(* paired_bet b w == w = [b b2 ~b2 ... bn ~bn b] *)
Inductive paired_bet (b : bool) : word -> Prop :=
paired_bet_bit :
forall w : word, paired_odd_r (negb b) w -> paired_bet b (bit b w).

(* Specifications: *)
Lemma paired_odd_r_from_bet :
forall (b : bool) (w : word),
paired_bet (negb b) w -> paired_odd_r b (bit b w).