We consider the following Problem:

  Strict Trs:
    {  f(x, y, w, w, a()) -> g1(x, x, y, w)
     , f(x, y, w, a(), a()) -> g1(y, x, x, w)
     , f(x, y, a(), a(), w) -> g2(x, y, y, w)
     , f(x, y, a(), w, w) -> g2(y, y, x, w)
     , g1(x, x, y, a()) -> h(x, y)
     , g1(y, x, x, a()) -> h(x, y)
     , g2(x, y, y, a()) -> h(x, y)
     , g2(y, y, x, a()) -> h(x, y)
     , h(x, x) -> x}
  StartTerms: basic terms
  Strategy: innermost

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

Proof:
  We consider the following Problem:
  
    Strict Trs:
      {  f(x, y, w, w, a()) -> g1(x, x, y, w)
       , f(x, y, w, a(), a()) -> g1(y, x, x, w)
       , f(x, y, a(), a(), w) -> g2(x, y, y, w)
       , f(x, y, a(), w, w) -> g2(y, y, x, w)
       , g1(x, x, y, a()) -> h(x, y)
       , g1(y, x, x, a()) -> h(x, y)
       , g2(x, y, y, a()) -> h(x, y)
       , g2(y, y, x, a()) -> h(x, y)
       , h(x, x) -> x}
    StartTerms: basic terms
    Strategy: innermost
  
  Certificate: YES(?,O(n^1))
  
  Proof:
    The weightgap principle applies, where following rules are oriented strictly:
    
    TRS Component:
      {  g2(x, y, y, a()) -> h(x, y)
       , g2(y, y, x, a()) -> h(x, y)}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      The following argument positions are usable:
        Uargs(f) = {}, Uargs(g1) = {}, Uargs(g2) = {}, Uargs(h) = {}
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       f(x1, x2, x3, x4, x5) = [1 1] x1 + [1 1] x2 + [0 0] x3 + [1 1] x4 + [0 0] x5 + [1]
                               [0 0]      [0 0]      [1 1]      [0 0]      [1 1]      [1]
       a() = [0]
             [0]
       g1(x1, x2, x3, x4) = [1 0] x1 + [0 1] x2 + [1 0] x3 + [0 0] x4 + [1]
                            [0 1]      [0 0]      [0 1]      [0 0]      [1]
       g2(x1, x2, x3, x4) = [1 1] x1 + [0 0] x2 + [1 1] x3 + [0 0] x4 + [1]
                            [0 0]      [1 1]      [0 0]      [0 0]      [1]
       h(x1, x2) = [1 1] x1 + [0 0] x2 + [0]
                   [0 0]      [1 1]      [0]
    
    The strictly oriented rules are moved into the weak component.
    
    We consider the following Problem:
    
      Strict Trs:
        {  f(x, y, w, w, a()) -> g1(x, x, y, w)
         , f(x, y, w, a(), a()) -> g1(y, x, x, w)
         , f(x, y, a(), a(), w) -> g2(x, y, y, w)
         , f(x, y, a(), w, w) -> g2(y, y, x, w)
         , g1(x, x, y, a()) -> h(x, y)
         , g1(y, x, x, a()) -> h(x, y)
         , h(x, x) -> x}
      Weak Trs:
        {  g2(x, y, y, a()) -> h(x, y)
         , g2(y, y, x, a()) -> h(x, y)}
      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) -> x}
      
      Interpretation of nonconstant growth:
      -------------------------------------
        The following argument positions are usable:
          Uargs(f) = {}, Uargs(g1) = {}, Uargs(g2) = {}, Uargs(h) = {}
        We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
        Interpretation Functions:
         f(x1, x2, x3, x4, x5) = [1 1] x1 + [1 1] x2 + [0 0] x3 + [1 1] x4 + [0 0] x5 + [1]
                                 [0 0]      [0 0]      [1 1]      [0 0]      [1 1]      [1]
         a() = [0]
               [0]
         g1(x1, x2, x3, x4) = [1 0] x1 + [0 1] x2 + [1 0] x3 + [0 0] x4 + [1]
                              [0 1]      [1 0]      [0 1]      [0 0]      [1]
         g2(x1, x2, x3, x4) = [1 1] x1 + [0 0] x2 + [1 1] x3 + [0 0] x4 + [1]
                              [0 0]      [1 1]      [0 0]      [0 0]      [1]
         h(x1, x2) = [1 1] x1 + [0 0] x2 + [1]
                     [0 0]      [1 1]      [0]
      
      The strictly oriented rules are moved into the weak component.
      
      We consider the following Problem:
      
        Strict Trs:
          {  f(x, y, w, w, a()) -> g1(x, x, y, w)
           , f(x, y, w, a(), a()) -> g1(y, x, x, w)
           , f(x, y, a(), a(), w) -> g2(x, y, y, w)
           , f(x, y, a(), w, w) -> g2(y, y, x, w)
           , g1(x, x, y, a()) -> h(x, y)
           , g1(y, x, x, a()) -> h(x, y)}
        Weak Trs:
          {  h(x, x) -> x
           , g2(x, y, y, a()) -> h(x, y)
           , g2(y, y, x, a()) -> h(x, y)}
        StartTerms: basic terms
        Strategy: innermost
      
      Certificate: YES(?,O(n^1))
      
      Proof:
        The weightgap principle applies, where following rules are oriented strictly:
        
        TRS Component:
          {  g1(x, x, y, a()) -> h(x, y)
           , g1(y, x, x, a()) -> h(x, y)}
        
        Interpretation of nonconstant growth:
        -------------------------------------
          The following argument positions are usable:
            Uargs(f) = {}, Uargs(g1) = {}, Uargs(g2) = {}, Uargs(h) = {}
          We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
          Interpretation Functions:
           f(x1, x2, x3, x4, x5) = [1 1] x1 + [1 1] x2 + [0 0] x3 + [1 1] x4 + [0 0] x5 + [1]
                                   [0 0]      [0 0]      [1 1]      [0 0]      [1 1]      [1]
           a() = [0]
                 [0]
           g1(x1, x2, x3, x4) = [1 1] x1 + [0 0] x2 + [1 1] x3 + [0 0] x4 + [1]
                                [0 0]      [1 1]      [0 0]      [0 0]      [1]
           g2(x1, x2, x3, x4) = [1 0] x1 + [0 1] x2 + [1 0] x3 + [0 0] x4 + [1]
                                [0 1]      [1 0]      [0 1]      [0 0]      [1]
           h(x1, x2) = [1 0] x1 + [1 1] x2 + [0]
                       [0 1]      [0 0]      [0]
        
        The strictly oriented rules are moved into the weak component.
        
        We consider the following Problem:
        
          Strict Trs:
            {  f(x, y, w, w, a()) -> g1(x, x, y, w)
             , f(x, y, w, a(), a()) -> g1(y, x, x, w)
             , f(x, y, a(), a(), w) -> g2(x, y, y, w)
             , f(x, y, a(), w, w) -> g2(y, y, x, w)}
          Weak Trs:
            {  g1(x, x, y, a()) -> h(x, y)
             , g1(y, x, x, a()) -> h(x, y)
             , h(x, x) -> x
             , g2(x, y, y, a()) -> h(x, y)
             , g2(y, y, x, a()) -> h(x, y)}
          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, a(), a(), w) -> g2(x, y, y, w)
             , f(x, y, a(), w, w) -> g2(y, y, x, w)}
          
          Interpretation of nonconstant growth:
          -------------------------------------
            The following argument positions are usable:
              Uargs(f) = {}, Uargs(g1) = {}, Uargs(g2) = {}, Uargs(h) = {}
            We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
            Interpretation Functions:
             f(x1, x2, x3, x4, x5) = [1 2] x1 + [1 0] x2 + [1 1] x3 + [0 0] x4 + [1 0] x5 + [1]
                                     [0 0]      [0 1]      [0 0]      [0 1]      [0 0]      [1]
             a() = [0]
                   [0]
             g1(x1, x2, x3, x4) = [1 0] x1 + [0 1] x2 + [1 0] x3 + [0 0] x4 + [1]
                                  [0 1]      [0 0]      [0 1]      [0 0]      [1]
             g2(x1, x2, x3, x4) = [1 0] x1 + [0 0] x2 + [1 0] x3 + [0 0] x4 + [0]
                                  [0 0]      [0 1]      [0 0]      [0 0]      [1]
             h(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                         [0 0]      [0 1]      [0]
          
          The strictly oriented rules are moved into the weak component.
          
          We consider the following Problem:
          
            Strict Trs:
              {  f(x, y, w, w, a()) -> g1(x, x, y, w)
               , f(x, y, w, a(), a()) -> g1(y, x, x, w)}
            Weak Trs:
              {  f(x, y, a(), a(), w) -> g2(x, y, y, w)
               , f(x, y, a(), w, w) -> g2(y, y, x, w)
               , g1(x, x, y, a()) -> h(x, y)
               , g1(y, x, x, a()) -> h(x, y)
               , h(x, x) -> x
               , g2(x, y, y, a()) -> h(x, y)
               , g2(y, y, x, a()) -> h(x, y)}
            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, w, w, a()) -> g1(x, x, y, w)
               , f(x, y, w, a(), a()) -> g1(y, x, x, w)}
            
            Interpretation of nonconstant growth:
            -------------------------------------
              The following argument positions are usable:
                Uargs(f) = {}, Uargs(g1) = {}, Uargs(g2) = {}, Uargs(h) = {}
              We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
              Interpretation Functions:
               f(x1, x2, x3, x4, x5) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [0 0] x4 + [1 0] x5 + [1]
                                       [1 1]      [1 1]      [1 1]      [0 0]      [1 1]      [1]
               a() = [0]
                     [0]
               g1(x1, x2, x3, x4) = [1 0] x1 + [0 0] x2 + [1 0] x3 + [0 0] x4 + [0]
                                    [1 1]      [0 0]      [1 1]      [0 0]      [1]
               g2(x1, x2, x3, x4) = [1 0] x1 + [0 0] x2 + [1 0] x3 + [0 0] x4 + [1]
                                    [1 1]      [0 0]      [1 1]      [0 0]      [1]
               h(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                           [0 1]      [1 1]      [0]
            
            The strictly oriented rules are moved into the weak component.
            
            We consider the following Problem:
            
              Weak Trs:
                {  f(x, y, w, w, a()) -> g1(x, x, y, w)
                 , f(x, y, w, a(), a()) -> g1(y, x, x, w)
                 , f(x, y, a(), a(), w) -> g2(x, y, y, w)
                 , f(x, y, a(), w, w) -> g2(y, y, x, w)
                 , g1(x, x, y, a()) -> h(x, y)
                 , g1(y, x, x, a()) -> h(x, y)
                 , h(x, x) -> x
                 , g2(x, y, y, a()) -> h(x, y)
                 , g2(y, y, x, a()) -> h(x, y)}
              StartTerms: basic terms
              Strategy: innermost
            
            Certificate: YES(O(1),O(1))
            
            Proof:
              We consider the following Problem:
              
                Weak Trs:
                  {  f(x, y, w, w, a()) -> g1(x, x, y, w)
                   , f(x, y, w, a(), a()) -> g1(y, x, x, w)
                   , f(x, y, a(), a(), w) -> g2(x, y, y, w)
                   , f(x, y, a(), w, w) -> g2(y, y, x, w)
                   , g1(x, x, y, a()) -> h(x, y)
                   , g1(y, x, x, a()) -> h(x, y)
                   , h(x, x) -> x
                   , g2(x, y, y, a()) -> h(x, y)
                   , g2(y, y, x, a()) -> h(x, y)}
                StartTerms: basic terms
                Strategy: innermost
              
              Certificate: YES(O(1),O(1))
              
              Proof:
                Empty rules are trivially bounded

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