We consider the following Problem:

  Strict Trs:
    {  a(b(x)) -> b(a(a(x)))
     , b(c(x)) -> c(b(b(x)))
     , c(a(x)) -> a(c(c(x)))
     , u(a(x)) -> x
     , v(b(x)) -> x
     , w(c(x)) -> x
     , a(u(x)) -> x
     , b(v(x)) -> x
     , c(w(x)) -> x}
  StartTerms: basic terms
  Strategy: innermost

Certificate: YES(?,O(n^1))

Proof:
  We consider the following Problem:
  
    Strict Trs:
      {  a(b(x)) -> b(a(a(x)))
       , b(c(x)) -> c(b(b(x)))
       , c(a(x)) -> a(c(c(x)))
       , u(a(x)) -> x
       , v(b(x)) -> x
       , w(c(x)) -> x
       , a(u(x)) -> x
       , b(v(x)) -> x
       , c(w(x)) -> x}
    StartTerms: basic terms
    Strategy: innermost
  
  Certificate: YES(?,O(n^1))
  
  Proof:
    The weightgap principle applies, where following rules are oriented strictly:
    
    TRS Component:
      {  w(c(x)) -> x
       , c(w(x)) -> x}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      The following argument positions are usable:
        Uargs(a) = {1}, Uargs(b) = {1}, Uargs(c) = {1}, Uargs(u) = {},
        Uargs(v) = {}, Uargs(w) = {}
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       a(x1) = [1 0] x1 + [1]
               [1 0]      [1]
       b(x1) = [1 0] x1 + [0]
               [0 0]      [1]
       c(x1) = [1 0] x1 + [0]
               [0 1]      [0]
       u(x1) = [1 0] x1 + [0]
               [0 0]      [1]
       v(x1) = [1 0] x1 + [1]
               [0 0]      [1]
       w(x1) = [1 0] x1 + [1]
               [0 1]      [1]
    
    The strictly oriented rules are moved into the weak component.
    
    We consider the following Problem:
    
      Strict Trs:
        {  a(b(x)) -> b(a(a(x)))
         , b(c(x)) -> c(b(b(x)))
         , c(a(x)) -> a(c(c(x)))
         , u(a(x)) -> x
         , v(b(x)) -> x
         , a(u(x)) -> x
         , b(v(x)) -> x}
      Weak Trs:
        {  w(c(x)) -> x
         , c(w(x)) -> x}
      StartTerms: basic terms
      Strategy: innermost
    
    Certificate: YES(?,O(n^1))
    
    Proof:
      The weightgap principle applies, where following rules are oriented strictly:
      
      TRS Component:
        {  v(b(x)) -> x
         , b(v(x)) -> x}
      
      Interpretation of nonconstant growth:
      -------------------------------------
        The following argument positions are usable:
          Uargs(a) = {1}, Uargs(b) = {1}, Uargs(c) = {1}, Uargs(u) = {},
          Uargs(v) = {}, Uargs(w) = {}
        We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
        Interpretation Functions:
         a(x1) = [1 0] x1 + [0]
                 [0 0]      [0]
         b(x1) = [1 0] x1 + [0]
                 [0 1]      [1]
         c(x1) = [1 0] x1 + [0]
                 [0 1]      [0]
         u(x1) = [1 0] x1 + [1]
                 [0 0]      [1]
         v(x1) = [1 0] x1 + [1]
                 [0 1]      [0]
         w(x1) = [1 0] x1 + [1]
                 [0 1]      [1]
      
      The strictly oriented rules are moved into the weak component.
      
      We consider the following Problem:
      
        Strict Trs:
          {  a(b(x)) -> b(a(a(x)))
           , b(c(x)) -> c(b(b(x)))
           , c(a(x)) -> a(c(c(x)))
           , u(a(x)) -> x
           , a(u(x)) -> x}
        Weak Trs:
          {  v(b(x)) -> x
           , b(v(x)) -> x
           , w(c(x)) -> x
           , c(w(x)) -> x}
        StartTerms: basic terms
        Strategy: innermost
      
      Certificate: YES(?,O(n^1))
      
      Proof:
        The weightgap principle applies, where following rules are oriented strictly:
        
        TRS Component:
          {  u(a(x)) -> x
           , a(u(x)) -> x}
        
        Interpretation of nonconstant growth:
        -------------------------------------
          The following argument positions are usable:
            Uargs(a) = {1}, Uargs(b) = {1}, Uargs(c) = {1}, Uargs(u) = {},
            Uargs(v) = {}, Uargs(w) = {}
          We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
          Interpretation Functions:
           a(x1) = [1 0] x1 + [0]
                   [0 1]      [1]
           b(x1) = [1 0] x1 + [0]
                   [0 1]      [0]
           c(x1) = [1 0] x1 + [0]
                   [0 1]      [0]
           u(x1) = [1 0] x1 + [1]
                   [0 1]      [0]
           v(x1) = [1 0] x1 + [1]
                   [0 1]      [0]
           w(x1) = [1 0] x1 + [1]
                   [0 1]      [1]
        
        The strictly oriented rules are moved into the weak component.
        
        We consider the following Problem:
        
          Strict Trs:
            {  a(b(x)) -> b(a(a(x)))
             , b(c(x)) -> c(b(b(x)))
             , c(a(x)) -> a(c(c(x)))}
          Weak Trs:
            {  u(a(x)) -> x
             , a(u(x)) -> x
             , v(b(x)) -> x
             , b(v(x)) -> x
             , w(c(x)) -> x
             , c(w(x)) -> x}
          StartTerms: basic terms
          Strategy: innermost
        
        Certificate: YES(?,O(n^1))
        
        Proof:
          We consider the following Problem:
          
            Strict Trs:
              {  a(b(x)) -> b(a(a(x)))
               , b(c(x)) -> c(b(b(x)))
               , c(a(x)) -> a(c(c(x)))}
            Weak Trs:
              {  u(a(x)) -> x
               , a(u(x)) -> x
               , v(b(x)) -> x
               , b(v(x)) -> x
               , w(c(x)) -> x
               , c(w(x)) -> x}
            StartTerms: basic terms
            Strategy: innermost
          
          Certificate: YES(?,O(n^1))
          
          Proof:
            The problem is match-bounded by 0.
            The enriched problem is compatible with the following automaton:
            {  a_0(2) -> 1
             , b_0(2) -> 1
             , c_0(2) -> 1
             , u_0(2) -> 1
             , v_0(2) -> 1
             , w_0(2) -> 1}

Hurray, we answered YES(?,O(n^1))