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

Details:         
  We have computed the following set of weak (innermost) dependency pairs:
   {  f^#(f(x)) -> c_0(f^#(c(f(x))))
    , f^#(f(x)) -> c_1(f^#(d(f(x))))
    , g^#(c(x)) -> c_2()
    , g^#(d(x)) -> c_3()
    , g^#(c(0())) -> c_4(g^#(d(1())))
    , g^#(c(1())) -> c_5(g^#(d(0())))}
  
  The usable rules are:
   {  f(f(x)) -> f(c(f(x)))
    , f(f(x)) -> f(d(f(x)))}
  
  The estimated dependency graph contains the following edges:
   {g^#(c(0())) -> c_4(g^#(d(1())))}
     ==> {g^#(d(x)) -> c_3()}
   {g^#(c(1())) -> c_5(g^#(d(0())))}
     ==> {g^#(d(x)) -> c_3()}
  
  We consider the following path(s):
   1) {f^#(f(x)) -> c_0(f^#(c(f(x))))}
      
      The usable rules for this path are the following:
      {  f(f(x)) -> f(c(f(x)))
       , f(f(x)) -> f(d(f(x)))}
      
        We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
        
          'Weight Gap Principle'
          ----------------------
          Answer:           YES(?,O(n^1))
          Input Problem:    innermost runtime-complexity with respect to
            Rules:
              {  f(f(x)) -> f(c(f(x)))
               , f(f(x)) -> f(d(f(x)))
               , f^#(f(x)) -> c_0(f^#(c(f(x))))}
          
          Details:         
            'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
            ------------------------------------------------------------------------------------------
            Answer:           YES(?,O(n^1))
            Input Problem:    innermost runtime-complexity with respect to
              Rules:
                {  f(f(x)) -> f(c(f(x)))
                 , f(f(x)) -> f(d(f(x)))
                 , f^#(f(x)) -> c_0(f^#(c(f(x))))}
            
            Details:         
              The problem was solved by processor 'Bounds with default enrichment':
              'Bounds with default enrichment'
              --------------------------------
              Answer:           YES(?,O(n^1))
              Input Problem:    innermost runtime-complexity with respect to
                Rules:
                  {  f(f(x)) -> f(c(f(x)))
                   , f(f(x)) -> f(d(f(x)))
                   , f^#(f(x)) -> c_0(f^#(c(f(x))))}
              
              Details:         
                The problem is Match-bounded by 0.
                The enriched problem is compatible with the following automaton:
                {  c_0(2) -> 2
                 , c_0(3) -> 2
                 , d_0(2) -> 3
                 , d_0(3) -> 3
                 , f^#_0(2) -> 7
                 , f^#_0(3) -> 7}
      
   2) {f^#(f(x)) -> c_1(f^#(d(f(x))))}
      
      The usable rules for this path are the following:
      {  f(f(x)) -> f(c(f(x)))
       , f(f(x)) -> f(d(f(x)))}
      
        We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
        
          'Weight Gap Principle'
          ----------------------
          Answer:           YES(?,O(n^1))
          Input Problem:    innermost runtime-complexity with respect to
            Rules:
              {  f(f(x)) -> f(c(f(x)))
               , f(f(x)) -> f(d(f(x)))
               , f^#(f(x)) -> c_1(f^#(d(f(x))))}
          
          Details:         
            'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
            ------------------------------------------------------------------------------------------
            Answer:           YES(?,O(n^1))
            Input Problem:    innermost runtime-complexity with respect to
              Rules:
                {  f(f(x)) -> f(c(f(x)))
                 , f(f(x)) -> f(d(f(x)))
                 , f^#(f(x)) -> c_1(f^#(d(f(x))))}
            
            Details:         
              The problem was solved by processor 'Bounds with default enrichment':
              'Bounds with default enrichment'
              --------------------------------
              Answer:           YES(?,O(n^1))
              Input Problem:    innermost runtime-complexity with respect to
                Rules:
                  {  f(f(x)) -> f(c(f(x)))
                   , f(f(x)) -> f(d(f(x)))
                   , f^#(f(x)) -> c_1(f^#(d(f(x))))}
              
              Details:         
                The problem is Match-bounded by 0.
                The enriched problem is compatible with the following automaton:
                {  c_0(2) -> 2
                 , d_0(2) -> 2
                 , f^#_0(2) -> 1}
      
   3) {g^#(c(1())) -> c_5(g^#(d(0())))}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           f(x1) = [0] x1 + [0]
           c(x1) = [0] x1 + [0]
           d(x1) = [0] x1 + [0]
           g(x1) = [0] x1 + [0]
           0() = [0]
           1() = [0]
           f^#(x1) = [0] x1 + [0]
           c_0(x1) = [0] x1 + [0]
           c_1(x1) = [0] x1 + [0]
           g^#(x1) = [0] x1 + [0]
           c_2() = [0]
           c_3() = [0]
           c_4(x1) = [0] x1 + [0]
           c_5(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: {g^#(c(1())) -> c_5(g^#(d(0())))}
            Weak Rules: {}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {g^#(c(1())) -> c_5(g^#(d(0())))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {g^#(c(1())) -> c_5(g^#(d(0())))}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [0] x1 + [0]
                  c(x1) = [1] x1 + [8]
                  d(x1) = [1] x1 + [0]
                  g(x1) = [0] x1 + [0]
                  0() = [4]
                  1() = [0]
                  f^#(x1) = [0] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  c_1(x1) = [0] x1 + [0]
                  g^#(x1) = [1] x1 + [12]
                  c_2() = [0]
                  c_3() = [0]
                  c_4(x1) = [0] x1 + [0]
                  c_5(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: {g^#(c(1())) -> c_5(g^#(d(0())))}
            
            Details:         
              The given problem does not contain any strict rules
      
   4) {  g^#(c(1())) -> c_5(g^#(d(0())))
       , g^#(d(x)) -> c_3()}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           f(x1) = [0] x1 + [0]
           c(x1) = [0] x1 + [0]
           d(x1) = [0] x1 + [0]
           g(x1) = [0] x1 + [0]
           0() = [0]
           1() = [0]
           f^#(x1) = [0] x1 + [0]
           c_0(x1) = [0] x1 + [0]
           c_1(x1) = [0] x1 + [0]
           g^#(x1) = [0] x1 + [0]
           c_2() = [0]
           c_3() = [0]
           c_4(x1) = [0] x1 + [0]
           c_5(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: {g^#(d(x)) -> c_3()}
            Weak Rules: {g^#(c(1())) -> c_5(g^#(d(0())))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {g^#(d(x)) -> c_3()}
            and weakly orienting the rules
            {g^#(c(1())) -> c_5(g^#(d(0())))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {g^#(d(x)) -> c_3()}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [0] x1 + [0]
                  c(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [0]
                  g(x1) = [0] x1 + [0]
                  0() = [0]
                  1() = [0]
                  f^#(x1) = [0] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  c_1(x1) = [0] x1 + [0]
                  g^#(x1) = [1] x1 + [1]
                  c_2() = [0]
                  c_3() = [0]
                  c_4(x1) = [0] x1 + [0]
                  c_5(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:
                {  g^#(d(x)) -> c_3()
                 , g^#(c(1())) -> c_5(g^#(d(0())))}
            
            Details:         
              The given problem does not contain any strict rules
      
   5) {g^#(c(0())) -> c_4(g^#(d(1())))}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           f(x1) = [0] x1 + [0]
           c(x1) = [0] x1 + [0]
           d(x1) = [0] x1 + [0]
           g(x1) = [0] x1 + [0]
           0() = [0]
           1() = [0]
           f^#(x1) = [0] x1 + [0]
           c_0(x1) = [0] x1 + [0]
           c_1(x1) = [0] x1 + [0]
           g^#(x1) = [0] x1 + [0]
           c_2() = [0]
           c_3() = [0]
           c_4(x1) = [0] x1 + [0]
           c_5(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: {g^#(c(0())) -> c_4(g^#(d(1())))}
            Weak Rules: {}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {g^#(c(0())) -> c_4(g^#(d(1())))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {g^#(c(0())) -> c_4(g^#(d(1())))}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [0] x1 + [0]
                  c(x1) = [1] x1 + [8]
                  d(x1) = [1] x1 + [0]
                  g(x1) = [0] x1 + [0]
                  0() = [0]
                  1() = [4]
                  f^#(x1) = [0] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  c_1(x1) = [0] x1 + [0]
                  g^#(x1) = [1] x1 + [12]
                  c_2() = [0]
                  c_3() = [0]
                  c_4(x1) = [1] x1 + [0]
                  c_5(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: {g^#(c(0())) -> c_4(g^#(d(1())))}
            
            Details:         
              The given problem does not contain any strict rules
      
   6) {  g^#(c(0())) -> c_4(g^#(d(1())))
       , g^#(d(x)) -> c_3()}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           f(x1) = [0] x1 + [0]
           c(x1) = [0] x1 + [0]
           d(x1) = [0] x1 + [0]
           g(x1) = [0] x1 + [0]
           0() = [0]
           1() = [0]
           f^#(x1) = [0] x1 + [0]
           c_0(x1) = [0] x1 + [0]
           c_1(x1) = [0] x1 + [0]
           g^#(x1) = [0] x1 + [0]
           c_2() = [0]
           c_3() = [0]
           c_4(x1) = [0] x1 + [0]
           c_5(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: {g^#(d(x)) -> c_3()}
            Weak Rules: {g^#(c(0())) -> c_4(g^#(d(1())))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {g^#(d(x)) -> c_3()}
            and weakly orienting the rules
            {g^#(c(0())) -> c_4(g^#(d(1())))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {g^#(d(x)) -> c_3()}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [0] x1 + [0]
                  c(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [0]
                  g(x1) = [0] x1 + [0]
                  0() = [0]
                  1() = [0]
                  f^#(x1) = [0] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  c_1(x1) = [0] x1 + [0]
                  g^#(x1) = [1] x1 + [1]
                  c_2() = [0]
                  c_3() = [0]
                  c_4(x1) = [1] x1 + [0]
                  c_5(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:
                {  g^#(d(x)) -> c_3()
                 , g^#(c(0())) -> c_4(g^#(d(1())))}
            
            Details:         
              The given problem does not contain any strict rules
      
   7) {g^#(c(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) = [0] x1 + [0]
           c(x1) = [0] x1 + [0]
           d(x1) = [0] x1 + [0]
           g(x1) = [0] x1 + [0]
           0() = [0]
           1() = [0]
           f^#(x1) = [0] x1 + [0]
           c_0(x1) = [0] x1 + [0]
           c_1(x1) = [0] x1 + [0]
           g^#(x1) = [0] x1 + [0]
           c_2() = [0]
           c_3() = [0]
           c_4(x1) = [0] x1 + [0]
           c_5(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: {g^#(c(x)) -> c_2()}
            Weak Rules: {}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {g^#(c(x)) -> c_2()}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {g^#(c(x)) -> c_2()}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [0] x1 + [0]
                  c(x1) = [1] x1 + [0]
                  d(x1) = [0] x1 + [0]
                  g(x1) = [0] x1 + [0]
                  0() = [0]
                  1() = [0]
                  f^#(x1) = [0] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  c_1(x1) = [0] x1 + [0]
                  g^#(x1) = [1] x1 + [1]
                  c_2() = [0]
                  c_3() = [0]
                  c_4(x1) = [0] x1 + [0]
                  c_5(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: {g^#(c(x)) -> c_2()}
            
            Details:         
              The given problem does not contain any strict rules