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

Details:         
  We have computed the following set of weak (innermost) dependency pairs:
   {  add^#(0(), x) -> c_0()
    , add^#(s(x), y) -> c_1(add^#(x, y))}
  
  The usable rules are:
   {}
  
  The estimated dependency graph contains the following edges:
   {add^#(s(x), y) -> c_1(add^#(x, y))}
     ==> {add^#(s(x), y) -> c_1(add^#(x, y))}
   {add^#(s(x), y) -> c_1(add^#(x, y))}
     ==> {add^#(0(), x) -> c_0()}
  
  We consider the following path(s):
   1) {  add^#(s(x), y) -> c_1(add^#(x, y))
       , add^#(0(), x) -> c_0()}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           add(x1, x2) = [0] x1 + [0] x2 + [0]
           0() = [0]
           s(x1) = [0] x1 + [0]
           add^#(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: {add^#(0(), x) -> c_0()}
            Weak Rules: {add^#(s(x), y) -> c_1(add^#(x, y))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {add^#(0(), x) -> c_0()}
            and weakly orienting the rules
            {add^#(s(x), y) -> c_1(add^#(x, y))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {add^#(0(), x) -> c_0()}
              
              Details:
                 Interpretation Functions:
                  add(x1, x2) = [0] x1 + [0] x2 + [0]
                  0() = [0]
                  s(x1) = [1] x1 + [0]
                  add^#(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:
                {  add^#(0(), x) -> c_0()
                 , add^#(s(x), y) -> c_1(add^#(x, y))}
            
            Details:         
              The given problem does not contain any strict rules
      
   2) {add^#(s(x), y) -> c_1(add^#(x, y))}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           add(x1, x2) = [0] x1 + [0] x2 + [0]
           0() = [0]
           s(x1) = [0] x1 + [0]
           add^#(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: {add^#(s(x), y) -> c_1(add^#(x, y))}
            Weak Rules: {}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {add^#(s(x), y) -> c_1(add^#(x, y))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {add^#(s(x), y) -> c_1(add^#(x, y))}
              
              Details:
                 Interpretation Functions:
                  add(x1, x2) = [0] x1 + [0] x2 + [0]
                  0() = [0]
                  s(x1) = [1] x1 + [8]
                  add^#(x1, x2) = [1] x1 + [1] x2 + [1]
                  c_0() = [0]
                  c_1(x1) = [1] x1 + [3]
              
            Finally we apply the subprocessor
            'Empty TRS'
            -----------
            Answer:           YES(?,O(1))
            Input Problem:    innermost DP runtime-complexity with respect to
              Strict Rules: {}
              Weak Rules: {add^#(s(x), y) -> c_1(add^#(x, y))}
            
            Details:         
              The given problem does not contain any strict rules