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

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