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))
  TDG 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))
   graph:
    implies#(not(x),not(y)) -> implies#(y,and(x,y)) ->
    implies#(not(x),not(y)) -> implies#(y,and(x,y))
    implies#(not(x),not(y)) -> implies#(y,and(x,y)) ->
    implies#(not(x),not(y)) -> and#(x,y)
    implies#(not(x),not(y)) -> implies#(y,and(x,y)) ->
    implies#(x,false()) -> not#(x)
    implies#(not(x),not(y)) -> implies#(y,and(x,y)) -> implies#(false(),y) -> not#(false())
   SCC Processor:
    #sccs: 1
    #rules: 1
    #arcs: 4/16
    DPs:
     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))
    Arctic Interpretation Processor:
     dimension: 1
     interpretation:
      [implies#](x0, x1) = 2x0 + x1,
      
      [implies](x0, x1) = 4x0 + 4x1,
      
      [not](x0) = 3x0,
      
      [and](x0, x1) = 2x0 + 2x1,
      
      [false] = 0
     orientation:
      implies#(not(x),not(y)) = 5x + 3y >= 2x + 2y = implies#(y,and(x,y))
      
      and(x,false()) = 2x + 2 >= 0 = false()
      
      and(x,not(false())) = 2x + 5 >= x = x
      
      not(not(x)) = 6x >= x = x
      
      implies(false(),y) = 4y + 4 >= 3 = not(false())
      
      implies(x,false()) = 4x + 4 >= 3x = not(x)
      
      implies(not(x),not(y)) = 7x + 7y >= 6x + 6y = implies(y,and(x,y))
     problem:
      DPs:
       
      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))
     Qed