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

Details:         
  We have computed the following set of weak (innermost) dependency pairs:
   {  half^#(0()) -> c_0()
    , half^#(s(s(x))) -> c_1(half^#(x))
    , log^#(s(0())) -> c_2()
    , log^#(s(s(x))) -> c_3(log^#(s(half(x))))}
  
  The usable rules are:
   {  half(0()) -> 0()
    , half(s(s(x))) -> s(half(x))}
  
  The estimated dependency graph contains the following edges:
   {half^#(s(s(x))) -> c_1(half^#(x))}
     ==> {half^#(s(s(x))) -> c_1(half^#(x))}
   {half^#(s(s(x))) -> c_1(half^#(x))}
     ==> {half^#(0()) -> c_0()}
   {log^#(s(s(x))) -> c_3(log^#(s(half(x))))}
     ==> {log^#(s(s(x))) -> c_3(log^#(s(half(x))))}
   {log^#(s(s(x))) -> c_3(log^#(s(half(x))))}
     ==> {log^#(s(0())) -> c_2()}
  
  We consider the following path(s):
   1) {  log^#(s(s(x))) -> c_3(log^#(s(half(x))))
       , log^#(s(0())) -> c_2()}
      
      The usable rules for this path are the following:
      {  half(0()) -> 0()
       , half(s(s(x))) -> s(half(x))}
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           half(x1) = [1] x1 + [1]
           0() = [0]
           s(x1) = [1] x1 + [8]
           log(x1) = [0] x1 + [0]
           half^#(x1) = [0] x1 + [0]
           c_0() = [0]
           c_1(x1) = [0] x1 + [0]
           log^#(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: {log^#(s(0())) -> c_2()}
            Weak Rules:
              {  half(0()) -> 0()
               , half(s(s(x))) -> s(half(x))
               , log^#(s(s(x))) -> c_3(log^#(s(half(x))))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {log^#(s(0())) -> c_2()}
            and weakly orienting the rules
            {  half(0()) -> 0()
             , half(s(s(x))) -> s(half(x))
             , log^#(s(s(x))) -> c_3(log^#(s(half(x))))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {log^#(s(0())) -> c_2()}
              
              Details:
                 Interpretation Functions:
                  half(x1) = [1] x1 + [0]
                  0() = [0]
                  s(x1) = [1] x1 + [0]
                  log(x1) = [0] x1 + [0]
                  half^#(x1) = [0] x1 + [0]
                  c_0() = [0]
                  c_1(x1) = [0] x1 + [0]
                  log^#(x1) = [1] x1 + [2]
                  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:
                {  log^#(s(0())) -> c_2()
                 , half(0()) -> 0()
                 , half(s(s(x))) -> s(half(x))
                 , log^#(s(s(x))) -> c_3(log^#(s(half(x))))}
            
            Details:         
              The given problem does not contain any strict rules
      
   2) {log^#(s(s(x))) -> c_3(log^#(s(half(x))))}
      
      The usable rules for this path are the following:
      {  half(0()) -> 0()
       , half(s(s(x))) -> s(half(x))}
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           half(x1) = [1] x1 + [1]
           0() = [0]
           s(x1) = [1] x1 + [8]
           log(x1) = [0] x1 + [0]
           half^#(x1) = [0] x1 + [0]
           c_0() = [0]
           c_1(x1) = [0] x1 + [0]
           log^#(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: {log^#(s(s(x))) -> c_3(log^#(s(half(x))))}
            Weak Rules:
              {  half(0()) -> 0()
               , half(s(s(x))) -> s(half(x))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {log^#(s(s(x))) -> c_3(log^#(s(half(x))))}
            and weakly orienting the rules
            {  half(0()) -> 0()
             , half(s(s(x))) -> s(half(x))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {log^#(s(s(x))) -> c_3(log^#(s(half(x))))}
              
              Details:
                 Interpretation Functions:
                  half(x1) = [1] x1 + [0]
                  0() = [0]
                  s(x1) = [1] x1 + [8]
                  log(x1) = [0] x1 + [0]
                  half^#(x1) = [0] x1 + [0]
                  c_0() = [0]
                  c_1(x1) = [0] x1 + [0]
                  log^#(x1) = [1] x1 + [8]
                  c_2() = [0]
                  c_3(x1) = [1] x1 + [1]
              
            Finally we apply the subprocessor
            'Empty TRS'
            -----------
            Answer:           YES(?,O(1))
            Input Problem:    innermost DP runtime-complexity with respect to
              Strict Rules: {}
              Weak Rules:
                {  log^#(s(s(x))) -> c_3(log^#(s(half(x))))
                 , half(0()) -> 0()
                 , half(s(s(x))) -> s(half(x))}
            
            Details:         
              The given problem does not contain any strict rules
      
   3) {  half^#(s(s(x))) -> c_1(half^#(x))
       , half^#(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:
           half(x1) = [0] x1 + [0]
           0() = [0]
           s(x1) = [0] x1 + [0]
           log(x1) = [0] x1 + [0]
           half^#(x1) = [0] x1 + [0]
           c_0() = [0]
           c_1(x1) = [0] x1 + [0]
           log^#(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: {half^#(0()) -> c_0()}
            Weak Rules: {half^#(s(s(x))) -> c_1(half^#(x))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {half^#(0()) -> c_0()}
            and weakly orienting the rules
            {half^#(s(s(x))) -> c_1(half^#(x))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {half^#(0()) -> c_0()}
              
              Details:
                 Interpretation Functions:
                  half(x1) = [0] x1 + [0]
                  0() = [0]
                  s(x1) = [1] x1 + [0]
                  log(x1) = [0] x1 + [0]
                  half^#(x1) = [1] x1 + [1]
                  c_0() = [0]
                  c_1(x1) = [1] x1 + [0]
                  log^#(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:
                {  half^#(0()) -> c_0()
                 , half^#(s(s(x))) -> c_1(half^#(x))}
            
            Details:         
              The given problem does not contain any strict rules
      
   4) {half^#(s(s(x))) -> c_1(half^#(x))}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           half(x1) = [0] x1 + [0]
           0() = [0]
           s(x1) = [0] x1 + [0]
           log(x1) = [0] x1 + [0]
           half^#(x1) = [0] x1 + [0]
           c_0() = [0]
           c_1(x1) = [0] x1 + [0]
           log^#(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: {half^#(s(s(x))) -> c_1(half^#(x))}
            Weak Rules: {}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {half^#(s(s(x))) -> c_1(half^#(x))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {half^#(s(s(x))) -> c_1(half^#(x))}
              
              Details:
                 Interpretation Functions:
                  half(x1) = [0] x1 + [0]
                  0() = [0]
                  s(x1) = [1] x1 + [8]
                  log(x1) = [0] x1 + [0]
                  half^#(x1) = [1] x1 + [0]
                  c_0() = [0]
                  c_1(x1) = [1] x1 + [11]
                  log^#(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: {half^#(s(s(x))) -> c_1(half^#(x))}
            
            Details:         
              The given problem does not contain any strict rules