Definition lt_eq_lt_dec : forall a b : A, {lt a b} + {a = b} + {lt b a} :=
    lt_eq_lt_dec.(* the natural order on booleans ( false < true ) *)

 Definition bool_le (b b' : bool) :=
   if b then if b' then True else False else True.Definition bool_lt (b b' : bool) :=
   if b then False else if b' then True else False.Definition A := bool.Definition le := bool_le.Definition lt := bool_lt.

Theorem lt_le_weak : forall a b : A, lt a b -> le a b.