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

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