proposition: Lemma wmsb_true_bound: forall sz (w: word (S sz)), wmsb w false = true -> (Npow2 sz <= wordToN w)%N.
proof: Proof. apply (induct_word_S (fun sz w => wmsb w false = true -> (Npow2 sz <= wordToN w)%N)); [simpl; intros; subst; reflexivity|]. intros; rewrite Npow2_S, Nmul_two. specialize (H H0). destruct b. - rewrite wordToN_WS_1. apply N.mul_le_mono_nonneg_l with (p:= 2%N) in H; [|compute; discriminate]. rewrite N.add_1_r. apply N.le_le_succ_r. assumption. - rewrite wordToN_WS_0. apply N.mul_le_mono_nonneg_l; [compute; discriminate|]. assumption. 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