proposition: Lemma wordToZ_ZToWord_full_post sz (H: (0 < sz)%nat) (z:Z) : wordToZ (ZToWord sz z) = ( wordToZ (ZToWord sz z) + 2 ^ (Z.of_nat sz - 1) ) mod (2 ^ Z.of_nat sz) - 2 ^ (Z.of_nat sz - 1).
proof: Proof. pose proof @wordToZ_size'' sz ltac:(trivial) (ZToWord _ z). case (wordToZ_ZToWord' sz z) as [y Hy]. assert ( wordToZ (ZToWord sz z) = (wordToZ (ZToWord sz z) + 2 ^ (Z.of_nat sz - 1)) mod (2*2 ^ (Z.of_nat sz - 1)) - 2 ^ (Z.of_nat sz - 1)) by (rewrite Z.mod_small; Lia.lia). enough (2^Z.of_nat sz = 2 * 2 ^ (Z.of_nat sz - 1)) by congruence; clear -H. rewrite pow2_times2; Lia.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