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

Details:         
  We have computed the following set of weak (innermost) dependency pairs:
   {  f^#(x, a()) -> c_0()
    , f^#(x, g(y)) -> c_1(f^#(g(x), y))}
  
  The usable rules are:
   {}
  
  The estimated dependency graph contains the following edges:
   {f^#(x, g(y)) -> c_1(f^#(g(x), y))}
     ==> {f^#(x, g(y)) -> c_1(f^#(g(x), y))}
   {f^#(x, g(y)) -> c_1(f^#(g(x), y))}
     ==> {f^#(x, a()) -> c_0()}
  
  We consider the following path(s):
   1) {f^#(x, g(y)) -> c_1(f^#(g(x), y))}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           f(x1, x2) = [0] x1 + [0] x2 + [0]
           a() = [0]
           g(x1) = [0] x1 + [0]
           f^#(x1, x2) = [0] x1 + [0] x2 + [0]
           c_0() = [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: {f^#(x, g(y)) -> c_1(f^#(g(x), y))}
            Weak Rules: {}
          
          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: {f^#(x, g(y)) -> c_1(f^#(g(x), y))}
              Weak Rules: {}
            
            Details:         
              The problem was solved by processor 'combine':
              'combine'
              ---------
              Answer:           YES(?,O(n^1))
              Input Problem:    innermost DP runtime-complexity with respect to
                Strict Rules: {f^#(x, g(y)) -> c_1(f^#(g(x), y))}
                Weak Rules: {}
              
              Details:         
                'sequentially if-then-else, sequentially'
                -----------------------------------------
                Answer:           YES(?,O(n^1))
                Input Problem:    innermost DP runtime-complexity with respect to
                  Strict Rules: {f^#(x, g(y)) -> c_1(f^#(g(x), y))}
                  Weak Rules: {}
                
                Details:         
                  'if Check whether the TRS is strict trs contains single rule then fastest else fastest'
                  ---------------------------------------------------------------------------------------
                  Answer:           YES(?,O(n^1))
                  Input Problem:    innermost DP runtime-complexity with respect to
                    Strict Rules: {f^#(x, g(y)) -> c_1(f^#(g(x), y))}
                    Weak Rules: {}
                  
                  Details:         
                    a) We first check the conditional [Success]:
                       We are considering a strict trs contains single rule TRS.
                    
                    b) We continue with the then-branch:
                       The problem was solved by processor 'fastest of 'Matrix Interpretation', 'Matrix Interpretation', 'Matrix Interpretation'':
                       'fastest of 'Matrix Interpretation', 'Matrix Interpretation', 'Matrix Interpretation''
                       --------------------------------------------------------------------------------------
                       Answer:           YES(?,O(n^1))
                       Input Problem:    innermost DP runtime-complexity with respect to
                         Strict Rules: {f^#(x, g(y)) -> c_1(f^#(g(x), y))}
                         Weak Rules: {}
                       
                       Details:         
                         The problem was solved by processor 'Matrix Interpretation':
                         'Matrix Interpretation'
                         -----------------------
                         Answer:           YES(?,O(n^1))
                         Input Problem:    innermost DP runtime-complexity with respect to
                           Strict Rules: {f^#(x, g(y)) -> c_1(f^#(g(x), y))}
                           Weak Rules: {}
                         
                         Details:         
                           Interpretation Functions:
                            f(x1, x2) = [0] x1 + [0] x2 + [0]
                            a() = [0]
                            g(x1) = [1] x1 + [2]
                            f^#(x1, x2) = [4] x1 + [5] x2 + [2]
                            c_0() = [0]
                            c_1(x1) = [1] x1 + [0]
      
   2) {  f^#(x, g(y)) -> c_1(f^#(g(x), y))
       , f^#(x, a()) -> 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, x2) = [0] x1 + [0] x2 + [0]
           a() = [0]
           g(x1) = [0] x1 + [0]
           f^#(x1, x2) = [0] x1 + [0] x2 + [0]
           c_0() = [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: {f^#(x, a()) -> c_0()}
            Weak Rules: {f^#(x, g(y)) -> c_1(f^#(g(x), y))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {f^#(x, a()) -> c_0()}
            and weakly orienting the rules
            {f^#(x, g(y)) -> c_1(f^#(g(x), y))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {f^#(x, a()) -> c_0()}
              
              Details:
                 Interpretation Functions:
                  f(x1, x2) = [0] x1 + [0] x2 + [0]
                  a() = [0]
                  g(x1) = [1] x1 + [0]
                  f^#(x1, x2) = [1] x1 + [1] x2 + [1]
                  c_0() = [0]
                  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:
                {  f^#(x, a()) -> c_0()
                 , f^#(x, g(y)) -> c_1(f^#(g(x), y))}
            
            Details:         
              The given problem does not contain any strict rules