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

Details:         
  We have computed the following set of weak (innermost) dependency pairs:
   {  +^#(0(), y) -> c_0()
    , +^#(s(x), y) -> c_1(+^#(x, y))
    , -^#(0(), y) -> c_2()
    , -^#(x, 0()) -> c_3()
    , -^#(s(x), s(y)) -> c_4(-^#(x, y))}
  
  The usable rules are:
   {}
  
  The estimated dependency graph contains the following edges:
   {+^#(s(x), y) -> c_1(+^#(x, y))}
     ==> {+^#(s(x), y) -> c_1(+^#(x, y))}
   {+^#(s(x), y) -> c_1(+^#(x, y))}
     ==> {+^#(0(), y) -> c_0()}
   {-^#(s(x), s(y)) -> c_4(-^#(x, y))}
     ==> {-^#(s(x), s(y)) -> c_4(-^#(x, y))}
   {-^#(s(x), s(y)) -> c_4(-^#(x, y))}
     ==> {-^#(x, 0()) -> c_3()}
   {-^#(s(x), s(y)) -> c_4(-^#(x, y))}
     ==> {-^#(0(), y) -> c_2()}
  
  We consider the following path(s):
   1) {-^#(s(x), s(y)) -> c_4(-^#(x, y))}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           +(x1, x2) = [0] x1 + [0] x2 + [0]
           0() = [0]
           s(x1) = [0] x1 + [0]
           -(x1, x2) = [0] x1 + [0] x2 + [0]
           +^#(x1, x2) = [0] x1 + [0] x2 + [0]
           c_0() = [0]
           c_1(x1) = [0] x1 + [0]
           -^#(x1, x2) = [0] x1 + [0] x2 + [0]
           c_2() = [0]
           c_3() = [0]
           c_4(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: {-^#(s(x), s(y)) -> c_4(-^#(x, y))}
            Weak Rules: {}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {-^#(s(x), s(y)) -> c_4(-^#(x, y))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {-^#(s(x), s(y)) -> c_4(-^#(x, y))}
              
              Details:
                 Interpretation Functions:
                  +(x1, x2) = [0] x1 + [0] x2 + [0]
                  0() = [0]
                  s(x1) = [1] x1 + [8]
                  -(x1, x2) = [0] x1 + [0] x2 + [0]
                  +^#(x1, x2) = [0] x1 + [0] x2 + [0]
                  c_0() = [0]
                  c_1(x1) = [0] x1 + [0]
                  -^#(x1, x2) = [1] x1 + [1] x2 + [0]
                  c_2() = [0]
                  c_3() = [0]
                  c_4(x1) = [1] x1 + [11]
              
            Finally we apply the subprocessor
            'Empty TRS'
            -----------
            Answer:           YES(?,O(1))
            Input Problem:    innermost DP runtime-complexity with respect to
              Strict Rules: {}
              Weak Rules: {-^#(s(x), s(y)) -> c_4(-^#(x, y))}
            
            Details:         
              The given problem does not contain any strict rules
      
   2) {  +^#(s(x), y) -> c_1(+^#(x, y))
       , +^#(0(), y) -> c_0()}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           +(x1, x2) = [0] x1 + [0] x2 + [0]
           0() = [0]
           s(x1) = [0] x1 + [0]
           -(x1, x2) = [0] x1 + [0] x2 + [0]
           +^#(x1, x2) = [0] x1 + [0] x2 + [0]
           c_0() = [0]
           c_1(x1) = [0] x1 + [0]
           -^#(x1, x2) = [0] x1 + [0] x2 + [0]
           c_2() = [0]
           c_3() = [0]
           c_4(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: {+^#(0(), y) -> c_0()}
            Weak Rules: {+^#(s(x), y) -> c_1(+^#(x, y))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {+^#(0(), y) -> c_0()}
            and weakly orienting the rules
            {+^#(s(x), y) -> c_1(+^#(x, y))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {+^#(0(), y) -> c_0()}
              
              Details:
                 Interpretation Functions:
                  +(x1, x2) = [0] x1 + [0] x2 + [0]
                  0() = [0]
                  s(x1) = [1] x1 + [0]
                  -(x1, x2) = [0] x1 + [0] x2 + [0]
                  +^#(x1, x2) = [1] x1 + [1] x2 + [1]
                  c_0() = [0]
                  c_1(x1) = [1] x1 + [0]
                  -^#(x1, x2) = [0] x1 + [0] x2 + [0]
                  c_2() = [0]
                  c_3() = [0]
                  c_4(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:
                {  +^#(0(), y) -> c_0()
                 , +^#(s(x), y) -> c_1(+^#(x, y))}
            
            Details:         
              The given problem does not contain any strict rules
      
   3) {  -^#(s(x), s(y)) -> c_4(-^#(x, y))
       , -^#(x, 0()) -> c_3()}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           +(x1, x2) = [0] x1 + [0] x2 + [0]
           0() = [0]
           s(x1) = [0] x1 + [0]
           -(x1, x2) = [0] x1 + [0] x2 + [0]
           +^#(x1, x2) = [0] x1 + [0] x2 + [0]
           c_0() = [0]
           c_1(x1) = [0] x1 + [0]
           -^#(x1, x2) = [0] x1 + [0] x2 + [0]
           c_2() = [0]
           c_3() = [0]
           c_4(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: {-^#(x, 0()) -> c_3()}
            Weak Rules: {-^#(s(x), s(y)) -> c_4(-^#(x, y))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {-^#(x, 0()) -> c_3()}
            and weakly orienting the rules
            {-^#(s(x), s(y)) -> c_4(-^#(x, y))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {-^#(x, 0()) -> c_3()}
              
              Details:
                 Interpretation Functions:
                  +(x1, x2) = [0] x1 + [0] x2 + [0]
                  0() = [0]
                  s(x1) = [1] x1 + [0]
                  -(x1, x2) = [0] x1 + [0] x2 + [0]
                  +^#(x1, x2) = [0] x1 + [0] x2 + [0]
                  c_0() = [0]
                  c_1(x1) = [0] x1 + [0]
                  -^#(x1, x2) = [1] x1 + [1] x2 + [1]
                  c_2() = [0]
                  c_3() = [0]
                  c_4(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:
                {  -^#(x, 0()) -> c_3()
                 , -^#(s(x), s(y)) -> c_4(-^#(x, y))}
            
            Details:         
              The given problem does not contain any strict rules
      
   4) {  -^#(s(x), s(y)) -> c_4(-^#(x, y))
       , -^#(0(), y) -> c_2()}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           +(x1, x2) = [0] x1 + [0] x2 + [0]
           0() = [0]
           s(x1) = [0] x1 + [0]
           -(x1, x2) = [0] x1 + [0] x2 + [0]
           +^#(x1, x2) = [0] x1 + [0] x2 + [0]
           c_0() = [0]
           c_1(x1) = [0] x1 + [0]
           -^#(x1, x2) = [0] x1 + [0] x2 + [0]
           c_2() = [0]
           c_3() = [0]
           c_4(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: {-^#(0(), y) -> c_2()}
            Weak Rules: {-^#(s(x), s(y)) -> c_4(-^#(x, y))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {-^#(0(), y) -> c_2()}
            and weakly orienting the rules
            {-^#(s(x), s(y)) -> c_4(-^#(x, y))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {-^#(0(), y) -> c_2()}
              
              Details:
                 Interpretation Functions:
                  +(x1, x2) = [0] x1 + [0] x2 + [0]
                  0() = [0]
                  s(x1) = [1] x1 + [0]
                  -(x1, x2) = [0] x1 + [0] x2 + [0]
                  +^#(x1, x2) = [0] x1 + [0] x2 + [0]
                  c_0() = [0]
                  c_1(x1) = [0] x1 + [0]
                  -^#(x1, x2) = [1] x1 + [1] x2 + [1]
                  c_2() = [0]
                  c_3() = [0]
                  c_4(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:
                {  -^#(0(), y) -> c_2()
                 , -^#(s(x), s(y)) -> c_4(-^#(x, y))}
            
            Details:         
              The given problem does not contain any strict rules
      
   5) {+^#(s(x), y) -> c_1(+^#(x, y))}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           +(x1, x2) = [0] x1 + [0] x2 + [0]
           0() = [0]
           s(x1) = [0] x1 + [0]
           -(x1, x2) = [0] x1 + [0] x2 + [0]
           +^#(x1, x2) = [0] x1 + [0] x2 + [0]
           c_0() = [0]
           c_1(x1) = [0] x1 + [0]
           -^#(x1, x2) = [0] x1 + [0] x2 + [0]
           c_2() = [0]
           c_3() = [0]
           c_4(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: {+^#(s(x), y) -> c_1(+^#(x, y))}
            Weak Rules: {}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {+^#(s(x), y) -> c_1(+^#(x, y))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {+^#(s(x), y) -> c_1(+^#(x, y))}
              
              Details:
                 Interpretation Functions:
                  +(x1, x2) = [0] x1 + [0] x2 + [0]
                  0() = [0]
                  s(x1) = [1] x1 + [8]
                  -(x1, x2) = [0] x1 + [0] x2 + [0]
                  +^#(x1, x2) = [1] x1 + [1] x2 + [1]
                  c_0() = [0]
                  c_1(x1) = [1] x1 + [3]
                  -^#(x1, x2) = [0] x1 + [0] x2 + [0]
                  c_2() = [0]
                  c_3() = [0]
                  c_4(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: {+^#(s(x), y) -> c_1(+^#(x, y))}
            
            Details:         
              The given problem does not contain any strict rules