YES(?,O(n^1))

Problem:
 f(g(i(a(),b(),b'()),c()),d()) -> if(e(),f(.(b(),c()),d'()),f(.(b'(),c()),d'()))
 f(g(h(a(),b()),c()),d()) -> if(e(),f(.(b(),g(h(a(),b()),c())),d()),f(c(),d'()))

Proof:
 RT Transformation Processor:
  strict:
   f(g(i(a(),b(),b'()),c()),d()) -> if(e(),f(.(b(),c()),d'()),f(.(b'(),c()),d'()))
   f(g(h(a(),b()),c()),d()) -> if(e(),f(.(b(),g(h(a(),b()),c())),d()),f(c(),d'()))
  weak:
   
  Bounds Processor:
   bound: 1
   enrichment: match-rt
   automaton:
    final states: {3,4}
    transitions:
     if1(44,43,36) -> 3*
     e1() -> 44*
     f1(46,34) -> 43*
     f1(42,37) -> 43*
     f1(35,34) -> 36*
     .1(38,35) -> 46*
     .1(38,41) -> 42*
     .1(45,35) -> 35*
     b1() -> 38*
     c1() -> 35*
     d'1() -> 34*
     b'1() -> 45*
     g1(40,35) -> 41*
     h1(39,38) -> 40*
     a1() -> 39*
     d1() -> 37*
     f0(3,3) -> 3*
     f0(4,4) -> 3*
     f0(3,4) -> 3*
     f0(4,3) -> 3*
     g0(3,3) -> 3*
     g0(4,4) -> 3*
     g0(3,4) -> 3*
     g0(4,3) -> 3*
     i0(3,3,4) -> 4*
     i0(4,4,4) -> 4*
     i0(3,4,3) -> 4*
     i0(4,3,3) -> 4*
     i0(3,4,4) -> 4*
     i0(4,3,4) -> 4*
     i0(3,3,3) -> 4*
     i0(4,4,3) -> 4*
     a0() -> 3*
     b0() -> 3*
     b'0() -> 3*
     c0() -> 3*
     d0() -> 3*
     if0(3,3,4) -> 3*
     if0(4,4,4) -> 3*
     if0(3,4,3) -> 3*
     if0(4,3,3) -> 3*
     if0(3,4,4) -> 3*
     if0(4,3,4) -> 3*
     if0(3,3,3) -> 3*
     if0(4,4,3) -> 3*
     e0() -> 3*
     .0(3,3) -> 3*
     .0(4,4) -> 3*
     .0(3,4) -> 3*
     .0(4,3) -> 3*
     d'0() -> 3*
     h0(3,3) -> 3*
     h0(4,4) -> 3*
     h0(3,4) -> 3*
     h0(4,3) -> 3*
   problem:
    strict:
     f(g(i(a(),b(),b'()),c()),d()) -> if(e(),f(.(b(),c()),d'()),f(.(b'(),c()),d'()))
    weak:
     f(g(h(a(),b()),c()),d()) -> if(e(),f(.(b(),g(h(a(),b()),c())),d()),f(c(),d'()))
   Bounds Processor:
    bound: 1
    enrichment: match-rt
    automaton:
     final states: {1,2}
     transitions:
      if1(13,31,25) -> 1*
      if1(13,12,9) -> 1*
      e1() -> 13*
      f1(8,5) -> 9*
      f1(30,26) -> 31*
      f1(11,5) -> 12*
      f1(6,5) -> 25*
      .1(10,29) -> 30*
      .1(10,6) -> 11*
      .1(7,6) -> 8*
      b1() -> 10*
      c1() -> 6*
      d'1() -> 5*
      b'1() -> 7*
      g1(28,6) -> 29*
      h1(27,10) -> 28*
      a1() -> 27*
      d1() -> 26*
      f0(1,2) -> 1*
      f0(2,1) -> 1*
      f0(1,1) -> 1*
      f0(2,2) -> 1*
      g0(1,2) -> 1*
      g0(2,1) -> 1*
      g0(1,1) -> 1*
      g0(2,2) -> 1*
      i0(1,1,1) -> 2*
      i0(2,2,1) -> 2*
      i0(1,1,2) -> 2*
      i0(2,2,2) -> 2*
      i0(1,2,1) -> 2*
      i0(2,1,1) -> 2*
      i0(1,2,2) -> 2*
      i0(2,1,2) -> 2*
      a0() -> 1*
      b0() -> 1*
      b'0() -> 1*
      c0() -> 1*
      d0() -> 1*
      if0(1,1,1) -> 1*
      if0(2,2,1) -> 1*
      if0(1,1,2) -> 1*
      if0(2,2,2) -> 1*
      if0(1,2,1) -> 1*
      if0(2,1,1) -> 1*
      if0(1,2,2) -> 1*
      if0(2,1,2) -> 1*
      e0() -> 1*
      .0(1,2) -> 1*
      .0(2,1) -> 1*
      .0(1,1) -> 1*
      .0(2,2) -> 1*
      d'0() -> 1*
      h0(1,2) -> 1*
      h0(2,1) -> 1*
      h0(1,1) -> 1*
      h0(2,2) -> 1*
    problem:
     strict:
      
     weak:
      f(g(i(a(),b(),b'()),c()),d()) -> if(e(),f(.(b(),c()),d'()),f(.(b'(),c()),d'()))
      f(g(h(a(),b()),c()),d()) -> if(e(),f(.(b(),g(h(a(),b()),c())),d()),f(c(),d'()))
    Qed