YES

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))

linear polynomial interpretations on N:

  and_A(x1,x2) = x1 + 1
  and#_A(x1,x2) = 1
  false_A = 1
  false#_A = 0
  not_A(x1) = x1 + 1
  not#_A(x1) = 0
  implies_A(x1,x2) = x1 + x2 + 1
  implies#_A(x1,x2) = x1 + x2

precedence:

  implies > and = not > false