YES(?,O(n^1))

Problem:
 f(a()) -> g(h(a()))
 h(g(x)) -> g(h(f(x)))
 k(x,h(x),a()) -> h(x)
 k(f(x),y,x) -> f(x)

Proof:
 Bounds Processor:
  bound: 2
  enrichment: match
  automaton:
   final states: {21,13,11,5,4,3}
   transitions:
    g0(20) -> 2*
    g0(12) -> 2*
    g0(2) -> 2*
    g0(1) -> 2*
    h0(20) -> 4*
    h0(12) -> 4*
    h0(2) -> 4*
    h0(1) -> 4*
    k0(20,12,1) -> 5*
    k0(20,2,20) -> 5*
    k0(20,20,1) -> 5*
    k0(20,1,12) -> 5*
    k0(1,1,1) -> 5*
    k0(20,12,20) -> 5*
    k0(1,1,20) -> 5*
    k0(20,2,2) -> 5*
    k0(12,2,1) -> 5*
    k0(1,2,12) -> 5*
    k0(2,2,1) -> 5*
    k0(20,12,2) -> 5*
    k0(12,12,1) -> 5*
    k0(1,12,12) -> 5*
    k0(2,12,1) -> 5*
    k0(12,2,20) -> 5*
    k0(2,2,20) -> 5*
    k0(20,20,2) -> 5*
    k0(12,20,1) -> 5*
    k0(12,1,12) -> 5*
    k0(1,20,12) -> 5*
    k0(2,20,1) -> 5*
    k0(2,1,12) -> 5*
    k0(1,1,2) -> 5*
    k0(12,12,20) -> 5*
    k0(20,20,20) -> 5*
    k0(2,12,20) -> 5*
    k0(12,2,2) -> 5*
    k0(2,2,2) -> 5*
    k0(12,12,2) -> 5*
    k0(2,12,2) -> 5*
    k0(20,1,1) -> 5*
    k0(12,20,2) -> 5*
    k0(2,20,2) -> 5*
    k0(12,20,20) -> 5*
    k0(2,20,20) -> 5*
    k0(20,1,20) -> 5*
    k0(20,2,12) -> 5*
    k0(1,2,1) -> 5*
    k0(20,12,12) -> 5*
    k0(1,12,1) -> 5*
    k0(20,20,12) -> 5*
    k0(1,2,20) -> 5*
    k0(20,1,2) -> 5*
    k0(12,1,1) -> 5*
    k0(1,20,1) -> 5*
    k0(1,1,12) -> 5*
    k0(2,1,1) -> 5*
    k0(1,12,20) -> 5*
    k0(12,1,20) -> 5*
    k0(2,1,20) -> 5*
    k0(12,2,12) -> 5*
    k0(2,2,12) -> 5*
    k0(1,2,2) -> 5*
    k0(12,12,12) -> 5*
    k0(2,12,12) -> 5*
    k0(1,12,2) -> 5*
    k0(12,20,12) -> 5*
    k0(12,1,2) -> 5*
    k0(2,20,12) -> 5*
    k0(1,20,2) -> 5*
    k0(2,1,2) -> 5*
    k0(1,20,20) -> 5*
    k0(20,2,1) -> 5*
    g1(7) -> 3*
    g1(9) -> 4*
    g1(21) -> 3*
    g1(13) -> 3*
    h1(20) -> 7*
    h1(12) -> 13*,4,7
    h1(11) -> 9*
    h1(6) -> 7*
    h1(8) -> 9*
    f1(20) -> 11*
    f1(12) -> 11*
    f1(2) -> 11*,3,8
    f1(1) -> 11*,3,8
    g2(15) -> 11*
    g2(21) -> 3,11*
    g2(23) -> 9*
    h2(20) -> 13,21*,7,4,15
    h2(22) -> 23*
    h2(14) -> 15*
    a2() -> 12,20*,6,1,14
    f2(15) -> 22*
    f2(21) -> 22*
  problem:
   
  Qed