YES

Problem:
 p(0(x1)) -> 0(s(s(p(x1))))
 p(s(x1)) -> x1
 p(p(s(x1))) -> p(x1)
 f(s(x1)) -> p(s(g(p(s(s(x1))))))
 g(s(x1)) -> p(p(s(s(s(j(s(p(s(p(s(x1)))))))))))
 j(s(x1)) -> p(s(s(p(s(f(p(s(p(p(s(x1)))))))))))
 half(0(x1)) -> 0(s(s(half(p(s(p(s(x1))))))))
 half(s(s(x1))) -> s(half(p(p(s(s(x1))))))
 rd(0(x1)) -> 0(s(0(0(0(0(s(0(rd(x1)))))))))

Proof:
 Bounds Processor:
  bound: 3
  enrichment: match
  automaton:
   final states: {38,35,31,22,12,6,3,2,1}
   transitions:
    f1(76) -> 77*
    g0(9) -> 10*
    p0(2) -> 3*
    p0(24) -> 25*
    p0(30) -> 22*
    p0(14) -> 15*
    p0(9) -> 36*
    p0(20) -> 21*
    p0(13) -> 23*
    p0(7) -> 13*
    p0(27) -> 28*
    p0(11) -> 6*
    p0(8) -> 9*
    p0(21) -> 12*
    f80() -> 2*
    s0(7) -> 8*
    s0(40) -> 41*
    s0(29) -> 30*
    s0(10) -> 11*
    s0(3) -> 4*
    s0(33) -> 34*
    s0(15) -> 16*
    s0(17) -> 18*
    s0(13) -> 14*
    s0(23) -> 24*
    s0(18) -> 19*
    s0(4) -> 5*
    s0(37) -> 35*
    s0(28) -> 29*
    s0(32) -> 33*
    s0(19) -> 20*
    s0(26) -> 27*
    s0(45) -> 46*
    s0(2) -> 7*
    00(34) -> 31*
    00(43) -> 44*
    00(5) -> 1*
    00(42) -> 43*
    00(39) -> 40*
    00(44) -> 45*
    00(41) -> 42*
    00(46) -> 38*
    f0(25) -> 26*
    j1(90) -> 91*
    j0(16) -> 17*
    f2(114) -> 115*
    half0(36) -> 37*
    half0(15) -> 32*
    p3(135) -> 136*
    s2(109) -> 110*
    s2(117) -> 118*
    s2(115) -> 116*
    s2(118) -> 119*
    s2(112) -> 113*
    rd0(2) -> 39*
    s1(91) -> 92*
    s1(87) -> 88*
    s1(89) -> 90*
    s1(92) -> 93*
    s1(74) -> 75*
    s1(93) -> 94*
    s1(71) -> 72*
    s1(80) -> 81*
    s1(77) -> 78*
    s1(79) -> 80*
    p1(55) -> 56*
    p1(94) -> 95*
    p1(88) -> 89*
    p1(78) -> 79*
    p1(81) -> 82*
    p1(53) -> 54*
    p1(72) -> 73*
    p1(75) -> 76*
    p1(95) -> 96*
    p1(47) -> 48*
    p1(73) -> 74*
    p2(111) -> 112*
    p2(119) -> 120*
    p2(107) -> 108*
    p2(83) -> 84*
    p2(110) -> 111*
    p2(116) -> 117*
    p2(113) -> 114*
    56 -> 36*
    19 -> 53,21
    92 -> 108,96
    74 -> 76*
    48 -> 23*
    115 -> 117*
    35 -> 37,32
    7 -> 55,9
    77 -> 79*
    112 -> 114*
    93 -> 107,95
    118 -> 120*
    2 -> 136,48,23,25,3,84,112,56,36,47,13
    82 -> 17*
    54 -> 12*
    109 -> 135,111
    6 -> 115,77,26,28,79,117
    38 -> 39*
    26 -> 28*
    73 -> 87*
    13 -> 15*
    71 -> 83,73
    1 -> 136,48,23,25,3,84,112
    31 -> 37,32
    108 -> 96*
    120 -> 91*
    18 -> 54,12
    136 -> 112*
    84 -> 74*
    96 -> 10,6
    15 -> 71*
    10 -> 6*
    80 -> 82*
    87 -> 89*
    89 -> 109*
    29 -> 22*
    23 -> 25*
  problem:
   
  Qed