YES

Problem:
 and(x,false()) -> false()
 and(x,not(false())) -> x
 not(not(x)) -> x
 implies(false(),y) -> not(false())
 implies(x,false()) -> not(x)
 implies(not(x),not(y)) -> implies(y,and(x,y))

Proof:
 DP Processor:
  DPs:
   implies#(false(),y) -> not#(false())
   implies#(x,false()) -> not#(x)
   implies#(not(x),not(y)) -> and#(x,y)
   implies#(not(x),not(y)) -> implies#(y,and(x,y))
  TRS:
   and(x,false()) -> false()
   and(x,not(false())) -> x
   not(not(x)) -> x
   implies(false(),y) -> not(false())
   implies(x,false()) -> not(x)
   implies(not(x),not(y)) -> implies(y,and(x,y))
  Usable Rule Processor:
   DPs:
    implies#(false(),y) -> not#(false())
    implies#(x,false()) -> not#(x)
    implies#(not(x),not(y)) -> and#(x,y)
    implies#(not(x),not(y)) -> implies#(y,and(x,y))
   TRS:
    and(x,false()) -> false()
    and(x,not(false())) -> x
   Arctic Interpretation Processor:
    dimension: 1
    usable rules:
     and(x,false()) -> false()
     and(x,not(false())) -> x
    interpretation:
     [implies#](x0, x1) = 1x0 + 2x1 + 0,
     
     [not#](x0) = x0 + -15,
     
     [and#](x0, x1) = 1x0 + 1x1,
     
     [not](x0) = 4x0 + -1,
     
     [and](x0, x1) = 1x0 + 1x1,
     
     [false] = 1
    orientation:
     implies#(false(),y) = 2y + 2 >= 1 = not#(false())
     
     implies#(x,false()) = 1x + 3 >= x + -15 = not#(x)
     
     implies#(not(x),not(y)) = 5x + 6y + 1 >= 1x + 1y = and#(x,y)
     
     implies#(not(x),not(y)) = 5x + 6y + 1 >= 3x + 3y + 0 = implies#(y,and(x,y))
     
     and(x,false()) = 1x + 2 >= 1 = false()
     
     and(x,not(false())) = 1x + 6 >= x = x
    problem:
     DPs:
      
     TRS:
      and(x,false()) -> false()
      and(x,not(false())) -> x
    Qed