We consider the following Problem:

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

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

Proof:
  Arguments of following rules are not normal-forms:
  {f(c(c(z, a(), a()), x, a())) -> z}
  
  All above mentioned rules can be savely removed.
  
  We consider the following Problem:
  
    Strict Trs:
      {  c(z, x, a()) -> f(b(b(f(z), z), x))
       , b(y, b(z, a())) -> f(b(c(f(a()), y, z), z))}
    StartTerms: basic terms
    Strategy: innermost
  
  Certificate: YES(?,O(n^1))
  
  Proof:
    The weightgap principle applies, where following rules are oriented strictly:
    
    TRS Component: {c(z, x, a()) -> f(b(b(f(z), z), x))}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      The following argument positions are usable:
        Uargs(c) = {}, Uargs(f) = {1}, Uargs(b) = {1}
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       c(x1, x2, x3) = [1 0] x1 + [0 0] x2 + [0 0] x3 + [1]
                       [0 0]      [0 0]      [0 0]      [1]
       a() = [0]
             [0]
       f(x1) = [1 0] x1 + [0]
               [0 0]      [1]
       b(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
                   [0 0]      [1 0]      [1]
    
    The strictly oriented rules are moved into the weak component.
    
    We consider the following Problem:
    
      Strict Trs: {b(y, b(z, a())) -> f(b(c(f(a()), y, z), z))}
      Weak Trs: {c(z, x, a()) -> f(b(b(f(z), z), x))}
      StartTerms: basic terms
      Strategy: innermost
    
    Certificate: YES(?,O(n^1))
    
    Proof:
      We consider the following Problem:
      
        Strict Trs: {b(y, b(z, a())) -> f(b(c(f(a()), y, z), z))}
        Weak Trs: {c(z, x, a()) -> f(b(b(f(z), z), x))}
        StartTerms: basic terms
        Strategy: innermost
      
      Certificate: YES(?,O(n^1))
      
      Proof:
        The following argument positions are usable:
          Uargs(c) = {}, Uargs(f) = {1}, Uargs(b) = {1}
        We have the following constructor-based EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
        Interpretation Functions:
         c(x1, x2, x3) = [2 0] x1 + [1 0] x2 + [0 2] x3 + [0]
                         [0 0]      [0 1]      [0 0]      [0]
         a() = [0]
               [2]
         f(x1) = [1 0] x1 + [0]
                 [0 0]      [0]
         b(x1, x2) = [1 2] x1 + [1 0] x2 + [1]
                     [0 0]      [0 0]      [0]

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