YES

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:
 DP Processor:
  DPs:
   f#(g(i(a(),b(),b'()),c()),d()) -> f#(.(b'(),c()),d'())
   f#(g(i(a(),b(),b'()),c()),d()) -> f#(.(b(),c()),d'())
   f#(g(h(a(),b()),c()),d()) -> f#(c(),d'())
   f#(g(h(a(),b()),c()),d()) -> f#(.(b(),g(h(a(),b()),c())),d())
  TRS:
   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'()))
  Usable Rule Processor:
   DPs:
    f#(g(i(a(),b(),b'()),c()),d()) -> f#(.(b'(),c()),d'())
    f#(g(i(a(),b(),b'()),c()),d()) -> f#(.(b(),c()),d'())
    f#(g(h(a(),b()),c()),d()) -> f#(c(),d'())
    f#(g(h(a(),b()),c()),d()) -> f#(.(b(),g(h(a(),b()),c())),d())
   TRS:
    
   Arctic Interpretation Processor:
    dimension: 1
    usable rules:
     
    interpretation:
     [f#](x0, x1) = x0 + 2x1 + 0,
     
     [h](x0, x1) = 2x1 + 0,
     
     [d'] = 2,
     
     [.](x0, x1) = x0 + 0,
     
     [d] = 2,
     
     [g](x0, x1) = x0 + 5x1 + 0,
     
     [c] = 0,
     
     [i](x0, x1, x2) = 2x0 + 7x1 + 3x2 + 0,
     
     [b'] = 0,
     
     [b] = 0,
     
     [a] = 1
    orientation:
     f#(g(i(a(),b(),b'()),c()),d()) = 7 >= 4 = f#(.(b'(),c()),d'())
     
     f#(g(i(a(),b(),b'()),c()),d()) = 7 >= 4 = f#(.(b(),c()),d'())
     
     f#(g(h(a(),b()),c()),d()) = 5 >= 4 = f#(c(),d'())
     
     f#(g(h(a(),b()),c()),d()) = 5 >= 4 = f#(.(b(),g(h(a(),b()),c())),d())
    problem:
     DPs:
      
     TRS:
      
    Qed