We consider the following Problem:

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

Certificate: YES(?,O(n^2))

Proof:
  We consider the following Problem:
  
    Strict Trs:
      {  :(:(x, y), z) -> :(x, :(y, z))
       , :(+(x, y), z) -> +(:(x, z), :(y, z))
       , :(z, +(x, f(y))) -> :(g(z, y), +(x, a()))}
    StartTerms: basic terms
    Strategy: innermost
  
  Certificate: YES(?,O(n^2))
  
  Proof:
    We consider the following Problem:
    
      Strict Trs:
        {  :(:(x, y), z) -> :(x, :(y, z))
         , :(+(x, y), z) -> +(:(x, z), :(y, z))
         , :(z, +(x, f(y))) -> :(g(z, y), +(x, a()))}
      StartTerms: basic terms
      Strategy: innermost
    
    Certificate: YES(?,O(n^2))
    
    Proof:
      The following argument positions are usable:
        Uargs(:) = {2}, Uargs(+) = {1, 2}, Uargs(f) = {}, Uargs(g) = {}
      We have the following restricted  polynomial interpretation:
      Interpretation Functions:
       [:](x1, x2) = 2 + x1 + 2*x1*x2 + x1^2 + x2
       [+](x1, x2) = 2 + x1 + x2
       [f](x1) = 3
       [g](x1, x2) = 0
       [a]() = 2

Hurray, we answered YES(?,O(n^2))