We consider the following Problem:

  Strict Trs:
    {  f(0(), 1(), x) -> f(x, x, x)
     , f(x, y, z) -> 2()
     , 0() -> 2()
     , 1() -> 2()
     , g(x, x, y) -> y
     , g(x, y, y) -> x}
  StartTerms: basic terms
  Strategy: innermost

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

Proof:
  Arguments of following rules are not normal-forms:
  {f(0(), 1(), x) -> f(x, x, x)}
  
  All above mentioned rules can be savely removed.
  
  We consider the following Problem:
  
    Strict Trs:
      {  f(x, y, z) -> 2()
       , 0() -> 2()
       , 1() -> 2()
       , g(x, x, y) -> y
       , g(x, y, y) -> x}
    StartTerms: basic terms
    Strategy: innermost
  
  Certificate: YES(?,O(n^1))
  
  Proof:
    The weightgap principle applies, where following rules are oriented strictly:
    
    TRS Component: {f(x, y, z) -> 2()}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      The following argument positions are usable:
        Uargs(f) = {}, Uargs(g) = {}
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [2]
                       [0 0]      [0 0]      [0 0]      [0]
       0() = [0]
             [0]
       1() = [0]
             [0]
       2() = [0]
             [0]
       g(x1, x2, x3) = [1 1] x1 + [0 0] x2 + [0 0] x3 + [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:
        {  0() -> 2()
         , 1() -> 2()
         , g(x, x, y) -> y
         , g(x, y, y) -> x}
      Weak Trs: {f(x, y, z) -> 2()}
      StartTerms: basic terms
      Strategy: innermost
    
    Certificate: YES(?,O(n^1))
    
    Proof:
      The weightgap principle applies, where following rules are oriented strictly:
      
      TRS Component: {0() -> 2()}
      
      Interpretation of nonconstant growth:
      -------------------------------------
        The following argument positions are usable:
          Uargs(f) = {}, Uargs(g) = {}
        We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
        Interpretation Functions:
         f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                         [0 0]      [0 0]      [0 0]      [0]
         0() = [2]
               [0]
         1() = [0]
               [0]
         2() = [0]
               [0]
         g(x1, x2, x3) = [1 1] x1 + [0 0] x2 + [0 0] x3 + [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:
          {  1() -> 2()
           , g(x, x, y) -> y
           , g(x, y, y) -> x}
        Weak Trs:
          {  0() -> 2()
           , f(x, y, z) -> 2()}
        StartTerms: basic terms
        Strategy: innermost
      
      Certificate: YES(?,O(n^1))
      
      Proof:
        The weightgap principle applies, where following rules are oriented strictly:
        
        TRS Component: {1() -> 2()}
        
        Interpretation of nonconstant growth:
        -------------------------------------
          The following argument positions are usable:
            Uargs(f) = {}, Uargs(g) = {}
          We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
          Interpretation Functions:
           f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                           [0 0]      [0 0]      [0 0]      [0]
           0() = [0]
                 [0]
           1() = [2]
                 [0]
           2() = [0]
                 [0]
           g(x1, x2, x3) = [1 1] x1 + [0 0] x2 + [0 0] x3 + [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, x, y) -> y
             , g(x, y, y) -> x}
          Weak Trs:
            {  1() -> 2()
             , 0() -> 2()
             , f(x, y, z) -> 2()}
          StartTerms: basic terms
          Strategy: innermost
        
        Certificate: YES(?,O(n^1))
        
        Proof:
          The weightgap principle applies, where following rules are oriented strictly:
          
          TRS Component: {g(x, x, y) -> y}
          
          Interpretation of nonconstant growth:
          -------------------------------------
            The following argument positions are usable:
              Uargs(f) = {}, Uargs(g) = {}
            We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
            Interpretation Functions:
             f(x1, x2, x3) = [1 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                             [0 1]      [0 0]      [0 0]      [0]
             0() = [0]
                   [0]
             1() = [0]
                   [0]
             2() = [0]
                   [0]
             g(x1, x2, x3) = [1 1] x1 + [0 0] x2 + [1 0] x3 + [2]
                             [0 0]      [0 0]      [0 1]      [0]
          
          The strictly oriented rules are moved into the weak component.
          
          We consider the following Problem:
          
            Strict Trs: {g(x, y, y) -> x}
            Weak Trs:
              {  g(x, x, y) -> y
               , 1() -> 2()
               , 0() -> 2()
               , f(x, y, z) -> 2()}
            StartTerms: basic terms
            Strategy: innermost
          
          Certificate: YES(?,O(n^1))
          
          Proof:
            The weightgap principle applies, where following rules are oriented strictly:
            
            TRS Component: {g(x, y, y) -> x}
            
            Interpretation of nonconstant growth:
            -------------------------------------
              The following argument positions are usable:
                Uargs(f) = {}, Uargs(g) = {}
              We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
              Interpretation Functions:
               f(x1, x2, x3) = [1 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                               [0 1]      [0 0]      [0 0]      [0]
               0() = [0]
                     [0]
               1() = [0]
                     [0]
               2() = [0]
                     [0]
               g(x1, x2, x3) = [1 0] x1 + [0 0] x2 + [1 0] x3 + [1]
                               [0 1]      [0 0]      [0 1]      [0]
            
            The strictly oriented rules are moved into the weak component.
            
            We consider the following Problem:
            
              Weak Trs:
                {  g(x, y, y) -> x
                 , g(x, x, y) -> y
                 , 1() -> 2()
                 , 0() -> 2()
                 , f(x, y, z) -> 2()}
              StartTerms: basic terms
              Strategy: innermost
            
            Certificate: YES(O(1),O(1))
            
            Proof:
              We consider the following Problem:
              
                Weak Trs:
                  {  g(x, y, y) -> x
                   , g(x, x, y) -> y
                   , 1() -> 2()
                   , 0() -> 2()
                   , f(x, y, z) -> 2()}
                StartTerms: basic terms
                Strategy: innermost
              
              Certificate: YES(O(1),O(1))
              
              Proof:
                Empty rules are trivially bounded

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