YES

Problem:
 or(x,x) -> x
 and(x,x) -> x
 not(not(x)) -> x
 not(and(x,y)) -> or(not(x),not(y))
 not(or(x,y)) -> and(not(x),not(y))

Proof:
 DP Processor:
  DPs:
   not#(and(x,y)) -> not#(y)
   not#(and(x,y)) -> not#(x)
   not#(and(x,y)) -> or#(not(x),not(y))
   not#(or(x,y)) -> not#(y)
   not#(or(x,y)) -> not#(x)
   not#(or(x,y)) -> and#(not(x),not(y))
  TRS:
   or(x,x) -> x
   and(x,x) -> x
   not(not(x)) -> x
   not(and(x,y)) -> or(not(x),not(y))
   not(or(x,y)) -> and(not(x),not(y))
  TDG Processor:
   DPs:
    not#(and(x,y)) -> not#(y)
    not#(and(x,y)) -> not#(x)
    not#(and(x,y)) -> or#(not(x),not(y))
    not#(or(x,y)) -> not#(y)
    not#(or(x,y)) -> not#(x)
    not#(or(x,y)) -> and#(not(x),not(y))
   TRS:
    or(x,x) -> x
    and(x,x) -> x
    not(not(x)) -> x
    not(and(x,y)) -> or(not(x),not(y))
    not(or(x,y)) -> and(not(x),not(y))
   graph:
    not#(and(x,y)) -> not#(y) -> not#(or(x,y)) -> and#(not(x),not(y))
    not#(and(x,y)) -> not#(y) -> not#(or(x,y)) -> not#(x)
    not#(and(x,y)) -> not#(y) -> not#(or(x,y)) -> not#(y)
    not#(and(x,y)) -> not#(y) -> not#(and(x,y)) -> or#(not(x),not(y))
    not#(and(x,y)) -> not#(y) -> not#(and(x,y)) -> not#(x)
    not#(and(x,y)) -> not#(y) -> not#(and(x,y)) -> not#(y)
    not#(and(x,y)) -> not#(x) -> not#(or(x,y)) -> and#(not(x),not(y))
    not#(and(x,y)) -> not#(x) -> not#(or(x,y)) -> not#(x)
    not#(and(x,y)) -> not#(x) -> not#(or(x,y)) -> not#(y)
    not#(and(x,y)) -> not#(x) -> not#(and(x,y)) -> or#(not(x),not(y))
    not#(and(x,y)) -> not#(x) -> not#(and(x,y)) -> not#(x)
    not#(and(x,y)) -> not#(x) -> not#(and(x,y)) -> not#(y)
    not#(or(x,y)) -> not#(y) -> not#(or(x,y)) -> and#(not(x),not(y))
    not#(or(x,y)) -> not#(y) -> not#(or(x,y)) -> not#(x)
    not#(or(x,y)) -> not#(y) -> not#(or(x,y)) -> not#(y)
    not#(or(x,y)) -> not#(y) -> not#(and(x,y)) -> or#(not(x),not(y))
    not#(or(x,y)) -> not#(y) -> not#(and(x,y)) -> not#(x)
    not#(or(x,y)) -> not#(y) -> not#(and(x,y)) -> not#(y)
    not#(or(x,y)) -> not#(x) -> not#(or(x,y)) -> and#(not(x),not(y))
    not#(or(x,y)) -> not#(x) -> not#(or(x,y)) -> not#(x)
    not#(or(x,y)) -> not#(x) -> not#(or(x,y)) -> not#(y)
    not#(or(x,y)) -> not#(x) -> not#(and(x,y)) -> or#(not(x),not(y))
    not#(or(x,y)) -> not#(x) -> not#(and(x,y)) -> not#(x)
    not#(or(x,y)) -> not#(x) -> not#(and(x,y)) -> not#(y)
   CDG Processor:
    DPs:
     not#(and(x,y)) -> not#(y)
     not#(and(x,y)) -> not#(x)
     not#(and(x,y)) -> or#(not(x),not(y))
     not#(or(x,y)) -> not#(y)
     not#(or(x,y)) -> not#(x)
     not#(or(x,y)) -> and#(not(x),not(y))
    TRS:
     or(x,x) -> x
     and(x,x) -> x
     not(not(x)) -> x
     not(and(x,y)) -> or(not(x),not(y))
     not(or(x,y)) -> and(not(x),not(y))
    graph:
     
    Qed