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(* Singletons *)
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Definition single (b : bool) := bit b empty.
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(* Parities of words *)
(*********************)
Inductive odd : word -> Prop :=
even_odd : forall w : word, even w -> forall b : bool, odd (bit b w)
with even : word -> Prop :=
| even_empty : even empty
| odd_even : forall w : word, odd w -> forall b : bool, even (bit b w).
Hint Resolve odd_even even_empty even_odd.

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(* Parities of subwords *)
(************************)
Lemma odd_even_conc :
forall u v w : word,
conc u v w ->
odd w /\ (odd u /\ even v \/ even u /\ odd v) \/
even w /\ (odd u /\ odd v \/ even u /\ even v).