YES

Problem:
 f(a()) -> b()
 f(c()) -> d()
 f(g(x,y)) -> g(f(x),f(y))
 f(h(x,y)) -> g(h(y,f(x)),h(x,f(y)))
 g(x,x) -> h(e(),x)

Proof:
 DP Processor:
  DPs:
   f#(g(x,y)) -> f#(y)
   f#(g(x,y)) -> f#(x)
   f#(g(x,y)) -> g#(f(x),f(y))
   f#(h(x,y)) -> f#(y)
   f#(h(x,y)) -> f#(x)
   f#(h(x,y)) -> g#(h(y,f(x)),h(x,f(y)))
  TRS:
   f(a()) -> b()
   f(c()) -> d()
   f(g(x,y)) -> g(f(x),f(y))
   f(h(x,y)) -> g(h(y,f(x)),h(x,f(y)))
   g(x,x) -> h(e(),x)
  TDG Processor:
   DPs:
    f#(g(x,y)) -> f#(y)
    f#(g(x,y)) -> f#(x)
    f#(g(x,y)) -> g#(f(x),f(y))
    f#(h(x,y)) -> f#(y)
    f#(h(x,y)) -> f#(x)
    f#(h(x,y)) -> g#(h(y,f(x)),h(x,f(y)))
   TRS:
    f(a()) -> b()
    f(c()) -> d()
    f(g(x,y)) -> g(f(x),f(y))
    f(h(x,y)) -> g(h(y,f(x)),h(x,f(y)))
    g(x,x) -> h(e(),x)
   graph:
    f#(h(x,y)) -> f#(y) -> f#(h(x,y)) -> g#(h(y,f(x)),h(x,f(y)))
    f#(h(x,y)) -> f#(y) -> f#(h(x,y)) -> f#(x)
    f#(h(x,y)) -> f#(y) -> f#(h(x,y)) -> f#(y)
    f#(h(x,y)) -> f#(y) -> f#(g(x,y)) -> g#(f(x),f(y))
    f#(h(x,y)) -> f#(y) -> f#(g(x,y)) -> f#(x)
    f#(h(x,y)) -> f#(y) -> f#(g(x,y)) -> f#(y)
    f#(h(x,y)) -> f#(x) -> f#(h(x,y)) -> g#(h(y,f(x)),h(x,f(y)))
    f#(h(x,y)) -> f#(x) -> f#(h(x,y)) -> f#(x)
    f#(h(x,y)) -> f#(x) -> f#(h(x,y)) -> f#(y)
    f#(h(x,y)) -> f#(x) -> f#(g(x,y)) -> g#(f(x),f(y))
    f#(h(x,y)) -> f#(x) -> f#(g(x,y)) -> f#(x)
    f#(h(x,y)) -> f#(x) -> f#(g(x,y)) -> f#(y)
    f#(g(x,y)) -> f#(y) -> f#(h(x,y)) -> g#(h(y,f(x)),h(x,f(y)))
    f#(g(x,y)) -> f#(y) -> f#(h(x,y)) -> f#(x)
    f#(g(x,y)) -> f#(y) -> f#(h(x,y)) -> f#(y)
    f#(g(x,y)) -> f#(y) -> f#(g(x,y)) -> g#(f(x),f(y))
    f#(g(x,y)) -> f#(y) -> f#(g(x,y)) -> f#(x)
    f#(g(x,y)) -> f#(y) -> f#(g(x,y)) -> f#(y)
    f#(g(x,y)) -> f#(x) -> f#(h(x,y)) -> g#(h(y,f(x)),h(x,f(y)))
    f#(g(x,y)) -> f#(x) -> f#(h(x,y)) -> f#(x)
    f#(g(x,y)) -> f#(x) -> f#(h(x,y)) -> f#(y)
    f#(g(x,y)) -> f#(x) -> f#(g(x,y)) -> g#(f(x),f(y))
    f#(g(x,y)) -> f#(x) -> f#(g(x,y)) -> f#(x)
    f#(g(x,y)) -> f#(x) -> f#(g(x,y)) -> f#(y)
   SCC Processor:
    #sccs: 1
    #rules: 4
    #arcs: 24/36
    DPs:
     f#(h(x,y)) -> f#(y)
     f#(g(x,y)) -> f#(y)
     f#(g(x,y)) -> f#(x)
     f#(h(x,y)) -> f#(x)
    TRS:
     f(a()) -> b()
     f(c()) -> d()
     f(g(x,y)) -> g(f(x),f(y))
     f(h(x,y)) -> g(h(y,f(x)),h(x,f(y)))
     g(x,x) -> h(e(),x)
    LPO Processor:
     argument filtering:
      pi(a) = []
      pi(f) = [0]
      pi(b) = []
      pi(c) = []
      pi(d) = []
      pi(g) = [0,1]
      pi(h) = [0,1]
      pi(e) = []
      pi(f#) = [0]
     precedence:
      f > g > h > f# ~ e ~ d ~ c ~ b ~ a
     problem:
      DPs:
       
      TRS:
       f(a()) -> b()
       f(c()) -> d()
       f(g(x,y)) -> g(f(x),f(y))
       f(h(x,y)) -> g(h(y,f(x)),h(x,f(y)))
       g(x,x) -> h(e(),x)
     Qed