We consider the following Problem:

  Strict Trs:
    {  f(f(x)) -> f(g(f(x), x))
     , f(f(x)) -> f(h(f(x), f(x)))
     , g(x, y) -> y
     , h(x, x) -> g(x, 0())}
  StartTerms: basic terms
  Strategy: innermost

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

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

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