We consider the following Problem:

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

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

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

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