proposition: Lemma natToWord_plus : forall sz n m, natToWord sz (n + m) = natToWord _ n ^+ natToWord _ m.
proof: Proof. destruct sz as [|sz]; intros n m; intuition. rewrite wplus_alt. unfold wplusN, wordBinN. destruct (wordToNat_natToWord (S sz) n); intuition. destruct (wordToNat_natToWord (S sz) m); intuition. do 2 match goal with H : _ |- _ => rewrite H; clear H end. match goal with | [ |- context[?n - ?x * pow2 (S ?sz) + (?m - ?x0 * pow2 (S ?sz))] ] => replace (n - x * pow2 (S sz) + (m - x0 * pow2 (S sz))) with (n + m - x * pow2 (S sz) - x0 * pow2 (S sz)) by lia end. repeat rewrite drop_sub; auto; lia. 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