We consider the following Problem:

  Strict Trs:
    {  f(0(), 1(), x) -> f(s(x), x, x)
     , f(x, y, s(z)) -> s(f(0(), 1(), z))}
  StartTerms: basic terms
  Strategy: innermost

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

Proof:
  We consider the following Problem:
  
    Strict Trs:
      {  f(0(), 1(), x) -> f(s(x), x, x)
       , f(x, y, s(z)) -> s(f(0(), 1(), 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(0(), 1(), x) -> f(s(x), x, x)}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      The following argument positions are usable:
        Uargs(f) = {}, Uargs(s) = {1}
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       f(x1, x2, x3) = [0 2] x1 + [0 0] x2 + [0 1] x3 + [1]
                       [0 0]      [0 0]      [0 0]      [1]
       0() = [0]
             [2]
       1() = [0]
             [0]
       s(x1) = [1 0] x1 + [0]
               [0 0]      [0]
    
    The strictly oriented rules are moved into the weak component.
    
    We consider the following Problem:
    
      Strict Trs: {f(x, y, s(z)) -> s(f(0(), 1(), z))}
      Weak Trs: {f(0(), 1(), x) -> f(s(x), x, x)}
      StartTerms: basic terms
      Strategy: innermost
    
    Certificate: YES(?,O(n^2))
    
    Proof:
      We consider the following Problem:
      
        Strict Trs: {f(x, y, s(z)) -> s(f(0(), 1(), z))}
        Weak Trs: {f(0(), 1(), x) -> f(s(x), x, x)}
        StartTerms: basic terms
        Strategy: innermost
      
      Certificate: YES(?,O(n^2))
      
      Proof:
        The following argument positions are usable:
          Uargs(f) = {}, Uargs(s) = {1}
        We have the following restricted  polynomial interpretation:
        Interpretation Functions:
         [f](x1, x2, x3) = x3^2
         [0]() = 2
         [1]() = 2
         [s](x1) = 2 + x1

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