YES(?,O(n^1))

Problem:
 f(f(x)) -> f(c(f(x)))
 f(f(x)) -> f(d(f(x)))
 g(c(x)) -> x
 g(d(x)) -> x
 g(c(0())) -> g(d(1()))
 g(c(1())) -> g(d(0()))

Proof:
 Bounds Processor:
  bound: 2
  enrichment: match
  automaton:
   final states: {7}
   transitions:
    g1(23) -> 24*
    d1(22) -> 23*
    d1(16) -> 17*
    01() -> 25*
    11() -> 22*
    f1(12) -> 13*
    f1(14) -> 15*
    c1(38) -> 39*
    c1(13) -> 14*
    f2(30) -> 31*
    f2(32) -> 33*
    f0(7) -> 7*
    d2(40) -> 41*
    c0(7) -> 7*
    c2(31) -> 32*
    d0(7) -> 7*
    g0(7) -> 7*
    00() -> 7*
    10() -> 7*
    7 -> 12*
    13 -> 16*
    14 -> 30*
    15 -> 13,38,16,7
    17 -> 14*
    22 -> 24,7
    24 -> 7*
    25 -> 22*
    31 -> 40*
    33 -> 13,16
    39 -> 14*
    41 -> 32*
  problem:
   
  Qed