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:
 Bounds Processor:
  bound: 2
  enrichment: top
  automaton:
   final states: {8,7,1}
   transitions:
    not1(7) -> 1*
    not1(1) -> 1*
    not1(8) -> 1*
    true2() -> 8*,1,5
    if2(7,7,8) -> 1*
    if2(7,7,5) -> 1*
    if2(8,6,8) -> 1*
    if2(8,6,5) -> 1*
    if2(1,7,8) -> 1*
    if2(1,7,5) -> 1*
    if2(7,6,8) -> 1*
    if2(7,6,5) -> 1*
    if2(8,7,8) -> 1*
    if2(8,7,5) -> 1*
    if2(1,6,8) -> 1*
    if2(1,6,5) -> 1*
    false2() -> 7*,1,6
    and0(8,1) -> 1*
    and0(8,7) -> 1*
    and0(1,8) -> 1*
    and0(7,1) -> 1*
    and0(7,7) -> 1*
    and0(8,8) -> 1*
    and0(1,1) -> 1*
    and0(1,7) -> 1*
    and0(7,8) -> 1*
    or0(8,1) -> 1*
    or0(8,7) -> 1*
    or0(1,8) -> 1*
    or0(7,1) -> 1*
    or0(7,7) -> 1*
    or0(8,8) -> 1*
    or0(1,1) -> 1*
    or0(1,7) -> 1*
    or0(7,8) -> 1*
    implies0(8,1) -> 1*
    implies0(8,7) -> 1*
    implies0(1,8) -> 1*
    implies0(7,1) -> 1*
    implies0(7,7) -> 1*
    implies0(8,8) -> 1*
    implies0(1,1) -> 1*
    implies0(1,7) -> 1*
    implies0(7,8) -> 1*
    =0(8,1) -> 1*
    =0(8,7) -> 1*
    =0(1,8) -> 1*
    =0(7,1) -> 1*
    =0(7,7) -> 1*
    =0(8,8) -> 1*
    =0(1,1) -> 1*
    =0(1,7) -> 1*
    =0(7,8) -> 1*
    if1(1,7,7) -> 1*
    if1(1,1,1) -> 1*
    if1(8,8,8) -> 1*
    if1(7,8,7) -> 1*
    if1(1,7,1) -> 1*
    if1(8,1,7) -> 1*
    if1(8,7,7) -> 1*
    if1(7,8,1) -> 1*
    if1(8,1,1) -> 1*
    if1(8,7,1) -> 1*
    if1(1,1,8) -> 1*
    if1(1,7,8) -> 1*
    if1(7,8,8) -> 1*
    if1(1,8,7) -> 1*
    if1(8,1,8) -> 1*
    if1(7,1,7) -> 1*
    if1(8,7,8) -> 1*
    if1(7,7,7) -> 1*
    if1(1,8,1) -> 1*
    if1(7,1,1) -> 1*
    if1(7,7,1) -> 1*
    if1(8,8,7) -> 1*
    if1(8,8,1) -> 1*
    if1(1,8,8) -> 1*
    if1(7,1,8) -> 1*
    if1(7,7,8) -> 1*
    if1(1,1,7) -> 1*
    5 -> 1*
    6 -> 1*
    7 -> 1*
    8 -> 1*
  problem:
   
  Qed