YES

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()))

linear polynomial interpretations on N:

  not_A(x1) = 2
  not#_A(x1) = x1 + 2
  if_A(x1,x2,x3) = x2 + x3
  if#_A(x1,x2,x3) = 1
  false_A = 1
  false#_A = 1
  true_A = 1
  true#_A = 0
  and_A(x1,x2) = x1 + x2 + 1
  and#_A(x1,x2) = x1 + 2
  or_A(x1,x2) = x1 + x2 + 1
  or#_A(x1,x2) = x1 + x2 + 2
  implies_A(x1,x2) = x1 + x2 + 1
  implies#_A(x1,x2) = 2
  =_A(x1,x2) = x1 + x2 + 2
  =#_A(x1,x2) = x1 + x2 + 3

precedence:

  = > not = and = or = implies > if = false > true