We consider the following Problem:

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

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

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

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