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()))

max/plus interpretations on N:

  not_A(x1) = max{3, 4 + x1}
  not#_A(x1) = max{0, 6}
  if_A(x1,x2,x3) = max{4, 2 + x1, x2, 1 + x3}
  if#_A(x1,x2,x3) = max{6, 1, 6, 6}
  false_A = 4
  false#_A = 3
  true_A = 3
  true#_A = 4
  and_A(x1,x2) = max{2, 3 + x1, 5 + x2}
  and#_A(x1,x2) = max{6, 2 + x1, 0}
  or_A(x1,x2) = max{2, 4 + x1, 1 + x2}
  or#_A(x1,x2) = max{6, 6, 5}
  implies_A(x1,x2) = max{3, 4 + x1, x2}
  implies#_A(x1,x2) = max{6, 6, 6}
  =_A(x1,x2) = max{2, 2 + x1, 5 + x2}
  =#_A(x1,x2) = max{7, 7, 7}

precedence:

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