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

Details:         
  We have computed the following set of weak (innermost) dependency pairs:
   {  p^#(m, n, s(r)) -> c_0(p^#(m, r, n))
    , p^#(m, s(n), 0()) -> c_1(p^#(0(), n, m))
    , p^#(m, 0(), 0()) -> c_2()}
  
  The usable rules are:
   {}
  
  The estimated dependency graph contains the following edges:
   {p^#(m, n, s(r)) -> c_0(p^#(m, r, n))}
     ==> {p^#(m, s(n), 0()) -> c_1(p^#(0(), n, m))}
   {p^#(m, n, s(r)) -> c_0(p^#(m, r, n))}
     ==> {p^#(m, 0(), 0()) -> c_2()}
   {p^#(m, n, s(r)) -> c_0(p^#(m, r, n))}
     ==> {p^#(m, n, s(r)) -> c_0(p^#(m, r, n))}
   {p^#(m, s(n), 0()) -> c_1(p^#(0(), n, m))}
     ==> {p^#(m, 0(), 0()) -> c_2()}
   {p^#(m, s(n), 0()) -> c_1(p^#(0(), n, m))}
     ==> {p^#(m, s(n), 0()) -> c_1(p^#(0(), n, m))}
   {p^#(m, s(n), 0()) -> c_1(p^#(0(), n, m))}
     ==> {p^#(m, n, s(r)) -> c_0(p^#(m, r, n))}
  
  We consider the following path(s):
   1) {  p^#(m, n, s(r)) -> c_0(p^#(m, r, n))
       , p^#(m, s(n), 0()) -> c_1(p^#(0(), n, m))}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           p(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
           s(x1) = [0] x1 + [0]
           0() = [0]
           p^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
           c_0(x1) = [0] x1 + [0]
           c_1(x1) = [0] x1 + [0]
           c_2() = [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^#(m, n, s(r)) -> c_0(p^#(m, r, n))
               , p^#(m, s(n), 0()) -> c_1(p^#(0(), n, m))}
            Weak Rules: {}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {  p^#(m, n, s(r)) -> c_0(p^#(m, r, n))
             , p^#(m, s(n), 0()) -> c_1(p^#(0(), n, m))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  p^#(m, n, s(r)) -> c_0(p^#(m, r, n))
               , p^#(m, s(n), 0()) -> c_1(p^#(0(), n, m))}
              
              Details:
                 Interpretation Functions:
                  p(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                  s(x1) = [1] x1 + [8]
                  0() = [8]
                  p^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
                  c_0(x1) = [1] x1 + [0]
                  c_1(x1) = [1] x1 + [0]
                  c_2() = [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^#(m, n, s(r)) -> c_0(p^#(m, r, n))
                 , p^#(m, s(n), 0()) -> c_1(p^#(0(), n, m))}
            
            Details:         
              The given problem does not contain any strict rules
      
   2) {  p^#(m, n, s(r)) -> c_0(p^#(m, r, n))
       , p^#(m, s(n), 0()) -> c_1(p^#(0(), n, m))
       , p^#(m, 0(), 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:
           p(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
           s(x1) = [0] x1 + [0]
           0() = [0]
           p^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
           c_0(x1) = [0] x1 + [0]
           c_1(x1) = [0] x1 + [0]
           c_2() = [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^#(m, 0(), 0()) -> c_2()}
            Weak Rules:
              {  p^#(m, n, s(r)) -> c_0(p^#(m, r, n))
               , p^#(m, s(n), 0()) -> c_1(p^#(0(), n, m))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {p^#(m, 0(), 0()) -> c_2()}
            and weakly orienting the rules
            {  p^#(m, n, s(r)) -> c_0(p^#(m, r, n))
             , p^#(m, s(n), 0()) -> c_1(p^#(0(), n, m))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {p^#(m, 0(), 0()) -> c_2()}
              
              Details:
                 Interpretation Functions:
                  p(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                  s(x1) = [1] x1 + [0]
                  0() = [0]
                  p^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
                  c_0(x1) = [1] x1 + [0]
                  c_1(x1) = [1] x1 + [0]
                  c_2() = [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^#(m, 0(), 0()) -> c_2()
                 , p^#(m, n, s(r)) -> c_0(p^#(m, r, n))
                 , p^#(m, s(n), 0()) -> c_1(p^#(0(), n, m))}
            
            Details:         
              The given problem does not contain any strict rules