YES

Problem:
 0(0(*(*(x1)))) -> *(*(1(1(x1))))
 1(1(*(*(x1)))) -> 0(0(#(#(x1))))
 #(#(0(0(x1)))) -> 0(0(#(#(x1))))
 #(#(1(1(x1)))) -> 1(1(#(#(x1))))
 #(#($($(x1)))) -> *(*($($(x1))))
 #(#(#(#(x1)))) -> #(#(x1))
 #(#(*(*(x1)))) -> *(*(x1))

Proof:
 DP Processor:
  DPs:
   0#(0(*(*(x1)))) -> 1#(x1)
   0#(0(*(*(x1)))) -> 1#(1(x1))
   1#(1(*(*(x1)))) -> ##(x1)
   1#(1(*(*(x1)))) -> ##(#(x1))
   1#(1(*(*(x1)))) -> 0#(#(#(x1)))
   1#(1(*(*(x1)))) -> 0#(0(#(#(x1))))
   ##(#(0(0(x1)))) -> ##(x1)
   ##(#(0(0(x1)))) -> ##(#(x1))
   ##(#(0(0(x1)))) -> 0#(#(#(x1)))
   ##(#(0(0(x1)))) -> 0#(0(#(#(x1))))
   ##(#(1(1(x1)))) -> ##(x1)
   ##(#(1(1(x1)))) -> ##(#(x1))
   ##(#(1(1(x1)))) -> 1#(#(#(x1)))
   ##(#(1(1(x1)))) -> 1#(1(#(#(x1))))
  TRS:
   0(0(*(*(x1)))) -> *(*(1(1(x1))))
   1(1(*(*(x1)))) -> 0(0(#(#(x1))))
   #(#(0(0(x1)))) -> 0(0(#(#(x1))))
   #(#(1(1(x1)))) -> 1(1(#(#(x1))))
   #(#($($(x1)))) -> *(*($($(x1))))
   #(#(#(#(x1)))) -> #(#(x1))
   #(#(*(*(x1)))) -> *(*(x1))
  KBO Processor:
   weight function:
    w0 = 1
    w(##) = w(1#) = w(0#) = w($) = w(#) = w(1) = w(0) = w(*) = 1
   precedence:
    ## ~ # > 1# ~ 1 > 0 > 0# ~ $ ~ *
   problem:
    DPs:
     
    TRS:
     
   Qed