We consider the following Problem:

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

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

Proof:
  We consider the following Problem:
  
    Strict Trs: {f(x, f(a(), y)) -> f(a(), f(f(f(a(), a()), y), x))}
    StartTerms: basic terms
    Strategy: innermost
  
  Certificate: YES(?,O(n^1))
  
  Proof:
    We consider the following Problem:
    
      Strict Trs: {f(x, f(a(), y)) -> f(a(), f(f(f(a(), a()), y), 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:
      {  f_0(2, 2) -> 1
       , a_0() -> 2}

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