We consider the following Problem:

  Strict Trs:
    {  if(true(), x, y) -> x
     , if(false(), x, y) -> y
     , if(x, y, y) -> y
     , if(if(x, y, z), u(), v()) ->
       if(x, if(y, u(), v()), if(z, u(), v()))}
  StartTerms: basic terms
  Strategy: innermost

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

Proof:
  We consider the following Problem:
  
    Strict Trs:
      {  if(true(), x, y) -> x
       , if(false(), x, y) -> y
       , if(x, y, y) -> y
       , if(if(x, y, z), u(), v()) ->
         if(x, if(y, u(), v()), if(z, u(), v()))}
    StartTerms: basic terms
    Strategy: innermost
  
  Certificate: YES(?,O(n^1))
  
  Proof:
    The weightgap principle applies, where following rules are oriented strictly:
    
    TRS Component:
      {  if(false(), x, y) -> y
       , if(x, y, y) -> y}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      The following argument positions are usable:
        Uargs(if) = {2, 3}
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       if(x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [1]
                        [0 1]      [0 0]      [1 1]      [1]
       true() = [0]
                [0]
       false() = [0]
                 [0]
       u() = [0]
             [0]
       v() = [0]
             [0]
    
    The strictly oriented rules are moved into the weak component.
    
    We consider the following Problem:
    
      Strict Trs:
        {  if(true(), x, y) -> x
         , if(if(x, y, z), u(), v()) ->
           if(x, if(y, u(), v()), if(z, u(), v()))}
      Weak Trs:
        {  if(false(), x, y) -> y
         , if(x, y, y) -> y}
      StartTerms: basic terms
      Strategy: innermost
    
    Certificate: YES(?,O(n^1))
    
    Proof:
      The weightgap principle applies, where following rules are oriented strictly:
      
      TRS Component: {if(true(), x, y) -> x}
      
      Interpretation of nonconstant growth:
      -------------------------------------
        The following argument positions are usable:
          Uargs(if) = {2, 3}
        We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
        Interpretation Functions:
         if(x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [1]
                          [0 1]      [0 1]      [0 1]      [0]
         true() = [0]
                  [1]
         false() = [0]
                   [0]
         u() = [0]
               [0]
         v() = [0]
               [0]
      
      The strictly oriented rules are moved into the weak component.
      
      We consider the following Problem:
      
        Strict Trs:
          {if(if(x, y, z), u(), v()) ->
           if(x, if(y, u(), v()), if(z, u(), v()))}
        Weak Trs:
          {  if(true(), x, y) -> x
           , if(false(), x, y) -> y
           , if(x, y, y) -> y}
        StartTerms: basic terms
        Strategy: innermost
      
      Certificate: YES(?,O(n^1))
      
      Proof:
        We consider the following Problem:
        
          Strict Trs:
            {if(if(x, y, z), u(), v()) ->
             if(x, if(y, u(), v()), if(z, u(), v()))}
          Weak Trs:
            {  if(true(), x, y) -> x
             , if(false(), x, y) -> y
             , if(x, y, y) -> y}
          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:
          {  if_0(2, 2, 2) -> 1
           , true_0() -> 1
           , true_0() -> 2
           , false_0() -> 1
           , false_0() -> 2
           , u_0() -> 1
           , u_0() -> 2
           , v_0() -> 1
           , v_0() -> 2}

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