'Weak Dependency Graph [60.0]'
------------------------------
Answer:           YES(?,O(n^1))
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  D(t()) -> 1()
     , D(constant()) -> 0()
     , D(+(x, y)) -> +(D(x), D(y))
     , D(*(x, y)) -> +(*(y, D(x)), *(x, D(y)))
     , D(-(x, y)) -> -(D(x), D(y))}

Details:         
  We have computed the following set of weak (innermost) dependency pairs:
   {  D^#(t()) -> c_0()
    , D^#(constant()) -> c_1()
    , D^#(+(x, y)) -> c_2(D^#(x), D^#(y))
    , D^#(*(x, y)) -> c_3(D^#(x), D^#(y))
    , D^#(-(x, y)) -> c_4(D^#(x), D^#(y))}
  
  The usable rules are:
   {}
  
  The estimated dependency graph contains the following edges:
   {D^#(+(x, y)) -> c_2(D^#(x), D^#(y))}
     ==> {D^#(-(x, y)) -> c_4(D^#(x), D^#(y))}
   {D^#(+(x, y)) -> c_2(D^#(x), D^#(y))}
     ==> {D^#(*(x, y)) -> c_3(D^#(x), D^#(y))}
   {D^#(+(x, y)) -> c_2(D^#(x), D^#(y))}
     ==> {D^#(+(x, y)) -> c_2(D^#(x), D^#(y))}
   {D^#(+(x, y)) -> c_2(D^#(x), D^#(y))}
     ==> {D^#(constant()) -> c_1()}
   {D^#(+(x, y)) -> c_2(D^#(x), D^#(y))}
     ==> {D^#(t()) -> c_0()}
   {D^#(*(x, y)) -> c_3(D^#(x), D^#(y))}
     ==> {D^#(-(x, y)) -> c_4(D^#(x), D^#(y))}
   {D^#(*(x, y)) -> c_3(D^#(x), D^#(y))}
     ==> {D^#(*(x, y)) -> c_3(D^#(x), D^#(y))}
   {D^#(*(x, y)) -> c_3(D^#(x), D^#(y))}
     ==> {D^#(+(x, y)) -> c_2(D^#(x), D^#(y))}
   {D^#(*(x, y)) -> c_3(D^#(x), D^#(y))}
     ==> {D^#(constant()) -> c_1()}
   {D^#(*(x, y)) -> c_3(D^#(x), D^#(y))}
     ==> {D^#(t()) -> c_0()}
   {D^#(-(x, y)) -> c_4(D^#(x), D^#(y))}
     ==> {D^#(-(x, y)) -> c_4(D^#(x), D^#(y))}
   {D^#(-(x, y)) -> c_4(D^#(x), D^#(y))}
     ==> {D^#(*(x, y)) -> c_3(D^#(x), D^#(y))}
   {D^#(-(x, y)) -> c_4(D^#(x), D^#(y))}
     ==> {D^#(+(x, y)) -> c_2(D^#(x), D^#(y))}
   {D^#(-(x, y)) -> c_4(D^#(x), D^#(y))}
     ==> {D^#(constant()) -> c_1()}
   {D^#(-(x, y)) -> c_4(D^#(x), D^#(y))}
     ==> {D^#(t()) -> c_0()}
  
  We consider the following path(s):
   1) {  D^#(+(x, y)) -> c_2(D^#(x), D^#(y))
       , D^#(-(x, y)) -> c_4(D^#(x), D^#(y))
       , D^#(*(x, y)) -> c_3(D^#(x), D^#(y))}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           D(x1) = [0] x1 + [0]
           t() = [0]
           1() = [0]
           constant() = [0]
           0() = [0]
           +(x1, x2) = [0] x1 + [0] x2 + [0]
           *(x1, x2) = [0] x1 + [0] x2 + [0]
           -(x1, x2) = [0] x1 + [0] x2 + [0]
           D^#(x1) = [0] x1 + [0]
           c_0() = [0]
           c_1() = [0]
           c_2(x1, x2) = [0] x1 + [0] x2 + [0]
           c_3(x1, x2) = [0] x1 + [0] x2 + [0]
           c_4(x1, x2) = [0] x1 + [0] x2 + [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:
              {  D^#(+(x, y)) -> c_2(D^#(x), D^#(y))
               , D^#(-(x, y)) -> c_4(D^#(x), D^#(y))
               , D^#(*(x, y)) -> c_3(D^#(x), D^#(y))}
            Weak Rules: {}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {D^#(*(x, y)) -> c_3(D^#(x), D^#(y))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {D^#(*(x, y)) -> c_3(D^#(x), D^#(y))}
              
              Details:
                 Interpretation Functions:
                  D(x1) = [0] x1 + [0]
                  t() = [0]
                  1() = [0]
                  constant() = [0]
                  0() = [0]
                  +(x1, x2) = [1] x1 + [1] x2 + [0]
                  *(x1, x2) = [1] x1 + [1] x2 + [4]
                  -(x1, x2) = [1] x1 + [1] x2 + [0]
                  D^#(x1) = [1] x1 + [1]
                  c_0() = [0]
                  c_1() = [0]
                  c_2(x1, x2) = [1] x1 + [1] x2 + [1]
                  c_3(x1, x2) = [1] x1 + [1] x2 + [1]
                  c_4(x1, x2) = [1] x1 + [1] x2 + [1]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {D^#(-(x, y)) -> c_4(D^#(x), D^#(y))}
            and weakly orienting the rules
            {D^#(*(x, y)) -> c_3(D^#(x), D^#(y))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {D^#(-(x, y)) -> c_4(D^#(x), D^#(y))}
              
              Details:
                 Interpretation Functions:
                  D(x1) = [0] x1 + [0]
                  t() = [0]
                  1() = [0]
                  constant() = [0]
                  0() = [0]
                  +(x1, x2) = [1] x1 + [1] x2 + [0]
                  *(x1, x2) = [1] x1 + [1] x2 + [8]
                  -(x1, x2) = [1] x1 + [1] x2 + [8]
                  D^#(x1) = [1] x1 + [1]
                  c_0() = [0]
                  c_1() = [0]
                  c_2(x1, x2) = [1] x1 + [1] x2 + [1]
                  c_3(x1, x2) = [1] x1 + [1] x2 + [7]
                  c_4(x1, x2) = [1] x1 + [1] x2 + [1]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {D^#(+(x, y)) -> c_2(D^#(x), D^#(y))}
            and weakly orienting the rules
            {  D^#(-(x, y)) -> c_4(D^#(x), D^#(y))
             , D^#(*(x, y)) -> c_3(D^#(x), D^#(y))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {D^#(+(x, y)) -> c_2(D^#(x), D^#(y))}
              
              Details:
                 Interpretation Functions:
                  D(x1) = [0] x1 + [0]
                  t() = [0]
                  1() = [0]
                  constant() = [0]
                  0() = [0]
                  +(x1, x2) = [1] x1 + [1] x2 + [8]
                  *(x1, x2) = [1] x1 + [1] x2 + [9]
                  -(x1, x2) = [1] x1 + [1] x2 + [8]
                  D^#(x1) = [1] x1 + [1]
                  c_0() = [0]
                  c_1() = [0]
                  c_2(x1, x2) = [1] x1 + [1] x2 + [1]
                  c_3(x1, x2) = [1] x1 + [1] x2 + [1]
                  c_4(x1, x2) = [1] x1 + [1] x2 + [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:
                {  D^#(+(x, y)) -> c_2(D^#(x), D^#(y))
                 , D^#(-(x, y)) -> c_4(D^#(x), D^#(y))
                 , D^#(*(x, y)) -> c_3(D^#(x), D^#(y))}
            
            Details:         
              The given problem does not contain any strict rules
      
   2) {  D^#(+(x, y)) -> c_2(D^#(x), D^#(y))
       , D^#(-(x, y)) -> c_4(D^#(x), D^#(y))
       , D^#(*(x, y)) -> c_3(D^#(x), D^#(y))
       , D^#(constant()) -> c_1()}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           D(x1) = [0] x1 + [0]
           t() = [0]
           1() = [0]
           constant() = [0]
           0() = [0]
           +(x1, x2) = [0] x1 + [0] x2 + [0]
           *(x1, x2) = [0] x1 + [0] x2 + [0]
           -(x1, x2) = [0] x1 + [0] x2 + [0]
           D^#(x1) = [0] x1 + [0]
           c_0() = [0]
           c_1() = [0]
           c_2(x1, x2) = [0] x1 + [0] x2 + [0]
           c_3(x1, x2) = [0] x1 + [0] x2 + [0]
           c_4(x1, x2) = [0] x1 + [0] x2 + [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: {D^#(constant()) -> c_1()}
            Weak Rules:
              {  D^#(+(x, y)) -> c_2(D^#(x), D^#(y))
               , D^#(-(x, y)) -> c_4(D^#(x), D^#(y))
               , D^#(*(x, y)) -> c_3(D^#(x), D^#(y))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {D^#(constant()) -> c_1()}
            and weakly orienting the rules
            {  D^#(+(x, y)) -> c_2(D^#(x), D^#(y))
             , D^#(-(x, y)) -> c_4(D^#(x), D^#(y))
             , D^#(*(x, y)) -> c_3(D^#(x), D^#(y))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {D^#(constant()) -> c_1()}
              
              Details:
                 Interpretation Functions:
                  D(x1) = [0] x1 + [0]
                  t() = [0]
                  1() = [0]
                  constant() = [0]
                  0() = [0]
                  +(x1, x2) = [1] x1 + [1] x2 + [8]
                  *(x1, x2) = [1] x1 + [1] x2 + [2]
                  -(x1, x2) = [1] x1 + [1] x2 + [8]
                  D^#(x1) = [1] x1 + [1]
                  c_0() = [0]
                  c_1() = [0]
                  c_2(x1, x2) = [1] x1 + [1] x2 + [3]
                  c_3(x1, x2) = [1] x1 + [1] x2 + [1]
                  c_4(x1, x2) = [1] x1 + [1] x2 + [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:
                {  D^#(constant()) -> c_1()
                 , D^#(+(x, y)) -> c_2(D^#(x), D^#(y))
                 , D^#(-(x, y)) -> c_4(D^#(x), D^#(y))
                 , D^#(*(x, y)) -> c_3(D^#(x), D^#(y))}
            
            Details:         
              The given problem does not contain any strict rules
      
   3) {  D^#(+(x, y)) -> c_2(D^#(x), D^#(y))
       , D^#(-(x, y)) -> c_4(D^#(x), D^#(y))
       , D^#(*(x, y)) -> c_3(D^#(x), D^#(y))
       , D^#(t()) -> c_0()}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           D(x1) = [0] x1 + [0]
           t() = [0]
           1() = [0]
           constant() = [0]
           0() = [0]
           +(x1, x2) = [0] x1 + [0] x2 + [0]
           *(x1, x2) = [0] x1 + [0] x2 + [0]
           -(x1, x2) = [0] x1 + [0] x2 + [0]
           D^#(x1) = [0] x1 + [0]
           c_0() = [0]
           c_1() = [0]
           c_2(x1, x2) = [0] x1 + [0] x2 + [0]
           c_3(x1, x2) = [0] x1 + [0] x2 + [0]
           c_4(x1, x2) = [0] x1 + [0] x2 + [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: {D^#(t()) -> c_0()}
            Weak Rules:
              {  D^#(+(x, y)) -> c_2(D^#(x), D^#(y))
               , D^#(-(x, y)) -> c_4(D^#(x), D^#(y))
               , D^#(*(x, y)) -> c_3(D^#(x), D^#(y))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {D^#(t()) -> c_0()}
            and weakly orienting the rules
            {  D^#(+(x, y)) -> c_2(D^#(x), D^#(y))
             , D^#(-(x, y)) -> c_4(D^#(x), D^#(y))
             , D^#(*(x, y)) -> c_3(D^#(x), D^#(y))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {D^#(t()) -> c_0()}
              
              Details:
                 Interpretation Functions:
                  D(x1) = [0] x1 + [0]
                  t() = [0]
                  1() = [0]
                  constant() = [0]
                  0() = [0]
                  +(x1, x2) = [1] x1 + [1] x2 + [8]
                  *(x1, x2) = [1] x1 + [1] x2 + [2]
                  -(x1, x2) = [1] x1 + [1] x2 + [8]
                  D^#(x1) = [1] x1 + [1]
                  c_0() = [0]
                  c_1() = [0]
                  c_2(x1, x2) = [1] x1 + [1] x2 + [3]
                  c_3(x1, x2) = [1] x1 + [1] x2 + [1]
                  c_4(x1, x2) = [1] x1 + [1] x2 + [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:
                {  D^#(t()) -> c_0()
                 , D^#(+(x, y)) -> c_2(D^#(x), D^#(y))
                 , D^#(-(x, y)) -> c_4(D^#(x), D^#(y))
                 , D^#(*(x, y)) -> c_3(D^#(x), D^#(y))}
            
            Details:         
              The given problem does not contain any strict rules