'Weak Dependency Graph [60.0]'
------------------------------
Answer:           YES(?,O(n^1))
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  and(tt(), X) -> X
     , plus(N, 0()) -> N
     , plus(N, s(M)) -> s(plus(N, M))}

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