The Coq Proof Assistant

# Library Coq.ZArith.Zbitwise

Require Import BinInt Lia Btauto. Local Open Scope Z_scope.
Import (ltac.notations) BinInt.Z.
Module Z.
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 Hint Rewrite
Z.b2z_bit0
Z.bits_opp
Z.lnot_spec
Z.testbit_ones_nonneg
Z.testbit_ones
Z.add_bit0
Z.shiftl_spec
Z.shiftr_spec
using solve [trivial] : bitwise.
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.
Bitwise operations alone
Lemma lnot_lnot x : Z.lnot (Z.lnot x) = x.
Lemma ldiff_lor_land x y : Z.ldiff (x .| y) (x .& y) = x .^ y.
Bitwise complement and +1/-1
Lemma succ_lnot x : Z.lnot x + 1 = - x.
Lemma lnot_pred x : Z.lnot (x - 1) = - x.
Lemma lnot_eq_pred_opp x : Z.lnot x = -x-1.
Lemma opp_lnot x : - (Z.lnot x) = x+1.
Lemma lnot_opp x : Z.lnot (-x) = x-1.
Bitwise complement as an input to + or -
Lemma sub_lnot_r x y : x - Z.lnot y = x + y + 1.
Lemma pred_sub_lnot_r x y : x - Z.lnot y - 1 = x + y.
Lemma add_lnot_r x y : x + Z.lnot y = x - y - 1.
Lemma succ_add_lnot_r x y : x + Z.lnot y + 1 = x - y.
Lemma lnot_sub x y : Z.lnot (x - y) = Z.lnot x + y.
Explicit formulas for carry bits during addition. Conceptually, the theory

# here matches the bitblasting rules for integers. However, the vector of

# carry bits is represented as a Z so it can be used in bitwise operations.

# The last three lemmas about addcarries are the main interface, but the

# generalization adccarries is provided as the same theory applies.

Definition adccarries (x y : Z) (c : bool) := (x + y + Z.b2z c) .^ (x .^ y).
Definition addcarries (x y : Z) := (x + y) .^ (x .^ y).
Lemma addcarries_ok x y : addcarries x y = adccarries x y false.
Lemma addcarryeqn (x y : Z) (c0 : bool)
(z := (x + y + Z.b2z c0))
(c := z .^ (x .^ y))
: Z.shiftr c 1 = (x .& y) .| (c .& (x .^ y)).
Lemma testbit_adccarries_0 x y c : Z.testbit (adccarries x y c) 0 = c.
Lemma shiftr_adccarries_1 x y c : Z.shiftr (adccarries x y c) 1 =
(x .& y) .| ((x .^ y) .& (adccarries x y c)).
Lemma testbit_adccarries_pos x y c i (Hi : 0 < i) : Z.testbit (adccarries x y c) i =
(x.[i-1] && y.[i-1] || (x.[i-1]^^y.[i-1]) && (adccarries x y c).[i-1])%bool.
Lemma testbit_addcarries_0 x y : Z.testbit (addcarries x y) 0 = false.
Lemma shiftr_addcarries_1 x y : Z.shiftr (addcarries x y) 1 =
(x .& y) .| ((x .^ y) .& (addcarries x y)).
Lemma testbit_addcarries_pos x y i (Hi : 0 < i) : Z.testbit (addcarries x y) i =
(x.[i-1] && y.[i-1] || (x.[i-1]^^y.[i-1]) && (addcarries x y).[i-1])%bool.
Local Hint Rewrite testbit_adccarries_0 testbit_addcarries_0 : bitwise.
Lemma addcarries_lor_land x y : addcarries (x .| y) (x .& y) = addcarries x y.
Bitwise operations, addition, and subtraction
Lemma sub_lor_land x y : (x .| y) - (x .& y) = (x .^ y).
Lemma add_lor_land x y : (x .| y) + (x .& y) = (x + y).
Lemma sub_lor_l_same_l x y : y .| x - y = x .& .~ y.
Lemma sub_lor_l_same_r x y : x .| y - y = x .& .~ y.
Lemma sub_landn_rlandn x y : x.&.~y - .~x .& y = x - y.
Lemma sub_land_same_l x y : x - x.&y = x .& .~y.
Bitwise operations, addition, subtraction, and doubling
Lemma add_lxor_2land x y : (x .^ y) + 2*(x .& y) = x + y.
Lemma sub_2lor_lxor x y : 2*(x .| y) - x .^ y = x + y.
Lemma sub_lxor_2rlandn x y : x .^ y - 2*(.~x .& y) = x - y.
Lemma sub_2landn_lxor x y : 2*(x.&.~y) - x.^y = x - y.
End Z.