We consider the following Problem:

  Strict Trs:
    {  times(x, plus(y, s(z))) ->
       plus(times(x, plus(y, times(s(z), 0()))), times(x, s(z)))
     , times(x, 0()) -> 0()
     , times(x, s(y)) -> plus(times(x, y), x)
     , plus(x, 0()) -> x
     , plus(x, s(y)) -> s(plus(x, y))}
  StartTerms: basic terms
  Strategy: innermost

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

Proof:
  Arguments of following rules are not normal-forms:
  {times(x, plus(y, s(z))) ->
   plus(times(x, plus(y, times(s(z), 0()))), times(x, s(z)))}
  
  All above mentioned rules can be savely removed.
  
  We consider the following Problem:
  
    Strict Trs:
      {  times(x, 0()) -> 0()
       , times(x, s(y)) -> plus(times(x, y), x)
       , plus(x, 0()) -> x
       , plus(x, s(y)) -> s(plus(x, y))}
    StartTerms: basic terms
    Strategy: innermost
  
  Certificate: YES(?,O(n^2))
  
  Proof:
    The weightgap principle applies, where following rules are oriented strictly:
    
    TRS Component: {times(x, 0()) -> 0()}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      The following argument positions are usable:
        Uargs(times) = {}, Uargs(plus) = {1}, Uargs(s) = {1}
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       times(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
                       [1 1]      [1 0]      [1]
       plus(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
                      [0 0]      [1 1]      [1]
       s(x1) = [1 0] x1 + [0]
               [0 0]      [0]
       0() = [0]
             [0]
    
    The strictly oriented rules are moved into the weak component.
    
    We consider the following Problem:
    
      Strict Trs:
        {  times(x, s(y)) -> plus(times(x, y), x)
         , plus(x, 0()) -> x
         , plus(x, s(y)) -> s(plus(x, y))}
      Weak Trs: {times(x, 0()) -> 0()}
      StartTerms: basic terms
      Strategy: innermost
    
    Certificate: YES(?,O(n^2))
    
    Proof:
      The weightgap principle applies, where following rules are oriented strictly:
      
      TRS Component: {plus(x, 0()) -> x}
      
      Interpretation of nonconstant growth:
      -------------------------------------
        The following argument positions are usable:
          Uargs(times) = {}, Uargs(plus) = {1}, Uargs(s) = {1}
        We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
        Interpretation Functions:
         times(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
                         [0 1]      [0 0]      [1]
         plus(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
                        [0 1]      [1 0]      [0]
         s(x1) = [1 0] x1 + [0]
                 [0 0]      [1]
         0() = [0]
               [0]
      
      The strictly oriented rules are moved into the weak component.
      
      We consider the following Problem:
      
        Strict Trs:
          {  times(x, s(y)) -> plus(times(x, y), x)
           , plus(x, s(y)) -> s(plus(x, y))}
        Weak Trs:
          {  plus(x, 0()) -> x
           , times(x, 0()) -> 0()}
        StartTerms: basic terms
        Strategy: innermost
      
      Certificate: YES(?,O(n^2))
      
      Proof:
        The weightgap principle applies, where following rules are oriented strictly:
        
        TRS Component: {times(x, s(y)) -> plus(times(x, y), x)}
        
        Interpretation of nonconstant growth:
        -------------------------------------
          The following argument positions are usable:
            Uargs(times) = {}, Uargs(plus) = {1}, Uargs(s) = {1}
          We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
          Interpretation Functions:
           times(x1, x2) = [1 0] x1 + [0 3] x2 + [0]
                           [0 1]      [0 0]      [1]
           plus(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
                          [0 1]      [0 0]      [0]
           s(x1) = [1 0] x1 + [0]
                   [0 1]      [3]
           0() = [0]
                 [0]
        
        The strictly oriented rules are moved into the weak component.
        
        We consider the following Problem:
        
          Strict Trs: {plus(x, s(y)) -> s(plus(x, y))}
          Weak Trs:
            {  times(x, s(y)) -> plus(times(x, y), x)
             , plus(x, 0()) -> x
             , times(x, 0()) -> 0()}
          StartTerms: basic terms
          Strategy: innermost
        
        Certificate: YES(?,O(n^2))
        
        Proof:
          We consider the following Problem:
          
            Strict Trs: {plus(x, s(y)) -> s(plus(x, y))}
            Weak Trs:
              {  times(x, s(y)) -> plus(times(x, y), x)
               , plus(x, 0()) -> x
               , times(x, 0()) -> 0()}
            StartTerms: basic terms
            Strategy: innermost
          
          Certificate: YES(?,O(n^2))
          
          Proof:
            The following argument positions are usable:
              Uargs(times) = {}, Uargs(plus) = {1}, Uargs(s) = {1}
            We have the following restricted  polynomial interpretation:
            Interpretation Functions:
             [times](x1, x2) = 2*x1*x2 + 2*x2^2
             [plus](x1, x2) = x1 + 2*x2
             [s](x1) = 2 + x1
             [0]() = 2

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