proposition: Lemma wordToZ_distr_diff_wmsb: forall sz (w1 w2: word sz), wmsb w1 false = negb (wmsb w2 false) -> wordToZ (w1 ^+ w2) = (wordToZ w1 + wordToZ w2)%Z.
proof: Proof. intros. destruct sz; [rewrite (shatter_word w1), (shatter_word w2); reflexivity|]. eapply eq_sym, wordToZ_wplus_bound. pose proof (wordToZ_size' w1). pose proof (wordToZ_size' w2). remember (wmsb w1 false) as msb1; destruct msb1. - apply eq_sym, wmsb_true_neg in Heqmsb1. apply eq_sym, negb_true_iff, wmsb_false_pos in H. destruct H0, H1. split; lia. - apply eq_sym, wmsb_false_pos in Heqmsb1. apply eq_sym, negb_false_iff, wmsb_true_neg in H. destruct H0, H1. split; 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