We consider the following Problem:

  Strict Trs:
    {  D(t()) -> 1()
     , D(constant()) -> 0()
     , D(+(x, y)) -> +(D(x), D(y))
     , D(*(x, y)) -> +(*(y, D(x)), *(x, D(y)))
     , D(-(x, y)) -> -(D(x), D(y))}
  StartTerms: basic terms
  Strategy: innermost

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

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

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