'Weak Dependency Graph [60.0]'
------------------------------
Answer:           YES(?,O(n^1))
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  sum(0()) -> 0()
     , sum(s(x)) -> +(sum(x), s(x))
     , sum1(0()) -> 0()
     , sum1(s(x)) -> s(+(sum1(x), +(x, x)))}

Details:         
  We have computed the following set of weak (innermost) dependency pairs:
   {  sum^#(0()) -> c_0()
    , sum^#(s(x)) -> c_1(sum^#(x))
    , sum1^#(0()) -> c_2()
    , sum1^#(s(x)) -> c_3(sum1^#(x))}
  
  The usable rules are:
   {}
  
  The estimated dependency graph contains the following edges:
   {sum^#(s(x)) -> c_1(sum^#(x))}
     ==> {sum^#(s(x)) -> c_1(sum^#(x))}
   {sum^#(s(x)) -> c_1(sum^#(x))}
     ==> {sum^#(0()) -> c_0()}
   {sum1^#(s(x)) -> c_3(sum1^#(x))}
     ==> {sum1^#(s(x)) -> c_3(sum1^#(x))}
   {sum1^#(s(x)) -> c_3(sum1^#(x))}
     ==> {sum1^#(0()) -> c_2()}
  
  We consider the following path(s):
   1) {sum^#(s(x)) -> c_1(sum^#(x))}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           sum(x1) = [0] x1 + [0]
           0() = [0]
           s(x1) = [0] x1 + [0]
           +(x1, x2) = [0] x1 + [0] x2 + [0]
           sum1(x1) = [0] x1 + [0]
           sum^#(x1) = [0] x1 + [0]
           c_0() = [0]
           c_1(x1) = [0] x1 + [0]
           sum1^#(x1) = [0] x1 + [0]
           c_2() = [0]
           c_3(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: {sum^#(s(x)) -> c_1(sum^#(x))}
            Weak Rules: {}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {sum^#(s(x)) -> c_1(sum^#(x))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {sum^#(s(x)) -> c_1(sum^#(x))}
              
              Details:
                 Interpretation Functions:
                  sum(x1) = [0] x1 + [0]
                  0() = [0]
                  s(x1) = [1] x1 + [8]
                  +(x1, x2) = [0] x1 + [0] x2 + [0]
                  sum1(x1) = [0] x1 + [0]
                  sum^#(x1) = [1] x1 + [1]
                  c_0() = [0]
                  c_1(x1) = [1] x1 + [3]
                  sum1^#(x1) = [0] x1 + [0]
                  c_2() = [0]
                  c_3(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: {sum^#(s(x)) -> c_1(sum^#(x))}
            
            Details:         
              The given problem does not contain any strict rules
      
   2) {  sum^#(s(x)) -> c_1(sum^#(x))
       , sum^#(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:
           sum(x1) = [0] x1 + [0]
           0() = [0]
           s(x1) = [0] x1 + [0]
           +(x1, x2) = [0] x1 + [0] x2 + [0]
           sum1(x1) = [0] x1 + [0]
           sum^#(x1) = [0] x1 + [0]
           c_0() = [0]
           c_1(x1) = [0] x1 + [0]
           sum1^#(x1) = [0] x1 + [0]
           c_2() = [0]
           c_3(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: {sum^#(0()) -> c_0()}
            Weak Rules: {sum^#(s(x)) -> c_1(sum^#(x))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {sum^#(0()) -> c_0()}
            and weakly orienting the rules
            {sum^#(s(x)) -> c_1(sum^#(x))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {sum^#(0()) -> c_0()}
              
              Details:
                 Interpretation Functions:
                  sum(x1) = [0] x1 + [0]
                  0() = [0]
                  s(x1) = [1] x1 + [0]
                  +(x1, x2) = [0] x1 + [0] x2 + [0]
                  sum1(x1) = [0] x1 + [0]
                  sum^#(x1) = [1] x1 + [1]
                  c_0() = [0]
                  c_1(x1) = [1] x1 + [0]
                  sum1^#(x1) = [0] x1 + [0]
                  c_2() = [0]
                  c_3(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:
                {  sum^#(0()) -> c_0()
                 , sum^#(s(x)) -> c_1(sum^#(x))}
            
            Details:         
              The given problem does not contain any strict rules
      
   3) {  sum1^#(s(x)) -> c_3(sum1^#(x))
       , sum1^#(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:
           sum(x1) = [0] x1 + [0]
           0() = [0]
           s(x1) = [0] x1 + [0]
           +(x1, x2) = [0] x1 + [0] x2 + [0]
           sum1(x1) = [0] x1 + [0]
           sum^#(x1) = [0] x1 + [0]
           c_0() = [0]
           c_1(x1) = [0] x1 + [0]
           sum1^#(x1) = [0] x1 + [0]
           c_2() = [0]
           c_3(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: {sum1^#(0()) -> c_2()}
            Weak Rules: {sum1^#(s(x)) -> c_3(sum1^#(x))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {sum1^#(0()) -> c_2()}
            and weakly orienting the rules
            {sum1^#(s(x)) -> c_3(sum1^#(x))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {sum1^#(0()) -> c_2()}
              
              Details:
                 Interpretation Functions:
                  sum(x1) = [0] x1 + [0]
                  0() = [0]
                  s(x1) = [1] x1 + [0]
                  +(x1, x2) = [0] x1 + [0] x2 + [0]
                  sum1(x1) = [0] x1 + [0]
                  sum^#(x1) = [0] x1 + [0]
                  c_0() = [0]
                  c_1(x1) = [0] x1 + [0]
                  sum1^#(x1) = [1] x1 + [1]
                  c_2() = [0]
                  c_3(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:
                {  sum1^#(0()) -> c_2()
                 , sum1^#(s(x)) -> c_3(sum1^#(x))}
            
            Details:         
              The given problem does not contain any strict rules
      
   4) {sum1^#(s(x)) -> c_3(sum1^#(x))}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           sum(x1) = [0] x1 + [0]
           0() = [0]
           s(x1) = [0] x1 + [0]
           +(x1, x2) = [0] x1 + [0] x2 + [0]
           sum1(x1) = [0] x1 + [0]
           sum^#(x1) = [0] x1 + [0]
           c_0() = [0]
           c_1(x1) = [0] x1 + [0]
           sum1^#(x1) = [0] x1 + [0]
           c_2() = [0]
           c_3(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: {sum1^#(s(x)) -> c_3(sum1^#(x))}
            Weak Rules: {}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {sum1^#(s(x)) -> c_3(sum1^#(x))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {sum1^#(s(x)) -> c_3(sum1^#(x))}
              
              Details:
                 Interpretation Functions:
                  sum(x1) = [0] x1 + [0]
                  0() = [0]
                  s(x1) = [1] x1 + [8]
                  +(x1, x2) = [0] x1 + [0] x2 + [0]
                  sum1(x1) = [0] x1 + [0]
                  sum^#(x1) = [0] x1 + [0]
                  c_0() = [0]
                  c_1(x1) = [0] x1 + [0]
                  sum1^#(x1) = [1] x1 + [1]
                  c_2() = [0]
                  c_3(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: {sum1^#(s(x)) -> c_3(sum1^#(x))}
            
            Details:         
              The given problem does not contain any strict rules