Ltac pre_word_omega :=
  unfold wzero, wone in *;
  repeat match goal with
           | H: @eq ?T ?a ?b |- _ =>
             match T with
               | word ?sz =>
                 apply (@wordToNat_eq1 sz a b) in H;
                   rewrite ?roundTrip_0, ?roundTrip_1, ?wones_pow2_minus_one in H;
                   simpl in H
             end
           | |- @eq ?T ?a ?b =>
             match T with
               | word ?sz =>
                 apply (@wordToNat_eq2 sz a b);
                   rewrite ?roundTrip_0, ?roundTrip_1, ?wones_pow2_minus_one;
                   simpl
             end
           | H: ?a < ?b |- _ =>
             apply wordToNat_lt1 in H;
               rewrite ?roundTrip_0, ?roundTrip_1, ?wones_pow2_minus_one in H;
               simpl in H
           | |- ?a < ?b =>
             apply wordToNat_lt2;
               rewrite ?roundTrip_0, ?roundTrip_1, ?wones_pow2_minus_one;
               simpl
           | H: ?a > ?b |- _ =>
             apply wordToNat_gt1 in H;
               rewrite ?roundTrip_0, ?roundTrip_1, ?wones_pow2_minus_one in H;
               simpl in H
           | |- ?a > ?b =>
             apply wordToNat_gt2;
               rewrite ?roundTrip_0, ?roundTrip_1, ?wones_pow2_minus_one;
               simpl
           | H: ?a <= ?b |- _ =>
             apply wordToNat_le1 in H;
               rewrite ?roundTrip_0, ?roundTrip_1, ?wones_pow2_minus_one in H;
               simpl in H
           | |- ?a <= ?b =>
             apply wordToNat_le2;
               rewrite ?roundTrip_0, ?roundTrip_1, ?wones_pow2_minus_one;
               simpl
           | H: ?a > ?b -> False |- _ =>
             apply wordToNat_le1 in H;
               rewrite ?roundTrip_0, ?roundTrip_1, ?wones_pow2_minus_one in H;
               simpl in H
           | |- ?a > ?b -> False =>
             apply wordToNat_le2;
               rewrite ?roundTrip_0, ?roundTrip_1, ?wones_pow2_minus_one;
               simpl
           | H: ?a < ?b -> False |- _ =>
             apply wordToNat_ge1 in H;
               rewrite ?roundTrip_0, ?roundTrip_1, ?wones_pow2_minus_one in H;
               simpl in H
           | |- ?a < ?b -> False =>
             apply wordToNat_ge2;
               rewrite ?roundTrip_0, ?roundTrip_1, ?wones_pow2_minus_one;
               simpl
           | H: not (@eq ?T ?a ?b) |- _ =>
             match T with
               | word ?sz =>
                 apply (@wordToNat_neq1 sz a b) in H;
                   rewrite ?roundTrip_0, ?roundTrip_1, ?wones_pow2_minus_one in H;
                   simpl in H
             end
           | |- not (@eq ?T ?a ?b) =>
             match T with
               | word ?sz =>
                 apply (@wordToNat_neq2 sz a b);
                   rewrite ?roundTrip_0, ?roundTrip_1, ?wones_pow2_minus_one;
                   simpl
             end
           | H: @eq ?T ?a ?b -> False |- _ =>
             match T with
               | word ?sz =>
                 apply (@wordToNat_neq1 sz a b) in H;
                   rewrite ?roundTrip_0, ?roundTrip_1, ?wones_pow2_minus_one in H;
                   simpl in H
             end
           | |- @eq ?T ?a ?b -> False =>
             match T with
               | word ?sz =>
                 apply (@wordToNat_neq2 sz a b);
                   rewrite ?roundTrip_0, ?roundTrip_1, ?wones_pow2_minus_one;
                   simpl
             end
           | H: (@eq ?T ?a ?b -> False) -> False |- _ =>
             match T with
               | word ?sz =>
                 apply (@wordNotNot sz a b) in H
             end
           | H: (not (@eq ?T ?a ?b)) -> False |- _ =>
             match T with
               | word ?sz =>
                 apply (@wordNotNot sz a b) in H
             end
           | H: not (@eq ?T ?a ?b -> False) |- _ =>
             match T with
               | word ?sz =>
                 apply (@wordNotNot sz a b) in H
             end
           | H: not (not (@eq ?T ?a ?b)) |- _ =>
             match T with
               | word ?sz =>
                 apply (@wordNotNot sz a b) in H
             end
         end.Ltac word_omega := pre_word_omega; omega.

Lemma pow2_lt_pow2_S: forall n, pow2 n < pow2 (n+1).