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

Details:         
  We have computed the following set of weak (innermost) dependency pairs:
   {  p^#(0()) -> c_0()
    , p^#(s(x)) -> c_1()
    , minus^#(x, 0()) -> c_2()
    , minus^#(x, s(y)) -> c_3(minus^#(p(x), y))}
  
  The usable rules are:
   {  p(0()) -> 0()
    , p(s(x)) -> x}
  
  The estimated dependency graph contains the following edges:
   {minus^#(x, s(y)) -> c_3(minus^#(p(x), y))}
     ==> {minus^#(x, s(y)) -> c_3(minus^#(p(x), y))}
   {minus^#(x, s(y)) -> c_3(minus^#(p(x), y))}
     ==> {minus^#(x, 0()) -> c_2()}
  
  We consider the following path(s):
   1) {minus^#(x, s(y)) -> c_3(minus^#(p(x), y))}
      
      The usable rules for this path are the following:
      {  p(0()) -> 0()
       , p(s(x)) -> x}
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           p(x1) = [1] x1 + [1]
           0() = [0]
           s(x1) = [1] x1 + [0]
           minus(x1, x2) = [0] x1 + [0] x2 + [0]
           p^#(x1) = [0] x1 + [0]
           c_0() = [0]
           c_1() = [0]
           minus^#(x1, x2) = [0] x1 + [0] x2 + [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: {minus^#(x, s(y)) -> c_3(minus^#(p(x), y))}
            Weak Rules:
              {  p(0()) -> 0()
               , p(s(x)) -> x}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {minus^#(x, s(y)) -> c_3(minus^#(p(x), y))}
            and weakly orienting the rules
            {  p(0()) -> 0()
             , p(s(x)) -> x}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {minus^#(x, s(y)) -> c_3(minus^#(p(x), y))}
              
              Details:
                 Interpretation Functions:
                  p(x1) = [1] x1 + [3]
                  0() = [12]
                  s(x1) = [1] x1 + [8]
                  minus(x1, x2) = [0] x1 + [0] x2 + [0]
                  p^#(x1) = [0] x1 + [0]
                  c_0() = [0]
                  c_1() = [0]
                  minus^#(x1, x2) = [1] x1 + [1] x2 + [9]
                  c_2() = [0]
                  c_3(x1) = [1] x1 + [4]
              
            Finally we apply the subprocessor
            'Empty TRS'
            -----------
            Answer:           YES(?,O(1))
            Input Problem:    innermost DP runtime-complexity with respect to
              Strict Rules: {}
              Weak Rules:
                {  minus^#(x, s(y)) -> c_3(minus^#(p(x), y))
                 , p(0()) -> 0()
                 , p(s(x)) -> x}
            
            Details:         
              The given problem does not contain any strict rules
      
   2) {  minus^#(x, s(y)) -> c_3(minus^#(p(x), y))
       , minus^#(x, 0()) -> c_2()}
      
      The usable rules for this path are the following:
      {  p(0()) -> 0()
       , p(s(x)) -> x}
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           p(x1) = [1] x1 + [1]
           0() = [0]
           s(x1) = [1] x1 + [0]
           minus(x1, x2) = [0] x1 + [0] x2 + [0]
           p^#(x1) = [0] x1 + [0]
           c_0() = [0]
           c_1() = [0]
           minus^#(x1, x2) = [0] x1 + [0] x2 + [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: {minus^#(x, 0()) -> c_2()}
            Weak Rules:
              {  p(0()) -> 0()
               , p(s(x)) -> x
               , minus^#(x, s(y)) -> c_3(minus^#(p(x), y))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {minus^#(x, 0()) -> c_2()}
            and weakly orienting the rules
            {  p(0()) -> 0()
             , p(s(x)) -> x
             , minus^#(x, s(y)) -> c_3(minus^#(p(x), y))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {minus^#(x, 0()) -> c_2()}
              
              Details:
                 Interpretation Functions:
                  p(x1) = [1] x1 + [1]
                  0() = [0]
                  s(x1) = [1] x1 + [8]
                  minus(x1, x2) = [0] x1 + [0] x2 + [0]
                  p^#(x1) = [0] x1 + [0]
                  c_0() = [0]
                  c_1() = [0]
                  minus^#(x1, x2) = [1] x1 + [1] x2 + [1]
                  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:
                {  minus^#(x, 0()) -> c_2()
                 , p(0()) -> 0()
                 , p(s(x)) -> x
                 , minus^#(x, s(y)) -> c_3(minus^#(p(x), y))}
            
            Details:         
              The given problem does not contain any strict rules
      
   3) {p^#(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:
           p(x1) = [0] x1 + [0]
           0() = [0]
           s(x1) = [0] x1 + [0]
           minus(x1, x2) = [0] x1 + [0] x2 + [0]
           p^#(x1) = [0] x1 + [0]
           c_0() = [0]
           c_1() = [0]
           minus^#(x1, x2) = [0] x1 + [0] x2 + [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: {p^#(0()) -> c_0()}
            Weak Rules: {}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {p^#(0()) -> c_0()}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {p^#(0()) -> c_0()}
              
              Details:
                 Interpretation Functions:
                  p(x1) = [0] x1 + [0]
                  0() = [0]
                  s(x1) = [0] x1 + [0]
                  minus(x1, x2) = [0] x1 + [0] x2 + [0]
                  p^#(x1) = [1] x1 + [1]
                  c_0() = [0]
                  c_1() = [0]
                  minus^#(x1, x2) = [0] x1 + [0] x2 + [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: {p^#(0()) -> c_0()}
            
            Details:         
              The given problem does not contain any strict rules
      
   4) {p^#(s(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:
           p(x1) = [0] x1 + [0]
           0() = [0]
           s(x1) = [0] x1 + [0]
           minus(x1, x2) = [0] x1 + [0] x2 + [0]
           p^#(x1) = [0] x1 + [0]
           c_0() = [0]
           c_1() = [0]
           minus^#(x1, x2) = [0] x1 + [0] x2 + [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: {p^#(s(x)) -> c_1()}
            Weak Rules: {}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {p^#(s(x)) -> c_1()}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {p^#(s(x)) -> c_1()}
              
              Details:
                 Interpretation Functions:
                  p(x1) = [0] x1 + [0]
                  0() = [0]
                  s(x1) = [1] x1 + [0]
                  minus(x1, x2) = [0] x1 + [0] x2 + [0]
                  p^#(x1) = [1] x1 + [1]
                  c_0() = [0]
                  c_1() = [0]
                  minus^#(x1, x2) = [0] x1 + [0] x2 + [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: {p^#(s(x)) -> c_1()}
            
            Details:         
              The given problem does not contain any strict rules