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:
    
   Restore Modifier:
    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'()))
    SCC Processor:
     #sccs: 1
     #rules: 4
     #arcs: 16/16
     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'()))
     Matrix Interpretation Processor:
      dimension: 1
      interpretation:
       [f#](x0, x1) = x0 + x1 + 1,
       
       [h](x0, x1) = x1,
       
       [if](x0, x1, x2) = 0,
       
       [d'] = 1,
       
       [.](x0, x1) = 1,
       
       [e] = 0,
       
       [f](x0, x1) = 0,
       
       [d] = 0,
       
       [g](x0, x1) = x0 + 1,
       
       [c] = 1,
       
       [i](x0, x1, x2) = x1,
       
       [b'] = 0,
       
       [b] = 1,
       
       [a] = 0
      orientation:
       f#(g(i(a(),b(),b'()),c()),d()) = 3 >= 3 = f#(.(b'(),c()),d'())
       
       f#(g(i(a(),b(),b'()),c()),d()) = 3 >= 3 = f#(.(b(),c()),d'())
       
       f#(g(h(a(),b()),c()),d()) = 3 >= 3 = f#(c(),d'())
       
       f#(g(h(a(),b()),c()),d()) = 3 >= 2 = f#(.(b(),g(h(a(),b()),c())),d())
       
       f(g(i(a(),b(),b'()),c()),d()) = 0 >= 0 = if(e(),f(.(b(),c()),d'()),f(.(b'(),c()),d'()))
       
       f(g(h(a(),b()),c()),d()) = 0 >= 0 = if(e(),f(.(b(),g(h(a(),b()),c())),d()),f(c(),d'()))
      problem:
       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'())
       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'()))
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [f#](x0, x1) = x0 + x1,
        
        [h](x0, x1) = 1,
        
        [if](x0, x1, x2) = x1,
        
        [d'] = 1,
        
        [.](x0, x1) = x1,
        
        [e] = 0,
        
        [f](x0, x1) = x0 + x1,
        
        [d] = 0,
        
        [g](x0, x1) = x0 + x1 + 1,
        
        [c] = 1,
        
        [i](x0, x1, x2) = 0,
        
        [b'] = 0,
        
        [b] = 0,
        
        [a] = 0
       orientation:
        f#(g(i(a(),b(),b'()),c()),d()) = 2 >= 2 = f#(.(b'(),c()),d'())
        
        f#(g(i(a(),b(),b'()),c()),d()) = 2 >= 2 = f#(.(b(),c()),d'())
        
        f#(g(h(a(),b()),c()),d()) = 3 >= 2 = f#(c(),d'())
        
        f(g(i(a(),b(),b'()),c()),d()) = 2 >= 2 = if(e(),f(.(b(),c()),d'()),f(.(b'(),c()),d'()))
        
        f(g(h(a(),b()),c()),d()) = 3 >= 3 = if(e(),f(.(b(),g(h(a(),b()),c())),d()),f(c(),d'()))
       problem:
        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'())
        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'()))
       Matrix Interpretation Processor:
        dimension: 1
        interpretation:
         [f#](x0, x1) = x0 + 1,
         
         [h](x0, x1) = 0,
         
         [if](x0, x1, x2) = 0,
         
         [d'] = 0,
         
         [.](x0, x1) = x0 + x1,
         
         [e] = 0,
         
         [f](x0, x1) = 0,
         
         [d] = 0,
         
         [g](x0, x1) = x0,
         
         [c] = 1,
         
         [i](x0, x1, x2) = x0 + 1,
         
         [b'] = 1,
         
         [b] = 0,
         
         [a] = 1
        orientation:
         f#(g(i(a(),b(),b'()),c()),d()) = 3 >= 3 = f#(.(b'(),c()),d'())
         
         f#(g(i(a(),b(),b'()),c()),d()) = 3 >= 2 = f#(.(b(),c()),d'())
         
         f(g(i(a(),b(),b'()),c()),d()) = 0 >= 0 = if(e(),f(.(b(),c()),d'()),f(.(b'(),c()),d'()))
         
         f(g(h(a(),b()),c()),d()) = 0 >= 0 = if(e(),f(.(b(),g(h(a(),b()),c())),d()),f(c(),d'()))
        problem:
         DPs:
          f#(g(i(a(),b(),b'()),c()),d()) -> f#(.(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'()))
        Matrix Interpretation Processor:
         dimension: 1
         interpretation:
          [f#](x0, x1) = x1,
          
          [h](x0, x1) = 1,
          
          [if](x0, x1, x2) = 0,
          
          [d'] = 0,
          
          [.](x0, x1) = x0,
          
          [e] = 0,
          
          [f](x0, x1) = 0,
          
          [d] = 1,
          
          [g](x0, x1) = x0 + 1,
          
          [c] = 0,
          
          [i](x0, x1, x2) = x1 + 1,
          
          [b'] = 0,
          
          [b] = 1,
          
          [a] = 0
         orientation:
          f#(g(i(a(),b(),b'()),c()),d()) = 1 >= 0 = f#(.(b'(),c()),d'())
          
          f(g(i(a(),b(),b'()),c()),d()) = 0 >= 0 = if(e(),f(.(b(),c()),d'()),f(.(b'(),c()),d'()))
          
          f(g(h(a(),b()),c()),d()) = 0 >= 0 = if(e(),f(.(b(),g(h(a(),b()),c())),d()),f(c(),d'()))
         problem:
          DPs:
           
          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'()))
         Qed