                 , mark_0(3) -> 10
                 , mark_0(4) -> 10
                 , mark_1(2) -> 11
                 , mark_1(3) -> 11
                 , mark_1(4) -> 11
                 , a__f^#_0(2) -> 6
                 , a__f^#_0(3) -> 6
                 , a__f^#_0(4) -> 6
                 , a__f^#_0(10) -> 9
                 , a__f^#_1(11) -> 12
                 , mark^#_0(2) -> 8
                 , mark^#_0(3) -> 8
                 , mark^#_0(4) -> 8
                 , c_1_0(9) -> 8
                 , c_1_1(12) -> 8
                 , c_4_0() -> 6
                 , c_4_0() -> 9
                 , c_4_1() -> 12}
      
   4) {mark^#(f(X)) -> c_1(a__f^#(mark(X)))}
      
      The usable rules for this path are the following:
      {  mark(f(X)) -> a__f(mark(X))
       , mark(a()) -> a()
       , mark(g(X)) -> g(X)
       , a__f(f(a())) -> a__f(g(f(a())))
       , a__f(X) -> f(X)}
      
        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:
              {  mark(f(X)) -> a__f(mark(X))
               , mark(a()) -> a()
               , mark(g(X)) -> g(X)
               , a__f(f(a())) -> a__f(g(f(a())))
               , a__f(X) -> f(X)
               , mark^#(f(X)) -> c_1(a__f^#(mark(X)))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {  mark(a()) -> a()
             , mark(g(X)) -> g(X)}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  mark(a()) -> a()
               , mark(g(X)) -> g(X)}
              
              Details:
                 Interpretation Functions:
                  a__f(x1) = [1] x1 + [0]
                  f(x1) = [1] x1 + [0]
                  a() = [0]
                  g(x1) = [1] x1 + [0]
                  mark(x1) = [1] x1 + [1]
                  a__f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  mark^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [0]
                  c_2() = [0]
                  c_3() = [0]
                  c_4() = [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {mark^#(f(X)) -> c_1(a__f^#(mark(X)))}
            and weakly orienting the rules
            {  mark(a()) -> a()
             , mark(g(X)) -> g(X)}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {mark^#(f(X)) -> c_1(a__f^#(mark(X)))}
              
              Details:
                 Interpretation Functions:
                  a__f(x1) = [1] x1 + [0]
                  f(x1) = [1] x1 + [0]
                  a() = [0]
                  g(x1) = [1] x1 + [0]
                  mark(x1) = [1] x1 + [1]
                  a__f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  mark^#(x1) = [1] x1 + [9]
                  c_1(x1) = [1] x1 + [0]
                  c_2() = [0]
                  c_3() = [0]
                  c_4() = [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {a__f(X) -> f(X)}
            and weakly orienting the rules
            {  mark^#(f(X)) -> c_1(a__f^#(mark(X)))
             , mark(a()) -> a()
             , mark(g(X)) -> g(X)}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {a__f(X) -> f(X)}
              
              Details:
                 Interpretation Functions:
                  a__f(x1) = [1] x1 + [1]
                  f(x1) = [1] x1 + [0]
                  a() = [2]
                  g(x1) = [1] x1 + [2]
                  mark(x1) = [1] x1 + [1]
                  a__f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  mark^#(x1) = [1] x1 + [13]
                  c_1(x1) = [1] x1 + [8]
                  c_2() = [0]
                  c_3() = [0]
                  c_4() = [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:
                {  mark(f(X)) -> a__f(mark(X))
                 , a__f(f(a())) -> a__f(g(f(a())))}
              Weak Rules:
                {  a__f(X) -> f(X)
                 , mark^#(f(X)) -> c_1(a__f^#(mark(X)))
                 , mark(a()) -> a()
                 , mark(g(X)) -> g(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:
                  {  mark(f(X)) -> a__f(mark(X))
                   , a__f(f(a())) -> a__f(g(f(a())))}
                Weak Rules:
                  {  a__f(X) -> f(X)
                   , mark^#(f(X)) -> c_1(a__f^#(mark(X)))
                   , mark(a()) -> a()
                   , mark(g(X)) -> g(X)}
              
              Details:         
                The problem is Match-bounded by 2.
                The enriched problem is compatible with the following automaton:
                {  a__f_1(11) -> 10
                 , a__f_1(11) -> 11
                 , a__f_2(13) -> 10
                 , a__f_2(13) -> 11
                 , f_0(2) -> 2
                 , f_0(3) -> 2
                 , f_0(4) -> 2
                 , f_1(11) -> 10
                 , f_1(11) -> 11
                 , f_2(13) -> 10
                 , f_2(13) -> 11
                 , f_2(15) -> 14
                 , a_0() -> 3
                 , a_0() -> 10
                 , a_1() -> 11
                 , a_2() -> 15
                 , g_0(2) -> 4
                 , g_0(2) -> 10
                 , g_0(3) -> 4
                 , g_0(3) -> 10
                 , g_0(4) -> 4
                 , g_0(4) -> 10
                 , g_1(2) -> 11
                 , g_1(3) -> 11
                 , g_1(4) -> 11
                 , g_2(14) -> 13
                 , mark_0(2) -> 10
                 , mark_0(3) -> 10
                 , mark_0(4) -> 10
                 , mark_1(2) -> 11
                 , mark_1(3) -> 11
                 , mark_1(4) -> 11
                 , a__f^#_0(2) -> 6
                 , a__f^#_0(3) -> 6
                 , a__f^#_0(4) -> 6
                 , a__f^#_0(10) -> 9
                 , a__f^#_1(11) -> 12
                 , mark^#_0(2) -> 8
                 , mark^#_0(3) -> 8
                 , mark^#_0(4) -> 8
                 , c_1_0(9) -> 8
                 , c_1_1(12) -> 8}
      
   5) {mark^#(g(X)) -> c_3()}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           a__f(x1) = [0] x1 + [0]
           f(x1) = [0] x1 + [0]
           a() = [0]
           g(x1) = [0] x1 + [0]
           mark(x1) = [0] x1 + [0]
           a__f^#(x1) = [0] x1 + [0]
           c_0(x1) = [0] x1 + [0]
           mark^#(x1) = [0] x1 + [0]
           c_1(x1) = [0] x1 + [0]
           c_2() = [0]
           c_3() = [0]
           c_4() = [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: {mark^#(g(X)) -> c_3()}
            Weak Rules: {}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {mark^#(g(X)) -> c_3()}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {mark^#(g(X)) -> c_3()}
              
              Details:
                 Interpretation Functions:
                  a__f(x1) = [0] x1 + [0]
                  f(x1) = [0] x1 + [0]
                  a() = [0]
                  g(x1) = [1] x1 + [0]
                  mark(x1) = [0] x1 + [0]
                  a__f^#(x1) = [0] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  mark^#(x1) = [1] x1 + [1]
                  c_1(x1) = [0] x1 + [0]
                  c_2() = [0]
                  c_3() = [0]
                  c_4() = [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: {mark^#(g(X)) -> c_3()}
            
            Details:         
              The given problem does not contain any strict rules
      
   6) {mark^#(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:
           a__f(x1) = [0] x1 + [0]
           f(x1) = [0] x1 + [0]
           a() = [0]
           g(x1) = [0] x1 + [0]
           mark(x1) = [0] x1 + [0]
           a__f^#(x1) = [0] x1 + [0]
           c_0(x1) = [0] x1 + [0]
           mark^#(x1) = [0] x1 + [0]
           c_1(x1) = [0] x1 + [0]
           c_2() = [0]
           c_3() = [0]
           c_4() = [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: {mark^#(a()) -> c_2()}
            Weak Rules: {}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {mark^#(a()) -> c_2()}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {mark^#(a()) -> c_2()}
              
              Details:
                 Interpretation Functions:
                  a__f(x1) = [0] x1 + [0]
                  f(x1) = [0] x1 + [0]
                  a() = [0]
                  g(x1) = [0] x1 + [0]
                  mark(x1) = [0] x1 + [0]
                  a__f^#(x1) = [0] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  mark^#(x1) = [1] x1 + [1]
                  c_1(x1) = [0] x1 + [0]
                  c_2() = [0]
                  c_3() = [0]
                  c_4() = [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: {mark^#(a()) -> c_2()}
            
            Details:         
              The given problem does not contain any strict rules