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

Details:         
  We have computed the following set of weak (innermost) dependency pairs:
   {  c^#(c(b(c(x)))) -> c_0(a^#(0(), c(x)))
    , c^#(c(x)) -> c_1(c^#(b(c(x))))
    , a^#(0(), x) -> c_2(c^#(c(x)))}
  
  The usable rules are:
   {  c(c(b(c(x)))) -> b(a(0(), c(x)))
    , c(c(x)) -> b(c(b(c(x))))
    , a(0(), x) -> c(c(x))}
  
  The estimated dependency graph contains the following edges:
   {c^#(c(b(c(x)))) -> c_0(a^#(0(), c(x)))}
     ==> {a^#(0(), x) -> c_2(c^#(c(x)))}
   {a^#(0(), x) -> c_2(c^#(c(x)))}
     ==> {c^#(c(x)) -> c_1(c^#(b(c(x))))}
   {a^#(0(), x) -> c_2(c^#(c(x)))}
     ==> {c^#(c(b(c(x)))) -> c_0(a^#(0(), c(x)))}
  
  We consider the following path(s):
   1) {  c^#(c(b(c(x)))) -> c_0(a^#(0(), c(x)))
       , a^#(0(), x) -> c_2(c^#(c(x)))}
      
      The usable rules for this path are the following:
      {  c(c(b(c(x)))) -> b(a(0(), c(x)))
       , c(c(x)) -> b(c(b(c(x))))
       , a(0(), x) -> c(c(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:
              {  c(c(b(c(x)))) -> b(a(0(), c(x)))
               , c(c(x)) -> b(c(b(c(x))))
               , a(0(), x) -> c(c(x))
               , c^#(c(b(c(x)))) -> c_0(a^#(0(), c(x)))
               , a^#(0(), x) -> c_2(c^#(c(x)))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {a^#(0(), x) -> c_2(c^#(c(x)))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {a^#(0(), x) -> c_2(c^#(c(x)))}
              
              Details:
                 Interpretation Functions:
                  c(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [0]
                  a(x1, x2) = [1] x1 + [1] x2 + [0]
                  0() = [0]
                  c^#(x1) = [1] x1 + [1]
                  c_0(x1) = [1] x1 + [1]
                  a^#(x1, x2) = [1] x1 + [1] x2 + [4]
                  c_1(x1) = [0] x1 + [0]
                  c_2(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {a(0(), x) -> c(c(x))}
            and weakly orienting the rules
            {a^#(0(), x) -> c_2(c^#(c(x)))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {a(0(), x) -> c(c(x))}
              
              Details:
                 Interpretation Functions:
                  c(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [0]
                  a(x1, x2) = [1] x1 + [1] x2 + [2]
                  0() = [0]
                  c^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [9]
                  a^#(x1, x2) = [1] x1 + [1] x2 + [8]
                  c_1(x1) = [0] x1 + [0]
                  c_2(x1) = [1] x1 + [1]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {c^#(c(b(c(x)))) -> c_0(a^#(0(), c(x)))}
            and weakly orienting the rules
            {  a(0(), x) -> c(c(x))
             , a^#(0(), x) -> c_2(c^#(c(x)))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c^#(c(b(c(x)))) -> c_0(a^#(0(), c(x)))}
              
              Details:
                 Interpretation Functions:
                  c(x1) = [1] x1 + [6]
                  b(x1) = [1] x1 + [10]
                  a(x1, x2) = [1] x1 + [1] x2 + [6]
                  0() = [12]
                  c^#(x1) = [1] x1 + [12]
                  c_0(x1) = [1] x1 + [1]
                  a^#(x1, x2) = [1] x1 + [1] x2 + [8]
                  c_1(x1) = [0] x1 + [0]
                  c_2(x1) = [1] x1 + [1]
              
            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(c(b(c(x)))) -> b(a(0(), c(x)))
                 , c(c(x)) -> b(c(b(c(x))))}
              Weak Rules:
                {  c^#(c(b(c(x)))) -> c_0(a^#(0(), c(x)))
                 , a(0(), x) -> c(c(x))
                 , a^#(0(), x) -> c_2(c^#(c(x)))}
            
            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(c(b(c(x)))) -> b(a(0(), c(x)))
                   , c(c(x)) -> b(c(b(c(x))))}
                Weak Rules:
                  {  c^#(c(b(c(x)))) -> c_0(a^#(0(), c(x)))
                   , a(0(), x) -> c(c(x))
                   , a^#(0(), x) -> c_2(c^#(c(x)))}
              
              Details:         
                The problem is Match-bounded by 0.
                The enriched problem is compatible with the following automaton:
                {  c_0(2) -> 9
                 , c_0(4) -> 9
                 , b_0(2) -> 2
                 , b_0(4) -> 2
                 , 0_0() -> 4
                 , c^#_0(2) -> 5
                 , c^#_0(4) -> 5
                 , c^#_0(9) -> 8
                 , a^#_0(2, 2) -> 7
                 , a^#_0(2, 4) -> 7
                 , a^#_0(4, 2) -> 7
                 , a^#_0(4, 4) -> 7
                 , c_2_0(8) -> 7}
      
   2) {  c^#(c(b(c(x)))) -> c_0(a^#(0(), c(x)))
       , a^#(0(), x) -> c_2(c^#(c(x)))
       , c^#(c(x)) -> c_1(c^#(b(c(x))))}
      
      The usable rules for this path are the following:
      {  c(c(b(c(x)))) -> b(a(0(), c(x)))
       , c(c(x)) -> b(c(b(c(x))))
       , a(0(), x) -> c(c(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:
              {  c(c(b(c(x)))) -> b(a(0(), c(x)))
               , c(c(x)) -> b(c(b(c(x))))
               , a(0(), x) -> c(c(x))
               , c^#(c(b(c(x)))) -> c_0(a^#(0(), c(x)))
               , a^#(0(), x) -> c_2(c^#(c(x)))
               , c^#(c(x)) -> c_1(c^#(b(c(x))))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {c^#(c(b(c(x)))) -> c_0(a^#(0(), c(x)))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c^#(c(b(c(x)))) -> c_0(a^#(0(), c(x)))}
              
              Details:
                 Interpretation Functions:
                  c(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [0]
                  a(x1, x2) = [1] x1 + [1] x2 + [0]
                  0() = [0]
                  c^#(x1) = [1] x1 + [1]
                  c_0(x1) = [1] x1 + [0]
                  a^#(x1, x2) = [1] x1 + [1] x2 + [0]
                  c_1(x1) = [1] x1 + [7]
                  c_2(x1) = [1] x1 + [5]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {c(c(b(c(x)))) -> b(a(0(), c(x)))}
            and weakly orienting the rules
            {c^#(c(b(c(x)))) -> c_0(a^#(0(), c(x)))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c(c(b(c(x)))) -> b(a(0(), c(x)))}
              
              Details:
                 Interpretation Functions:
                  c(x1) = [1] x1 + [14]
                  b(x1) = [1] x1 + [0]
                  a(x1, x2) = [1] x1 + [1] x2 + [6]
                  0() = [12]
                  c^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [0]
                  a^#(x1, x2) = [1] x1 + [1] x2 + [0]
                  c_1(x1) = [1] x1 + [0]
                  c_2(x1) = [1] x1 + [1]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {a^#(0(), x) -> c_2(c^#(c(x)))}
            and weakly orienting the rules
            {  c(c(b(c(x)))) -> b(a(0(), c(x)))
             , c^#(c(b(c(x)))) -> c_0(a^#(0(), c(x)))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {a^#(0(), x) -> c_2(c^#(c(x)))}
              
              Details:
                 Interpretation Functions:
                  c(x1) = [1] x1 + [13]
                  b(x1) = [1] x1 + [4]
                  a(x1, x2) = [1] x1 + [1] x2 + [8]
                  0() = [13]
                  c^#(x1) = [1] x1 + [1]
                  c_0(x1) = [1] x1 + [3]
                  a^#(x1, x2) = [1] x1 + [1] x2 + [2]
                  c_1(x1) = [1] x1 + [1]
                  c_2(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(c(x)) -> b(c(b(c(x))))
                 , a(0(), x) -> c(c(x))
                 , c^#(c(x)) -> c_1(c^#(b(c(x))))}
              Weak Rules:
                {  a^#(0(), x) -> c_2(c^#(c(x)))
                 , c(c(b(c(x)))) -> b(a(0(), c(x)))
                 , c^#(c(b(c(x)))) -> c_0(a^#(0(), c(x)))}
            
            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(c(x)) -> b(c(b(c(x))))
                   , a(0(), x) -> c(c(x))
                   , c^#(c(x)) -> c_1(c^#(b(c(x))))}
                Weak Rules:
                  {  a^#(0(), x) -> c_2(c^#(c(x)))
                   , c(c(b(c(x)))) -> b(a(0(), c(x)))
                   , c^#(c(b(c(x)))) -> c_0(a^#(0(), c(x)))}
              
              Details:         
                The problem is Match-bounded by 2.
                The enriched problem is compatible with the following automaton:
                {  c_0(2) -> 4
                 , c_1(2) -> 7
                 , c_2(2) -> 11
                 , b_0(2) -> 2
                 , b_1(7) -> 6
                 , b_2(11) -> 10
                 , 0_0() -> 2
                 , c^#_0(2) -> 1
                 , c^#_0(4) -> 3
                 , c^#_1(6) -> 5
                 , c^#_1(7) -> 8
                 , c^#_2(10) -> 9
                 , a^#_0(2, 2) -> 1
                 , c_1_1(5) -> 3
                 , c_1_2(9) -> 8
                 , c_2_0(3) -> 1
                 , c_2_1(8) -> 1}