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

Details:         
  We have computed the following set of weak (innermost) dependency pairs:
   {  f^#(x1) -> c_0(n^#(c(n(a(x1)))))
    , c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))
    , n^#(a(x1)) -> c_2(c^#(x1))
    , c^#(c(x1)) -> c_3(c^#(x1))
    , n^#(s(x1)) -> c_4(f^#(s(s(x1))))
    , n^#(f(x1)) -> c_5(f^#(n(x1)))}
  
  The usable rules are:
   {  c(f(x1)) -> f(n(a(c(x1))))
    , n(a(x1)) -> c(x1)
    , c(c(x1)) -> c(x1)
    , n(s(x1)) -> f(s(s(x1)))
    , n(f(x1)) -> f(n(x1))
    , f(x1) -> n(c(n(a(x1))))}
  
  The estimated dependency graph contains the following edges:
   {f^#(x1) -> c_0(n^#(c(n(a(x1)))))}
     ==> {n^#(f(x1)) -> c_5(f^#(n(x1)))}
   {c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))}
     ==> {f^#(x1) -> c_0(n^#(c(n(a(x1)))))}
   {n^#(a(x1)) -> c_2(c^#(x1))}
     ==> {c^#(c(x1)) -> c_3(c^#(x1))}
   {n^#(a(x1)) -> c_2(c^#(x1))}
     ==> {c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))}
   {c^#(c(x1)) -> c_3(c^#(x1))}
     ==> {c^#(c(x1)) -> c_3(c^#(x1))}
   {c^#(c(x1)) -> c_3(c^#(x1))}
     ==> {c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))}
   {n^#(s(x1)) -> c_4(f^#(s(s(x1))))}
     ==> {f^#(x1) -> c_0(n^#(c(n(a(x1)))))}
   {n^#(f(x1)) -> c_5(f^#(n(x1)))}
     ==> {f^#(x1) -> c_0(n^#(c(n(a(x1)))))}
  
  We consider the following path(s):
   1) {  n^#(a(x1)) -> c_2(c^#(x1))
       , c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))
       , f^#(x1) -> c_0(n^#(c(n(a(x1)))))
       , n^#(f(x1)) -> c_5(f^#(n(x1)))}
      
      The usable rules for this path are the following:
      {  c(f(x1)) -> f(n(a(c(x1))))
       , n(a(x1)) -> c(x1)
       , c(c(x1)) -> c(x1)
       , n(s(x1)) -> f(s(s(x1)))
       , n(f(x1)) -> f(n(x1))
       , f(x1) -> n(c(n(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:
              {  c(f(x1)) -> f(n(a(c(x1))))
               , n(a(x1)) -> c(x1)
               , c(c(x1)) -> c(x1)
               , n(s(x1)) -> f(s(s(x1)))
               , n(f(x1)) -> f(n(x1))
               , f(x1) -> n(c(n(a(x1))))
               , c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))
               , n^#(a(x1)) -> c_2(c^#(x1))
               , f^#(x1) -> c_0(n^#(c(n(a(x1)))))
               , n^#(f(x1)) -> c_5(f^#(n(x1)))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {c(c(x1)) -> c(x1)}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c(c(x1)) -> c(x1)}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  n(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [1]
                  a(x1) = [1] x1 + [0]
                  s(x1) = [1] x1 + [0]
                  f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [1]
                  n^#(x1) = [1] x1 + [1]
                  c^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [0]
                  c_2(x1) = [1] x1 + [0]
                  c_3(x1) = [0] x1 + [0]
                  c_4(x1) = [0] x1 + [0]
                  c_5(x1) = [1] x1 + [1]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {n^#(f(x1)) -> c_5(f^#(n(x1)))}
            and weakly orienting the rules
            {c(c(x1)) -> c(x1)}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {n^#(f(x1)) -> c_5(f^#(n(x1)))}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  n(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [1]
                  a(x1) = [1] x1 + [0]
                  s(x1) = [1] x1 + [0]
                  f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [1]
                  n^#(x1) = [1] x1 + [1]
                  c^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [0]
                  c_2(x1) = [1] x1 + [0]
                  c_3(x1) = [0] x1 + [0]
                  c_4(x1) = [0] x1 + [0]
                  c_5(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {n^#(a(x1)) -> c_2(c^#(x1))}
            and weakly orienting the rules
            {  n^#(f(x1)) -> c_5(f^#(n(x1)))
             , c(c(x1)) -> c(x1)}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {n^#(a(x1)) -> c_2(c^#(x1))}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  n(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [1]
                  a(x1) = [1] x1 + [0]
                  s(x1) = [1] x1 + [0]
                  f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [1]
                  n^#(x1) = [1] x1 + [9]
                  c^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [8]
                  c_2(x1) = [1] x1 + [7]
                  c_3(x1) = [0] x1 + [0]
                  c_4(x1) = [0] x1 + [0]
                  c_5(x1) = [1] x1 + [3]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))}
            and weakly orienting the rules
            {  n^#(a(x1)) -> c_2(c^#(x1))
             , n^#(f(x1)) -> c_5(f^#(n(x1)))
             , c(c(x1)) -> c(x1)}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  n(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [1]
                  a(x1) = [1] x1 + [0]
                  s(x1) = [1] x1 + [0]
                  f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [1]
                  n^#(x1) = [1] x1 + [15]
                  c^#(x1) = [1] x1 + [12]
                  c_1(x1) = [1] x1 + [7]
                  c_2(x1) = [1] x1 + [1]
                  c_3(x1) = [0] x1 + [0]
                  c_4(x1) = [0] x1 + [0]
                  c_5(x1) = [1] x1 + [1]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {  n(a(x1)) -> c(x1)
             , n(s(x1)) -> f(s(s(x1)))}
            and weakly orienting the rules
            {  c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))
             , n^#(a(x1)) -> c_2(c^#(x1))
             , n^#(f(x1)) -> c_5(f^#(n(x1)))
             , c(c(x1)) -> c(x1)}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  n(a(x1)) -> c(x1)
               , n(s(x1)) -> f(s(s(x1)))}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  n(x1) = [1] x1 + [8]
                  c(x1) = [1] x1 + [1]
                  a(x1) = [1] x1 + [0]
                  s(x1) = [1] x1 + [0]
                  f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [0]
                  n^#(x1) = [1] x1 + [14]
                  c^#(x1) = [1] x1 + [12]
                  c_1(x1) = [1] x1 + [0]
                  c_2(x1) = [1] x1 + [1]
                  c_3(x1) = [0] x1 + [0]
                  c_4(x1) = [0] x1 + [0]
                  c_5(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(f(x1)) -> f(n(a(c(x1))))
                 , n(f(x1)) -> f(n(x1))
                 , f(x1) -> n(c(n(a(x1))))
                 , f^#(x1) -> c_0(n^#(c(n(a(x1)))))}
              Weak Rules:
                {  n(a(x1)) -> c(x1)
                 , n(s(x1)) -> f(s(s(x1)))
                 , c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))
                 , n^#(a(x1)) -> c_2(c^#(x1))
                 , n^#(f(x1)) -> c_5(f^#(n(x1)))
                 , c(c(x1)) -> 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(f(x1)) -> f(n(a(c(x1))))
                   , n(f(x1)) -> f(n(x1))
                   , f(x1) -> n(c(n(a(x1))))
                   , f^#(x1) -> c_0(n^#(c(n(a(x1)))))}
                Weak Rules:
                  {  n(a(x1)) -> c(x1)
                   , n(s(x1)) -> f(s(s(x1)))
                   , c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))
                   , n^#(a(x1)) -> c_2(c^#(x1))
                   , n^#(f(x1)) -> c_5(f^#(n(x1)))
                   , c(c(x1)) -> c(x1)}
              
              Details:         
                The problem is Match-bounded by 1.
                The enriched problem is compatible with the following automaton:
                {  n_1(13) -> 12
                 , c_1(4) -> 11
                 , c_1(4) -> 12
                 , c_1(5) -> 11
                 , c_1(5) -> 12
                 , c_1(12) -> 11
                 , a_0(4) -> 4
                 , a_0(5) -> 4
                 , a_1(4) -> 13
                 , a_1(5) -> 13
                 , s_0(4) -> 5
                 , s_0(5) -> 5
                 , f^#_0(4) -> 6
                 , f^#_0(5) -> 6
                 , c_0_1(10) -> 6
                 , n^#_0(4) -> 8
                 , n^#_0(5) -> 8
                 , n^#_1(11) -> 10
                 , c^#_0(4) -> 9
                 , c^#_0(5) -> 9
                 , c_2_0(9) -> 8}
      
   2) {  n^#(a(x1)) -> c_2(c^#(x1))
       , c^#(c(x1)) -> c_3(c^#(x1))
       , c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))
       , f^#(x1) -> c_0(n^#(c(n(a(x1)))))
       , n^#(f(x1)) -> c_5(f^#(n(x1)))}
      
      The usable rules for this path are the following:
      {  c(f(x1)) -> f(n(a(c(x1))))
       , n(a(x1)) -> c(x1)
       , c(c(x1)) -> c(x1)
       , n(s(x1)) -> f(s(s(x1)))
       , n(f(x1)) -> f(n(x1))
       , f(x1) -> n(c(n(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:
              {  c(f(x1)) -> f(n(a(c(x1))))
               , n(a(x1)) -> c(x1)
               , c(c(x1)) -> c(x1)
               , n(s(x1)) -> f(s(s(x1)))
               , n(f(x1)) -> f(n(x1))
               , f(x1) -> n(c(n(a(x1))))
               , c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))
               , c^#(c(x1)) -> c_3(c^#(x1))
               , n^#(a(x1)) -> c_2(c^#(x1))
               , f^#(x1) -> c_0(n^#(c(n(a(x1)))))
               , n^#(f(x1)) -> c_5(f^#(n(x1)))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {  c(c(x1)) -> c(x1)
             , c^#(c(x1)) -> c_3(c^#(x1))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  c(c(x1)) -> c(x1)
               , c^#(c(x1)) -> c_3(c^#(x1))}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  n(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [1]
                  a(x1) = [1] x1 + [0]
                  s(x1) = [1] x1 + [0]
                  f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [1]
                  n^#(x1) = [1] x1 + [1]
                  c^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [0]
                  c_2(x1) = [1] x1 + [0]
                  c_3(x1) = [1] x1 + [0]
                  c_4(x1) = [0] x1 + [0]
                  c_5(x1) = [1] x1 + [1]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {  n^#(a(x1)) -> c_2(c^#(x1))
             , n^#(f(x1)) -> c_5(f^#(n(x1)))}
            and weakly orienting the rules
            {  c(c(x1)) -> c(x1)
             , c^#(c(x1)) -> c_3(c^#(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  n^#(a(x1)) -> c_2(c^#(x1))
               , n^#(f(x1)) -> c_5(f^#(n(x1)))}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  n(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [1]
                  a(x1) = [1] x1 + [0]
                  s(x1) = [1] x1 + [0]
                  f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [1]
                  n^#(x1) = [1] x1 + [9]
                  c^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [8]
                  c_2(x1) = [1] x1 + [7]
                  c_3(x1) = [1] x1 + [0]
                  c_4(x1) = [0] x1 + [0]
                  c_5(x1) = [1] x1 + [5]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))}
            and weakly orienting the rules
            {  n^#(a(x1)) -> c_2(c^#(x1))
             , n^#(f(x1)) -> c_5(f^#(n(x1)))
             , c(c(x1)) -> c(x1)
             , c^#(c(x1)) -> c_3(c^#(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  n(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [1]
                  a(x1) = [1] x1 + [0]
                  s(x1) = [1] x1 + [0]
                  f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [8]
                  n^#(x1) = [1] x1 + [2]
                  c^#(x1) = [1] x1 + [2]
                  c_1(x1) = [1] x1 + [0]
                  c_2(x1) = [1] x1 + [0]
                  c_3(x1) = [1] x1 + [1]
                  c_4(x1) = [0] x1 + [0]
                  c_5(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {  n(a(x1)) -> c(x1)
             , n(s(x1)) -> f(s(s(x1)))}
            and weakly orienting the rules
            {  c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))
             , n^#(a(x1)) -> c_2(c^#(x1))
             , n^#(f(x1)) -> c_5(f^#(n(x1)))
             , c(c(x1)) -> c(x1)
             , c^#(c(x1)) -> c_3(c^#(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  n(a(x1)) -> c(x1)
               , n(s(x1)) -> f(s(s(x1)))}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  n(x1) = [1] x1 + [2]
                  c(x1) = [1] x1 + [1]
                  a(x1) = [1] x1 + [0]
                  s(x1) = [1] x1 + [1]
                  f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [0]
                  n^#(x1) = [1] x1 + [8]
                  c^#(x1) = [1] x1 + [8]
                  c_1(x1) = [1] x1 + [0]
                  c_2(x1) = [1] x1 + [0]
                  c_3(x1) = [1] x1 + [0]
                  c_4(x1) = [0] x1 + [0]
                  c_5(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(f(x1)) -> f(n(a(c(x1))))
                 , n(f(x1)) -> f(n(x1))
                 , f(x1) -> n(c(n(a(x1))))
                 , f^#(x1) -> c_0(n^#(c(n(a(x1)))))}
              Weak Rules:
                {  n(a(x1)) -> c(x1)
                 , n(s(x1)) -> f(s(s(x1)))
                 , c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))
                 , n^#(a(x1)) -> c_2(c^#(x1))
                 , n^#(f(x1)) -> c_5(f^#(n(x1)))
                 , c(c(x1)) -> c(x1)
                 , c^#(c(x1)) -> c_3(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(f(x1)) -> f(n(a(c(x1))))
                   , n(f(x1)) -> f(n(x1))
                   , f(x1) -> n(c(n(a(x1))))
                   , f^#(x1) -> c_0(n^#(c(n(a(x1)))))}
                Weak Rules:
                  {  n(a(x1)) -> c(x1)
                   , n(s(x1)) -> f(s(s(x1)))
                   , c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))
                   , n^#(a(x1)) -> c_2(c^#(x1))
                   , n^#(f(x1)) -> c_5(f^#(n(x1)))
                   , c(c(x1)) -> c(x1)
                   , c^#(c(x1)) -> c_3(c^#(x1))}
              
              Details:         
                The problem is Match-bounded by 1.
                The enriched problem is compatible with the following automaton:
                {  n_1(13) -> 12
                 , c_1(4) -> 11
                 , c_1(4) -> 12
                 , c_1(5) -> 11
                 , c_1(5) -> 12
                 , c_1(12) -> 11
                 , a_0(4) -> 4
                 , a_0(5) -> 4
                 , a_1(4) -> 13
                 , a_1(5) -> 13
                 , s_0(4) -> 5
                 , s_0(5) -> 5
                 , f^#_0(4) -> 6
                 , f^#_0(5) -> 6
                 , c_0_1(10) -> 6
                 , n^#_0(4) -> 8
                 , n^#_0(5) -> 8
                 , n^#_1(11) -> 10
                 , c^#_0(4) -> 9
                 , c^#_0(5) -> 9
                 , c_2_0(9) -> 8}
      
   3) {  n^#(s(x1)) -> c_4(f^#(s(s(x1))))
       , f^#(x1) -> c_0(n^#(c(n(a(x1)))))
       , n^#(f(x1)) -> c_5(f^#(n(x1)))}
      
      The usable rules for this path are the following:
      {  c(f(x1)) -> f(n(a(c(x1))))
       , n(a(x1)) -> c(x1)
       , c(c(x1)) -> c(x1)
       , n(s(x1)) -> f(s(s(x1)))
       , n(f(x1)) -> f(n(x1))
       , f(x1) -> n(c(n(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:
              {  c(f(x1)) -> f(n(a(c(x1))))
               , n(a(x1)) -> c(x1)
               , c(c(x1)) -> c(x1)
               , n(s(x1)) -> f(s(s(x1)))
               , n(f(x1)) -> f(n(x1))
               , f(x1) -> n(c(n(a(x1))))
               , n^#(s(x1)) -> c_4(f^#(s(s(x1))))
               , f^#(x1) -> c_0(n^#(c(n(a(x1)))))
               , n^#(f(x1)) -> c_5(f^#(n(x1)))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {c(c(x1)) -> c(x1)}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c(c(x1)) -> c(x1)}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  n(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [1]
                  a(x1) = [1] x1 + [0]
                  s(x1) = [1] x1 + [0]
                  f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [1]
                  n^#(x1) = [1] x1 + [1]
                  c^#(x1) = [0] x1 + [0]
                  c_1(x1) = [0] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
                  c_3(x1) = [0] x1 + [0]
                  c_4(x1) = [1] x1 + [1]
                  c_5(x1) = [1] x1 + [1]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {n^#(f(x1)) -> c_5(f^#(n(x1)))}
            and weakly orienting the rules
            {c(c(x1)) -> c(x1)}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {n^#(f(x1)) -> c_5(f^#(n(x1)))}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  n(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [1]
                  a(x1) = [1] x1 + [0]
                  s(x1) = [1] x1 + [0]
                  f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [1]
                  n^#(x1) = [1] x1 + [1]
                  c^#(x1) = [0] x1 + [0]
                  c_1(x1) = [0] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
                  c_3(x1) = [0] x1 + [0]
                  c_4(x1) = [1] x1 + [1]
                  c_5(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {n^#(s(x1)) -> c_4(f^#(s(s(x1))))}
            and weakly orienting the rules
            {  n^#(f(x1)) -> c_5(f^#(n(x1)))
             , c(c(x1)) -> c(x1)}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {n^#(s(x1)) -> c_4(f^#(s(s(x1))))}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  n(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [1]
                  a(x1) = [1] x1 + [0]
                  s(x1) = [1] x1 + [0]
                  f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [1]
                  n^#(x1) = [1] x1 + [1]
                  c^#(x1) = [0] x1 + [0]
                  c_1(x1) = [0] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
                  c_3(x1) = [0] x1 + [0]
                  c_4(x1) = [1] x1 + [0]
                  c_5(x1) = [1] x1 + [1]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {n(a(x1)) -> c(x1)}
            and weakly orienting the rules
            {  n^#(s(x1)) -> c_4(f^#(s(s(x1))))
             , n^#(f(x1)) -> c_5(f^#(n(x1)))
             , c(c(x1)) -> c(x1)}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {n(a(x1)) -> c(x1)}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [9]
                  n(x1) = [1] x1 + [2]
                  c(x1) = [1] x1 + [0]
                  a(x1) = [1] x1 + [9]
                  s(x1) = [1] x1 + [3]
                  f^#(x1) = [1] x1 + [1]
                  c_0(x1) = [1] x1 + [0]
                  n^#(x1) = [1] x1 + [6]
                  c^#(x1) = [0] x1 + [0]
                  c_1(x1) = [0] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
                  c_3(x1) = [0] x1 + [0]
                  c_4(x1) = [1] x1 + [0]
                  c_5(x1) = [1] x1 + [2]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {n(s(x1)) -> f(s(s(x1)))}
            and weakly orienting the rules
            {  n(a(x1)) -> c(x1)
             , n^#(s(x1)) -> c_4(f^#(s(s(x1))))
             , n^#(f(x1)) -> c_5(f^#(n(x1)))
             , c(c(x1)) -> c(x1)}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {n(s(x1)) -> f(s(s(x1)))}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  n(x1) = [1] x1 + [5]
                  c(x1) = [1] x1 + [9]
                  a(x1) = [1] x1 + [4]
                  s(x1) = [1] x1 + [0]
                  f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [0]
                  n^#(x1) = [1] x1 + [13]
                  c^#(x1) = [0] x1 + [0]
                  c_1(x1) = [0] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
                  c_3(x1) = [0] x1 + [0]
                  c_4(x1) = [1] x1 + [1]
                  c_5(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:
                {  c(f(x1)) -> f(n(a(c(x1))))
                 , n(f(x1)) -> f(n(x1))
                 , f(x1) -> n(c(n(a(x1))))
                 , f^#(x1) -> c_0(n^#(c(n(a(x1)))))}
              Weak Rules:
                {  n(s(x1)) -> f(s(s(x1)))
                 , n(a(x1)) -> c(x1)
                 , n^#(s(x1)) -> c_4(f^#(s(s(x1))))
                 , n^#(f(x1)) -> c_5(f^#(n(x1)))
                 , c(c(x1)) -> 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(f(x1)) -> f(n(a(c(x1))))
                   , n(f(x1)) -> f(n(x1))
                   , f(x1) -> n(c(n(a(x1))))
                   , f^#(x1) -> c_0(n^#(c(n(a(x1)))))}
                Weak Rules:
                  {  n(s(x1)) -> f(s(s(x1)))
                   , n(a(x1)) -> c(x1)
                   , n^#(s(x1)) -> c_4(f^#(s(s(x1))))
                   , n^#(f(x1)) -> c_5(f^#(n(x1)))
                   , c(c(x1)) -> c(x1)}
              
              Details:         
                The problem is Match-bounded by 2.
                The enriched problem is compatible with the following automaton:
                {  n_1(12) -> 11
                 , n_2(19) -> 18
                 , c_1(4) -> 10
                 , c_1(4) -> 11
                 , c_1(5) -> 10
                 , c_1(5) -> 11
                 , c_1(11) -> 10
                 , c_2(14) -> 17
                 , c_2(14) -> 18
                 , c_2(18) -> 17
                 , a_0(4) -> 4
                 , a_0(5) -> 4
                 , a_1(4) -> 12
                 , a_1(5) -> 12
                 , a_2(14) -> 19
                 , s_0(4) -> 5
                 , s_0(5) -> 5
                 , s_1(4) -> 15
                 , s_1(5) -> 15
                 , s_1(15) -> 14
                 , f^#_0(4) -> 6
                 , f^#_0(5) -> 6
                 , f^#_1(14) -> 13
                 , c_0_1(9) -> 6
                 , c_0_2(16) -> 13
                 , n^#_0(4) -> 8
                 , n^#_0(5) -> 8
                 , n^#_1(10) -> 9
                 , n^#_2(17) -> 16
                 , c_4_0(6) -> 8
                 , c_4_1(13) -> 8}
      
   4) {  n^#(a(x1)) -> c_2(c^#(x1))
       , c^#(c(x1)) -> c_3(c^#(x1))
       , c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))}
      
      The usable rules for this path are the following:
      {  c(f(x1)) -> f(n(a(c(x1))))
       , n(a(x1)) -> c(x1)
       , c(c(x1)) -> c(x1)
       , n(s(x1)) -> f(s(s(x1)))
       , n(f(x1)) -> f(n(x1))
       , f(x1) -> n(c(n(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:
              {  c(f(x1)) -> f(n(a(c(x1))))
               , n(a(x1)) -> c(x1)
               , c(c(x1)) -> c(x1)
               , n(s(x1)) -> f(s(s(x1)))
               , n(f(x1)) -> f(n(x1))
               , f(x1) -> n(c(n(a(x1))))
               , c^#(c(x1)) -> c_3(c^#(x1))
               , n^#(a(x1)) -> c_2(c^#(x1))
               , c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {c(c(x1)) -> c(x1)}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c(c(x1)) -> c(x1)}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  n(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [1]
                  a(x1) = [1] x1 + [0]
                  s(x1) = [1] x1 + [0]
                  f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  n^#(x1) = [1] x1 + [1]
                  c^#(x1) = [1] x1 + [0]
                  c_1(x1) = [1] x1 + [3]
                  c_2(x1) = [1] x1 + [1]
                  c_3(x1) = [1] x1 + [1]
                  c_4(x1) = [0] x1 + [0]
                  c_5(x1) = [0] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {n^#(a(x1)) -> c_2(c^#(x1))}
            and weakly orienting the rules
            {c(c(x1)) -> c(x1)}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {n^#(a(x1)) -> c_2(c^#(x1))}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  n(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [1]
                  a(x1) = [1] x1 + [0]
                  s(x1) = [1] x1 + [0]
                  f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  n^#(x1) = [1] x1 + [1]
                  c^#(x1) = [1] x1 + [0]
                  c_1(x1) = [1] x1 + [0]
                  c_2(x1) = [1] x1 + [0]
                  c_3(x1) = [1] x1 + [1]
                  c_4(x1) = [0] x1 + [0]
                  c_5(x1) = [0] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {c^#(c(x1)) -> c_3(c^#(x1))}
            and weakly orienting the rules
            {  n^#(a(x1)) -> c_2(c^#(x1))
             , c(c(x1)) -> c(x1)}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c^#(c(x1)) -> c_3(c^#(x1))}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  n(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [1]
                  a(x1) = [1] x1 + [0]
                  s(x1) = [1] x1 + [0]
                  f^#(x1) = [1] x1 + [3]
                  c_0(x1) = [0] x1 + [0]
                  n^#(x1) = [1] x1 + [1]
                  c^#(x1) = [1] x1 + [0]
                  c_1(x1) = [1] x1 + [1]
                  c_2(x1) = [1] x1 + [1]
                  c_3(x1) = [1] x1 + [0]
                  c_4(x1) = [0] x1 + [0]
                  c_5(x1) = [0] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))}
            and weakly orienting the rules
            {  c^#(c(x1)) -> c_3(c^#(x1))
             , n^#(a(x1)) -> c_2(c^#(x1))
             , c(c(x1)) -> c(x1)}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  n(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [1]
                  a(x1) = [1] x1 + [0]
                  s(x1) = [1] x1 + [0]
                  f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  n^#(x1) = [1] x1 + [9]
                  c^#(x1) = [1] x1 + [9]
                  c_1(x1) = [1] x1 + [0]
                  c_2(x1) = [1] x1 + [0]
                  c_3(x1) = [1] x1 + [0]
                  c_4(x1) = [0] x1 + [0]
                  c_5(x1) = [0] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {f(x1) -> n(c(n(a(x1))))}
            and weakly orienting the rules
            {  c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))
             , c^#(c(x1)) -> c_3(c^#(x1))
             , n^#(a(x1)) -> c_2(c^#(x1))
             , c(c(x1)) -> c(x1)}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {f(x1) -> n(c(n(a(x1))))}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [4]
                  n(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [1]
                  a(x1) = [1] x1 + [0]
                  s(x1) = [1] x1 + [0]
                  f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  n^#(x1) = [1] x1 + [1]
                  c^#(x1) = [1] x1 + [0]
                  c_1(x1) = [1] x1 + [0]
                  c_2(x1) = [1] x1 + [1]
                  c_3(x1) = [1] x1 + [1]
                  c_4(x1) = [0] x1 + [0]
                  c_5(x1) = [0] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {n(a(x1)) -> c(x1)}
            and weakly orienting the rules
            {  f(x1) -> n(c(n(a(x1))))
             , c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))
             , c^#(c(x1)) -> c_3(c^#(x1))
             , n^#(a(x1)) -> c_2(c^#(x1))
             , c(c(x1)) -> c(x1)}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {n(a(x1)) -> c(x1)}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [10]
                  n(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [4]
                  a(x1) = [1] x1 + [5]
                  s(x1) = [1] x1 + [0]
                  f^#(x1) = [1] x1 + [13]
                  c_0(x1) = [0] x1 + [0]
                  n^#(x1) = [1] x1 + [12]
                  c^#(x1) = [1] x1 + [13]
                  c_1(x1) = [1] x1 + [1]
                  c_2(x1) = [1] x1 + [2]
                  c_3(x1) = [1] x1 + [4]
                  c_4(x1) = [0] x1 + [0]
                  c_5(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:
                {  c(f(x1)) -> f(n(a(c(x1))))
                 , n(s(x1)) -> f(s(s(x1)))
                 , n(f(x1)) -> f(n(x1))}
              Weak Rules:
                {  n(a(x1)) -> c(x1)
                 , f(x1) -> n(c(n(a(x1))))
                 , c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))
                 , c^#(c(x1)) -> c_3(c^#(x1))
                 , n^#(a(x1)) -> c_2(c^#(x1))
                 , c(c(x1)) -> 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(f(x1)) -> f(n(a(c(x1))))
                   , n(s(x1)) -> f(s(s(x1)))
                   , n(f(x1)) -> f(n(x1))}
                Weak Rules:
                  {  n(a(x1)) -> c(x1)
                   , f(x1) -> n(c(n(a(x1))))
                   , c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))
                   , c^#(c(x1)) -> c_3(c^#(x1))
                   , n^#(a(x1)) -> c_2(c^#(x1))
                   , c(c(x1)) -> c(x1)}
              
              Details:         
                The problem is Match-bounded by 0.
                The enriched problem is compatible with the following automaton:
                {  a_0(4) -> 4
                 , a_0(5) -> 4
                 , s_0(4) -> 5
                 , s_0(5) -> 5
                 , f^#_0(4) -> 6
                 , f^#_0(5) -> 6
                 , n^#_0(4) -> 8
                 , n^#_0(5) -> 8
                 , c^#_0(4) -> 9
                 , c^#_0(5) -> 9
                 , c_2_0(9) -> 8}
      
   5) {  n^#(a(x1)) -> c_2(c^#(x1))
       , c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))}
      
      The usable rules for this path are the following:
      {  c(f(x1)) -> f(n(a(c(x1))))
       , n(a(x1)) -> c(x1)
       , c(c(x1)) -> c(x1)
       , n(s(x1)) -> f(s(s(x1)))
       , n(f(x1)) -> f(n(x1))
       , f(x1) -> n(c(n(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:
              {  c(f(x1)) -> f(n(a(c(x1))))
               , n(a(x1)) -> c(x1)
               , c(c(x1)) -> c(x1)
               , n(s(x1)) -> f(s(s(x1)))
               , n(f(x1)) -> f(n(x1))
               , f(x1) -> n(c(n(a(x1))))
               , n^#(a(x1)) -> c_2(c^#(x1))
               , c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {c(c(x1)) -> c(x1)}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c(c(x1)) -> c(x1)}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  n(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [1]
                  a(x1) = [1] x1 + [0]
                  s(x1) = [1] x1 + [0]
                  f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  n^#(x1) = [1] x1 + [1]
                  c^#(x1) = [1] x1 + [0]
                  c_1(x1) = [1] x1 + [0]
                  c_2(x1) = [1] x1 + [1]
                  c_3(x1) = [0] x1 + [0]
                  c_4(x1) = [0] x1 + [0]
                  c_5(x1) = [0] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))}
            and weakly orienting the rules
            {c(c(x1)) -> c(x1)}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  n(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [1]
                  a(x1) = [1] x1 + [0]
                  s(x1) = [1] x1 + [0]
                  f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  n^#(x1) = [1] x1 + [1]
                  c^#(x1) = [1] x1 + [8]
                  c_1(x1) = [1] x1 + [0]
                  c_2(x1) = [1] x1 + [1]
                  c_3(x1) = [0] x1 + [0]
                  c_4(x1) = [0] x1 + [0]
                  c_5(x1) = [0] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {n^#(a(x1)) -> c_2(c^#(x1))}
            and weakly orienting the rules
            {  c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))
             , c(c(x1)) -> c(x1)}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {n^#(a(x1)) -> c_2(c^#(x1))}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  n(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [1]
                  a(x1) = [1] x1 + [0]
                  s(x1) = [1] x1 + [0]
                  f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  n^#(x1) = [1] x1 + [9]
                  c^#(x1) = [1] x1 + [5]
                  c_1(x1) = [1] x1 + [0]
                  c_2(x1) = [1] x1 + [0]
                  c_3(x1) = [0] x1 + [0]
                  c_4(x1) = [0] x1 + [0]
                  c_5(x1) = [0] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {n(a(x1)) -> c(x1)}
            and weakly orienting the rules
            {  n^#(a(x1)) -> c_2(c^#(x1))
             , c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))
             , c(c(x1)) -> c(x1)}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {n(a(x1)) -> c(x1)}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  n(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [1]
                  a(x1) = [1] x1 + [2]
                  s(x1) = [1] x1 + [1]
                  f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  n^#(x1) = [1] x1 + [5]
                  c^#(x1) = [1] x1 + [3]
                  c_1(x1) = [1] x1 + [0]
                  c_2(x1) = [1] x1 + [0]
                  c_3(x1) = [0] x1 + [0]
                  c_4(x1) = [0] x1 + [0]
                  c_5(x1) = [0] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {n(s(x1)) -> f(s(s(x1)))}
            and weakly orienting the rules
            {  n(a(x1)) -> c(x1)
             , n^#(a(x1)) -> c_2(c^#(x1))
             , c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))
             , c(c(x1)) -> c(x1)}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {n(s(x1)) -> f(s(s(x1)))}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [1] x1 + [0]
                  n(x1) = [1] x1 + [4]
                  c(x1) = [1] x1 + [1]
                  a(x1) = [1] x1 + [0]
                  s(x1) = [1] x1 + [0]
                  f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  n^#(x1) = [1] x1 + [13]
                  c^#(x1) = [1] x1 + [8]
                  c_1(x1) = [1] x1 + [0]
                  c_2(x1) = [1] x1 + [1]
                  c_3(x1) = [0] x1 + [0]
                  c_4(x1) = [0] x1 + [0]
                  c_5(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:
                {  c(f(x1)) -> f(n(a(c(x1))))
                 , n(f(x1)) -> f(n(x1))
                 , f(x1) -> n(c(n(a(x1))))}
              Weak Rules:
                {  n(s(x1)) -> f(s(s(x1)))
                 , n(a(x1)) -> c(x1)
                 , n^#(a(x1)) -> c_2(c^#(x1))
                 , c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))
                 , c(c(x1)) -> 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(f(x1)) -> f(n(a(c(x1))))
                   , n(f(x1)) -> f(n(x1))
                   , f(x1) -> n(c(n(a(x1))))}
                Weak Rules:
                  {  n(s(x1)) -> f(s(s(x1)))
                   , n(a(x1)) -> c(x1)
                   , n^#(a(x1)) -> c_2(c^#(x1))
                   , c^#(f(x1)) -> c_1(f^#(n(a(c(x1)))))
                   , c(c(x1)) -> c(x1)}
              
              Details:         
                The problem is Match-bounded by 0.
                The enriched problem is compatible with the following automaton:
                {  a_0(4) -> 4
                 , a_0(5) -> 4
                 , s_0(4) -> 5
                 , s_0(5) -> 5
                 , f^#_0(4) -> 6
                 , f^#_0(5) -> 6
                 , n^#_0(4) -> 8
                 , n^#_0(5) -> 8
                 , c^#_0(4) -> 9
                 , c^#_0(5) -> 9
                 , c_2_0(9) -> 8}
      
   6) {  n^#(a(x1)) -> c_2(c^#(x1))
       , c^#(c(x1)) -> c_3(c^#(x1))}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           f(x1) = [0] x1 + [0]
           n(x1) = [0] x1 + [0]
           c(x1) = [0] x1 + [0]
           a(x1) = [0] x1 + [0]
           s(x1) = [0] x1 + [0]
           f^#(x1) = [0] x1 + [0]
           c_0(x1) = [0] x1 + [0]
           n^#(x1) = [0] x1 + [0]
           c^#(x1) = [0] x1 + [0]
           c_1(x1) = [0] x1 + [0]
           c_2(x1) = [0] x1 + [0]
           c_3(x1) = [0] x1 + [0]
           c_4(x1) = [0] x1 + [0]
           c_5(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^#(c(x1)) -> c_3(c^#(x1))}
            Weak Rules: {n^#(a(x1)) -> c_2(c^#(x1))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {c^#(c(x1)) -> c_3(c^#(x1))}
            and weakly orienting the rules
            {n^#(a(x1)) -> c_2(c^#(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c^#(c(x1)) -> c_3(c^#(x1))}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [0] x1 + [0]
                  n(x1) = [0] x1 + [0]
                  c(x1) = [1] x1 + [8]
                  a(x1) = [1] x1 + [0]
                  s(x1) = [0] x1 + [0]
                  f^#(x1) = [0] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  n^#(x1) = [1] x1 + [8]
                  c^#(x1) = [1] x1 + [1]
                  c_1(x1) = [0] x1 + [0]
                  c_2(x1) = [1] x1 + [1]
                  c_3(x1) = [1] x1 + [3]
                  c_4(x1) = [0] x1 + [0]
                  c_5(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^#(c(x1)) -> c_3(c^#(x1))
                 , n^#(a(x1)) -> c_2(c^#(x1))}
            
            Details:         
              The given problem does not contain any strict rules
      
   7) {n^#(s(x1)) -> c_4(f^#(s(s(x1))))}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           f(x1) = [0] x1 + [0]
           n(x1) = [0] x1 + [0]
           c(x1) = [0] x1 + [0]
           a(x1) = [0] x1 + [0]
           s(x1) = [0] x1 + [0]
           f^#(x1) = [0] x1 + [0]
           c_0(x1) = [0] x1 + [0]
           n^#(x1) = [0] x1 + [0]
           c^#(x1) = [0] x1 + [0]
           c_1(x1) = [0] x1 + [0]
           c_2(x1) = [0] x1 + [0]
           c_3(x1) = [0] x1 + [0]
           c_4(x1) = [0] x1 + [0]
           c_5(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: {n^#(s(x1)) -> c_4(f^#(s(s(x1))))}
            Weak Rules: {}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {n^#(s(x1)) -> c_4(f^#(s(s(x1))))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {n^#(s(x1)) -> c_4(f^#(s(s(x1))))}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [0] x1 + [0]
                  n(x1) = [0] x1 + [0]
                  c(x1) = [0] x1 + [0]
                  a(x1) = [0] x1 + [0]
                  s(x1) = [1] x1 + [0]
                  f^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  n^#(x1) = [1] x1 + [1]
                  c^#(x1) = [0] x1 + [0]
                  c_1(x1) = [0] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
                  c_3(x1) = [0] x1 + [0]
                  c_4(x1) = [1] x1 + [0]
                  c_5(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: {n^#(s(x1)) -> c_4(f^#(s(s(x1))))}
            
            Details:         
              The given problem does not contain any strict rules
      
   8) {n^#(a(x1)) -> c_2(c^#(x1))}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           f(x1) = [0] x1 + [0]
           n(x1) = [0] x1 + [0]
           c(x1) = [0] x1 + [0]
           a(x1) = [0] x1 + [0]
           s(x1) = [0] x1 + [0]
           f^#(x1) = [0] x1 + [0]
           c_0(x1) = [0] x1 + [0]
           n^#(x1) = [0] x1 + [0]
           c^#(x1) = [0] x1 + [0]
           c_1(x1) = [0] x1 + [0]
           c_2(x1) = [0] x1 + [0]
           c_3(x1) = [0] x1 + [0]
           c_4(x1) = [0] x1 + [0]
           c_5(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: {n^#(a(x1)) -> c_2(c^#(x1))}
            Weak Rules: {}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {n^#(a(x1)) -> c_2(c^#(x1))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {n^#(a(x1)) -> c_2(c^#(x1))}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [0] x1 + [0]
                  n(x1) = [0] x1 + [0]
                  c(x1) = [0] x1 + [0]
                  a(x1) = [1] x1 + [0]
                  s(x1) = [0] x1 + [0]
                  f^#(x1) = [0] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  n^#(x1) = [1] x1 + [1]
                  c^#(x1) = [1] x1 + [0]
                  c_1(x1) = [0] x1 + [0]
                  c_2(x1) = [1] x1 + [0]
                  c_3(x1) = [0] x1 + [0]
                  c_4(x1) = [0] x1 + [0]
                  c_5(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: {n^#(a(x1)) -> c_2(c^#(x1))}
            
            Details:         
              The given problem does not contain any strict rules