We consider the following Problem:

  Strict Trs:
    {  a(x, y) -> b(x, b(0(), c(y)))
     , c(b(y, c(x))) -> c(c(b(a(0(), 0()), y)))
     , b(y, 0()) -> y}
  StartTerms: basic terms
  Strategy: innermost

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

Proof:
  We consider the following Problem:
  
    Strict Trs:
      {  a(x, y) -> b(x, b(0(), c(y)))
       , c(b(y, c(x))) -> c(c(b(a(0(), 0()), y)))
       , b(y, 0()) -> y}
    StartTerms: basic terms
    Strategy: innermost
  
  Certificate: YES(?,O(n^1))
  
  Proof:
    The weightgap principle applies, where following rules are oriented strictly:
    
    TRS Component: {a(x, y) -> b(x, b(0(), c(y)))}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      The following argument positions are usable:
        Uargs(a) = {}, Uargs(b) = {1, 2}, Uargs(c) = {1}
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       a(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                   [0 1]      [0 1]      [1]
       b(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                   [0 1]      [0 0]      [1]
       0() = [0]
             [3]
       c(x1) = [1 0] x1 + [0]
               [0 1]      [0]
    
    The strictly oriented rules are moved into the weak component.
    
    We consider the following Problem:
    
      Strict Trs:
        {  c(b(y, c(x))) -> c(c(b(a(0(), 0()), y)))
         , b(y, 0()) -> y}
      Weak Trs: {a(x, y) -> b(x, b(0(), c(y)))}
      StartTerms: basic terms
      Strategy: innermost
    
    Certificate: YES(?,O(n^1))
    
    Proof:
      The weightgap principle applies, where following rules are oriented strictly:
      
      TRS Component: {b(y, 0()) -> y}
      
      Interpretation of nonconstant growth:
      -------------------------------------
        The following argument positions are usable:
          Uargs(a) = {}, Uargs(b) = {1, 2}, Uargs(c) = {1}
        We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
        Interpretation Functions:
         a(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                     [0 1]      [0 0]      [2]
         b(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                     [0 1]      [0 0]      [1]
         0() = [1]
               [0]
         c(x1) = [1 0] x1 + [0]
                 [0 0]      [1]
      
      The strictly oriented rules are moved into the weak component.
      
      We consider the following Problem:
      
        Strict Trs: {c(b(y, c(x))) -> c(c(b(a(0(), 0()), y)))}
        Weak Trs:
          {  b(y, 0()) -> y
           , a(x, y) -> b(x, b(0(), c(y)))}
        StartTerms: basic terms
        Strategy: innermost
      
      Certificate: YES(?,O(n^1))
      
      Proof:
        We consider the following Problem:
        
          Strict Trs: {c(b(y, c(x))) -> c(c(b(a(0(), 0()), y)))}
          Weak Trs:
            {  b(y, 0()) -> y
             , a(x, y) -> b(x, b(0(), c(y)))}
          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(3, 3) -> 1
           , b_0(3, 3) -> 2
           , b_0(3, 4) -> 5
           , b_0(3, 5) -> 1
           , 0_0() -> 2
           , 0_0() -> 3
           , c_0(3) -> 4}

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