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

Details:         
  We have computed the following set of weak (innermost) dependency pairs:
   {  a^#(x1) -> c_0()
    , a^#(a(x1)) -> c_1(c^#(x1))
    , c^#(x1) -> c_2()
    , c^#(b(x1)) -> c_3(c^#(a(x1)))}
  
  The usable rules are:
   {  a(x1) -> x1
    , a(a(x1)) -> b(b(c(x1)))
    , c(x1) -> x1
    , c(b(x1)) -> b(c(a(x1)))}
  
  The estimated dependency graph contains the following edges:
   {a^#(a(x1)) -> c_1(c^#(x1))}
     ==> {c^#(b(x1)) -> c_3(c^#(a(x1)))}
   {a^#(a(x1)) -> c_1(c^#(x1))}
     ==> {c^#(x1) -> c_2()}
   {c^#(b(x1)) -> c_3(c^#(a(x1)))}
     ==> {c^#(b(x1)) -> c_3(c^#(a(x1)))}
   {c^#(b(x1)) -> c_3(c^#(a(x1)))}
     ==> {c^#(x1) -> c_2()}
  
  We consider the following path(s):
   1) {  a^#(a(x1)) -> c_1(c^#(x1))
       , c^#(b(x1)) -> c_3(c^#(a(x1)))
       , c^#(x1) -> c_2()}
      
      The usable rules for this path are the following:
      {  a(x1) -> x1
       , a(a(x1)) -> b(b(c(x1)))
       , c(x1) -> x1
       , c(b(x1)) -> b(c(a(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(x1) -> x1
               , a(a(x1)) -> b(b(c(x1)))
               , c(x1) -> x1
               , c(b(x1)) -> b(c(a(x1)))
               , c^#(b(x1)) -> c_3(c^#(a(x1)))
               , a^#(a(x1)) -> c_1(c^#(x1))
               , c^#(x1) -> c_2()}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {c^#(x1) -> c_2()}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c^#(x1) -> c_2()}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [0]
                  a^#(x1) = [1] x1 + [1]
                  c_0() = [0]
                  c_1(x1) = [1] x1 + [0]
                  c^#(x1) = [1] x1 + [1]
                  c_2() = [0]
                  c_3(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {a^#(a(x1)) -> c_1(c^#(x1))}
            and weakly orienting the rules
            {c^#(x1) -> c_2()}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {a^#(a(x1)) -> c_1(c^#(x1))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [0]
                  a^#(x1) = [1] x1 + [9]
                  c_0() = [0]
                  c_1(x1) = [1] x1 + [1]
                  c^#(x1) = [1] x1 + [1]
                  c_2() = [0]
                  c_3(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {c^#(b(x1)) -> c_3(c^#(a(x1)))}
            and weakly orienting the rules
            {  a^#(a(x1)) -> c_1(c^#(x1))
             , c^#(x1) -> c_2()}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c^#(b(x1)) -> c_3(c^#(a(x1)))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [8]
                  c(x1) = [1] x1 + [0]
                  a^#(x1) = [1] x1 + [13]
                  c_0() = [0]
                  c_1(x1) = [1] x1 + [4]
                  c^#(x1) = [1] x1 + [8]
                  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) -> x1}
            and weakly orienting the rules
            {  c^#(b(x1)) -> c_3(c^#(a(x1)))
             , a^#(a(x1)) -> c_1(c^#(x1))
             , c^#(x1) -> c_2()}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c(x1) -> x1}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [8]
                  a^#(x1) = [1] x1 + [9]
                  c_0() = [0]
                  c_1(x1) = [1] x1 + [0]
                  c^#(x1) = [1] x1 + [1]
                  c_2() = [0]
                  c_3(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {a(x1) -> x1}
            and weakly orienting the rules
            {  c(x1) -> x1
             , c^#(b(x1)) -> c_3(c^#(a(x1)))
             , a^#(a(x1)) -> c_1(c^#(x1))
             , c^#(x1) -> c_2()}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {a(x1) -> x1}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [8]
                  b(x1) = [1] x1 + [8]
                  c(x1) = [1] x1 + [0]
                  a^#(x1) = [1] x1 + [9]
                  c_0() = [0]
                  c_1(x1) = [1] x1 + [9]
                  c^#(x1) = [1] x1 + [0]
                  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:
                {  a(a(x1)) -> b(b(c(x1)))
                 , c(b(x1)) -> b(c(a(x1)))}
              Weak Rules:
                {  a(x1) -> x1
                 , c(x1) -> x1
                 , c^#(b(x1)) -> c_3(c^#(a(x1)))
                 , a^#(a(x1)) -> c_1(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:
                  {  a(a(x1)) -> b(b(c(x1)))
                   , c(b(x1)) -> b(c(a(x1)))}
                Weak Rules:
                  {  a(x1) -> x1
                   , c(x1) -> x1
                   , c^#(b(x1)) -> c_3(c^#(a(x1)))
                   , a^#(a(x1)) -> c_1(c^#(x1))
                   , c^#(x1) -> c_2()}
              
              Details:         
                The problem is Match-bounded by 0.
                The enriched problem is compatible with the following automaton:
                {  a_0(2) -> 9
                 , b_0(2) -> 2
                 , b_0(2) -> 9
                 , a^#_0(2) -> 4
                 , c^#_0(2) -> 7
                 , c^#_0(9) -> 8
                 , c_2_0() -> 7
                 , c_2_0() -> 8
                 , c_3_0(8) -> 7
                 , c_3_0(8) -> 8}
      
   2) {  a^#(a(x1)) -> c_1(c^#(x1))
       , c^#(b(x1)) -> c_3(c^#(a(x1)))}
      
      The usable rules for this path are the following:
      {  a(x1) -> x1
       , a(a(x1)) -> b(b(c(x1)))
       , c(x1) -> x1
       , c(b(x1)) -> b(c(a(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(x1) -> x1
               , a(a(x1)) -> b(b(c(x1)))
               , c(x1) -> x1
               , c(b(x1)) -> b(c(a(x1)))
               , a^#(a(x1)) -> c_1(c^#(x1))
               , c^#(b(x1)) -> c_3(c^#(a(x1)))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {a^#(a(x1)) -> c_1(c^#(x1))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {a^#(a(x1)) -> c_1(c^#(x1))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [0]
                  a^#(x1) = [1] x1 + [1]
                  c_0() = [0]
                  c_1(x1) = [1] x1 + [0]
                  c^#(x1) = [1] x1 + [0]
                  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^#(b(x1)) -> c_3(c^#(a(x1)))}
            and weakly orienting the rules
            {a^#(a(x1)) -> c_1(c^#(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c^#(b(x1)) -> c_3(c^#(a(x1)))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [8]
                  c(x1) = [1] x1 + [0]
                  a^#(x1) = [1] x1 + [5]
                  c_0() = [0]
                  c_1(x1) = [1] x1 + [0]
                  c^#(x1) = [1] x1 + [0]
                  c_2() = [0]
                  c_3(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {c(x1) -> x1}
            and weakly orienting the rules
            {  c^#(b(x1)) -> c_3(c^#(a(x1)))
             , a^#(a(x1)) -> c_1(c^#(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c(x1) -> x1}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [8]
                  a^#(x1) = [1] x1 + [9]
                  c_0() = [0]
                  c_1(x1) = [1] x1 + [2]
                  c^#(x1) = [1] x1 + [3]
                  c_2() = [0]
                  c_3(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {a(x1) -> x1}
            and weakly orienting the rules
            {  c(x1) -> x1
             , c^#(b(x1)) -> c_3(c^#(a(x1)))
             , a^#(a(x1)) -> c_1(c^#(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {a(x1) -> x1}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [8]
                  b(x1) = [1] x1 + [8]
                  c(x1) = [1] x1 + [0]
                  a^#(x1) = [1] x1 + [9]
                  c_0() = [0]
                  c_1(x1) = [1] x1 + [7]
                  c^#(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:
                {  a(a(x1)) -> b(b(c(x1)))
                 , c(b(x1)) -> b(c(a(x1)))}
              Weak Rules:
                {  a(x1) -> x1
                 , c(x1) -> x1
                 , c^#(b(x1)) -> c_3(c^#(a(x1)))
                 , a^#(a(x1)) -> c_1(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:
                  {  a(a(x1)) -> b(b(c(x1)))
                   , c(b(x1)) -> b(c(a(x1)))}
                Weak Rules:
                  {  a(x1) -> x1
                   , c(x1) -> x1
                   , c^#(b(x1)) -> c_3(c^#(a(x1)))
                   , a^#(a(x1)) -> c_1(c^#(x1))}
              
              Details:         
                The problem is Match-bounded by 0.
                The enriched problem is compatible with the following automaton:
                {  a_0(2) -> 4
                 , b_0(2) -> 2
                 , b_0(2) -> 4
                 , a^#_0(2) -> 1
                 , c^#_0(2) -> 1
                 , c^#_0(4) -> 3
                 , c_3_0(3) -> 1
                 , c_3_0(3) -> 3}
      
   3) {a^#(a(x1)) -> c_1(c^#(x1))}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           a(x1) = [0] x1 + [0]
           b(x1) = [0] x1 + [0]
           c(x1) = [0] x1 + [0]
           a^#(x1) = [0] x1 + [0]
           c_0() = [0]
           c_1(x1) = [0] x1 + [0]
           c^#(x1) = [0] x1 + [0]
           c_2() = [0]
           c_3(x1) = [0] x1 + [0]
        
        We have applied the subprocessor on the resulting DP-problem:
        
          'Weight Gap Principle'
          ----------------------
          Answer:           YES(?,O(n^1))
          Input Problem:    innermost DP runtime-complexity with respect to
            Strict Rules: {a^#(a(x1)) -> c_1(c^#(x1))}
            Weak Rules: {}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {a^#(a(x1)) -> c_1(c^#(x1))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {a^#(a(x1)) -> c_1(c^#(x1))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [0]
                  b(x1) = [0] x1 + [0]
                  c(x1) = [0] x1 + [0]
                  a^#(x1) = [1] x1 + [1]
                  c_0() = [0]
                  c_1(x1) = [1] x1 + [0]
                  c^#(x1) = [1] x1 + [0]
                  c_2() = [0]
                  c_3(x1) = [0] x1 + [0]
              
            Finally we apply the subprocessor
            'Empty TRS'
            -----------
            Answer:           YES(?,O(1))
            Input Problem:    innermost DP runtime-complexity with respect to
              Strict Rules: {}
              Weak Rules: {a^#(a(x1)) -> c_1(c^#(x1))}
            
            Details:         
              The given problem does not contain any strict rules
      
   4) {  a^#(a(x1)) -> c_1(c^#(x1))
       , c^#(x1) -> c_2()}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           a(x1) = [0] x1 + [0]
           b(x1) = [0] x1 + [0]
           c(x1) = [0] x1 + [0]
           a^#(x1) = [0] x1 + [0]
           c_0() = [0]
           c_1(x1) = [0] x1 + [0]
           c^#(x1) = [0] x1 + [0]
           c_2() = [0]
           c_3(x1) = [0] x1 + [0]
        
        We have applied the subprocessor on the resulting DP-problem:
        
          'Weight Gap Principle'
          ----------------------
          Answer:           YES(?,O(n^1))
          Input Problem:    innermost DP runtime-complexity with respect to
            Strict Rules: {c^#(x1) -> c_2()}
            Weak Rules: {a^#(a(x1)) -> c_1(c^#(x1))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {c^#(x1) -> c_2()}
            and weakly orienting the rules
            {a^#(a(x1)) -> c_1(c^#(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c^#(x1) -> c_2()}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [0]
                  b(x1) = [0] x1 + [0]
                  c(x1) = [0] x1 + [0]
                  a^#(x1) = [1] x1 + [1]
                  c_0() = [0]
                  c_1(x1) = [1] x1 + [0]
                  c^#(x1) = [1] x1 + [1]
                  c_2() = [0]
                  c_3(x1) = [0] x1 + [0]
              
            Finally we apply the subprocessor
            'Empty TRS'
            -----------
            Answer:           YES(?,O(1))
            Input Problem:    innermost DP runtime-complexity with respect to
              Strict Rules: {}
              Weak Rules:
                {  c^#(x1) -> c_2()
                 , a^#(a(x1)) -> c_1(c^#(x1))}
            
            Details:         
              The given problem does not contain any strict rules
      
   5) {a^#(x1) -> c_0()}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           a(x1) = [0] x1 + [0]
           b(x1) = [0] x1 + [0]
           c(x1) = [0] x1 + [0]
           a^#(x1) = [0] x1 + [0]
           c_0() = [0]
           c_1(x1) = [0] x1 + [0]
           c^#(x1) = [0] x1 + [0]
           c_2() = [0]
           c_3(x1) = [0] x1 + [0]
        
        We have applied the subprocessor on the resulting DP-problem:
        
          'Weight Gap Principle'
          ----------------------
          Answer:           YES(?,O(n^1))
          Input Problem:    innermost DP runtime-complexity with respect to
            Strict Rules: {a^#(x1) -> c_0()}
            Weak Rules: {}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {a^#(x1) -> c_0()}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {a^#(x1) -> c_0()}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [0] x1 + [0]
                  b(x1) = [0] x1 + [0]
                  c(x1) = [0] x1 + [0]
                  a^#(x1) = [1] x1 + [4]
                  c_0() = [0]
                  c_1(x1) = [0] x1 + [0]
                  c^#(x1) = [0] x1 + [0]
                  c_2() = [0]
                  c_3(x1) = [0] x1 + [0]
              
            Finally we apply the subprocessor
            'Empty TRS'
            -----------
            Answer:           YES(?,O(1))
            Input Problem:    innermost DP runtime-complexity with respect to
              Strict Rules: {}
              Weak Rules: {a^#(x1) -> c_0()}
            
            Details:         
              The given problem does not contain any strict rules