proposition: Theorem weqb_true_iff : forall sz x y, @weqb sz x y = true <-> x = y.
proof: Proof. induction x as [|b ? x IHx]; simpl; intros y. { split; auto. } { rewrite (shatter_word y) in *. simpl in *. case_eq (eqb b (whd y)); intros H. case_eq (weqb x (wtl y)); intros H0. split; auto; intros. rewrite eqb_true_iff in H. f_equal; eauto. eapply IHx; eauto. split; intros H1; try congruence. inversion H1; clear H1; subst. eapply inj_pair2_eq_dec in H4. eapply IHx in H4. congruence. eapply Peano_dec.eq_nat_dec. split; intros; try congruence. inversion H0. apply eqb_false_iff in H. congruence. } Qed.
imports: Coq.Classes.Morphisms Coq.Classes.Morphisms_Prop Coq.Classes.Equivalence Coq.Arith.PeanoNat Coq.ZArith.BinInt  Coq.Arith.Arith Coq.NArith.NArith Coq.Bool.Bool Coq.ZArith.ZArith Coq.Logic.Eqdep_dec Coq.Logic.EqdepFacts Coq.Program.Tactics Coq.Program.Equality Coq.setoid_ring.Ring Coq.setoid_ring.Ring_polynom bbv.Nomega bbv.ZLib bbv.NatLib bbv.NLib bbv.N_Z_nat_conversions bbv.DepEq bbv.DepEqNat bbv.ReservedNotations Coq.micromega.Lia