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

Details:         
  We have computed the following set of weak (innermost) dependency pairs:
   {  f^#(f(x)) -> c_0(f^#(x))
    , f^#(s(x)) -> c_1(f^#(x))
    , g^#(s(0())) -> c_2(g^#(f(s(0()))))}
  
  The usable rules are:
   {  f(f(x)) -> f(x)
    , f(s(x)) -> f(x)}
  
  The estimated dependency graph contains the following edges:
   {f^#(f(x)) -> c_0(f^#(x))}
     ==> {f^#(s(x)) -> c_1(f^#(x))}
   {f^#(f(x)) -> c_0(f^#(x))}
     ==> {f^#(f(x)) -> c_0(f^#(x))}
   {f^#(s(x)) -> c_1(f^#(x))}
     ==> {f^#(s(x)) -> c_1(f^#(x))}
   {f^#(s(x)) -> c_1(f^#(x))}
     ==> {f^#(f(x)) -> c_0(f^#(x))}
  
  We consider the following path(s):
   1) {  f^#(f(x)) -> c_0(f^#(x))
       , f^#(s(x)) -> c_1(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]
           s(x1) = [0] x1 + [0]
           g(x1) = [0] x1 + [0]
           0() = [0]
           f^#(x1) = [0] x1 + [0]
           c_0(x1) = [0] x1 + [0]
           c_1(x1) = [0] x1 + [0]
           g^#(x1) = [0] x1 + [0]
           c_2(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^#(f(x)) -> c_0(f^#(x))
               , f^#(s(x)) -> c_1(f^#(x))}
            Weak Rules: {}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {f^#(s(x)) -> c_1(f^#(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(f^#(x))}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  s(x1) = [1] x1 + [8]
                  g(x1) = [0] x1 + [0]
                  0() = [0]
                  f^#(x1) = [1] x1 + [1]
                  c_0(x1) = [1] x1 + [0]
                  c_1(x1) = [1] x1 + [1]
                  g^#(x1) = [0] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {f^#(f(x)) -> c_0(f^#(x))}
            and weakly orienting the rules
            {f^#(s(x)) -> c_1(f^#(x))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {f^#(f(x)) -> c_0(f^#(x))}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [8]
                  s(x1) = [1] x1 + [8]
                  g(x1) = [0] x1 + [0]
                  0() = [0]
                  f^#(x1) = [1] x1 + [1]
                  c_0(x1) = [1] x1 + [3]
                  c_1(x1) = [1] x1 + [1]
                  g^#(x1) = [0] x1 + [0]
                  c_2(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:
                {  f^#(f(x)) -> c_0(f^#(x))
                 , f^#(s(x)) -> c_1(f^#(x))}
            
            Details:         
              The given problem does not contain any strict rules
      
   2) {g^#(s(0())) -> c_2(g^#(f(s(0()))))}
      
      The usable rules for this path are the following:
      {  f(f(x)) -> f(x)
       , f(s(x)) -> f(x)}
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           f(x1) = [1] x1 + [8]
           s(x1) = [1] x1 + [2]
           g(x1) = [0] x1 + [0]
           0() = [0]
           f^#(x1) = [0] x1 + [0]
           c_0(x1) = [0] x1 + [0]
           c_1(x1) = [0] x1 + [0]
           g^#(x1) = [0] x1 + [0]
           c_2(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_2(g^#(f(s(0()))))}
            Weak Rules:
              {  f(f(x)) -> f(x)
               , f(s(x)) -> f(x)}
          
          Details:         
            'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
            ------------------------------------------------------------------------------------------
            Answer:           YES(?,O(n^1))
            Input Problem:    innermost DP runtime-complexity with respect to
              Strict Rules: {g^#(s(0())) -> c_2(g^#(f(s(0()))))}
              Weak Rules:
                {  f(f(x)) -> f(x)
                 , f(s(x)) -> f(x)}
            
            Details:         
              The problem was solved by processor 'Bounds with default enrichment':
              'Bounds with default enrichment'
              --------------------------------
              Answer:           YES(?,O(n^1))
              Input Problem:    innermost DP runtime-complexity with respect to
                Strict Rules: {g^#(s(0())) -> c_2(g^#(f(s(0()))))}
                Weak Rules:
                  {  f(f(x)) -> f(x)
                   , f(s(x)) -> f(x)}
              
              Details:         
                The problem is Match-bounded by 1.
                The enriched problem is compatible with the following automaton:
                {  f_1(11) -> 10
                 , f_1(12) -> 10
                 , s_0(2) -> 2
                 , s_0(4) -> 2
                 , s_1(12) -> 11
                 , 0_0() -> 4
                 , 0_1() -> 12
                 , g^#_0(2) -> 8
                 , g^#_0(4) -> 8
                 , g^#_1(10) -> 9
                 , c_2_1(9) -> 8}