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

Details:         
  We have computed the following set of weak (innermost) dependency pairs:
   {  a^#(b(c(x1))) -> c_0()
    , c^#(b(b(x1))) -> c_1(a^#(x1))
    , c^#(x1) -> c_2()
    , a^#(a(x1)) -> c_3(c^#(b(a(c(x1)))))}
  
  The usable rules are:
   {  a(b(c(x1))) -> b(x1)
    , c(b(b(x1))) -> a(x1)
    , c(x1) -> b(x1)
    , a(a(x1)) -> c(b(a(c(x1))))}
  
  The estimated dependency graph contains the following edges:
   {c^#(b(b(x1))) -> c_1(a^#(x1))}
     ==> {a^#(a(x1)) -> c_3(c^#(b(a(c(x1)))))}
   {c^#(b(b(x1))) -> c_1(a^#(x1))}
     ==> {a^#(b(c(x1))) -> c_0()}
   {a^#(a(x1)) -> c_3(c^#(b(a(c(x1)))))}
     ==> {c^#(x1) -> c_2()}
   {a^#(a(x1)) -> c_3(c^#(b(a(c(x1)))))}
     ==> {c^#(b(b(x1))) -> c_1(a^#(x1))}
  
  We consider the following path(s):
   1) {  c^#(b(b(x1))) -> c_1(a^#(x1))
       , a^#(a(x1)) -> c_3(c^#(b(a(c(x1)))))}
      
      The usable rules for this path are the following:
      {  a(b(c(x1))) -> b(x1)
       , c(b(b(x1))) -> a(x1)
       , c(x1) -> b(x1)
       , a(a(x1)) -> c(b(a(c(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:
              {  a(b(c(x1))) -> b(x1)
               , c(b(b(x1))) -> a(x1)
               , c(x1) -> b(x1)
               , a(a(x1)) -> c(b(a(c(x1))))
               , c^#(b(b(x1))) -> c_1(a^#(x1))
               , a^#(a(x1)) -> c_3(c^#(b(a(c(x1)))))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {  a(b(c(x1))) -> b(x1)
             , a(a(x1)) -> c(b(a(c(x1))))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  a(b(c(x1))) -> b(x1)
               , a(a(x1)) -> c(b(a(c(x1))))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [1]
                  b(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [0]
                  a^#(x1) = [1] x1 + [0]
                  c_0() = [0]
                  c^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [1]
                  c_2() = [0]
                  c_3(x1) = [1] x1 + [1]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {c(x1) -> b(x1)}
            and weakly orienting the rules
            {  a(b(c(x1))) -> b(x1)
             , a(a(x1)) -> c(b(a(c(x1))))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c(x1) -> b(x1)}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [8]
                  b(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [1]
                  a^#(x1) = [1] x1 + [0]
                  c_0() = [0]
                  c^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [1]
                  c_2() = [0]
                  c_3(x1) = [1] x1 + [1]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {a^#(a(x1)) -> c_3(c^#(b(a(c(x1)))))}
            and weakly orienting the rules
            {  c(x1) -> b(x1)
             , a(b(c(x1))) -> b(x1)
             , a(a(x1)) -> c(b(a(c(x1))))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {a^#(a(x1)) -> c_3(c^#(b(a(c(x1)))))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [3]
                  b(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [1]
                  a^#(x1) = [1] x1 + [8]
                  c_0() = [0]
                  c^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [1]
                  c_2() = [0]
                  c_3(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:
                {  c(b(b(x1))) -> a(x1)
                 , c^#(b(b(x1))) -> c_1(a^#(x1))}
              Weak Rules:
                {  a^#(a(x1)) -> c_3(c^#(b(a(c(x1)))))
                 , c(x1) -> b(x1)
                 , a(b(c(x1))) -> b(x1)
                 , a(a(x1)) -> c(b(a(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:
                  {  c(b(b(x1))) -> a(x1)
                   , c^#(b(b(x1))) -> c_1(a^#(x1))}
                Weak Rules:
                  {  a^#(a(x1)) -> c_3(c^#(b(a(c(x1)))))
                   , c(x1) -> b(x1)
                   , a(b(c(x1))) -> b(x1)
                   , a(a(x1)) -> c(b(a(c(x1))))}
              
              Details:         
                The problem is Match-bounded by 1.
                The enriched problem is compatible with the following automaton:
                {  b_0(2) -> 2
                 , a^#_0(2) -> 4
                 , a^#_1(2) -> 7
                 , c^#_0(2) -> 6
                 , c_1_1(7) -> 6}
      
   2) {  c^#(b(b(x1))) -> c_1(a^#(x1))
       , a^#(a(x1)) -> c_3(c^#(b(a(c(x1)))))
       , c^#(x1) -> c_2()}
      
      The usable rules for this path are the following:
      {  a(b(c(x1))) -> b(x1)
       , c(b(b(x1))) -> a(x1)
       , c(x1) -> b(x1)
       , a(a(x1)) -> c(b(a(c(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:
              {  a(b(c(x1))) -> b(x1)
               , c(b(b(x1))) -> a(x1)
               , c(x1) -> b(x1)
               , a(a(x1)) -> c(b(a(c(x1))))
               , c^#(b(b(x1))) -> c_1(a^#(x1))
               , a^#(a(x1)) -> c_3(c^#(b(a(c(x1)))))
               , c^#(x1) -> c_2()}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {  a(b(c(x1))) -> b(x1)
             , a(a(x1)) -> c(b(a(c(x1))))
             , c^#(x1) -> c_2()}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  a(b(c(x1))) -> b(x1)
               , a(a(x1)) -> c(b(a(c(x1))))
               , c^#(x1) -> c_2()}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [1]
                  b(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [0]
                  a^#(x1) = [1] x1 + [0]
                  c_0() = [0]
                  c^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [1]
                  c_2() = [0]
                  c_3(x1) = [1] x1 + [1]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {c(x1) -> b(x1)}
            and weakly orienting the rules
            {  a(b(c(x1))) -> b(x1)
             , a(a(x1)) -> c(b(a(c(x1))))
             , c^#(x1) -> c_2()}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c(x1) -> b(x1)}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [8]
                  b(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [1]
                  a^#(x1) = [1] x1 + [0]
                  c_0() = [0]
                  c^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [1]
                  c_2() = [0]
                  c_3(x1) = [1] x1 + [1]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {a^#(a(x1)) -> c_3(c^#(b(a(c(x1)))))}
            and weakly orienting the rules
            {  c(x1) -> b(x1)
             , a(b(c(x1))) -> b(x1)
             , a(a(x1)) -> c(b(a(c(x1))))
             , c^#(x1) -> c_2()}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {a^#(a(x1)) -> c_3(c^#(b(a(c(x1)))))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [7]
                  b(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [1]
                  a^#(x1) = [1] x1 + [8]
                  c_0() = [0]
                  c^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [1]
                  c_2() = [0]
                  c_3(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:
                {  c(b(b(x1))) -> a(x1)
                 , c^#(b(b(x1))) -> c_1(a^#(x1))}
              Weak Rules:
                {  a^#(a(x1)) -> c_3(c^#(b(a(c(x1)))))
                 , c(x1) -> b(x1)
                 , a(b(c(x1))) -> b(x1)
                 , a(a(x1)) -> c(b(a(c(x1))))
                 , c^#(x1) -> c_2()}
            
            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:
                  {  c(b(b(x1))) -> a(x1)
                   , c^#(b(b(x1))) -> c_1(a^#(x1))}
                Weak Rules:
                  {  a^#(a(x1)) -> c_3(c^#(b(a(c(x1)))))
                   , c(x1) -> b(x1)
                   , a(b(c(x1))) -> b(x1)
                   , a(a(x1)) -> c(b(a(c(x1))))
                   , c^#(x1) -> c_2()}
              
              Details:         
                The problem is Match-bounded by 1.
                The enriched problem is compatible with the following automaton:
                {  b_0(2) -> 2
                 , a^#_0(2) -> 4
                 , a^#_1(2) -> 7
                 , c^#_0(2) -> 6
                 , c_1_1(7) -> 6
                 , c_2_0() -> 6}
      
   3) {  c^#(b(b(x1))) -> c_1(a^#(x1))
       , a^#(a(x1)) -> c_3(c^#(b(a(c(x1)))))
       , a^#(b(c(x1))) -> c_0()}
      
      The usable rules for this path are the following:
      {  a(b(c(x1))) -> b(x1)
       , c(b(b(x1))) -> a(x1)
       , c(x1) -> b(x1)
       , a(a(x1)) -> c(b(a(c(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:
              {  a(b(c(x1))) -> b(x1)
               , c(b(b(x1))) -> a(x1)
               , c(x1) -> b(x1)
               , a(a(x1)) -> c(b(a(c(x1))))
               , c^#(b(b(x1))) -> c_1(a^#(x1))
               , a^#(a(x1)) -> c_3(c^#(b(a(c(x1)))))
               , a^#(b(c(x1))) -> c_0()}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {  a(b(c(x1))) -> b(x1)
             , a(a(x1)) -> c(b(a(c(x1))))
             , c^#(b(b(x1))) -> c_1(a^#(x1))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  a(b(c(x1))) -> b(x1)
               , a(a(x1)) -> c(b(a(c(x1))))
               , c^#(b(b(x1))) -> c_1(a^#(x1))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [1]
                  b(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [0]
                  a^#(x1) = [1] x1 + [0]
                  c_0() = [0]
                  c^#(x1) = [1] x1 + [7]
                  c_1(x1) = [1] x1 + [1]
                  c_2() = [0]
                  c_3(x1) = [1] x1 + [1]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {  c(x1) -> b(x1)
             , a^#(b(c(x1))) -> c_0()}
            and weakly orienting the rules
            {  a(b(c(x1))) -> b(x1)
             , a(a(x1)) -> c(b(a(c(x1))))
             , c^#(b(b(x1))) -> c_1(a^#(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  c(x1) -> b(x1)
               , a^#(b(c(x1))) -> c_0()}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [8]
                  b(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [1]
                  a^#(x1) = [1] x1 + [1]
                  c_0() = [0]
                  c^#(x1) = [1] x1 + [8]
                  c_1(x1) = [1] x1 + [4]
                  c_2() = [0]
                  c_3(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:
                {  c(b(b(x1))) -> a(x1)
                 , a^#(a(x1)) -> c_3(c^#(b(a(c(x1)))))}
              Weak Rules:
                {  c(x1) -> b(x1)
                 , a^#(b(c(x1))) -> c_0()
                 , a(b(c(x1))) -> b(x1)
                 , a(a(x1)) -> c(b(a(c(x1))))
                 , c^#(b(b(x1))) -> c_1(a^#(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:
                  {  c(b(b(x1))) -> a(x1)
                   , a^#(a(x1)) -> c_3(c^#(b(a(c(x1)))))}
                Weak Rules:
                  {  c(x1) -> b(x1)
                   , a^#(b(c(x1))) -> c_0()
                   , a(b(c(x1))) -> b(x1)
                   , a(a(x1)) -> c(b(a(c(x1))))
                   , c^#(b(b(x1))) -> c_1(a^#(x1))}
              
              Details:         
                The problem is Match-bounded by 0.
                The enriched problem is compatible with the following automaton:
                {  b_0(2) -> 2
                 , a^#_0(2) -> 4
                 , c^#_0(2) -> 6
                 , c_1_0(4) -> 6}