'Weak Dependency Graph [60.0]'
------------------------------
Answer:           YES(?,O(n^1))
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  g(0()) -> 0()
     , g(s(x)) -> f(g(x))
     , f(0()) -> 0()}

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