We consider the following Problem:

  Strict Trs:
    {  :(x, x) -> e()
     , :(x, e()) -> x
     , i(:(x, y)) -> :(y, x)
     , :(:(x, y), z) -> :(x, :(z, i(y)))
     , :(e(), x) -> i(x)
     , i(i(x)) -> x
     , i(e()) -> e()
     , :(x, :(y, i(x))) -> i(y)
     , :(x, :(y, :(i(x), z))) -> :(i(z), y)
     , :(i(x), :(y, x)) -> i(y)
     , :(i(x), :(y, :(x, z))) -> :(i(z), y)}
  StartTerms: basic terms
  Strategy: innermost

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

Proof:
  We consider the following Problem:
  
    Strict Trs:
      {  :(x, x) -> e()
       , :(x, e()) -> x
       , i(:(x, y)) -> :(y, x)
       , :(:(x, y), z) -> :(x, :(z, i(y)))
       , :(e(), x) -> i(x)
       , i(i(x)) -> x
       , i(e()) -> e()
       , :(x, :(y, i(x))) -> i(y)
       , :(x, :(y, :(i(x), z))) -> :(i(z), y)
       , :(i(x), :(y, x)) -> i(y)
       , :(i(x), :(y, :(x, z))) -> :(i(z), y)}
    StartTerms: basic terms
    Strategy: innermost
  
  Certificate: YES(?,O(n^1))
  
  Proof:
    The weightgap principle applies, where following rules are oriented strictly:
    
    TRS Component:
      {  :(x, x) -> e()
       , :(x, e()) -> x}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      The following argument positions are usable:
        Uargs(:) = {1, 2}, Uargs(i) = {}
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       :(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                   [0 1]      [0 0]      [1]
       e() = [0]
             [0]
       i(x1) = [1 0] x1 + [0]
               [1 0]      [0]
    
    The strictly oriented rules are moved into the weak component.
    
    We consider the following Problem:
    
      Strict Trs:
        {  i(:(x, y)) -> :(y, x)
         , :(:(x, y), z) -> :(x, :(z, i(y)))
         , :(e(), x) -> i(x)
         , i(i(x)) -> x
         , i(e()) -> e()
         , :(x, :(y, i(x))) -> i(y)
         , :(x, :(y, :(i(x), z))) -> :(i(z), y)
         , :(i(x), :(y, x)) -> i(y)
         , :(i(x), :(y, :(x, z))) -> :(i(z), y)}
      Weak Trs:
        {  :(x, x) -> e()
         , :(x, e()) -> x}
      StartTerms: basic terms
      Strategy: innermost
    
    Certificate: YES(?,O(n^1))
    
    Proof:
      The weightgap principle applies, where following rules are oriented strictly:
      
      TRS Component: {i(e()) -> e()}
      
      Interpretation of nonconstant growth:
      -------------------------------------
        The following argument positions are usable:
          Uargs(:) = {1, 2}, Uargs(i) = {}
        We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
        Interpretation Functions:
         :(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                     [0 1]      [0 0]      [0]
         e() = [0]
               [0]
         i(x1) = [1 0] x1 + [1]
                 [1 0]      [1]
      
      The strictly oriented rules are moved into the weak component.
      
      We consider the following Problem:
      
        Strict Trs:
          {  i(:(x, y)) -> :(y, x)
           , :(:(x, y), z) -> :(x, :(z, i(y)))
           , :(e(), x) -> i(x)
           , i(i(x)) -> x
           , :(x, :(y, i(x))) -> i(y)
           , :(x, :(y, :(i(x), z))) -> :(i(z), y)
           , :(i(x), :(y, x)) -> i(y)
           , :(i(x), :(y, :(x, z))) -> :(i(z), y)}
        Weak Trs:
          {  i(e()) -> e()
           , :(x, x) -> e()
           , :(x, e()) -> x}
        StartTerms: basic terms
        Strategy: innermost
      
      Certificate: YES(?,O(n^1))
      
      Proof:
        The weightgap principle applies, where following rules are oriented strictly:
        
        TRS Component:
          {  :(e(), x) -> i(x)
           , :(x, :(y, i(x))) -> i(y)
           , :(x, :(y, :(i(x), z))) -> :(i(z), y)
           , :(i(x), :(y, x)) -> i(y)
           , :(i(x), :(y, :(x, z))) -> :(i(z), y)}
        
        Interpretation of nonconstant growth:
        -------------------------------------
          The following argument positions are usable:
            Uargs(:) = {1, 2}, Uargs(i) = {}
          We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
          Interpretation Functions:
           :(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
                       [0 1]      [1 0]      [0]
           e() = [2]
                 [0]
           i(x1) = [1 0] x1 + [0]
                   [1 0]      [0]
        
        The strictly oriented rules are moved into the weak component.
        
        We consider the following Problem:
        
          Strict Trs:
            {  i(:(x, y)) -> :(y, x)
             , :(:(x, y), z) -> :(x, :(z, i(y)))
             , i(i(x)) -> x}
          Weak Trs:
            {  :(e(), x) -> i(x)
             , :(x, :(y, i(x))) -> i(y)
             , :(x, :(y, :(i(x), z))) -> :(i(z), y)
             , :(i(x), :(y, x)) -> i(y)
             , :(i(x), :(y, :(x, z))) -> :(i(z), y)
             , i(e()) -> e()
             , :(x, x) -> e()
             , :(x, e()) -> x}
          StartTerms: basic terms
          Strategy: innermost
        
        Certificate: YES(?,O(n^1))
        
        Proof:
          The weightgap principle applies, where following rules are oriented strictly:
          
          TRS Component: {i(i(x)) -> x}
          
          Interpretation of nonconstant growth:
          -------------------------------------
            The following argument positions are usable:
              Uargs(:) = {1, 2}, Uargs(i) = {}
            We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
            Interpretation Functions:
             :(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                         [0 1]      [1 1]      [3]
             e() = [0]
                   [0]
             i(x1) = [1 0] x1 + [1]
                     [0 1]      [0]
          
          The strictly oriented rules are moved into the weak component.
          
          We consider the following Problem:
          
            Strict Trs:
              {  i(:(x, y)) -> :(y, x)
               , :(:(x, y), z) -> :(x, :(z, i(y)))}
            Weak Trs:
              {  i(i(x)) -> x
               , :(e(), x) -> i(x)
               , :(x, :(y, i(x))) -> i(y)
               , :(x, :(y, :(i(x), z))) -> :(i(z), y)
               , :(i(x), :(y, x)) -> i(y)
               , :(i(x), :(y, :(x, z))) -> :(i(z), y)
               , i(e()) -> e()
               , :(x, x) -> e()
               , :(x, e()) -> x}
            StartTerms: basic terms
            Strategy: innermost
          
          Certificate: YES(?,O(n^1))
          
          Proof:
            The weightgap principle applies, where following rules are oriented strictly:
            
            TRS Component: {i(:(x, y)) -> :(y, x)}
            
            Interpretation of nonconstant growth:
            -------------------------------------
              The following argument positions are usable:
                Uargs(:) = {1, 2}, Uargs(i) = {}
              We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
              Interpretation Functions:
               :(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
                           [0 1]      [0 1]      [0]
               e() = [2]
                     [0]
               i(x1) = [1 0] x1 + [1]
                       [0 1]      [0]
            
            The strictly oriented rules are moved into the weak component.
            
            We consider the following Problem:
            
              Strict Trs: {:(:(x, y), z) -> :(x, :(z, i(y)))}
              Weak Trs:
                {  i(:(x, y)) -> :(y, x)
                 , i(i(x)) -> x
                 , :(e(), x) -> i(x)
                 , :(x, :(y, i(x))) -> i(y)
                 , :(x, :(y, :(i(x), z))) -> :(i(z), y)
                 , :(i(x), :(y, x)) -> i(y)
                 , :(i(x), :(y, :(x, z))) -> :(i(z), y)
                 , i(e()) -> e()
                 , :(x, x) -> e()
                 , :(x, e()) -> x}
              StartTerms: basic terms
              Strategy: innermost
            
            Certificate: YES(?,O(n^1))
            
            Proof:
              We consider the following Problem:
              
                Strict Trs: {:(:(x, y), z) -> :(x, :(z, i(y)))}
                Weak Trs:
                  {  i(:(x, y)) -> :(y, x)
                   , i(i(x)) -> x
                   , :(e(), x) -> i(x)
                   , :(x, :(y, i(x))) -> i(y)
                   , :(x, :(y, :(i(x), z))) -> :(i(z), y)
                   , :(i(x), :(y, x)) -> i(y)
                   , :(i(x), :(y, :(x, z))) -> :(i(z), y)
                   , i(e()) -> e()
                   , :(x, x) -> e()
                   , :(x, e()) -> x}
                StartTerms: basic terms
                Strategy: innermost
              
              Certificate: YES(?,O(n^1))
              
              Proof:
                The problem is match-bounded by 0.
                The enriched problem is compatible with the following automaton:
                {  :_0(2, 2) -> 1
                 , e_0() -> 1
                 , e_0() -> 2
                 , i_0(2) -> 1}

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