We consider the following Problem:

  Strict Trs:
    {  +(x, 0()) -> x
     , +(x, i(x)) -> 0()
     , +(+(x, y), z) -> +(x, +(y, z))
     , *(x, +(y, z)) -> +(*(x, y), *(x, z))
     , *(+(x, y), z) -> +(*(x, z), *(y, z))}
  StartTerms: basic terms
  Strategy: innermost

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

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

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