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

Details:         
  We have computed the following set of weak (innermost) dependency pairs:
   {  b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))
    , a^#(y, x) -> c_1()
    , a^#(y, c(b(a(0(), x), 0()))) ->
      c_2(b^#(a(c(b(0(), y)), x), 0()))}
  
  The usable rules are:
   {  b(x, y) -> c(a(c(y), a(0(), x)))
    , a(y, x) -> y
    , a(y, c(b(a(0(), x), 0()))) -> b(a(c(b(0(), y)), x), 0())}
  
  The estimated dependency graph contains the following edges:
   {b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))}
     ==> {a^#(y, c(b(a(0(), x), 0()))) ->
          c_2(b^#(a(c(b(0(), y)), x), 0()))}
   {b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))}
     ==> {a^#(y, x) -> c_1()}
   {a^#(y, c(b(a(0(), x), 0()))) -> c_2(b^#(a(c(b(0(), y)), x), 0()))}
     ==> {b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))}
  
  We consider the following path(s):
   1) {  b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))
       , a^#(y, c(b(a(0(), x), 0()))) ->
         c_2(b^#(a(c(b(0(), y)), x), 0()))}
      
      The usable rules for this path are the following:
      {  b(x, y) -> c(a(c(y), a(0(), x)))
       , a(y, x) -> y
       , a(y, c(b(a(0(), x), 0()))) -> b(a(c(b(0(), y)), x), 0())}
      
        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(x, y) -> c(a(c(y), a(0(), x)))
               , a(y, x) -> y
               , a(y, c(b(a(0(), x), 0()))) -> b(a(c(b(0(), y)), x), 0())
               , b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))
               , a^#(y, c(b(a(0(), x), 0()))) ->
                 c_2(b^#(a(c(b(0(), y)), x), 0()))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))}
              
              Details:
                 Interpretation Functions:
                  b(x1, x2) = [1] x1 + [1] x2 + [0]
                  c(x1) = [1] x1 + [0]
                  a(x1, x2) = [1] x1 + [1] x2 + [0]
                  0() = [0]
                  b^#(x1, x2) = [1] x1 + [1] x2 + [8]
                  c_0(x1) = [1] x1 + [1]
                  a^#(x1, x2) = [1] x1 + [1] x2 + [0]
                  c_1() = [0]
                  c_2(x1) = [1] x1 + [1]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {b(x, y) -> c(a(c(y), a(0(), x)))}
            and weakly orienting the rules
            {b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {b(x, y) -> c(a(c(y), a(0(), x)))}
              
              Details:
                 Interpretation Functions:
                  b(x1, x2) = [1] x1 + [1] x2 + [8]
                  c(x1) = [1] x1 + [0]
                  a(x1, x2) = [1] x1 + [1] x2 + [0]
                  0() = [0]
                  b^#(x1, x2) = [1] x1 + [1] x2 + [8]
                  c_0(x1) = [1] x1 + [0]
                  a^#(x1, x2) = [1] x1 + [1] x2 + [8]
                  c_1() = [0]
                  c_2(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {a(y, x) -> y}
            and weakly orienting the rules
            {  b(x, y) -> c(a(c(y), a(0(), x)))
             , b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {a(y, x) -> y}
              
              Details:
                 Interpretation Functions:
                  b(x1, x2) = [1] x1 + [1] x2 + [4]
                  c(x1) = [1] x1 + [0]
                  a(x1, x2) = [1] x1 + [1] x2 + [1]
                  0() = [1]
                  b^#(x1, x2) = [1] x1 + [1] x2 + [8]
                  c_0(x1) = [1] x1 + [0]
                  a^#(x1, x2) = [1] x1 + [1] x2 + [6]
                  c_1() = [0]
                  c_2(x1) = [1] x1 + [8]
              
            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:
                {  a(y, c(b(a(0(), x), 0()))) -> b(a(c(b(0(), y)), x), 0())
                 , a^#(y, c(b(a(0(), x), 0()))) ->
                   c_2(b^#(a(c(b(0(), y)), x), 0()))}
              Weak Rules:
                {  a(y, x) -> y
                 , b(x, y) -> c(a(c(y), a(0(), x)))
                 , b^#(x, y) -> c_0(a^#(c(y), a(0(), 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:
                  {  a(y, c(b(a(0(), x), 0()))) -> b(a(c(b(0(), y)), x), 0())
                   , a^#(y, c(b(a(0(), x), 0()))) ->
                     c_2(b^#(a(c(b(0(), y)), x), 0()))}
                Weak Rules:
                  {  a(y, x) -> y
                   , b(x, y) -> c(a(c(y), a(0(), x)))
                   , b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))}
              
              Details:         
                The problem is Match-bounded by 0.
                The enriched problem is compatible with the following automaton:
                {  c_0(2) -> 2
                 , c_0(4) -> 2
                 , a_0(4, 2) -> 9
                 , a_0(4, 4) -> 9
                 , 0_0() -> 4
                 , 0_0() -> 9
                 , b^#_0(2, 2) -> 5
                 , b^#_0(2, 4) -> 5
                 , b^#_0(4, 2) -> 5
                 , b^#_0(4, 4) -> 5
                 , c_0_0(8) -> 5
                 , a^#_0(2, 2) -> 7
                 , a^#_0(2, 4) -> 7
                 , a^#_0(2, 9) -> 8
                 , a^#_0(4, 2) -> 7
                 , a^#_0(4, 4) -> 7}
      
   2) {  b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))
       , a^#(y, c(b(a(0(), x), 0()))) -> c_2(b^#(a(c(b(0(), y)), x), 0()))
       , a^#(y, x) -> c_1()}
      
      The usable rules for this path are the following:
      {  b(x, y) -> c(a(c(y), a(0(), x)))
       , a(y, x) -> y
       , a(y, c(b(a(0(), x), 0()))) -> b(a(c(b(0(), y)), x), 0())}
      
        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(x, y) -> c(a(c(y), a(0(), x)))
               , a(y, x) -> y
               , a(y, c(b(a(0(), x), 0()))) -> b(a(c(b(0(), y)), x), 0())
               , b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))
               , a^#(y, c(b(a(0(), x), 0()))) -> c_2(b^#(a(c(b(0(), y)), x), 0()))
               , a^#(y, x) -> c_1()}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {b(x, y) -> c(a(c(y), a(0(), x)))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {b(x, y) -> c(a(c(y), a(0(), x)))}
              
              Details:
                 Interpretation Functions:
                  b(x1, x2) = [1] x1 + [1] x2 + [1]
                  c(x1) = [1] x1 + [0]
                  a(x1, x2) = [1] x1 + [1] x2 + [0]
                  0() = [0]
                  b^#(x1, x2) = [1] x1 + [1] x2 + [0]
                  c_0(x1) = [1] x1 + [1]
                  a^#(x1, x2) = [1] x1 + [1] x2 + [0]
                  c_1() = [0]
                  c_2(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))}
            and weakly orienting the rules
            {b(x, y) -> c(a(c(y), a(0(), x)))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))}
              
              Details:
                 Interpretation Functions:
                  b(x1, x2) = [1] x1 + [1] x2 + [0]
                  c(x1) = [1] x1 + [0]
                  a(x1, x2) = [1] x1 + [1] x2 + [0]
                  0() = [0]
                  b^#(x1, x2) = [1] x1 + [1] x2 + [8]
                  c_0(x1) = [1] x1 + [1]
                  a^#(x1, x2) = [1] x1 + [1] x2 + [0]
                  c_1() = [0]
                  c_2(x1) = [1] x1 + [1]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {a^#(y, x) -> c_1()}
            and weakly orienting the rules
            {  b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))
             , b(x, y) -> c(a(c(y), a(0(), x)))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {a^#(y, x) -> c_1()}
              
              Details:
                 Interpretation Functions:
                  b(x1, x2) = [1] x1 + [1] x2 + [0]
                  c(x1) = [1] x1 + [0]
                  a(x1, x2) = [1] x1 + [1] x2 + [0]
                  0() = [0]
                  b^#(x1, x2) = [1] x1 + [1] x2 + [8]
                  c_0(x1) = [1] x1 + [1]
                  a^#(x1, x2) = [1] x1 + [1] x2 + [1]
                  c_1() = [0]
                  c_2(x1) = [1] x1 + [1]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {a(y, x) -> y}
            and weakly orienting the rules
            {  a^#(y, x) -> c_1()
             , b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))
             , b(x, y) -> c(a(c(y), a(0(), x)))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {a(y, x) -> y}
              
              Details:
                 Interpretation Functions:
                  b(x1, x2) = [1] x1 + [1] x2 + [11]
                  c(x1) = [1] x1 + [0]
                  a(x1, x2) = [1] x1 + [1] x2 + [3]
                  0() = [1]
                  b^#(x1, x2) = [1] x1 + [1] x2 + [8]
                  c_0(x1) = [1] x1 + [2]
                  a^#(x1, x2) = [1] x1 + [1] x2 + [0]
                  c_1() = [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:
                {  a(y, c(b(a(0(), x), 0()))) -> b(a(c(b(0(), y)), x), 0())
                 , a^#(y, c(b(a(0(), x), 0()))) ->
                   c_2(b^#(a(c(b(0(), y)), x), 0()))}
              Weak Rules:
                {  a(y, x) -> y
                 , a^#(y, x) -> c_1()
                 , b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))
                 , b(x, y) -> c(a(c(y), a(0(), 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:
                  {  a(y, c(b(a(0(), x), 0()))) -> b(a(c(b(0(), y)), x), 0())
                   , a^#(y, c(b(a(0(), x), 0()))) ->
                     c_2(b^#(a(c(b(0(), y)), x), 0()))}
                Weak Rules:
                  {  a(y, x) -> y
                   , a^#(y, x) -> c_1()
                   , b^#(x, y) -> c_0(a^#(c(y), a(0(), x)))
                   , b(x, y) -> c(a(c(y), a(0(), x)))}
              
              Details:         
                The problem is Match-bounded by 0.
                The enriched problem is compatible with the following automaton:
                {  c_0(2) -> 2
                 , c_0(4) -> 2
                 , a_0(4, 2) -> 9
                 , a_0(4, 4) -> 9
                 , 0_0() -> 4
                 , 0_0() -> 9
                 , b^#_0(2, 2) -> 5
                 , b^#_0(2, 4) -> 5
                 , b^#_0(4, 2) -> 5
                 , b^#_0(4, 4) -> 5
                 , c_0_0(8) -> 5
                 , a^#_0(2, 2) -> 7
                 , a^#_0(2, 4) -> 7
                 , a^#_0(2, 9) -> 8
                 , a^#_0(4, 2) -> 7
                 , a^#_0(4, 4) -> 7
                 , c_1_0() -> 7
                 , c_1_0() -> 8}