YES

Problem:
 f(a(),b()) -> f(a(),c())
 f(c(),d()) -> f(b(),d())

Proof:
 DP Processor:
  DPs:
   f#(a(),b()) -> f#(a(),c())
   f#(c(),d()) -> f#(b(),d())
  TRS:
   f(a(),b()) -> f(a(),c())
   f(c(),d()) -> f(b(),d())
  Usable Rule Processor:
   DPs:
    f#(a(),b()) -> f#(a(),c())
    f#(c(),d()) -> f#(b(),d())
   TRS:
    
   Restore Modifier:
    DPs:
     f#(a(),b()) -> f#(a(),c())
     f#(c(),d()) -> f#(b(),d())
    TRS:
     f(a(),b()) -> f(a(),c())
     f(c(),d()) -> f(b(),d())
    SCC Processor:
     #sccs: 1
     #rules: 2
     #arcs: 4/4
     DPs:
      f#(a(),b()) -> f#(a(),c())
      f#(c(),d()) -> f#(b(),d())
     TRS:
      f(a(),b()) -> f(a(),c())
      f(c(),d()) -> f(b(),d())
     Matrix Interpretation Processor:
      dimension: 1
      interpretation:
       [f#](x0, x1) = x0,
       
       [d] = 0,
       
       [c] = 1,
       
       [f](x0, x1) = 0,
       
       [b] = 0,
       
       [a] = 0
      orientation:
       f#(a(),b()) = 0 >= 0 = f#(a(),c())
       
       f#(c(),d()) = 1 >= 0 = f#(b(),d())
       
       f(a(),b()) = 0 >= 0 = f(a(),c())
       
       f(c(),d()) = 0 >= 0 = f(b(),d())
      problem:
       DPs:
        f#(a(),b()) -> f#(a(),c())
       TRS:
        f(a(),b()) -> f(a(),c())
        f(c(),d()) -> f(b(),d())
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [f#](x0, x1) = x1,
        
        [d] = 0,
        
        [c] = 0,
        
        [f](x0, x1) = 0,
        
        [b] = 1,
        
        [a] = 0
       orientation:
        f#(a(),b()) = 1 >= 0 = f#(a(),c())
        
        f(a(),b()) = 0 >= 0 = f(a(),c())
        
        f(c(),d()) = 0 >= 0 = f(b(),d())
       problem:
        DPs:
         
        TRS:
         f(a(),b()) -> f(a(),c())
         f(c(),d()) -> f(b(),d())
       Qed