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))

max/plus interpretations on N:

  and_A(x1,x2) = max{6, 1 + x1, 8}
  and#_A(x1,x2) = max{0, 1, 15}
  false_A = 8
  false#_A = 2
  not_A(x1) = max{15, 11 + x1}
  not#_A(x1) = max{0, 10}
  implies_A(x1,x2) = max{29, 11 + x1, 21 + x2}
  implies#_A(x1,x2) = max{10, 1 + x1, 3 + x2}

precedence:

  implies > and = not > false