We consider the following Problem:

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

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

Proof:
  We consider the following Problem:
  
    Strict Trs:
      {  +(+(x, y), z) -> +(x, +(y, z))
       , +(f(x), f(y)) -> f(+(x, y))
       , +(f(x), +(f(y), z)) -> +(f(+(x, y)), z)}
    StartTerms: basic terms
    Strategy: innermost
  
  Certificate: YES(?,O(n^1))
  
  Proof:
    The weightgap principle applies, where following rules are oriented strictly:
    
    TRS Component:
      {  +(f(x), f(y)) -> f(+(x, y))
       , +(f(x), +(f(y), z)) -> +(f(+(x, y)), z)}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       +(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                   [0 1]      [0 0]      [0]
       f(x1) = [1 0] x1 + [2]
               [0 0]      [0]
    
    The strictly oriented rules are moved into the weak component.
    
    We consider the following Problem:
    
      Strict Trs: {+(+(x, y), z) -> +(x, +(y, z))}
      Weak Trs:
        {  +(f(x), f(y)) -> f(+(x, y))
         , +(f(x), +(f(y), z)) -> +(f(+(x, y)), z)}
      StartTerms: basic terms
      Strategy: innermost
    
    Certificate: YES(?,O(n^1))
    
    Proof:
      We consider the following Problem:
      
        Strict Trs: {+(+(x, y), z) -> +(x, +(y, z))}
        Weak Trs:
          {  +(f(x), f(y)) -> f(+(x, y))
           , +(f(x), +(f(y), z)) -> +(f(+(x, y)), z)}
        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:
        {  +_0(2, 2) -> 1
         , f_0(1) -> 1
         , f_0(2) -> 2}

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