YES

Problem:
 not(x) -> if(x,false(),true())
 and(x,y) -> if(x,y,false())
 or(x,y) -> if(x,true(),y)
 implies(x,y) -> if(x,y,true())
 =(x,x) -> true()
 =(x,y) -> if(x,y,not(y))
 if(true(),x,y) -> x
 if(false(),x,y) -> y
 if(x,x,if(x,false(),true())) -> true()
 =(x,y) -> if(x,y,if(y,false(),true()))

Proof:
 DP Processor:
  DPs:
   not#(x) -> if#(x,false(),true())
   and#(x,y) -> if#(x,y,false())
   or#(x,y) -> if#(x,true(),y)
   implies#(x,y) -> if#(x,y,true())
   =#(x,y) -> not#(y)
   =#(x,y) -> if#(x,y,not(y))
   =#(x,y) -> if#(y,false(),true())
   =#(x,y) -> if#(x,y,if(y,false(),true()))
  TRS:
   not(x) -> if(x,false(),true())
   and(x,y) -> if(x,y,false())
   or(x,y) -> if(x,true(),y)
   implies(x,y) -> if(x,y,true())
   =(x,x) -> true()
   =(x,y) -> if(x,y,not(y))
   if(true(),x,y) -> x
   if(false(),x,y) -> y
   if(x,x,if(x,false(),true())) -> true()
   =(x,y) -> if(x,y,if(y,false(),true()))
  Usable Rule Processor:
   DPs:
    not#(x) -> if#(x,false(),true())
    and#(x,y) -> if#(x,y,false())
    or#(x,y) -> if#(x,true(),y)
    implies#(x,y) -> if#(x,y,true())
    =#(x,y) -> not#(y)
    =#(x,y) -> if#(x,y,not(y))
    =#(x,y) -> if#(y,false(),true())
    =#(x,y) -> if#(x,y,if(y,false(),true()))
   TRS:
    not(x) -> if(x,false(),true())
    if(true(),x,y) -> x
    if(false(),x,y) -> y
   Arctic Interpretation Processor:
    dimension: 1
    usable rules:
     
    interpretation:
     [=#](x0, x1) = x0 + 1x1 + 13,
     
     [implies#](x0, x1) = x0 + 1x1 + 2,
     
     [or#](x0, x1) = x0 + x1 + 8,
     
     [and#](x0, x1) = x0 + 1x1 + 2,
     
     [if#](x0, x1, x2) = x1 + 0,
     
     [not#](x0) = 8,
     
     [if](x0, x1, x2) = x0 + x1 + 1x2 + 0,
     
     [true] = 5,
     
     [false] = 7,
     
     [not](x0) = x0 + 3
    orientation:
     not#(x) = 8 >= 7 = if#(x,false(),true())
     
     and#(x,y) = x + 1y + 2 >= y + 0 = if#(x,y,false())
     
     or#(x,y) = x + y + 8 >= 5 = if#(x,true(),y)
     
     implies#(x,y) = x + 1y + 2 >= y + 0 = if#(x,y,true())
     
     =#(x,y) = x + 1y + 13 >= 8 = not#(y)
     
     =#(x,y) = x + 1y + 13 >= y + 0 = if#(x,y,not(y))
     
     =#(x,y) = x + 1y + 13 >= 7 = if#(y,false(),true())
     
     =#(x,y) = x + 1y + 13 >= y + 0 = if#(x,y,if(y,false(),true()))
     
     not(x) = x + 3 >= x + 7 = if(x,false(),true())
     
     if(true(),x,y) = x + 1y + 5 >= x = x
     
     if(false(),x,y) = x + 1y + 7 >= y = y
    problem:
     DPs:
      
     TRS:
      not(x) -> if(x,false(),true())
      if(true(),x,y) -> x
      if(false(),x,y) -> y
    Qed