YES(?,O(n^1))

Problem:
 f(h(x)) -> f(i(x))
 g(i(x)) -> g(h(x))
 h(a()) -> b()
 i(a()) -> b()

Proof:
 RT Transformation Processor:
  strict:
   f(h(x)) -> f(i(x))
   g(i(x)) -> g(h(x))
   h(a()) -> b()
   i(a()) -> b()
  weak:
   
  Matrix Interpretation Processor:
   dimension: 1
   interpretation:
    [b] = 0,
    
    [a] = 24,
    
    [g](x0) = x0 + 7,
    
    [i](x0) = x0 + 13,
    
    [f](x0) = x0 + 28,
    
    [h](x0) = x0 + 10
   orientation:
    f(h(x)) = x + 38 >= x + 41 = f(i(x))
    
    g(i(x)) = x + 20 >= x + 17 = g(h(x))
    
    h(a()) = 34 >= 0 = b()
    
    i(a()) = 37 >= 0 = b()
   problem:
    strict:
     f(h(x)) -> f(i(x))
    weak:
     g(i(x)) -> g(h(x))
     h(a()) -> b()
     i(a()) -> b()
   Bounds Processor:
    bound: 1
    enrichment: match-rt
    automaton:
     final states: {7,8}
     transitions:
      f1(13) -> 14*
      i1(15) -> 16*
      i1(12) -> 13*
      b1() -> 21*
      f0(7) -> 7*
      f0(8) -> 7*
      h0(7) -> 7*
      h0(8) -> 7*
      i0(7) -> 7*
      i0(8) -> 7*
      g0(7) -> 7*
      g0(8) -> 7*
      a0() -> 8*
      b0() -> 7*
      7 -> 12*
      8 -> 15*
      14 -> 7*
      16 -> 13*
      21 -> 16,13
    problem:
     strict:
      
     weak:
      f(h(x)) -> f(i(x))
      g(i(x)) -> g(h(x))
      h(a()) -> b()
      i(a()) -> b()
    Qed