'Weak Dependency Graph [60.0]'
------------------------------
Answer:           YES(?,O(n^1))
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  f(f(x, y, z), u, f(x, y, v)) -> f(x, y, f(z, u, v))
     , f(x, y, y) -> y
     , f(x, y, g(y)) -> x
     , f(x, x, y) -> x
     , f(g(x), x, y) -> y}

Details:         
  We have computed the following set of weak (innermost) dependency pairs:
   {  f^#(f(x, y, z), u, f(x, y, v)) -> c_0(f^#(x, y, f(z, u, v)))
    , f^#(x, y, y) -> c_1()
    , f^#(x, y, g(y)) -> c_2()
    , f^#(x, x, y) -> c_3()
    , f^#(g(x), x, y) -> c_4()}
  
  The usable rules are:
   {  f(f(x, y, z), u, f(x, y, v)) -> f(x, y, f(z, u, v))
    , f(x, y, y) -> y
    , f(x, y, g(y)) -> x
    , f(x, x, y) -> x
    , f(g(x), x, y) -> y}
  
  The estimated dependency graph contains the following edges:
   {f^#(f(x, y, z), u, f(x, y, v)) -> c_0(f^#(x, y, f(z, u, v)))}
     ==> {f^#(g(x), x, y) -> c_4()}
   {f^#(f(x, y, z), u, f(x, y, v)) -> c_0(f^#(x, y, f(z, u, v)))}
     ==> {f^#(x, x, y) -> c_3()}
   {f^#(f(x, y, z), u, f(x, y, v)) -> c_0(f^#(x, y, f(z, u, v)))}
     ==> {f^#(x, y, g(y)) -> c_2()}
   {f^#(f(x, y, z), u, f(x, y, v)) -> c_0(f^#(x, y, f(z, u, v)))}
     ==> {f^#(x, y, y) -> c_1()}
   {f^#(f(x, y, z), u, f(x, y, v)) -> c_0(f^#(x, y, f(z, u, v)))}
     ==> {f^#(f(x, y, z), u, f(x, y, v)) -> c_0(f^#(x, y, f(z, u, v)))}
  
  We consider the following path(s):
   1) {  f^#(f(x, y, z), u, f(x, y, v)) -> c_0(f^#(x, y, f(z, u, v)))
       , f^#(g(x), x, y) -> c_4()}
      
      The usable rules for this path are the following:
      {  f(f(x, y, z), u, f(x, y, v)) -> f(x, y, f(z, u, v))
       , f(x, y, y) -> y
       , f(x, y, g(y)) -> x
       , f(x, x, y) -> x
       , f(g(x), x, y) -> y}
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [8]
           g(x1) = [1] x1 + [0]
           f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
           c_0(x1) = [0] x1 + [0]
           c_1() = [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: {f^#(g(x), x, y) -> c_4()}
            Weak Rules:
              {  f(f(x, y, z), u, f(x, y, v)) -> f(x, y, f(z, u, v))
               , f(x, y, y) -> y
               , f(x, y, g(y)) -> x
               , f(x, x, y) -> x
               , f(g(x), x, y) -> y
               , f^#(f(x, y, z), u, f(x, y, v)) -> c_0(f^#(x, y, f(z, u, v)))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {f^#(g(x), x, y) -> c_4()}
            and weakly orienting the rules
            {  f(f(x, y, z), u, f(x, y, v)) -> f(x, y, f(z, u, v))
             , f(x, y, y) -> y
             , f(x, y, g(y)) -> x
             , f(x, x, y) -> x
             , f(g(x), x, y) -> y
             , f^#(f(x, y, z), u, f(x, y, v)) -> c_0(f^#(x, y, f(z, u, v)))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {f^#(g(x), x, y) -> c_4()}
              
              Details:
                 Interpretation Functions:
                  f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
                  g(x1) = [1] x1 + [0]
                  f^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
                  c_0(x1) = [1] x1 + [0]
                  c_1() = [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:
                {  f^#(g(x), x, y) -> c_4()
                 , f(f(x, y, z), u, f(x, y, v)) -> f(x, y, f(z, u, v))
                 , f(x, y, y) -> y
                 , f(x, y, g(y)) -> x
                 , f(x, x, y) -> x
                 , f(g(x), x, y) -> y
                 , f^#(f(x, y, z), u, f(x, y, v)) -> c_0(f^#(x, y, f(z, u, v)))}
            
            Details:         
              The given problem does not contain any strict rules
      
   2) {  f^#(f(x, y, z), u, f(x, y, v)) -> c_0(f^#(x, y, f(z, u, v)))
       , f^#(x, x, y) -> c_3()}
      
      The usable rules for this path are the following:
      {  f(f(x, y, z), u, f(x, y, v)) -> f(x, y, f(z, u, v))
       , f(x, y, y) -> y
       , f(x, y, g(y)) -> x
       , f(x, x, y) -> x
       , f(g(x), x, y) -> y}
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [8]
           g(x1) = [1] x1 + [0]
           f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
           c_0(x1) = [0] x1 + [0]
           c_1() = [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: {f^#(x, x, y) -> c_3()}
            Weak Rules:
              {  f(f(x, y, z), u, f(x, y, v)) -> f(x, y, f(z, u, v))
               , f(x, y, y) -> y
               , f(x, y, g(y)) -> x
               , f(x, x, y) -> x
               , f(g(x), x, y) -> y
               , f^#(f(x, y, z), u, f(x, y, v)) -> c_0(f^#(x, y, f(z, u, v)))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {f^#(x, x, y) -> c_3()}
            and weakly orienting the rules
            {  f(f(x, y, z), u, f(x, y, v)) -> f(x, y, f(z, u, v))
             , f(x, y, y) -> y
             , f(x, y, g(y)) -> x
             , f(x, x, y) -> x
             , f(g(x), x, y) -> y
             , f^#(f(x, y, z), u, f(x, y, v)) -> c_0(f^#(x, y, f(z, u, v)))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {f^#(x, x, y) -> c_3()}
              
              Details:
                 Interpretation Functions:
                  f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
                  g(x1) = [1] x1 + [0]
                  f^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
                  c_0(x1) = [1] x1 + [0]
                  c_1() = [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:
                {  f^#(x, x, y) -> c_3()
                 , f(f(x, y, z), u, f(x, y, v)) -> f(x, y, f(z, u, v))
                 , f(x, y, y) -> y
                 , f(x, y, g(y)) -> x
                 , f(x, x, y) -> x
                 , f(g(x), x, y) -> y
                 , f^#(f(x, y, z), u, f(x, y, v)) -> c_0(f^#(x, y, f(z, u, v)))}
            
            Details:         
              The given problem does not contain any strict rules
      
   3) {  f^#(f(x, y, z), u, f(x, y, v)) -> c_0(f^#(x, y, f(z, u, v)))
       , f^#(x, y, y) -> c_1()}
      
      The usable rules for this path are the following:
      {  f(f(x, y, z), u, f(x, y, v)) -> f(x, y, f(z, u, v))
       , f(x, y, y) -> y
       , f(x, y, g(y)) -> x
       , f(x, x, y) -> x
       , f(g(x), x, y) -> y}
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [8]
           g(x1) = [1] x1 + [0]
           f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
           c_0(x1) = [0] x1 + [0]
           c_1() = [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: {f^#(x, y, y) -> c_1()}
            Weak Rules:
              {  f(f(x, y, z), u, f(x, y, v)) -> f(x, y, f(z, u, v))
               , f(x, y, y) -> y
               , f(x, y, g(y)) -> x
               , f(x, x, y) -> x
               , f(g(x), x, y) -> y
               , f^#(f(x, y, z), u, f(x, y, v)) -> c_0(f^#(x, y, f(z, u, v)))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {f^#(x, y, y) -> c_1()}
            and weakly orienting the rules
            {  f(f(x, y, z), u, f(x, y, v)) -> f(x, y, f(z, u, v))
             , f(x, y, y) -> y
             , f(x, y, g(y)) -> x
             , f(x, x, y) -> x
             , f(g(x), x, y) -> y
             , f^#(f(x, y, z), u, f(x, y, v)) -> c_0(f^#(x, y, f(z, u, v)))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {f^#(x, y, y) -> c_1()}
              
              Details:
                 Interpretation Functions:
                  f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
                  g(x1) = [1] x1 + [0]
                  f^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
                  c_0(x1) = [1] x1 + [0]
                  c_1() = [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:
                {  f^#(x, y, y) -> c_1()
                 , f(f(x, y, z), u, f(x, y, v)) -> f(x, y, f(z, u, v))
                 , f(x, y, y) -> y
                 , f(x, y, g(y)) -> x
                 , f(x, x, y) -> x
                 , f(g(x), x, y) -> y
                 , f^#(f(x, y, z), u, f(x, y, v)) -> c_0(f^#(x, y, f(z, u, v)))}
            
            Details:         
              The given problem does not contain any strict rules
      
   4) {  f^#(f(x, y, z), u, f(x, y, v)) -> c_0(f^#(x, y, f(z, u, v)))
       , f^#(x, y, g(y)) -> c_2()}
      
      The usable rules for this path are the following:
      {  f(f(x, y, z), u, f(x, y, v)) -> f(x, y, f(z, u, v))
       , f(x, y, y) -> y
       , f(x, y, g(y)) -> x
       , f(x, x, y) -> x
       , f(g(x), x, y) -> y}
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [8]
           g(x1) = [1] x1 + [0]
           f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
           c_0(x1) = [0] x1 + [0]
           c_1() = [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: {f^#(x, y, g(y)) -> c_2()}
            Weak Rules:
              {  f(f(x, y, z), u, f(x, y, v)) -> f(x, y, f(z, u, v))
               , f(x, y, y) -> y
               , f(x, y, g(y)) -> x
               , f(x, x, y) -> x
               , f(g(x), x, y) -> y
               , f^#(f(x, y, z), u, f(x, y, v)) -> c_0(f^#(x, y, f(z, u, v)))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {f^#(x, y, g(y)) -> c_2()}
            and weakly orienting the rules
            {  f(f(x, y, z), u, f(x, y, v)) -> f(x, y, f(z, u, v))
             , f(x, y, y) -> y
             , f(x, y, g(y)) -> x
             , f(x, x, y) -> x
             , f(g(x), x, y) -> y
             , f^#(f(x, y, z), u, f(x, y, v)) -> c_0(f^#(x, y, f(z, u, v)))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {f^#(x, y, g(y)) -> c_2()}
              
              Details:
                 Interpretation Functions:
                  f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
                  g(x1) = [1] x1 + [0]
                  f^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
                  c_0(x1) = [1] x1 + [0]
                  c_1() = [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:
                {  f^#(x, y, g(y)) -> c_2()
                 , f(f(x, y, z), u, f(x, y, v)) -> f(x, y, f(z, u, v))
                 , f(x, y, y) -> y
                 , f(x, y, g(y)) -> x
                 , f(x, x, y) -> x
                 , f(g(x), x, y) -> y
                 , f^#(f(x, y, z), u, f(x, y, v)) -> c_0(f^#(x, y, f(z, u, v)))}
            
            Details:         
              The given problem does not contain any strict rules
      
   5) {f^#(f(x, y, z), u, f(x, y, v)) -> c_0(f^#(x, y, f(z, u, v)))}
      
      The usable rules for this path are the following:
      {  f(f(x, y, z), u, f(x, y, v)) -> f(x, y, f(z, u, v))
       , f(x, y, y) -> y
       , f(x, y, g(y)) -> x
       , f(x, x, y) -> x
       , f(g(x), x, y) -> y}
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [8]
           g(x1) = [1] x1 + [0]
           f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
           c_0(x1) = [0] x1 + [0]
           c_1() = [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:
              {f^#(f(x, y, z), u, f(x, y, v)) -> c_0(f^#(x, y, f(z, u, v)))}
            Weak Rules:
              {  f(f(x, y, z), u, f(x, y, v)) -> f(x, y, f(z, u, v))
               , f(x, y, y) -> y
               , f(x, y, g(y)) -> x
               , f(x, x, y) -> x
               , f(g(x), x, y) -> y}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {f^#(f(x, y, z), u, f(x, y, v)) -> c_0(f^#(x, y, f(z, u, v)))}
            and weakly orienting the rules
            {  f(f(x, y, z), u, f(x, y, v)) -> f(x, y, f(z, u, v))
             , f(x, y, y) -> y
             , f(x, y, g(y)) -> x
             , f(x, x, y) -> x
             , f(g(x), x, y) -> y}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {f^#(f(x, y, z), u, f(x, y, v)) -> c_0(f^#(x, y, f(z, u, v)))}
              
              Details:
                 Interpretation Functions:
                  f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [8]
                  g(x1) = [1] x1 + [0]
                  f^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
                  c_0(x1) = [1] x1 + [4]
                  c_1() = [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:
                {  f^#(f(x, y, z), u, f(x, y, v)) -> c_0(f^#(x, y, f(z, u, v)))
                 , f(f(x, y, z), u, f(x, y, v)) -> f(x, y, f(z, u, v))
                 , f(x, y, y) -> y
                 , f(x, y, g(y)) -> x
                 , f(x, x, y) -> x
                 , f(g(x), x, y) -> y}
            
            Details:         
              The given problem does not contain any strict rules