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

Details:         
  We have computed the following set of weak (innermost) dependency pairs:
   {  c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
    , d^#(b(x1)) -> c_1(c^#(c(x1)))
    , b^#(c(x1)) -> c_2(b^#(a(c(x1))))
    , c^#(x1) -> c_3()
    , d^#(x1) -> c_4(b^#(c(x1)))}
  
  The usable rules are:
   {  c(c(c(a(x1)))) -> d(d(x1))
    , d(b(x1)) -> c(c(x1))
    , c(x1) -> a(a(x1))
    , d(x1) -> b(c(x1))
    , b(c(x1)) -> b(a(c(x1)))}
  
  The estimated dependency graph contains the following edges:
   {c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))}
     ==> {d^#(x1) -> c_4(b^#(c(x1)))}
   {c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))}
     ==> {d^#(b(x1)) -> c_1(c^#(c(x1)))}
   {d^#(b(x1)) -> c_1(c^#(c(x1)))}
     ==> {c^#(x1) -> c_3()}
   {d^#(b(x1)) -> c_1(c^#(c(x1)))}
     ==> {c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))}
   {d^#(x1) -> c_4(b^#(c(x1)))}
     ==> {b^#(c(x1)) -> c_2(b^#(a(c(x1))))}
  
  We consider the following path(s):
   1) {  c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
       , d^#(b(x1)) -> c_1(c^#(c(x1)))}
      
      The usable rules for this path are the following:
      {  c(c(c(a(x1)))) -> d(d(x1))
       , d(b(x1)) -> c(c(x1))
       , c(x1) -> a(a(x1))
       , d(x1) -> b(c(x1))
       , b(c(x1)) -> 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:
              {  c(c(c(a(x1)))) -> d(d(x1))
               , d(b(x1)) -> c(c(x1))
               , c(x1) -> a(a(x1))
               , d(x1) -> b(c(x1))
               , b(c(x1)) -> b(a(c(x1)))
               , c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
               , d^#(b(x1)) -> c_1(c^#(c(x1)))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {d^#(b(x1)) -> c_1(c^#(c(x1)))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {d^#(b(x1)) -> c_1(c^#(c(x1)))}
              
              Details:
                 Interpretation Functions:
                  c(x1) = [1] x1 + [0]
                  a(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [0]
                  c^#(x1) = [1] x1 + [1]
                  c_0(x1) = [1] x1 + [1]
                  d^#(x1) = [1] x1 + [9]
                  c_1(x1) = [1] x1 + [0]
                  b^#(x1) = [0] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
                  c_3() = [0]
                  c_4(x1) = [0] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {d(b(x1)) -> c(c(x1))}
            and weakly orienting the rules
            {d^#(b(x1)) -> c_1(c^#(c(x1)))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {d(b(x1)) -> c(c(x1))}
              
              Details:
                 Interpretation Functions:
                  c(x1) = [1] x1 + [0]
                  a(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [8]
                  c^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [1]
                  d^#(x1) = [1] x1 + [0]
                  c_1(x1) = [1] x1 + [0]
                  b^#(x1) = [0] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
                  c_3() = [0]
                  c_4(x1) = [0] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))}
            and weakly orienting the rules
            {  d(b(x1)) -> c(c(x1))
             , d^#(b(x1)) -> c_1(c^#(c(x1)))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))}
              
              Details:
                 Interpretation Functions:
                  c(x1) = [1] x1 + [0]
                  a(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [1]
                  c^#(x1) = [1] x1 + [4]
                  c_0(x1) = [1] x1 + [0]
                  d^#(x1) = [1] x1 + [3]
                  c_1(x1) = [1] x1 + [0]
                  b^#(x1) = [0] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
                  c_3() = [0]
                  c_4(x1) = [0] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {c(c(c(a(x1)))) -> d(d(x1))}
            and weakly orienting the rules
            {  c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
             , d(b(x1)) -> c(c(x1))
             , d^#(b(x1)) -> c_1(c^#(c(x1)))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c(c(c(a(x1)))) -> d(d(x1))}
              
              Details:
                 Interpretation Functions:
                  c(x1) = [1] x1 + [6]
                  a(x1) = [1] x1 + [14]
                  d(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [13]
                  c^#(x1) = [1] x1 + [12]
                  c_0(x1) = [1] x1 + [8]
                  d^#(x1) = [1] x1 + [12]
                  c_1(x1) = [1] x1 + [0]
                  b^#(x1) = [0] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
                  c_3() = [0]
                  c_4(x1) = [0] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {c(x1) -> a(a(x1))}
            and weakly orienting the rules
            {  c(c(c(a(x1)))) -> d(d(x1))
             , c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
             , d(b(x1)) -> c(c(x1))
             , d^#(b(x1)) -> c_1(c^#(c(x1)))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c(x1) -> a(a(x1))}
              
              Details:
                 Interpretation Functions:
                  c(x1) = [1] x1 + [8]
                  a(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [8]
                  b(x1) = [1] x1 + [9]
                  c^#(x1) = [1] x1 + [9]
                  c_0(x1) = [1] x1 + [0]
                  d^#(x1) = [1] x1 + [15]
                  c_1(x1) = [1] x1 + [7]
                  b^#(x1) = [0] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
                  c_3() = [0]
                  c_4(x1) = [0] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {d(x1) -> b(c(x1))}
            and weakly orienting the rules
            {  c(x1) -> a(a(x1))
             , c(c(c(a(x1)))) -> d(d(x1))
             , c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
             , d(b(x1)) -> c(c(x1))
             , d^#(b(x1)) -> c_1(c^#(c(x1)))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {d(x1) -> b(c(x1))}
              
              Details:
                 Interpretation Functions:
                  c(x1) = [1] x1 + [4]
                  a(x1) = [1] x1 + [2]
                  d(x1) = [1] x1 + [7]
                  b(x1) = [1] x1 + [1]
                  c^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [0]
                  d^#(x1) = [1] x1 + [3]
                  c_1(x1) = [1] x1 + [0]
                  b^#(x1) = [0] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
                  c_3() = [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(c(x1)) -> b(a(c(x1)))}
              Weak Rules:
                {  d(x1) -> b(c(x1))
                 , c(x1) -> a(a(x1))
                 , c(c(c(a(x1)))) -> d(d(x1))
                 , c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
                 , d(b(x1)) -> c(c(x1))
                 , d^#(b(x1)) -> c_1(c^#(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(c(x1)) -> b(a(c(x1)))}
                Weak Rules:
                  {  d(x1) -> b(c(x1))
                   , c(x1) -> a(a(x1))
                   , c(c(c(a(x1)))) -> d(d(x1))
                   , c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
                   , d(b(x1)) -> c(c(x1))
                   , d^#(b(x1)) -> c_1(c^#(c(x1)))}
              
              Details:         
                The problem is Match-bounded by 0.
                The enriched problem is compatible with the following automaton:
                {  a_0(2) -> 2
                 , c^#_0(2) -> 5
                 , d^#_0(2) -> 7}
      
   2) {  c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
       , d^#(b(x1)) -> c_1(c^#(c(x1)))
       , d^#(x1) -> c_4(b^#(c(x1)))}
      
      The usable rules for this path are the following:
      {  c(c(c(a(x1)))) -> d(d(x1))
       , d(b(x1)) -> c(c(x1))
       , c(x1) -> a(a(x1))
       , d(x1) -> b(c(x1))
       , b(c(x1)) -> 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:
              {  c(c(c(a(x1)))) -> d(d(x1))
               , d(b(x1)) -> c(c(x1))
               , c(x1) -> a(a(x1))
               , d(x1) -> b(c(x1))
               , b(c(x1)) -> b(a(c(x1)))
               , c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
               , d^#(b(x1)) -> c_1(c^#(c(x1)))
               , d^#(x1) -> c_4(b^#(c(x1)))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))}
              
              Details:
                 Interpretation Functions:
                  c(x1) = [1] x1 + [0]
                  a(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [0]
                  c^#(x1) = [1] x1 + [1]
                  c_0(x1) = [1] x1 + [0]
                  d^#(x1) = [1] x1 + [0]
                  c_1(x1) = [1] x1 + [0]
                  b^#(x1) = [1] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
                  c_3() = [0]
                  c_4(x1) = [1] x1 + [1]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {d^#(x1) -> c_4(b^#(c(x1)))}
            and weakly orienting the rules
            {c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {d^#(x1) -> c_4(b^#(c(x1)))}
              
              Details:
                 Interpretation Functions:
                  c(x1) = [1] x1 + [0]
                  a(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [0]
                  c^#(x1) = [1] x1 + [8]
                  c_0(x1) = [1] x1 + [7]
                  d^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [1]
                  b^#(x1) = [1] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
                  c_3() = [0]
                  c_4(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {  d(b(x1)) -> c(c(x1))
             , d^#(b(x1)) -> c_1(c^#(c(x1)))}
            and weakly orienting the rules
            {  d^#(x1) -> c_4(b^#(c(x1)))
             , c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  d(b(x1)) -> c(c(x1))
               , d^#(b(x1)) -> c_1(c^#(c(x1)))}
              
              Details:
                 Interpretation Functions:
                  c(x1) = [1] x1 + [0]
                  a(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [8]
                  c^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [0]
                  d^#(x1) = [1] x1 + [0]
                  c_1(x1) = [1] x1 + [1]
                  b^#(x1) = [1] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
                  c_3() = [0]
                  c_4(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {c(c(c(a(x1)))) -> d(d(x1))}
            and weakly orienting the rules
            {  d(b(x1)) -> c(c(x1))
             , d^#(b(x1)) -> c_1(c^#(c(x1)))
             , d^#(x1) -> c_4(b^#(c(x1)))
             , c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c(c(c(a(x1)))) -> d(d(x1))}
              
              Details:
                 Interpretation Functions:
                  c(x1) = [1] x1 + [0]
                  a(x1) = [1] x1 + [8]
                  d(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [8]
                  c^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [0]
                  d^#(x1) = [1] x1 + [8]
                  c_1(x1) = [1] x1 + [1]
                  b^#(x1) = [1] x1 + [4]
                  c_2(x1) = [0] x1 + [0]
                  c_3() = [0]
                  c_4(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {c(x1) -> a(a(x1))}
            and weakly orienting the rules
            {  c(c(c(a(x1)))) -> d(d(x1))
             , d(b(x1)) -> c(c(x1))
             , d^#(b(x1)) -> c_1(c^#(c(x1)))
             , d^#(x1) -> c_4(b^#(c(x1)))
             , c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c(x1) -> a(a(x1))}
              
              Details:
                 Interpretation Functions:
                  c(x1) = [1] x1 + [5]
                  a(x1) = [1] x1 + [1]
                  d(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [11]
                  c^#(x1) = [1] x1 + [2]
                  c_0(x1) = [1] x1 + [1]
                  d^#(x1) = [1] x1 + [9]
                  c_1(x1) = [1] x1 + [9]
                  b^#(x1) = [1] x1 + [2]
                  c_2(x1) = [0] x1 + [0]
                  c_3() = [0]
                  c_4(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {d(x1) -> b(c(x1))}
            and weakly orienting the rules
            {  c(x1) -> a(a(x1))
             , c(c(c(a(x1)))) -> d(d(x1))
             , d(b(x1)) -> c(c(x1))
             , d^#(b(x1)) -> c_1(c^#(c(x1)))
             , d^#(x1) -> c_4(b^#(c(x1)))
             , c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {d(x1) -> b(c(x1))}
              
              Details:
                 Interpretation Functions:
                  c(x1) = [1] x1 + [4]
                  a(x1) = [1] x1 + [2]
                  d(x1) = [1] x1 + [7]
                  b(x1) = [1] x1 + [1]
                  c^#(x1) = [1] x1 + [12]
                  c_0(x1) = [1] x1 + [0]
                  d^#(x1) = [1] x1 + [15]
                  c_1(x1) = [1] x1 + [0]
                  b^#(x1) = [1] x1 + [4]
                  c_2(x1) = [0] x1 + [0]
                  c_3() = [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(c(x1)) -> b(a(c(x1)))}
              Weak Rules:
                {  d(x1) -> b(c(x1))
                 , c(x1) -> a(a(x1))
                 , c(c(c(a(x1)))) -> d(d(x1))
                 , d(b(x1)) -> c(c(x1))
                 , d^#(b(x1)) -> c_1(c^#(c(x1)))
                 , d^#(x1) -> c_4(b^#(c(x1)))
                 , c^#(c(c(a(x1)))) -> c_0(d^#(d(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(c(x1)) -> b(a(c(x1)))}
                Weak Rules:
                  {  d(x1) -> b(c(x1))
                   , c(x1) -> a(a(x1))
                   , c(c(c(a(x1)))) -> d(d(x1))
                   , d(b(x1)) -> c(c(x1))
                   , d^#(b(x1)) -> c_1(c^#(c(x1)))
                   , d^#(x1) -> c_4(b^#(c(x1)))
                   , c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))}
              
              Details:         
                The problem is Match-bounded by 0.
                The enriched problem is compatible with the following automaton:
                {  c_0(2) -> 11
                 , a_0(2) -> 2
                 , a_0(2) -> 11
                 , c^#_0(2) -> 5
                 , d^#_0(2) -> 7
                 , b^#_0(2) -> 9
                 , b^#_0(11) -> 10
                 , c_4_0(10) -> 7}
      
   3) {  c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
       , d^#(b(x1)) -> c_1(c^#(c(x1)))
       , d^#(x1) -> c_4(b^#(c(x1)))
       , b^#(c(x1)) -> c_2(b^#(a(c(x1))))}
      
      The usable rules for this path are the following:
      {  c(c(c(a(x1)))) -> d(d(x1))
       , d(b(x1)) -> c(c(x1))
       , c(x1) -> a(a(x1))
       , d(x1) -> b(c(x1))
       , b(c(x1)) -> 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:
              {  c(c(c(a(x1)))) -> d(d(x1))
               , d(b(x1)) -> c(c(x1))
               , c(x1) -> a(a(x1))
               , d(x1) -> b(c(x1))
               , b(c(x1)) -> b(a(c(x1)))
               , d^#(x1) -> c_4(b^#(c(x1)))
               , c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
               , d^#(b(x1)) -> c_1(c^#(c(x1)))
               , b^#(c(x1)) -> c_2(b^#(a(c(x1))))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {  d^#(x1) -> c_4(b^#(c(x1)))
             , d^#(b(x1)) -> c_1(c^#(c(x1)))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  d^#(x1) -> c_4(b^#(c(x1)))
               , d^#(b(x1)) -> c_1(c^#(c(x1)))}
              
              Details:
                 Interpretation Functions:
                  c(x1) = [1] x1 + [0]
                  a(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [0]
                  c^#(x1) = [1] x1 + [1]
                  c_0(x1) = [1] x1 + [3]
                  d^#(x1) = [1] x1 + [4]
                  c_1(x1) = [1] x1 + [0]
                  b^#(x1) = [1] x1 + [1]
                  c_2(x1) = [1] x1 + [7]
                  c_3() = [0]
                  c_4(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {d(b(x1)) -> c(c(x1))}
            and weakly orienting the rules
            {  d^#(x1) -> c_4(b^#(c(x1)))
             , d^#(b(x1)) -> c_1(c^#(c(x1)))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {d(b(x1)) -> c(c(x1))}
              
              Details:
                 Interpretation Functions:
                  c(x1) = [1] x1 + [0]
                  a(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [5]
                  b(x1) = [1] x1 + [8]
                  c^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [0]
                  d^#(x1) = [1] x1 + [8]
                  c_1(x1) = [1] x1 + [1]
                  b^#(x1) = [1] x1 + [0]
                  c_2(x1) = [1] x1 + [8]
                  c_3() = [0]
                  c_4(x1) = [1] x1 + [1]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))}
            and weakly orienting the rules
            {  d(b(x1)) -> c(c(x1))
             , d^#(x1) -> c_4(b^#(c(x1)))
             , d^#(b(x1)) -> c_1(c^#(c(x1)))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))}
              
              Details:
                 Interpretation Functions:
                  c(x1) = [1] x1 + [0]
                  a(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [1]
                  c^#(x1) = [1] x1 + [1]
                  c_0(x1) = [1] x1 + [0]
                  d^#(x1) = [1] x1 + [0]
                  c_1(x1) = [1] x1 + [0]
                  b^#(x1) = [1] x1 + [0]
                  c_2(x1) = [1] x1 + [1]
                  c_3() = [0]
                  c_4(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {  c(c(c(a(x1)))) -> d(d(x1))
             , d(x1) -> b(c(x1))}
            and weakly orienting the rules
            {  c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
             , d(b(x1)) -> c(c(x1))
             , d^#(x1) -> c_4(b^#(c(x1)))
             , d^#(b(x1)) -> c_1(c^#(c(x1)))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  c(c(c(a(x1)))) -> d(d(x1))
               , d(x1) -> b(c(x1))}
              
              Details:
                 Interpretation Functions:
                  c(x1) = [1] x1 + [6]
                  a(x1) = [1] x1 + [8]
                  d(x1) = [1] x1 + [11]
                  b(x1) = [1] x1 + [3]
                  c^#(x1) = [1] x1 + [11]
                  c_0(x1) = [1] x1 + [0]
                  d^#(x1) = [1] x1 + [14]
                  c_1(x1) = [1] x1 + [0]
                  b^#(x1) = [1] x1 + [2]
                  c_2(x1) = [1] x1 + [0]
                  c_3() = [0]
                  c_4(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {c(x1) -> a(a(x1))}
            and weakly orienting the rules
            {  c(c(c(a(x1)))) -> d(d(x1))
             , d(x1) -> b(c(x1))
             , c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
             , d(b(x1)) -> c(c(x1))
             , d^#(x1) -> c_4(b^#(c(x1)))
             , d^#(b(x1)) -> c_1(c^#(c(x1)))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c(x1) -> a(a(x1))}
              
              Details:
                 Interpretation Functions:
                  c(x1) = [1] x1 + [10]
                  a(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [15]
                  b(x1) = [1] x1 + [5]
                  c^#(x1) = [1] x1 + [10]
                  c_0(x1) = [1] x1 + [0]
                  d^#(x1) = [1] x1 + [15]
                  c_1(x1) = [1] x1 + [0]
                  b^#(x1) = [1] x1 + [2]
                  c_2(x1) = [1] x1 + [0]
                  c_3() = [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(c(x1)) -> b(a(c(x1)))
                 , b^#(c(x1)) -> c_2(b^#(a(c(x1))))}
              Weak Rules:
                {  c(x1) -> a(a(x1))
                 , c(c(c(a(x1)))) -> d(d(x1))
                 , d(x1) -> b(c(x1))
                 , c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
                 , d(b(x1)) -> c(c(x1))
                 , d^#(x1) -> c_4(b^#(c(x1)))
                 , d^#(b(x1)) -> c_1(c^#(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(c(x1)) -> b(a(c(x1)))
                   , b^#(c(x1)) -> c_2(b^#(a(c(x1))))}
                Weak Rules:
                  {  c(x1) -> a(a(x1))
                   , c(c(c(a(x1)))) -> d(d(x1))
                   , d(x1) -> b(c(x1))
                   , c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
                   , d(b(x1)) -> c(c(x1))
                   , d^#(x1) -> c_4(b^#(c(x1)))
                   , d^#(b(x1)) -> c_1(c^#(c(x1)))}
              
              Details:         
                The problem is Match-bounded by 2.
                The enriched problem is compatible with the following automaton:
                {  c_0(2) -> 11
                 , c_1(2) -> 14
                 , c_2(2) -> 19
                 , a_0(2) -> 2
                 , a_1(2) -> 15
                 , a_1(14) -> 13
                 , a_1(15) -> 11
                 , a_1(15) -> 14
                 , a_2(2) -> 20
                 , a_2(19) -> 18
                 , a_2(20) -> 14
                 , a_2(20) -> 19
                 , c^#_0(2) -> 5
                 , d^#_0(2) -> 7
                 , b^#_0(2) -> 9
                 , b^#_0(11) -> 10
                 , b^#_1(13) -> 12
                 , b^#_1(14) -> 16
                 , b^#_2(18) -> 17
                 , c_2_1(12) -> 10
                 , c_2_2(17) -> 16
                 , c_4_0(10) -> 7
                 , c_4_1(16) -> 7}
      
   4) {  c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
       , d^#(b(x1)) -> c_1(c^#(c(x1)))
       , c^#(x1) -> c_3()}
      
      The usable rules for this path are the following:
      {  c(c(c(a(x1)))) -> d(d(x1))
       , d(b(x1)) -> c(c(x1))
       , c(x1) -> a(a(x1))
       , d(x1) -> b(c(x1))
       , b(c(x1)) -> 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:
              {  c(c(c(a(x1)))) -> d(d(x1))
               , d(b(x1)) -> c(c(x1))
               , c(x1) -> a(a(x1))
               , d(x1) -> b(c(x1))
               , b(c(x1)) -> b(a(c(x1)))
               , c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
               , d^#(b(x1)) -> c_1(c^#(c(x1)))
               , c^#(x1) -> c_3()}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {  c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
             , c^#(x1) -> c_3()}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
               , c^#(x1) -> c_3()}
              
              Details:
                 Interpretation Functions:
                  c(x1) = [1] x1 + [0]
                  a(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [0]
                  c^#(x1) = [1] x1 + [9]
                  c_0(x1) = [1] x1 + [1]
                  d^#(x1) = [1] x1 + [4]
                  c_1(x1) = [1] x1 + [0]
                  b^#(x1) = [0] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
                  c_3() = [0]
                  c_4(x1) = [0] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {d(b(x1)) -> c(c(x1))}
            and weakly orienting the rules
            {  c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
             , c^#(x1) -> c_3()}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {d(b(x1)) -> c(c(x1))}
              
              Details:
                 Interpretation Functions:
                  c(x1) = [1] x1 + [0]
                  a(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [8]
                  c^#(x1) = [1] x1 + [9]
                  c_0(x1) = [1] x1 + [0]
                  d^#(x1) = [1] x1 + [0]
                  c_1(x1) = [1] x1 + [8]
                  b^#(x1) = [0] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
                  c_3() = [0]
                  c_4(x1) = [0] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {d^#(b(x1)) -> c_1(c^#(c(x1)))}
            and weakly orienting the rules
            {  d(b(x1)) -> c(c(x1))
             , c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
             , c^#(x1) -> c_3()}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {d^#(b(x1)) -> c_1(c^#(c(x1)))}
              
              Details:
                 Interpretation Functions:
                  c(x1) = [1] x1 + [0]
                  a(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [1]
                  c^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [0]
                  d^#(x1) = [1] x1 + [0]
                  c_1(x1) = [1] x1 + [0]
                  b^#(x1) = [0] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
                  c_3() = [0]
                  c_4(x1) = [0] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {c(c(c(a(x1)))) -> d(d(x1))}
            and weakly orienting the rules
            {  d^#(b(x1)) -> c_1(c^#(c(x1)))
             , d(b(x1)) -> c(c(x1))
             , c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
             , c^#(x1) -> c_3()}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c(c(c(a(x1)))) -> d(d(x1))}
              
              Details:
                 Interpretation Functions:
                  c(x1) = [1] x1 + [2]
                  a(x1) = [1] x1 + [12]
                  d(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [7]
                  c^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [1]
                  d^#(x1) = [1] x1 + [2]
                  c_1(x1) = [1] x1 + [1]
                  b^#(x1) = [0] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
                  c_3() = [0]
                  c_4(x1) = [0] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {c(x1) -> a(a(x1))}
            and weakly orienting the rules
            {  c(c(c(a(x1)))) -> d(d(x1))
             , d^#(b(x1)) -> c_1(c^#(c(x1)))
             , d(b(x1)) -> c(c(x1))
             , c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
             , c^#(x1) -> c_3()}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c(x1) -> a(a(x1))}
              
              Details:
                 Interpretation Functions:
                  c(x1) = [1] x1 + [8]
                  a(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [8]
                  b(x1) = [1] x1 + [9]
                  c^#(x1) = [1] x1 + [1]
                  c_0(x1) = [1] x1 + [0]
                  d^#(x1) = [1] x1 + [8]
                  c_1(x1) = [1] x1 + [0]
                  b^#(x1) = [0] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
                  c_3() = [0]
                  c_4(x1) = [0] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {d(x1) -> b(c(x1))}
            and weakly orienting the rules
            {  c(x1) -> a(a(x1))
             , c(c(c(a(x1)))) -> d(d(x1))
             , d^#(b(x1)) -> c_1(c^#(c(x1)))
             , d(b(x1)) -> c(c(x1))
             , c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
             , c^#(x1) -> c_3()}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {d(x1) -> b(c(x1))}
              
              Details:
                 Interpretation Functions:
                  c(x1) = [1] x1 + [3]
                  a(x1) = [1] x1 + [1]
                  d(x1) = [1] x1 + [5]
                  b(x1) = [1] x1 + [1]
                  c^#(x1) = [1] x1 + [1]
                  c_0(x1) = [1] x1 + [0]
                  d^#(x1) = [1] x1 + [3]
                  c_1(x1) = [1] x1 + [0]
                  b^#(x1) = [0] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
                  c_3() = [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(c(x1)) -> b(a(c(x1)))}
              Weak Rules:
                {  d(x1) -> b(c(x1))
                 , c(x1) -> a(a(x1))
                 , c(c(c(a(x1)))) -> d(d(x1))
                 , d^#(b(x1)) -> c_1(c^#(c(x1)))
                 , d(b(x1)) -> c(c(x1))
                 , c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
                 , c^#(x1) -> c_3()}
            
            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(c(x1)) -> b(a(c(x1)))}
                Weak Rules:
                  {  d(x1) -> b(c(x1))
                   , c(x1) -> a(a(x1))
                   , c(c(c(a(x1)))) -> d(d(x1))
                   , d^#(b(x1)) -> c_1(c^#(c(x1)))
                   , d(b(x1)) -> c(c(x1))
                   , c^#(c(c(a(x1)))) -> c_0(d^#(d(x1)))
                   , c^#(x1) -> c_3()}
              
              Details:         
                The problem is Match-bounded by 0.
                The enriched problem is compatible with the following automaton:
                {  a_0(2) -> 2
                 , c^#_0(2) -> 5
                 , d^#_0(2) -> 7
                 , c_3_0() -> 5}