diff options
Diffstat (limited to 'lib/Integers.v')
| -rw-r--r-- | lib/Integers.v | 928 |
1 files changed, 64 insertions, 864 deletions
diff --git a/lib/Integers.v b/lib/Integers.v index 0e506208..f4213332 100644 --- a/lib/Integers.v +++ b/lib/Integers.v @@ -16,7 +16,7 @@ (** Formalizations of machine integers modulo $2^N$ #2<sup>N</sup>#. *) Require Import Eqdep_dec Zquot Zwf. -Require Import Coqlib. +Require Import Coqlib Zbits. Require Archi. (** * Comparisons *) @@ -80,11 +80,19 @@ Proof. unfold modulus. apply two_power_nat_two_p. Qed. +Remark modulus_gt_one: modulus > 1. +Proof. + rewrite modulus_power. apply Z.lt_gt. apply (two_p_monotone_strict 0). + generalize wordsize_pos; omega. +Qed. + Remark modulus_pos: modulus > 0. Proof. - rewrite modulus_power. apply two_p_gt_ZERO. generalize wordsize_pos; omega. + generalize modulus_gt_one; omega. Qed. +Hint Resolve modulus_pos: ints. + (** * Representation of machine integers *) (** A machine integer (type [int]) is represented as a Coq arbitrary-precision @@ -95,17 +103,6 @@ Record int: Type := mkint { intval: Z; intrange: -1 < intval < modulus }. (** Fast normalization modulo [2^wordsize] *) -Fixpoint P_mod_two_p (p: positive) (n: nat) {struct n} : Z := - match n with - | O => 0 - | S m => - match p with - | xH => 1 - | xO q => Z.double (P_mod_two_p q m) - | xI q => Z.succ_double (P_mod_two_p q m) - end - end. - Definition Z_mod_modulus (x: Z) : Z := match x with | Z0 => 0 @@ -113,51 +110,9 @@ Definition Z_mod_modulus (x: Z) : Z := | Zneg p => let r := P_mod_two_p p wordsize in if zeq r 0 then 0 else modulus - r end. -Lemma P_mod_two_p_range: - forall n p, 0 <= P_mod_two_p p n < two_power_nat n. -Proof. - induction n; simpl; intros. - - rewrite two_power_nat_O. omega. - - rewrite two_power_nat_S. destruct p. - + generalize (IHn p). rewrite Z.succ_double_spec. omega. - + generalize (IHn p). rewrite Z.double_spec. omega. - + generalize (two_power_nat_pos n). omega. -Qed. - -Lemma P_mod_two_p_eq: - forall n p, P_mod_two_p p n = (Zpos p) mod (two_power_nat n). -Proof. - assert (forall n p, exists y, Zpos p = y * two_power_nat n + P_mod_two_p p n). - { - induction n; simpl; intros. - - rewrite two_power_nat_O. exists (Zpos p). ring. - - rewrite two_power_nat_S. destruct p. - + destruct (IHn p) as [y EQ]. exists y. - change (Zpos p~1) with (2 * Zpos p + 1). rewrite EQ. - rewrite Z.succ_double_spec. ring. - + destruct (IHn p) as [y EQ]. exists y. - change (Zpos p~0) with (2 * Zpos p). rewrite EQ. - rewrite (Z.double_spec (P_mod_two_p p n)). ring. - + exists 0; omega. - } - intros. - destruct (H n p) as [y EQ]. - symmetry. apply Zmod_unique with y. auto. apply P_mod_two_p_range. -Qed. - Lemma Z_mod_modulus_range: forall x, 0 <= Z_mod_modulus x < modulus. -Proof. - intros; unfold Z_mod_modulus. - destruct x. - - generalize modulus_pos; intuition. - - apply P_mod_two_p_range. - - set (r := P_mod_two_p p wordsize). - assert (0 <= r < modulus) by apply P_mod_two_p_range. - destruct (zeq r 0). - + generalize modulus_pos; intuition. - + Psatz.lia. -Qed. +Proof (Z_mod_two_p_range wordsize). Lemma Z_mod_modulus_range': forall x, -1 < Z_mod_modulus x < modulus. @@ -167,22 +122,7 @@ Qed. Lemma Z_mod_modulus_eq: forall x, Z_mod_modulus x = x mod modulus. -Proof. - intros. unfold Z_mod_modulus. destruct x. - - rewrite Zmod_0_l. auto. - - apply P_mod_two_p_eq. - - generalize (P_mod_two_p_range wordsize p) (P_mod_two_p_eq wordsize p). - fold modulus. intros A B. - exploit (Z_div_mod_eq (Zpos p) modulus). apply modulus_pos. intros C. - set (q := Zpos p / modulus) in *. - set (r := P_mod_two_p p wordsize) in *. - rewrite <- B in C. - change (Z.neg p) with (- (Z.pos p)). destruct (zeq r 0). - + symmetry. apply Zmod_unique with (-q). rewrite C; rewrite e. Psatz.lia. - intuition. - + symmetry. apply Zmod_unique with (-q - 1). rewrite C. Psatz.lia. - intuition. -Qed. +Proof (Z_mod_two_p_eq wordsize). (** The [unsigned] and [signed] functions return the Coq integer corresponding to the given machine integer, interpreted as unsigned or signed @@ -317,63 +257,20 @@ Definition shr_carry (x y: int) : int := (** Zero and sign extensions *) -Definition Zshiftin (b: bool) (x: Z) : Z := - if b then Z.succ_double x else Z.double x. - -(** In pseudo-code: -<< - Fixpoint Zzero_ext (n: Z) (x: Z) : Z := - if zle n 0 then - 0 - else - Zshiftin (Z.odd x) (Zzero_ext (Z.pred n) (Z.div2 x)). - Fixpoint Zsign_ext (n: Z) (x: Z) : Z := - if zle n 1 then - if Z.odd x then -1 else 0 - else - Zshiftin (Z.odd x) (Zzero_ext (Z.pred n) (Z.div2 x)). ->> - We encode this [nat]-like recursion using the [Z.iter] iteration - function, in order to make the [Zzero_ext] and [Zsign_ext] - functions efficiently executable within Coq. -*) - -Definition Zzero_ext (n: Z) (x: Z) : Z := - Z.iter n - (fun rec x => Zshiftin (Z.odd x) (rec (Z.div2 x))) - (fun x => 0) - x. - -Definition Zsign_ext (n: Z) (x: Z) : Z := - Z.iter (Z.pred n) - (fun rec x => Zshiftin (Z.odd x) (rec (Z.div2 x))) - (fun x => if Z.odd x then -1 else 0) - x. - Definition zero_ext (n: Z) (x: int) : int := repr (Zzero_ext n (unsigned x)). - Definition sign_ext (n: Z) (x: int) : int := repr (Zsign_ext n (unsigned x)). (** Decomposition of a number as a sum of powers of two. *) -Fixpoint Z_one_bits (n: nat) (x: Z) (i: Z) {struct n}: list Z := - match n with - | O => nil - | S m => - if Z.odd x - then i :: Z_one_bits m (Z.div2 x) (i+1) - else Z_one_bits m (Z.div2 x) (i+1) - end. - Definition one_bits (x: int) : list int := List.map repr (Z_one_bits wordsize (unsigned x) 0). (** Recognition of powers of two. *) Definition is_power2 (x: int) : option int := - match Z_one_bits wordsize (unsigned x) 0 with - | i :: nil => Some (repr i) - | _ => None + match Z_is_power2 (unsigned x) with + | Some i => Some (repr i) + | None => None end. (** Comparisons. *) @@ -497,101 +394,7 @@ Qed. (** ** Modulo arithmetic *) -(** We define and state properties of equality and arithmetic modulo a - positive integer. *) - -Section EQ_MODULO. - -Variable modul: Z. -Hypothesis modul_pos: modul > 0. - -Definition eqmod (x y: Z) : Prop := exists k, x = k * modul + y. - -Lemma eqmod_refl: forall x, eqmod x x. -Proof. - intros; red. exists 0. omega. -Qed. - -Lemma eqmod_refl2: forall x y, x = y -> eqmod x y. -Proof. - intros. subst y. apply eqmod_refl. -Qed. - -Lemma eqmod_sym: forall x y, eqmod x y -> eqmod y x. -Proof. - intros x y [k EQ]; red. exists (-k). subst x. ring. -Qed. - -Lemma eqmod_trans: forall x y z, eqmod x y -> eqmod y z -> eqmod x z. -Proof. - intros x y z [k1 EQ1] [k2 EQ2]; red. - exists (k1 + k2). subst x; subst y. ring. -Qed. - -Lemma eqmod_small_eq: - forall x y, eqmod x y -> 0 <= x < modul -> 0 <= y < modul -> x = y. -Proof. - intros x y [k EQ] I1 I2. - generalize (Zdiv_unique _ _ _ _ EQ I2). intro. - rewrite (Zdiv_small x modul I1) in H. subst k. omega. -Qed. - -Lemma eqmod_mod_eq: - forall x y, eqmod x y -> x mod modul = y mod modul. -Proof. - intros x y [k EQ]. subst x. - rewrite Z.add_comm. apply Z_mod_plus. auto. -Qed. - -Lemma eqmod_mod: - forall x, eqmod x (x mod modul). -Proof. - intros; red. exists (x / modul). - rewrite Z.mul_comm. apply Z_div_mod_eq. auto. -Qed. - -Lemma eqmod_add: - forall a b c d, eqmod a b -> eqmod c d -> eqmod (a + c) (b + d). -Proof. - intros a b c d [k1 EQ1] [k2 EQ2]; red. - subst a; subst c. exists (k1 + k2). ring. -Qed. - -Lemma eqmod_neg: - forall x y, eqmod x y -> eqmod (-x) (-y). -Proof. - intros x y [k EQ]; red. exists (-k). rewrite EQ. ring. -Qed. - -Lemma eqmod_sub: - forall a b c d, eqmod a b -> eqmod c d -> eqmod (a - c) (b - d). -Proof. - intros a b c d [k1 EQ1] [k2 EQ2]; red. - subst a; subst c. exists (k1 - k2). ring. -Qed. - -Lemma eqmod_mult: - forall a b c d, eqmod a c -> eqmod b d -> eqmod (a * b) (c * d). -Proof. - intros a b c d [k1 EQ1] [k2 EQ2]; red. - subst a; subst b. - exists (k1 * k2 * modul + c * k2 + k1 * d). - ring. -Qed. - -End EQ_MODULO. - -Lemma eqmod_divides: - forall n m x y, eqmod n x y -> Z.divide m n -> eqmod m x y. -Proof. - intros. destruct H as [k1 EQ1]. destruct H0 as [k2 EQ2]. - exists (k1*k2). rewrite <- Z.mul_assoc. rewrite <- EQ2. auto. -Qed. - -(** We then specialize these definitions to equality modulo - $2^{wordsize}$ #2<sup>wordsize</sup>#. *) - -Hint Resolve modulus_pos: ints. +(** [eqm] is equality modulo $2^{wordsize}$ #2<sup>wordsize</sup>#. *) Definition eqm := eqmod modulus. @@ -637,6 +440,19 @@ Lemma eqm_mult: Proof (eqmod_mult modulus). Hint Resolve eqm_mult: ints. +Lemma eqm_same_bits: + forall x y, + (forall i, 0 <= i < zwordsize -> Z.testbit x i = Z.testbit y i) -> + eqm x y. +Proof (eqmod_same_bits wordsize). + +Lemma same_bits_eqm: + forall x y i, + eqm x y -> + 0 <= i < zwordsize -> + Z.testbit x i = Z.testbit y i. +Proof (same_bits_eqmod wordsize). + (** ** Properties of the coercions between [Z] and [int] *) Lemma eqm_samerepr: forall x y, eqm x y -> repr x = repr y. @@ -706,7 +522,7 @@ Theorem repr_unsigned: forall i, repr (unsigned i) = i. Proof. destruct i; simpl. unfold repr. apply mkint_eq. - rewrite Z_mod_modulus_eq. apply Zmod_small; omega. + rewrite Z_mod_modulus_eq. apply Z.mod_small; omega. Qed. Hint Resolve repr_unsigned: ints. @@ -729,7 +545,7 @@ Theorem unsigned_repr: forall z, 0 <= z <= max_unsigned -> unsigned (repr z) = z. Proof. intros. rewrite unsigned_repr_eq. - apply Zmod_small. unfold max_unsigned in H. omega. + apply Z.mod_small. unfold max_unsigned in H. omega. Qed. Hint Resolve unsigned_repr: ints. @@ -776,7 +592,7 @@ Qed. Theorem unsigned_one: unsigned one = 1. Proof. - unfold one; rewrite unsigned_repr_eq. apply Zmod_small. split. omega. + unfold one; rewrite unsigned_repr_eq. apply Z.mod_small. split. omega. unfold modulus. replace wordsize with (S(Init.Nat.pred wordsize)). rewrite two_power_nat_S. generalize (two_power_nat_pos (Init.Nat.pred wordsize)). omega. @@ -787,7 +603,7 @@ Theorem unsigned_mone: unsigned mone = modulus - 1. Proof. unfold mone; rewrite unsigned_repr_eq. replace (-1) with ((modulus - 1) + (-1) * modulus). - rewrite Z_mod_plus_full. apply Zmod_small. + rewrite Z_mod_plus_full. apply Z.mod_small. generalize modulus_pos. omega. omega. Qed. @@ -819,7 +635,7 @@ Qed. Theorem unsigned_repr_wordsize: unsigned iwordsize = zwordsize. Proof. - unfold iwordsize; rewrite unsigned_repr_eq. apply Zmod_small. + unfold iwordsize; rewrite unsigned_repr_eq. apply Z.mod_small. generalize wordsize_pos wordsize_max_unsigned; unfold max_unsigned; omega. Qed. @@ -1298,298 +1114,6 @@ Qed. (** ** Bit-level properties *) -(** ** Properties of bit-level operations over [Z] *) - -Remark Ztestbit_0: forall n, Z.testbit 0 n = false. -Proof Z.testbit_0_l. - -Remark Ztestbit_1: forall n, Z.testbit 1 n = zeq n 0. -Proof. - intros. destruct n; simpl; auto. -Qed. - -Remark Ztestbit_m1: forall n, 0 <= n -> Z.testbit (-1) n = true. -Proof. - intros. destruct n; simpl; auto. -Qed. - -Remark Zshiftin_spec: - forall b x, Zshiftin b x = 2 * x + (if b then 1 else 0). -Proof. - unfold Zshiftin; intros. destruct b. - - rewrite Z.succ_double_spec. omega. - - rewrite Z.double_spec. omega. -Qed. - -Remark Zshiftin_inj: - forall b1 x1 b2 x2, - Zshiftin b1 x1 = Zshiftin b2 x2 -> b1 = b2 /\ x1 = x2. -Proof. - intros. rewrite !Zshiftin_spec in H. - destruct b1; destruct b2. - split; [auto|omega]. - omegaContradiction. - omegaContradiction. - split; [auto|omega]. -Qed. - -Remark Zdecomp: - forall x, x = Zshiftin (Z.odd x) (Z.div2 x). -Proof. - intros. destruct x; simpl. - - auto. - - destruct p; auto. - - destruct p; auto. simpl. rewrite Pos.pred_double_succ. auto. -Qed. - -Remark Ztestbit_shiftin: - forall b x n, - 0 <= n -> - Z.testbit (Zshiftin b x) n = if zeq n 0 then b else Z.testbit x (Z.pred n). -Proof. - intros. rewrite Zshiftin_spec. destruct (zeq n 0). - - subst n. destruct b. - + apply Z.testbit_odd_0. - + rewrite Z.add_0_r. apply Z.testbit_even_0. - - assert (0 <= Z.pred n) by omega. - set (n' := Z.pred n) in *. - replace n with (Z.succ n') by (unfold n'; omega). - destruct b. - + apply Z.testbit_odd_succ; auto. - + rewrite Z.add_0_r. apply Z.testbit_even_succ; auto. -Qed. - -Remark Ztestbit_shiftin_base: - forall b x, Z.testbit (Zshiftin b x) 0 = b. -Proof. - intros. rewrite Ztestbit_shiftin. apply zeq_true. omega. -Qed. - -Remark Ztestbit_shiftin_succ: - forall b x n, 0 <= n -> Z.testbit (Zshiftin b x) (Z.succ n) = Z.testbit x n. -Proof. - intros. rewrite Ztestbit_shiftin. rewrite zeq_false. rewrite Z.pred_succ. auto. - omega. omega. -Qed. - -Remark Ztestbit_eq: - forall n x, 0 <= n -> - Z.testbit x n = if zeq n 0 then Z.odd x else Z.testbit (Z.div2 x) (Z.pred n). -Proof. - intros. rewrite (Zdecomp x) at 1. apply Ztestbit_shiftin; auto. -Qed. - -Remark Ztestbit_base: - forall x, Z.testbit x 0 = Z.odd x. -Proof. - intros. rewrite Ztestbit_eq. apply zeq_true. omega. -Qed. - -Remark Ztestbit_succ: - forall n x, 0 <= n -> Z.testbit x (Z.succ n) = Z.testbit (Z.div2 x) n. -Proof. - intros. rewrite Ztestbit_eq. rewrite zeq_false. rewrite Z.pred_succ. auto. - omega. omega. -Qed. - -Lemma eqmod_same_bits: - forall n x y, - (forall i, 0 <= i < Z.of_nat n -> Z.testbit x i = Z.testbit y i) -> - eqmod (two_power_nat n) x y. -Proof. - induction n; intros. - - change (two_power_nat 0) with 1. exists (x-y); ring. - - rewrite two_power_nat_S. - assert (eqmod (two_power_nat n) (Z.div2 x) (Z.div2 y)). - apply IHn. intros. rewrite <- !Ztestbit_succ. apply H. rewrite Nat2Z.inj_succ; omega. - omega. omega. - destruct H0 as [k EQ]. - exists k. rewrite (Zdecomp x). rewrite (Zdecomp y). - replace (Z.odd y) with (Z.odd x). - rewrite EQ. rewrite !Zshiftin_spec. ring. - exploit (H 0). rewrite Nat2Z.inj_succ; omega. - rewrite !Ztestbit_base. auto. -Qed. - -Lemma eqm_same_bits: - forall x y, - (forall i, 0 <= i < zwordsize -> Z.testbit x i = Z.testbit y i) -> - eqm x y. -Proof (eqmod_same_bits wordsize). - -Lemma same_bits_eqmod: - forall n x y i, - eqmod (two_power_nat n) x y -> 0 <= i < Z.of_nat n -> - Z.testbit x i = Z.testbit y i. -Proof. - induction n; intros. - - simpl in H0. omegaContradiction. - - rewrite Nat2Z.inj_succ in H0. rewrite two_power_nat_S in H. - rewrite !(Ztestbit_eq i); intuition. - destruct H as [k EQ]. - assert (EQ': Zshiftin (Z.odd x) (Z.div2 x) = - Zshiftin (Z.odd y) (k * two_power_nat n + Z.div2 y)). - { - rewrite (Zdecomp x) in EQ. rewrite (Zdecomp y) in EQ. - rewrite EQ. rewrite !Zshiftin_spec. ring. - } - exploit Zshiftin_inj; eauto. intros [A B]. - destruct (zeq i 0). - + auto. - + apply IHn. exists k; auto. omega. -Qed. - -Lemma same_bits_eqm: - forall x y i, - eqm x y -> - 0 <= i < zwordsize -> - Z.testbit x i = Z.testbit y i. -Proof (same_bits_eqmod wordsize). - -Remark two_power_nat_infinity: - forall x, 0 <= x -> exists n, x < two_power_nat n. -Proof. - intros x0 POS0; pattern x0; apply natlike_ind; auto. - exists O. compute; auto. - intros. destruct H0 as [n LT]. exists (S n). rewrite two_power_nat_S. - generalize (two_power_nat_pos n). omega. -Qed. - -Lemma equal_same_bits: - forall x y, - (forall i, 0 <= i -> Z.testbit x i = Z.testbit y i) -> - x = y. -Proof. - intros. - set (z := if zlt x y then y - x else x - y). - assert (0 <= z). - unfold z; destruct (zlt x y); omega. - exploit (two_power_nat_infinity z); auto. intros [n LT]. - assert (eqmod (two_power_nat n) x y). - apply eqmod_same_bits. intros. apply H. tauto. - assert (eqmod (two_power_nat n) z 0). - unfold z. destruct (zlt x y). - replace 0 with (y - y) by omega. apply eqmod_sub. apply eqmod_refl. auto. - replace 0 with (x - x) by omega. apply eqmod_sub. apply eqmod_refl. apply eqmod_sym; auto. - assert (z = 0). - apply eqmod_small_eq with (two_power_nat n). auto. omega. generalize (two_power_nat_pos n); omega. - unfold z in H3. destruct (zlt x y); omega. -Qed. - -Lemma Z_one_complement: - forall i, 0 <= i -> - forall x, Z.testbit (-x-1) i = negb (Z.testbit x i). -Proof. - intros i0 POS0. pattern i0. apply Zlt_0_ind; auto. - intros i IND POS x. - rewrite (Zdecomp x). set (y := Z.div2 x). - replace (- Zshiftin (Z.odd x) y - 1) - with (Zshiftin (negb (Z.odd x)) (- y - 1)). - rewrite !Ztestbit_shiftin; auto. - destruct (zeq i 0). auto. apply IND. omega. - rewrite !Zshiftin_spec. destruct (Z.odd x); simpl negb; ring. -Qed. - -Lemma Ztestbit_above: - forall n x i, - 0 <= x < two_power_nat n -> - i >= Z.of_nat n -> - Z.testbit x i = false. -Proof. - induction n; intros. - - change (two_power_nat 0) with 1 in H. - replace x with 0 by omega. - apply Z.testbit_0_l. - - rewrite Nat2Z.inj_succ in H0. rewrite Ztestbit_eq. rewrite zeq_false. - apply IHn. rewrite two_power_nat_S in H. rewrite (Zdecomp x) in H. - rewrite Zshiftin_spec in H. destruct (Z.odd x); omega. - omega. omega. omega. -Qed. - -Lemma Ztestbit_above_neg: - forall n x i, - -two_power_nat n <= x < 0 -> - i >= Z.of_nat n -> - Z.testbit x i = true. -Proof. - intros. set (y := -x-1). - assert (Z.testbit y i = false). - apply Ztestbit_above with n. - unfold y; omega. auto. - unfold y in H1. rewrite Z_one_complement in H1. - change true with (negb false). rewrite <- H1. rewrite negb_involutive; auto. - omega. -Qed. - -Lemma Zsign_bit: - forall n x, - 0 <= x < two_power_nat (S n) -> - Z.testbit x (Z.of_nat n) = if zlt x (two_power_nat n) then false else true. -Proof. - induction n; intros. - - change (two_power_nat 1) with 2 in H. - assert (x = 0 \/ x = 1) by omega. - destruct H0; subst x; reflexivity. - - rewrite Nat2Z.inj_succ. rewrite Ztestbit_eq. rewrite zeq_false. rewrite Z.pred_succ. - rewrite IHn. rewrite two_power_nat_S. - destruct (zlt (Z.div2 x) (two_power_nat n)); rewrite (Zdecomp x); rewrite Zshiftin_spec. - rewrite zlt_true. auto. destruct (Z.odd x); omega. - rewrite zlt_false. auto. destruct (Z.odd x); omega. - rewrite (Zdecomp x) in H; rewrite Zshiftin_spec in H. - rewrite two_power_nat_S in H. destruct (Z.odd x); omega. - omega. omega. -Qed. - -Lemma Zshiftin_ind: - forall (P: Z -> Prop), - P 0 -> - (forall b x, 0 <= x -> P x -> P (Zshiftin b x)) -> - forall x, 0 <= x -> P x. -Proof. - intros. destruct x. - - auto. - - induction p. - + change (P (Zshiftin true (Z.pos p))). auto. - + change (P (Zshiftin false (Z.pos p))). auto. - + change (P (Zshiftin true 0)). apply H0. omega. auto. - - compute in H1. intuition congruence. -Qed. - -Lemma Zshiftin_pos_ind: - forall (P: Z -> Prop), - P 1 -> - (forall b x, 0 < x -> P x -> P (Zshiftin b x)) -> - forall x, 0 < x -> P x. -Proof. - intros. destruct x; simpl in H1; try discriminate. - induction p. - + change (P (Zshiftin true (Z.pos p))). auto. - + change (P (Zshiftin false (Z.pos p))). auto. - + auto. -Qed. - -Lemma Ztestbit_le: - forall x y, - 0 <= y -> - (forall i, 0 <= i -> Z.testbit x i = true -> Z.testbit y i = true) -> - x <= y. -Proof. - intros x y0 POS0; revert x; pattern y0; apply Zshiftin_ind; auto; intros. - - replace x with 0. omega. apply equal_same_bits; intros. - rewrite Ztestbit_0. destruct (Z.testbit x i) as [] eqn:E; auto. - exploit H; eauto. rewrite Ztestbit_0. auto. - - assert (Z.div2 x0 <= x). - { apply H0. intros. exploit (H1 (Z.succ i)). - omega. rewrite Ztestbit_succ; auto. rewrite Ztestbit_shiftin_succ; auto. - } - rewrite (Zdecomp x0). rewrite !Zshiftin_spec. - destruct (Z.odd x0) as [] eqn:E1; destruct b as [] eqn:E2; try omega. - exploit (H1 0). omega. rewrite Ztestbit_base; auto. - rewrite Ztestbit_shiftin_base. congruence. -Qed. - -(** ** Bit-level reasoning over type [int] *) - Definition testbit (x: int) (i: Z) : bool := Z.testbit (unsigned x) i. Lemma testbit_repr: @@ -1888,7 +1412,7 @@ Proof. rewrite bits_or; auto. rewrite H0; auto. Qed. -(** Properties of bitwise complement.*) +(** ** Properties of bitwise complement.*) Theorem not_involutive: forall (x: int), not (not x) = x. @@ -2007,7 +1531,7 @@ Proof. rewrite xor_idem. rewrite unsigned_one, unsigned_zero; auto. Qed. -(** Connections between [add] and bitwise logical operations. *) +(** ** Connections between [add] and bitwise logical operations. *) Lemma Z_add_is_or: forall i, 0 <= i -> @@ -2458,7 +1982,7 @@ Proof. - rewrite andb_false_r; auto. - generalize (unsigned_range n); intros. rewrite bits_mone. rewrite andb_true_r. f_equal. - symmetry. apply Zmod_small. omega. + symmetry. apply Z.mod_small. omega. omega. Qed. @@ -2485,7 +2009,7 @@ Theorem rol_zero: rol x zero = x. Proof. bit_solve. f_equal. rewrite unsigned_zero. rewrite Z.sub_0_r. - apply Zmod_small; auto. + apply Z.mod_small; auto. Qed. Lemma bitwise_binop_rol: @@ -2610,65 +2134,31 @@ Proof. rewrite !testbit_repr; auto. rewrite !Z.lor_spec. rewrite orb_comm. f_equal; apply same_bits_eqm; auto. - apply eqm_unsigned_repr_r. apply eqm_refl2. f_equal. - rewrite Zmod_small; auto. + rewrite Z.mod_small; auto. assert (unsigned (add y z) = zwordsize). rewrite H1. apply unsigned_repr_wordsize. unfold add in H5. rewrite unsigned_repr in H5. omega. generalize two_wordsize_max_unsigned; omega. - apply eqm_unsigned_repr_r. apply eqm_refl2. f_equal. - apply Zmod_small; auto. + apply Z.mod_small; auto. Qed. -(** ** Properties of [Z_one_bits] and [is_power2]. *) - -Fixpoint powerserie (l: list Z): Z := - match l with - | nil => 0 - | x :: xs => two_p x + powerserie xs - end. +(** ** Properties of [is_power2]. *) -Lemma Z_one_bits_powerserie: - forall x, 0 <= x < modulus -> x = powerserie (Z_one_bits wordsize x 0). -Proof. - assert (forall n x i, - 0 <= i -> - 0 <= x < two_power_nat n -> - x * two_p i = powerserie (Z_one_bits n x i)). - { - induction n; intros. - simpl. rewrite two_power_nat_O in H0. - assert (x = 0) by omega. subst x. omega. - rewrite two_power_nat_S in H0. simpl Z_one_bits. - rewrite (Zdecomp x) in H0. rewrite Zshiftin_spec in H0. - assert (EQ: Z.div2 x * two_p (i + 1) = powerserie (Z_one_bits n (Z.div2 x) (i + 1))). - apply IHn. omega. - destruct (Z.odd x); omega. - rewrite two_p_is_exp in EQ. change (two_p 1) with 2 in EQ. - rewrite (Zdecomp x) at 1. rewrite Zshiftin_spec. - destruct (Z.odd x); simpl powerserie; rewrite <- EQ; ring. - omega. omega. - } - intros. rewrite <- H. change (two_p 0) with 1. omega. - omega. exact H0. -Qed. - -Lemma Z_one_bits_range: - forall x i, In i (Z_one_bits wordsize x 0) -> 0 <= i < zwordsize. +Remark is_power2_inv: + forall n logn, + is_power2 n = Some logn -> + Z_is_power2 (unsigned n) = Some (unsigned logn) /\ 0 <= unsigned logn < zwordsize. Proof. - assert (forall n x i j, - In j (Z_one_bits n x i) -> i <= j < i + Z.of_nat n). - { - induction n; simpl In. - tauto. - intros x i j. rewrite Nat2Z.inj_succ. - assert (In j (Z_one_bits n (Z.div2 x) (i + 1)) -> i <= j < i + Z.succ (Z.of_nat n)). - intros. exploit IHn; eauto. omega. - destruct (Z.odd x); simpl. - intros [A|B]. subst j. omega. auto. - auto. - } - intros. generalize (H wordsize x 0 i H0). fold zwordsize; omega. + unfold is_power2; intros. + destruct (Z_is_power2 (unsigned n)) as [i|] eqn:E; inv H. + assert (0 <= i < zwordsize). + { apply Z_is_power2_range with (unsigned n). + generalize wordsize_pos; omega. + rewrite <- modulus_power. apply unsigned_range. + auto. } + rewrite unsigned_repr; auto. generalize wordsize_max_unsigned; omega. Qed. Lemma is_power2_rng: @@ -2676,16 +2166,7 @@ Lemma is_power2_rng: is_power2 n = Some logn -> 0 <= unsigned logn < zwordsize. Proof. - intros n logn. unfold is_power2. - generalize (Z_one_bits_range (unsigned n)). - destruct (Z_one_bits wordsize (unsigned n) 0). - intros; discriminate. - destruct l. - intros. injection H0; intro; subst logn; clear H0. - assert (0 <= z < zwordsize). - apply H. auto with coqlib. - rewrite unsigned_repr. auto. generalize wordsize_max_unsigned; omega. - intros; discriminate. + intros. apply (is_power2_inv n logn); auto. Qed. Theorem is_power2_range: @@ -2701,18 +2182,8 @@ Lemma is_power2_correct: is_power2 n = Some logn -> unsigned n = two_p (unsigned logn). Proof. - intros n logn. unfold is_power2. - generalize (Z_one_bits_powerserie (unsigned n) (unsigned_range n)). - generalize (Z_one_bits_range (unsigned n)). - destruct (Z_one_bits wordsize (unsigned n) 0). - intros; discriminate. - destruct l. - intros. simpl in H0. injection H1; intros; subst logn; clear H1. - rewrite unsigned_repr. replace (two_p z) with (two_p z + 0). - auto. omega. elim (H z); intros. - generalize wordsize_max_unsigned; omega. - auto with coqlib. - intros; discriminate. + intros. apply is_power2_inv in H. destruct H as [P Q]. + apply Z_is_power2_sound in P. tauto. Qed. Remark two_p_range: @@ -2727,34 +2198,12 @@ Proof. unfold max_unsigned, modulus. omega. Qed. -Remark Z_one_bits_zero: - forall n i, Z_one_bits n 0 i = nil. -Proof. - induction n; intros; simpl; auto. -Qed. - -Remark Z_one_bits_two_p: - forall n x i, - 0 <= x < Z.of_nat n -> - Z_one_bits n (two_p x) i = (i + x) :: nil. -Proof. - induction n; intros; simpl. simpl in H. omegaContradiction. - rewrite Nat2Z.inj_succ in H. - assert (x = 0 \/ 0 < x) by omega. destruct H0. - subst x; simpl. decEq. omega. apply Z_one_bits_zero. - assert (Z.odd (two_p x) = false /\ Z.div2 (two_p x) = two_p (x-1)). - apply Zshiftin_inj. rewrite <- Zdecomp. rewrite !Zshiftin_spec. - rewrite <- two_p_S. rewrite Z.add_0_r. f_equal; omega. omega. - destruct H1 as [A B]; rewrite A; rewrite B. - rewrite IHn. f_equal; omega. omega. -Qed. - Lemma is_power2_two_p: forall n, 0 <= n < zwordsize -> is_power2 (repr (two_p n)) = Some (repr n). Proof. intros. unfold is_power2. rewrite unsigned_repr. - rewrite Z_one_bits_two_p. auto. auto. + rewrite Z_is_power2_complete by omega; auto. apply two_p_range. auto. Qed. @@ -2762,19 +2211,6 @@ Qed. (** Left shifts and multiplications by powers of 2. *) -Lemma Zshiftl_mul_two_p: - forall x n, 0 <= n -> Z.shiftl x n = x * two_p n. -Proof. - intros. destruct n; simpl. - - omega. - - pattern p. apply Pos.peano_ind. - + change (two_power_pos 1) with 2. simpl. ring. - + intros. rewrite Pos.iter_succ. rewrite H0. - rewrite Pplus_one_succ_l. rewrite two_power_pos_is_exp. - change (two_power_pos 1) with 2. ring. - - compute in H. congruence. -Qed. - Lemma shl_mul_two_p: forall x y, shl x y = mul x (repr (two_p (unsigned y))). @@ -2834,21 +2270,6 @@ Qed. (** Unsigned right shifts and unsigned divisions by powers of 2. *) -Lemma Zshiftr_div_two_p: - forall x n, 0 <= n -> Z.shiftr x n = x / two_p n. -Proof. - intros. destruct n; unfold Z.shiftr; simpl. - - rewrite Zdiv_1_r. auto. - - pattern p. apply Pos.peano_ind. - + change (two_power_pos 1) with 2. simpl. apply Zdiv2_div. - + intros. rewrite Pos.iter_succ. rewrite H0. - rewrite Pplus_one_succ_l. rewrite two_power_pos_is_exp. - change (two_power_pos 1) with 2. - rewrite Zdiv2_div. rewrite Z.mul_comm. apply Zdiv_Zdiv. - rewrite two_power_pos_nat. apply two_power_nat_pos. omega. - - compute in H. congruence. -Qed. - Lemma shru_div_two_p: forall x y, shru x y = repr (unsigned x / two_p (unsigned y)). @@ -2891,43 +2312,6 @@ Qed. (** Unsigned modulus over [2^n] is masking with [2^n-1]. *) -Lemma Ztestbit_mod_two_p: - forall n x i, - 0 <= n -> 0 <= i -> - Z.testbit (x mod (two_p n)) i = if zlt i n then Z.testbit x i else false. -Proof. - intros n0 x i N0POS. revert x i; pattern n0; apply natlike_ind; auto. - - intros. change (two_p 0) with 1. rewrite Zmod_1_r. rewrite Z.testbit_0_l. - rewrite zlt_false; auto. omega. - - intros. rewrite two_p_S; auto. - replace (x0 mod (2 * two_p x)) - with (Zshiftin (Z.odd x0) (Z.div2 x0 mod two_p x)). - rewrite Ztestbit_shiftin; auto. rewrite (Ztestbit_eq i x0); auto. destruct (zeq i 0). - + rewrite zlt_true; auto. omega. - + rewrite H0. destruct (zlt (Z.pred i) x). - * rewrite zlt_true; auto. omega. - * rewrite zlt_false; auto. omega. - * omega. - + rewrite (Zdecomp x0) at 3. set (x1 := Z.div2 x0). symmetry. - apply Zmod_unique with (x1 / two_p x). - rewrite !Zshiftin_spec. rewrite Z.add_assoc. f_equal. - transitivity (2 * (two_p x * (x1 / two_p x) + x1 mod two_p x)). - f_equal. apply Z_div_mod_eq. apply two_p_gt_ZERO; auto. - ring. - rewrite Zshiftin_spec. exploit (Z_mod_lt x1 (two_p x)). apply two_p_gt_ZERO; auto. - destruct (Z.odd x0); omega. -Qed. - -Corollary Ztestbit_two_p_m1: - forall n i, 0 <= n -> 0 <= i -> - Z.testbit (two_p n - 1) i = if zlt i n then true else false. -Proof. - intros. replace (two_p n - 1) with ((-1) mod (two_p n)). - rewrite Ztestbit_mod_two_p; auto. destruct (zlt i n); auto. apply Ztestbit_m1; auto. - apply Zmod_unique with (-1). ring. - exploit (two_p_gt_ZERO n). auto. omega. -Qed. - Theorem modu_and: forall x n logn, is_power2 n = Some logn -> @@ -2949,21 +2333,6 @@ Qed. (** ** Properties of [shrx] (signed division by a power of 2) *) -Lemma Zquot_Zdiv: - forall x y, - y > 0 -> - Z.quot x y = if zlt x 0 then (x + y - 1) / y else x / y. -Proof. - intros. destruct (zlt x 0). - - symmetry. apply Zquot_unique_full with ((x + y - 1) mod y - (y - 1)). - + red. right; split. omega. - exploit (Z_mod_lt (x + y - 1) y); auto. - rewrite Z.abs_eq. omega. omega. - + transitivity ((y * ((x + y - 1) / y) + (x + y - 1) mod y) - (y-1)). - rewrite <- Z_div_mod_eq. ring. auto. ring. - - apply Zquot_Zdiv_pos; omega. -Qed. - Theorem shrx_zero: forall x, zwordsize > 1 -> shrx x zero = x. Proof. @@ -3042,17 +2411,6 @@ Proof. bit_solve. destruct (zlt (i + unsigned (sub iwordsize y)) zwordsize); auto. Qed. -Lemma Zdiv_shift: - forall x y, y > 0 -> - (x + (y - 1)) / y = x / y + if zeq (Z.modulo x y) 0 then 0 else 1. -Proof. - intros. generalize (Z_div_mod_eq x y H). generalize (Z_mod_lt x y H). - set (q := x / y). set (r := x mod y). intros. - destruct (zeq r 0). - apply Zdiv_unique with (y - 1). rewrite H1. rewrite e. ring. omega. - apply Zdiv_unique with (r - 1). rewrite H1. ring. omega. -Qed. - Theorem shrx_carry: forall x y, ltu y (repr (zwordsize - 1)) = true -> @@ -3143,51 +2501,6 @@ Qed. (** ** Properties of integer zero extension and sign extension. *) -Lemma Ziter_base: - forall (A: Type) n (f: A -> A) x, n <= 0 -> Z.iter n f x = x. -Proof. - intros. unfold Z.iter. destruct n; auto. compute in H. elim H; auto. -Qed. - -Lemma Ziter_succ: - forall (A: Type) n (f: A -> A) x, - 0 <= n -> Z.iter (Z.succ n) f x = f (Z.iter n f x). -Proof. - intros. destruct n; simpl. - - auto. - - rewrite Pos.add_1_r. apply Pos.iter_succ. - - compute in H. elim H; auto. -Qed. - -Lemma Znatlike_ind: - forall (P: Z -> Prop), - (forall n, n <= 0 -> P n) -> - (forall n, 0 <= n -> P n -> P (Z.succ n)) -> - forall n, P n. -Proof. - intros. destruct (zle 0 n). - apply natlike_ind; auto. apply H; omega. - apply H. omega. -Qed. - -Lemma Zzero_ext_spec: - forall n x i, 0 <= i -> - Z.testbit (Zzero_ext n x) i = if zlt i n then Z.testbit x i else false. -Proof. - unfold Zzero_ext. induction n using Znatlike_ind. - - intros. rewrite Ziter_base; auto. - rewrite zlt_false. rewrite Ztestbit_0; auto. omega. - - intros. rewrite Ziter_succ; auto. - rewrite Ztestbit_shiftin; auto. - rewrite (Ztestbit_eq i x); auto. - destruct (zeq i 0). - + subst i. rewrite zlt_true; auto. omega. - + rewrite IHn. destruct (zlt (Z.pred i) n). - rewrite zlt_true; auto. omega. - rewrite zlt_false; auto. omega. - omega. -Qed. - Lemma bits_zero_ext: forall n x i, 0 <= i -> testbit (zero_ext n x) i = if zlt i n then testbit x i else false. @@ -3197,35 +2510,6 @@ Proof. rewrite !bits_above; auto. destruct (zlt i n); auto. Qed. -Lemma Zsign_ext_spec: - forall n x i, 0 <= i -> 0 < n -> - Z.testbit (Zsign_ext n x) i = Z.testbit x (if zlt i n then i else n - 1). -Proof. - intros n0 x i I0 N0. - revert x i I0. pattern n0. apply Zlt_lower_bound_ind with (z := 1). - - unfold Zsign_ext. intros. - destruct (zeq x 1). - + subst x; simpl. - replace (if zlt i 1 then i else 0) with 0. - rewrite Ztestbit_base. - destruct (Z.odd x0). - apply Ztestbit_m1; auto. - apply Ztestbit_0. - destruct (zlt i 1); omega. - + set (x1 := Z.pred x). replace x1 with (Z.succ (Z.pred x1)). - rewrite Ziter_succ. rewrite Ztestbit_shiftin. - destruct (zeq i 0). - * subst i. rewrite zlt_true. rewrite Ztestbit_base; auto. omega. - * rewrite H. unfold x1. destruct (zlt (Z.pred i) (Z.pred x)). - rewrite zlt_true. rewrite (Ztestbit_eq i x0); auto. rewrite zeq_false; auto. omega. - rewrite zlt_false. rewrite (Ztestbit_eq (x - 1) x0). rewrite zeq_false; auto. - omega. omega. omega. unfold x1; omega. omega. - * omega. - * unfold x1; omega. - * omega. - - omega. -Qed. - Lemma bits_sign_ext: forall n x i, 0 <= i < zwordsize -> 0 < n -> testbit (sign_ext n x) i = testbit x (if zlt i n then i else n - 1). @@ -3527,7 +2811,7 @@ Proof. auto with ints. decEq. apply Z_one_bits_powerserie. auto with ints. unfold one_bits. - generalize (Z_one_bits_range (unsigned x)). + generalize (Z_one_bits_range wordsize (unsigned x)). generalize (Z_one_bits wordsize (unsigned x) 0). induction l. intros; reflexivity. @@ -3535,7 +2819,8 @@ Proof. apply eqm_add. rewrite shl_mul_two_p. rewrite mul_commut. rewrite mul_one. apply eqm_unsigned_repr_r. rewrite unsigned_repr. auto with ints. - generalize (H a (in_eq _ _)). generalize wordsize_max_unsigned. omega. + generalize (H a (in_eq _ _)). change (Z.of_nat wordsize) with zwordsize. + generalize wordsize_max_unsigned. omega. auto with ints. intros; apply H; auto with coqlib. Qed. @@ -3735,8 +3020,7 @@ Proof. intros. rewrite <- negb_orb. rewrite <- not_ltu. rewrite negb_involutive. auto. Qed. - -(** Non-overlapping test *) +(** ** Non-overlapping test *) Definition no_overlap (ofs1: int) (sz1: Z) (ofs2: int) (sz2: Z) : bool := let x1 := unsigned ofs1 in let x2 := unsigned ofs2 in @@ -3762,94 +3046,10 @@ Proof. intros [C|C] [D|D]; omega. Qed. -(** Size of integers, in bits. *) - -Definition Zsize (x: Z) : Z := - match x with - | Zpos p => Zpos (Pos.size p) - | _ => 0 - end. +(** ** Size of integers, in bits. *) Definition size (x: int) : Z := Zsize (unsigned x). -Remark Zsize_pos: forall x, 0 <= Zsize x. -Proof. - destruct x; simpl. omega. compute; intuition congruence. omega. -Qed. - -Remark Zsize_pos': forall x, 0 < x -> 0 < Zsize x. -Proof. - destruct x; simpl; intros; try discriminate. compute; auto. -Qed. - -Lemma Zsize_shiftin: - forall b x, 0 < x -> Zsize (Zshiftin b x) = Z.succ (Zsize x). -Proof. - intros. destruct x; compute in H; try discriminate. - destruct b. - change (Zshiftin true (Zpos p)) with (Zpos (p~1)). - simpl. f_equal. rewrite Pos.add_1_r; auto. - change (Zshiftin false (Zpos p)) with (Zpos (p~0)). - simpl. f_equal. rewrite Pos.add_1_r; auto. -Qed. - -Lemma Ztestbit_size_1: - forall x, 0 < x -> Z.testbit x (Z.pred (Zsize x)) = true. -Proof. - intros x0 POS0; pattern x0; apply Zshiftin_pos_ind; auto. - intros. rewrite Zsize_shiftin; auto. - replace (Z.pred (Z.succ (Zsize x))) with (Z.succ (Z.pred (Zsize x))) by omega. - rewrite Ztestbit_shiftin_succ. auto. generalize (Zsize_pos' x H); omega. -Qed. - -Lemma Ztestbit_size_2: - forall x, 0 <= x -> forall i, i >= Zsize x -> Z.testbit x i = false. -Proof. - intros x0 POS0. destruct (zeq x0 0). - - subst x0; intros. apply Ztestbit_0. - - pattern x0; apply Zshiftin_pos_ind. - + simpl. intros. change 1 with (Zshiftin true 0). rewrite Ztestbit_shiftin. - rewrite zeq_false. apply Ztestbit_0. omega. omega. - + intros. rewrite Zsize_shiftin in H1; auto. - generalize (Zsize_pos' _ H); intros. - rewrite Ztestbit_shiftin. rewrite zeq_false. apply H0. omega. - omega. omega. - + omega. -Qed. - -Lemma Zsize_interval_1: - forall x, 0 <= x -> 0 <= x < two_p (Zsize x). -Proof. - intros. - assert (x = x mod (two_p (Zsize x))). - apply equal_same_bits; intros. - rewrite Ztestbit_mod_two_p; auto. - destruct (zlt i (Zsize x)). auto. apply Ztestbit_size_2; auto. - apply Zsize_pos; auto. - rewrite H0 at 1. rewrite H0 at 3. apply Z_mod_lt. apply two_p_gt_ZERO. apply Zsize_pos; auto. -Qed. - -Lemma Zsize_interval_2: - forall x n, 0 <= n -> 0 <= x < two_p n -> n >= Zsize x. -Proof. - intros. set (N := Z.to_nat n). - assert (Z.of_nat N = n) by (apply Z2Nat.id; auto). - rewrite <- H1 in H0. rewrite <- two_power_nat_two_p in H0. - destruct (zeq x 0). - subst x; simpl; omega. - destruct (zlt n (Zsize x)); auto. - exploit (Ztestbit_above N x (Z.pred (Zsize x))). auto. omega. - rewrite Ztestbit_size_1. congruence. omega. -Qed. - -Lemma Zsize_monotone: - forall x y, 0 <= x <= y -> Zsize x <= Zsize y. -Proof. - intros. apply Z.ge_le. apply Zsize_interval_2. apply Zsize_pos. - exploit (Zsize_interval_1 y). omega. - omega. -Qed. - Theorem size_zero: size zero = 0. Proof. unfold size; rewrite unsigned_zero; auto. @@ -4315,7 +3515,7 @@ Theorem one_bits'_range: Proof. intros. destruct (list_in_map_inv _ _ _ H) as [i0 [EQ IN]]. - exploit Z_one_bits_range; eauto. intros R. + exploit Z_one_bits_range; eauto. fold zwordsize. intros R. unfold Int.ltu. rewrite EQ. rewrite Int.unsigned_repr. change (Int.unsigned iwordsize') with zwordsize. apply zlt_true. omega. assert (zwordsize < Int.max_unsigned) by reflexivity. omega. @@ -4374,7 +3574,7 @@ Lemma is_power2'_correct: Proof. unfold is_power2'; intros. destruct (Z_one_bits wordsize (unsigned n) 0) as [ | i [ | ? ?]] eqn:B; inv H. - rewrite (Z_one_bits_powerserie (unsigned n)) by (apply unsigned_range). + rewrite (Z_one_bits_powerserie wordsize (unsigned n)) by (apply unsigned_range). rewrite Int.unsigned_repr. rewrite B; simpl. omega. assert (0 <= i < zwordsize). { apply Z_one_bits_range with (unsigned n). rewrite B; auto with coqlib. } |