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

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