proposition: Lemma combine_ge sz1 sz2 (x1 y1: word sz1) (x2 y2: word sz2): (combine x1 x2 <= combine y1 y2 -> x2 < y2 \/ (x2 = y2 /\ x1 <= y1))%word.
proof: Proof. intros. pre_word_omega. rewrite ?wordToNat_combine in *. pose proof (pow2_zero sz1). pose proof (pow2_zero sz2). destruct (weq x2 y2); subst. - right; split; auto. word_lia. - left. pose proof (wordToNat_bound x1). pose proof (wordToNat_bound y1). pre_word_omega. nia. 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