We consider the following Problem:

  Strict Trs:
    {  or(true(), y) -> true()
     , or(x, true()) -> true()
     , or(false(), false()) -> false()
     , mem(x, nil()) -> false()
     , mem(x, set(y)) -> =(x, y)
     , mem(x, union(y, z)) -> or(mem(x, y), mem(x, z))}
  StartTerms: basic terms
  Strategy: innermost

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

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