We consider the following Problem:

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

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

Proof:
  We consider the following Problem:
  
    Strict Trs:
      {  +(*(x, y), *(a(), y)) -> *(+(x, a()), y)
       , *(*(x, y), z) -> *(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: {+(*(x, y), *(a(), y)) -> *(+(x, a()), y)}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      The following argument positions are usable:
        Uargs(+) = {}, Uargs(*) = {2}
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       +(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
                   [0 0]      [1 0]      [1]
       *(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                   [0 0]      [0 0]      [1]
       a() = [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: {+(*(x, y), *(a(), y)) -> *(+(x, a()), y)}
      StartTerms: basic terms
      Strategy: innermost
    
    Certificate: YES(?,O(n^1))
    
    Proof:
      The weightgap principle applies, where following rules are oriented strictly:
      
      TRS Component: {*(*(x, y), z) -> *(x, *(y, z))}
      
      Interpretation of nonconstant growth:
      -------------------------------------
        The following argument positions are usable:
          Uargs(+) = {}, Uargs(*) = {2}
        We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
        Interpretation Functions:
         +(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
                     [0 0]      [0 1]      [0]
         *(x1, x2) = [0 3] x1 + [1 0] x2 + [0]
                     [0 1]      [0 1]      [3]
         a() = [0]
               [0]
      
      The strictly oriented rules are moved into the weak component.
      
      We consider the following Problem:
      
        Weak Trs:
          {  *(*(x, y), z) -> *(x, *(y, z))
           , +(*(x, y), *(a(), y)) -> *(+(x, a()), y)}
        StartTerms: basic terms
        Strategy: innermost
      
      Certificate: YES(O(1),O(1))
      
      Proof:
        We consider the following Problem:
        
          Weak Trs:
            {  *(*(x, y), z) -> *(x, *(y, z))
             , +(*(x, y), *(a(), y)) -> *(+(x, a()), y)}
          StartTerms: basic terms
          Strategy: innermost
        
        Certificate: YES(O(1),O(1))
        
        Proof:
          Empty rules are trivially bounded

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