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

Details:         
  We have computed the following set of weak (innermost) dependency pairs:
   {  f^#(s(X), X) -> c_0(f^#(X, a(X)))
    , f^#(X, c(X)) -> c_1(f^#(s(X), X))
    , f^#(X, X) -> c_2()}
  
  The usable rules are:
   {}
  
  The estimated dependency graph contains the following edges:
   {f^#(s(X), X) -> c_0(f^#(X, a(X)))}
     ==> {f^#(X, X) -> c_2()}
   {f^#(s(X), X) -> c_0(f^#(X, a(X)))}
     ==> {f^#(s(X), X) -> c_0(f^#(X, a(X)))}
   {f^#(X, c(X)) -> c_1(f^#(s(X), X))}
     ==> {f^#(X, X) -> c_2()}
   {f^#(X, c(X)) -> c_1(f^#(s(X), X))}
     ==> {f^#(X, c(X)) -> c_1(f^#(s(X), X))}
   {f^#(X, c(X)) -> c_1(f^#(s(X), X))}
     ==> {f^#(s(X), X) -> c_0(f^#(X, a(X)))}
  
  We consider the following path(s):
   1) {  f^#(X, c(X)) -> c_1(f^#(s(X), X))
       , f^#(s(X), X) -> c_0(f^#(X, a(X)))
       , f^#(X, X) -> c_2()}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           f(x1, x2) = [0] x1 + [0] x2 + [0]
           s(x1) = [0] x1 + [0]
           a(x1) = [0] x1 + [0]
           c(x1) = [0] x1 + [0]
           f^#(x1, x2) = [0] x1 + [0] x2 + [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: {f^#(X, X) -> c_2()}
            Weak Rules:
              {  f^#(s(X), X) -> c_0(f^#(X, a(X)))
               , f^#(X, c(X)) -> c_1(f^#(s(X), X))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {f^#(X, X) -> c_2()}
            and weakly orienting the rules
            {  f^#(s(X), X) -> c_0(f^#(X, a(X)))
             , f^#(X, c(X)) -> c_1(f^#(s(X), X))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {f^#(X, X) -> c_2()}
              
              Details:
                 Interpretation Functions:
                  f(x1, x2) = [0] x1 + [0] x2 + [0]
                  s(x1) = [1] x1 + [0]
                  a(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [0]
                  f^#(x1, x2) = [1] x1 + [1] x2 + [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:
                {  f^#(X, X) -> c_2()
                 , f^#(s(X), X) -> c_0(f^#(X, a(X)))
                 , f^#(X, c(X)) -> c_1(f^#(s(X), X))}
            
            Details:         
              The given problem does not contain any strict rules
      
   2) {  f^#(X, c(X)) -> c_1(f^#(s(X), X))
       , f^#(s(X), X) -> c_0(f^#(X, a(X)))}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           f(x1, x2) = [0] x1 + [0] x2 + [0]
           s(x1) = [0] x1 + [0]
           a(x1) = [0] x1 + [0]
           c(x1) = [0] x1 + [0]
           f^#(x1, x2) = [0] x1 + [0] x2 + [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: {f^#(s(X), X) -> c_0(f^#(X, a(X)))}
            Weak Rules: {f^#(X, c(X)) -> c_1(f^#(s(X), X))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {f^#(s(X), X) -> c_0(f^#(X, a(X)))}
            and weakly orienting the rules
            {f^#(X, c(X)) -> c_1(f^#(s(X), X))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {f^#(s(X), X) -> c_0(f^#(X, a(X)))}
              
              Details:
                 Interpretation Functions:
                  f(x1, x2) = [0] x1 + [0] x2 + [0]
                  s(x1) = [1] x1 + [8]
                  a(x1) = [1] x1 + [3]
                  c(x1) = [1] x1 + [12]
                  f^#(x1, x2) = [1] x1 + [1] x2 + [9]
                  c_0(x1) = [1] x1 + [4]
                  c_1(x1) = [1] x1 + [3]
                  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:
                {  f^#(s(X), X) -> c_0(f^#(X, a(X)))
                 , f^#(X, c(X)) -> c_1(f^#(s(X), X))}
            
            Details:         
              The given problem does not contain any strict rules
      
   3) {  f^#(X, c(X)) -> c_1(f^#(s(X), X))
       , f^#(X, X) -> c_2()}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           f(x1, x2) = [0] x1 + [0] x2 + [0]
           s(x1) = [0] x1 + [0]
           a(x1) = [0] x1 + [0]
           c(x1) = [0] x1 + [0]
           f^#(x1, x2) = [0] x1 + [0] x2 + [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: {f^#(X, X) -> c_2()}
            Weak Rules: {f^#(X, c(X)) -> c_1(f^#(s(X), X))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {f^#(X, X) -> c_2()}
            and weakly orienting the rules
            {f^#(X, c(X)) -> c_1(f^#(s(X), X))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {f^#(X, X) -> c_2()}
              
              Details:
                 Interpretation Functions:
                  f(x1, x2) = [0] x1 + [0] x2 + [0]
                  s(x1) = [1] x1 + [0]
                  a(x1) = [0] x1 + [0]
                  c(x1) = [1] x1 + [0]
                  f^#(x1, x2) = [1] x1 + [1] x2 + [1]
                  c_0(x1) = [0] 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:
                {  f^#(X, X) -> c_2()
                 , f^#(X, c(X)) -> c_1(f^#(s(X), X))}
            
            Details:         
              The given problem does not contain any strict rules
      
   4) {f^#(X, c(X)) -> c_1(f^#(s(X), X))}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           f(x1, x2) = [0] x1 + [0] x2 + [0]
           s(x1) = [0] x1 + [0]
           a(x1) = [0] x1 + [0]
           c(x1) = [0] x1 + [0]
           f^#(x1, x2) = [0] x1 + [0] x2 + [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: {f^#(X, c(X)) -> c_1(f^#(s(X), X))}
            Weak Rules: {}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {f^#(X, c(X)) -> c_1(f^#(s(X), X))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {f^#(X, c(X)) -> c_1(f^#(s(X), X))}
              
              Details:
                 Interpretation Functions:
                  f(x1, x2) = [0] x1 + [0] x2 + [0]
                  s(x1) = [1] x1 + [3]
                  a(x1) = [0] x1 + [0]
                  c(x1) = [1] x1 + [8]
                  f^#(x1, x2) = [1] x1 + [1] x2 + [9]
                  c_0(x1) = [0] x1 + [0]
                  c_1(x1) = [1] x1 + [4]
                  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: {f^#(X, c(X)) -> c_1(f^#(s(X), X))}
            
            Details:         
              The given problem does not contain any strict rules