'Weak Dependency Graph [60.0]'
------------------------------
Answer:           YES(?,O(n^1))
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  a(b(x1)) -> d(x1)
     , b(a(x1)) -> a(b(x1))
     , d(c(x1)) -> f(a(b(b(c(x1)))))
     , d(f(x1)) -> f(a(b(x1)))
     , a(f(x1)) -> a(x1)}

Details:         
  We have computed the following set of weak (innermost) dependency pairs:
   {  a^#(b(x1)) -> c_0(d^#(x1))
    , b^#(a(x1)) -> c_1(a^#(b(x1)))
    , d^#(c(x1)) -> c_2(a^#(b(b(c(x1)))))
    , d^#(f(x1)) -> c_3(a^#(b(x1)))
    , a^#(f(x1)) -> c_4(a^#(x1))}
  
  The usable rules are:
   {  b(a(x1)) -> a(b(x1))
    , a(b(x1)) -> d(x1)
    , a(f(x1)) -> a(x1)
    , d(c(x1)) -> f(a(b(b(c(x1)))))
    , d(f(x1)) -> f(a(b(x1)))}
  
  The estimated dependency graph contains the following edges:
   {a^#(b(x1)) -> c_0(d^#(x1))}
     ==> {d^#(f(x1)) -> c_3(a^#(b(x1)))}
   {a^#(b(x1)) -> c_0(d^#(x1))}
     ==> {d^#(c(x1)) -> c_2(a^#(b(b(c(x1)))))}
   {b^#(a(x1)) -> c_1(a^#(b(x1)))}
     ==> {a^#(f(x1)) -> c_4(a^#(x1))}
   {b^#(a(x1)) -> c_1(a^#(b(x1)))}
     ==> {a^#(b(x1)) -> c_0(d^#(x1))}
   {d^#(c(x1)) -> c_2(a^#(b(b(c(x1)))))}
     ==> {a^#(b(x1)) -> c_0(d^#(x1))}
   {d^#(f(x1)) -> c_3(a^#(b(x1)))}
     ==> {a^#(f(x1)) -> c_4(a^#(x1))}
   {d^#(f(x1)) -> c_3(a^#(b(x1)))}
     ==> {a^#(b(x1)) -> c_0(d^#(x1))}
   {a^#(f(x1)) -> c_4(a^#(x1))}
     ==> {a^#(f(x1)) -> c_4(a^#(x1))}
   {a^#(f(x1)) -> c_4(a^#(x1))}
     ==> {a^#(b(x1)) -> c_0(d^#(x1))}
  
  We consider the following path(s):
   1) {  b^#(a(x1)) -> c_1(a^#(b(x1)))
       , a^#(b(x1)) -> c_0(d^#(x1))
       , a^#(f(x1)) -> c_4(a^#(x1))
       , d^#(f(x1)) -> c_3(a^#(b(x1)))
       , d^#(c(x1)) -> c_2(a^#(b(b(c(x1)))))}
      
      The usable rules for this path are the following:
      {  b(a(x1)) -> a(b(x1))
       , a(b(x1)) -> d(x1)
       , a(f(x1)) -> a(x1)
       , d(c(x1)) -> f(a(b(b(c(x1)))))
       , d(f(x1)) -> f(a(b(x1)))}
      
        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:
              {  b(a(x1)) -> a(b(x1))
               , a(b(x1)) -> d(x1)
               , a(f(x1)) -> a(x1)
               , d(c(x1)) -> f(a(b(b(c(x1)))))
               , d(f(x1)) -> f(a(b(x1)))
               , b^#(a(x1)) -> c_1(a^#(b(x1)))
               , a^#(b(x1)) -> c_0(d^#(x1))
               , a^#(f(x1)) -> c_4(a^#(x1))
               , d^#(f(x1)) -> c_3(a^#(b(x1)))
               , d^#(c(x1)) -> c_2(a^#(b(b(c(x1)))))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {  d^#(f(x1)) -> c_3(a^#(b(x1)))
             , d^#(c(x1)) -> c_2(a^#(b(b(c(x1)))))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  d^#(f(x1)) -> c_3(a^#(b(x1)))
               , d^#(c(x1)) -> c_2(a^#(b(b(c(x1)))))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [1]
                  d(x1) = [1] x1 + [1]
                  c(x1) = [1] x1 + [0]
                  f(x1) = [1] x1 + [0]
                  a^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [1]
                  d^#(x1) = [1] x1 + [8]
                  b^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [0]
                  c_2(x1) = [1] x1 + [1]
                  c_3(x1) = [1] x1 + [0]
                  c_4(x1) = [1] x1 + [1]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {a(b(x1)) -> d(x1)}
            and weakly orienting the rules
            {  d^#(f(x1)) -> c_3(a^#(b(x1)))
             , d^#(c(x1)) -> c_2(a^#(b(b(c(x1)))))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {a(b(x1)) -> d(x1)}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [1]
                  d(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [0]
                  f(x1) = [1] x1 + [0]
                  a^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [1]
                  d^#(x1) = [1] x1 + [4]
                  b^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [2]
                  c_2(x1) = [1] x1 + [2]
                  c_3(x1) = [1] x1 + [1]
                  c_4(x1) = [1] x1 + [12]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {b^#(a(x1)) -> c_1(a^#(b(x1)))}
            and weakly orienting the rules
            {  a(b(x1)) -> d(x1)
             , d^#(f(x1)) -> c_3(a^#(b(x1)))
             , d^#(c(x1)) -> c_2(a^#(b(b(c(x1)))))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {b^#(a(x1)) -> c_1(a^#(b(x1)))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [1]
                  d(x1) = [1] x1 + [1]
                  c(x1) = [1] x1 + [0]
                  f(x1) = [1] x1 + [0]
                  a^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [0]
                  d^#(x1) = [1] x1 + [3]
                  b^#(x1) = [1] x1 + [5]
                  c_1(x1) = [1] x1 + [0]
                  c_2(x1) = [1] x1 + [1]
                  c_3(x1) = [1] x1 + [0]
                  c_4(x1) = [1] x1 + [4]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {  a(f(x1)) -> a(x1)
             , a^#(f(x1)) -> c_4(a^#(x1))}
            and weakly orienting the rules
            {  b^#(a(x1)) -> c_1(a^#(b(x1)))
             , a(b(x1)) -> d(x1)
             , d^#(f(x1)) -> c_3(a^#(b(x1)))
             , d^#(c(x1)) -> c_2(a^#(b(b(c(x1)))))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  a(f(x1)) -> a(x1)
               , a^#(f(x1)) -> c_4(a^#(x1))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [1]
                  d(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [1]
                  f(x1) = [1] x1 + [12]
                  a^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [0]
                  d^#(x1) = [1] x1 + [3]
                  b^#(x1) = [1] x1 + [9]
                  c_1(x1) = [1] x1 + [4]
                  c_2(x1) = [1] x1 + [0]
                  c_3(x1) = [1] x1 + [0]
                  c_4(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
            ------------------------------------------------------------------------------------------
            Answer:           YES(?,O(n^1))
            Input Problem:    innermost relative runtime-complexity with respect to
              Strict Rules:
                {  b(a(x1)) -> a(b(x1))
                 , d(c(x1)) -> f(a(b(b(c(x1)))))
                 , d(f(x1)) -> f(a(b(x1)))
                 , a^#(b(x1)) -> c_0(d^#(x1))}
              Weak Rules:
                {  a(f(x1)) -> a(x1)
                 , a^#(f(x1)) -> c_4(a^#(x1))
                 , b^#(a(x1)) -> c_1(a^#(b(x1)))
                 , a(b(x1)) -> d(x1)
                 , d^#(f(x1)) -> c_3(a^#(b(x1)))
                 , d^#(c(x1)) -> c_2(a^#(b(b(c(x1)))))}
            
            Details:         
              The problem was solved by processor 'Bounds with default enrichment':
              'Bounds with default enrichment'
              --------------------------------
              Answer:           YES(?,O(n^1))
              Input Problem:    innermost relative runtime-complexity with respect to
                Strict Rules:
                  {  b(a(x1)) -> a(b(x1))
                   , d(c(x1)) -> f(a(b(b(c(x1)))))
                   , d(f(x1)) -> f(a(b(x1)))
                   , a^#(b(x1)) -> c_0(d^#(x1))}
                Weak Rules:
                  {  a(f(x1)) -> a(x1)
                   , a^#(f(x1)) -> c_4(a^#(x1))
                   , b^#(a(x1)) -> c_1(a^#(b(x1)))
                   , a(b(x1)) -> d(x1)
                   , d^#(f(x1)) -> c_3(a^#(b(x1)))
                   , d^#(c(x1)) -> c_2(a^#(b(b(c(x1)))))}
              
              Details:         
                The problem is Match-bounded by 2.
                The enriched problem is compatible with the following automaton:
                {  b_0(2) -> 4
                 , b_0(4) -> 6
                 , b_1(2) -> 10
                 , b_1(13) -> 12
                 , b_1(14) -> 13
                 , c_0(2) -> 2
                 , c_1(2) -> 14
                 , f_0(2) -> 2
                 , a^#_0(2) -> 1
                 , a^#_0(4) -> 3
                 , a^#_0(6) -> 5
                 , a^#_1(10) -> 9
                 , a^#_1(12) -> 11
                 , c_0_1(7) -> 3
                 , c_0_1(8) -> 5
                 , c_0_2(15) -> 9
                 , c_0_2(16) -> 11
                 , d^#_0(2) -> 1
                 , d^#_1(2) -> 7
                 , d^#_1(4) -> 8
                 , d^#_2(2) -> 15
                 , d^#_2(13) -> 16
                 , b^#_0(2) -> 1
                 , c_2_0(5) -> 1
                 , c_2_1(11) -> 1
                 , c_2_1(11) -> 7
                 , c_2_1(11) -> 15
                 , c_3_0(3) -> 1
                 , c_3_1(9) -> 1
                 , c_3_1(9) -> 7
                 , c_3_1(9) -> 15
                 , c_4_0(1) -> 1}
      
   2) {b^#(a(x1)) -> c_1(a^#(b(x1)))}
      
      The usable rules for this path are the following:
      {  b(a(x1)) -> a(b(x1))
       , a(b(x1)) -> d(x1)
       , a(f(x1)) -> a(x1)
       , d(c(x1)) -> f(a(b(b(c(x1)))))
       , d(f(x1)) -> f(a(b(x1)))}
      
        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:
              {  b(a(x1)) -> a(b(x1))
               , a(b(x1)) -> d(x1)
               , a(f(x1)) -> a(x1)
               , d(c(x1)) -> f(a(b(b(c(x1)))))
               , d(f(x1)) -> f(a(b(x1)))
               , b^#(a(x1)) -> c_1(a^#(b(x1)))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {b^#(a(x1)) -> c_1(a^#(b(x1)))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {b^#(a(x1)) -> c_1(a^#(b(x1)))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [1]
                  d(x1) = [1] x1 + [1]
                  c(x1) = [1] x1 + [0]
                  f(x1) = [1] x1 + [0]
                  a^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  d^#(x1) = [0] x1 + [0]
                  b^#(x1) = [1] x1 + [5]
                  c_1(x1) = [1] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
                  c_3(x1) = [0] x1 + [0]
                  c_4(x1) = [0] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {a(b(x1)) -> d(x1)}
            and weakly orienting the rules
            {b^#(a(x1)) -> c_1(a^#(b(x1)))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {a(b(x1)) -> d(x1)}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [1]
                  d(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [0]
                  f(x1) = [1] x1 + [0]
                  a^#(x1) = [1] x1 + [3]
                  c_0(x1) = [0] x1 + [0]
                  d^#(x1) = [0] x1 + [0]
                  b^#(x1) = [1] x1 + [9]
                  c_1(x1) = [1] x1 + [1]
                  c_2(x1) = [0] x1 + [0]
                  c_3(x1) = [0] x1 + [0]
                  c_4(x1) = [0] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {a(f(x1)) -> a(x1)}
            and weakly orienting the rules
            {  a(b(x1)) -> d(x1)
             , b^#(a(x1)) -> c_1(a^#(b(x1)))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {a(f(x1)) -> a(x1)}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [1]
                  d(x1) = [1] x1 + [1]
                  c(x1) = [1] x1 + [0]
                  f(x1) = [1] x1 + [8]
                  a^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  d^#(x1) = [0] x1 + [0]
                  b^#(x1) = [1] x1 + [9]
                  c_1(x1) = [1] x1 + [1]
                  c_2(x1) = [0] x1 + [0]
                  c_3(x1) = [0] x1 + [0]
                  c_4(x1) = [0] x1 + [0]
              
            Finally we apply the subprocessor
            'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
            ------------------------------------------------------------------------------------------
            Answer:           YES(?,O(n^1))
            Input Problem:    innermost relative runtime-complexity with respect to
              Strict Rules:
                {  b(a(x1)) -> a(b(x1))
                 , d(c(x1)) -> f(a(b(b(c(x1)))))
                 , d(f(x1)) -> f(a(b(x1)))}
              Weak Rules:
                {  a(f(x1)) -> a(x1)
                 , a(b(x1)) -> d(x1)
                 , b^#(a(x1)) -> c_1(a^#(b(x1)))}
            
            Details:         
              The problem was solved by processor 'Bounds with default enrichment':
              'Bounds with default enrichment'
              --------------------------------
              Answer:           YES(?,O(n^1))
              Input Problem:    innermost relative runtime-complexity with respect to
                Strict Rules:
                  {  b(a(x1)) -> a(b(x1))
                   , d(c(x1)) -> f(a(b(b(c(x1)))))
                   , d(f(x1)) -> f(a(b(x1)))}
                Weak Rules:
                  {  a(f(x1)) -> a(x1)
                   , a(b(x1)) -> d(x1)
                   , b^#(a(x1)) -> c_1(a^#(b(x1)))}
              
              Details:         
                The problem is Match-bounded by 0.
                The enriched problem is compatible with the following automaton:
                {  c_0(4) -> 4
                 , c_0(5) -> 4
                 , f_0(4) -> 5
                 , f_0(5) -> 5
                 , a^#_0(4) -> 6
                 , a^#_0(5) -> 6
                 , b^#_0(4) -> 9
                 , b^#_0(5) -> 9}