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()))
  Arctic Interpretation Processor:
   dimension: 1
   interpretation:
    [=#](x0, x1) = x0 + 8x1 + 7,
    
    [implies#](x0, x1) = x0 + x1 + 4,
    
    [or#](x0, x1) = x0 + 2x1 + 3,
    
    [and#](x0, x1) = x0 + x1 + 3,
    
    [if#](x0, x1, x2) = 1x2 + 2,
    
    [not#](x0) = 4,
    
    [=](x0, x1) = 4x0 + 13x1 + 7,
    
    [implies](x0, x1) = 4x0 + 2x1 + 4,
    
    [or](x0, x1) = 4x0 + 2x1 + 4,
    
    [and](x0, x1) = 4x0 + 8x1 + 3,
    
    [if](x0, x1, x2) = 3x0 + 1x1 + 1x2 + 2,
    
    [true] = 2,
    
    [false] = 0,
    
    [not](x0) = 4x0 + 5
   orientation:
    not#(x) = 4 >= 3 = if#(x,false(),true())
    
    and#(x,y) = x + y + 3 >= 2 = if#(x,y,false())
    
    or#(x,y) = x + 2y + 3 >= 1y + 2 = if#(x,true(),y)
    
    implies#(x,y) = x + y + 4 >= 3 = if#(x,y,true())
    
    =#(x,y) = x + 8y + 7 >= 4 = not#(y)
    
    =#(x,y) = x + 8y + 7 >= 5y + 6 = if#(x,y,not(y))
    
    =#(x,y) = x + 8y + 7 >= 3 = if#(y,false(),true())
    
    =#(x,y) = x + 8y + 7 >= 4y + 4 = if#(x,y,if(y,false(),true()))
    
    not(x) = 4x + 5 >= 3x + 3 = if(x,false(),true())
    
    and(x,y) = 4x + 8y + 3 >= 3x + 1y + 2 = if(x,y,false())
    
    or(x,y) = 4x + 2y + 4 >= 3x + 1y + 3 = if(x,true(),y)
    
    implies(x,y) = 4x + 2y + 4 >= 3x + 1y + 3 = if(x,y,true())
    
    =(x,x) = 13x + 7 >= 2 = true()
    
    =(x,y) = 4x + 13y + 7 >= 3x + 5y + 6 = if(x,y,not(y))
    
    if(true(),x,y) = 1x + 1y + 5 >= x = x
    
    if(false(),x,y) = 1x + 1y + 3 >= y = y
    
    if(x,x,if(x,false(),true())) = 4x + 4 >= 2 = true()
    
    =(x,y) = 4x + 13y + 7 >= 3x + 4y + 4 = if(x,y,if(y,false(),true()))
   problem:
    DPs:
     
    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()))
   Qed