We consider the following Problem:

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

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

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