Local Infix ".|" := Z.lor (at level 40).Local Infix ".&" := Z.land (at level 40).Local Infix ".^" := Z.lxor (at level 40).Local Notation ".~ x" := (Z.lnot x) (at level 35).Local Notation "x .[ i ]" := (Z.testbit x i) (at level 9, format "x .[ i ]").Local Infix "^^" := xorb (at level 40).Local Ltac to_bitwise :=
  let i := fresh "i" in
  bitwise as i ?Hi;
  repeat match goal with H : ?a = ?b :> Z |- _ =>
    apply (f_equal (fun x => Z.testbit x i)) in H end.Local Ltac prove_bitwise :=
  apply Bool.eqb_true_iff;
  repeat match goal with H : _ = _ :> bool |- _ =>
    apply Bool.eqb_true_iff in H; revert H end;
  rewrite <-?Bool.negb_xorb, <-?Bool.implb_true_iff, ?Bool.implb_orb;
  autorewrite with bitwise;
  btauto.Local Ltac p := to_bitwise; prove_bitwise.

Lemma ldiff_lor_land x y : Z.ldiff (x .| y) (x .& y) = x .^ y.