proposition: Lemma combine_shiftl_plus_n n x: x < pow2 n -> (combine (natToWord n x) WO~1) = (natToWord (n + 1) (pow2 n)) ^+ natToWord (n + 1) x.
proof: Proof. intros. apply wordToNat_eq2. rewrite ?wordToNat_combine. rewrite ?wordToNat_natToWord_idempotent'; simpl; auto. rewrite <- wordToNat_plus. pose proof (pow2_lt_pow2_S n) as sth. rewrite ?wordToNat_natToWord_idempotent'; simpl; try lia. rewrite ?wordToNat_natToWord_idempotent'; simpl; try lia. apply Nat.lt_add_lt_sub_l. rewrite Nat.add_1_r. simpl. 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