diff options
| -rw-r--r-- | Makefile | 2 | ||||
| -rw-r--r-- | arm/Asmgenproof1.v | 3 | ||||
| -rw-r--r-- | backend/NeedDomain.v | 31 | ||||
| -rw-r--r-- | backend/SelectDivproof.v | 6 | ||||
| -rw-r--r-- | backend/ValueDomain.v | 4 | ||||
| -rw-r--r-- | common/Memdata.v | 9 | ||||
| -rw-r--r-- | lib/Floats.v | 7 | ||||
| -rw-r--r-- | lib/Integers.v | 868 | ||||
| -rw-r--r-- | lib/Zbits.v | 945 | ||||
| -rw-r--r-- | powerpc/Asmgenproof1.v | 9 | ||||
| -rw-r--r-- | powerpc/SelectLongproof.v | 10 | ||||
| -rw-r--r-- | riscV/Asmgenproof1.v | 16 | ||||
| -rw-r--r-- | riscV/SelectOpproof.v | 5 |
13 files changed, 1031 insertions, 884 deletions
@@ -53,7 +53,7 @@ FLOCQ=\ # General-purpose libraries (in lib/) VLIB=Axioms.v Coqlib.v Intv.v Maps.v Heaps.v Lattice.v Ordered.v \ - Iteration.v Integers.v Archi.v Fappli_IEEE_extra.v Floats.v \ + Iteration.v Zbits.v Integers.v Archi.v Fappli_IEEE_extra.v Floats.v \ Parmov.v UnionFind.v Wfsimpl.v \ Postorder.v FSetAVLplus.v IntvSets.v Decidableplus.v BoolEqual.v diff --git a/arm/Asmgenproof1.v b/arm/Asmgenproof1.v index 98cd5eea..edf35bb8 100644 --- a/arm/Asmgenproof1.v +++ b/arm/Asmgenproof1.v @@ -16,6 +16,7 @@ Require Import Coqlib. Require Import Errors. Require Import Maps. Require Import AST. +Require Import Zbits. Require Import Integers. Require Import Floats. Require Import Values. @@ -355,7 +356,7 @@ Proof. rewrite Int.and_assoc. change 65535 with (two_p 16 - 1). rewrite Int.and_idem. apply Int.same_bits_eq; intros. rewrite Int.bits_or, Int.bits_and, Int.bits_shl, Int.testbit_repr by auto. - rewrite Int.Ztestbit_two_p_m1 by omega. change (Int.unsigned (Int.repr 16)) with 16. + rewrite Ztestbit_two_p_m1 by omega. change (Int.unsigned (Int.repr 16)) with 16. destruct (zlt i 16). rewrite andb_true_r, orb_false_r; auto. rewrite andb_false_r; simpl. rewrite Int.bits_shru by omega. diff --git a/backend/NeedDomain.v b/backend/NeedDomain.v index 692b4f9b..5d19e8f6 100644 --- a/backend/NeedDomain.v +++ b/backend/NeedDomain.v @@ -16,6 +16,7 @@ Require Import Coqlib. Require Import Maps. Require Import IntvSets. Require Import AST. +Require Import Zbits. Require Import Integers. Require Import Floats. Require Import Values. @@ -300,13 +301,13 @@ Proof. rewrite Int.bits_ror. replace (((i - Int.unsigned amount) mod Int.zwordsize + Int.unsigned amount) mod Int.zwordsize) with i. auto. - apply Int.eqmod_small_eq with Int.zwordsize; auto. - apply Int.eqmod_trans with ((i - Int.unsigned amount) + Int.unsigned amount). - apply Int.eqmod_refl2; omega. - eapply Int.eqmod_trans. 2: apply Int.eqmod_mod; auto. - apply Int.eqmod_add. - apply Int.eqmod_mod; auto. - apply Int.eqmod_refl. + apply eqmod_small_eq with Int.zwordsize; auto. + apply eqmod_trans with ((i - Int.unsigned amount) + Int.unsigned amount). + apply eqmod_refl2; omega. + eapply eqmod_trans. 2: apply eqmod_mod; auto. + apply eqmod_add. + apply eqmod_mod; auto. + apply eqmod_refl. apply Z_mod_lt; auto. apply Z_mod_lt; auto. Qed. @@ -324,7 +325,7 @@ Qed. Lemma eqmod_iagree: forall m x y, - Int.eqmod (two_p (Int.size m)) x y -> + eqmod (two_p (Int.size m)) x y -> iagree (Int.repr x) (Int.repr y) m. Proof. intros. set (p := Z.to_nat (Int.size m)). @@ -333,7 +334,7 @@ Proof. rewrite EQ in H; rewrite <- two_power_nat_two_p in H. red; intros. rewrite ! Int.testbit_repr by auto. destruct (zlt i (Int.size m)). - eapply Int.same_bits_eqmod; eauto. omega. + eapply same_bits_eqmod; eauto. omega. assert (Int.testbit m i = false) by (eapply Int.bits_size_2; omega). congruence. Qed. @@ -343,13 +344,13 @@ Definition complete_mask (m: int) := Int.zero_ext (Int.size m) Int.mone. Lemma iagree_eqmod: forall x y m, iagree x y (complete_mask m) -> - Int.eqmod (two_p (Int.size m)) (Int.unsigned x) (Int.unsigned y). + eqmod (two_p (Int.size m)) (Int.unsigned x) (Int.unsigned y). Proof. intros. set (p := Z.to_nat (Int.size m)). generalize (Int.size_range m); intros RANGE. assert (EQ: Int.size m = Z.of_nat p). { symmetry; apply Z2Nat.id. omega. } rewrite EQ; rewrite <- two_power_nat_two_p. - apply Int.eqmod_same_bits. intros. apply H. omega. + apply eqmod_same_bits. intros. apply H. omega. unfold complete_mask. rewrite Int.bits_zero_ext by omega. rewrite zlt_true by omega. rewrite Int.bits_mone by omega. auto. Qed. @@ -362,7 +363,7 @@ Proof. + assert (Int.unsigned m <> 0). { red; intros; elim n. rewrite <- (Int.repr_unsigned m). rewrite H; auto. } assert (0 < Int.size m). - { apply Int.Zsize_pos'. generalize (Int.unsigned_range m); omega. } + { apply Zsize_pos'. generalize (Int.unsigned_range m); omega. } generalize (Int.size_range m); intros. f_equal. apply Int.bits_size_4. tauto. rewrite Int.bits_zero_ext by omega. rewrite zlt_true by omega. @@ -610,7 +611,7 @@ Proof. unfold modarith; intros. destruct x; simpl in *. - auto. - unfold Val.add; InvAgree. - apply eqmod_iagree. apply Int.eqmod_add; apply iagree_eqmod; auto. + apply eqmod_iagree. apply eqmod_add; apply iagree_eqmod; auto. - inv H; auto. inv H0; auto. destruct w1; auto. Qed. @@ -626,7 +627,7 @@ Lemma mul_sound: Proof. unfold mul, add; intros. destruct x; simpl in *. - auto. -- unfold Val.mul; InvAgree. apply eqmod_iagree. apply Int.eqmod_mult; apply iagree_eqmod; auto. +- unfold Val.mul; InvAgree. apply eqmod_iagree. apply eqmod_mult; apply iagree_eqmod; auto. - inv H; auto. inv H0; auto. destruct w1; auto. Qed. @@ -638,7 +639,7 @@ Proof. intros; destruct x; simpl in *. - auto. - unfold Val.neg; InvAgree. - apply eqmod_iagree. apply Int.eqmod_neg. apply iagree_eqmod; auto. + apply eqmod_iagree. apply eqmod_neg. apply iagree_eqmod; auto. - inv H; simpl; auto. Qed. diff --git a/backend/SelectDivproof.v b/backend/SelectDivproof.v index e660677a..f4ff2c86 100644 --- a/backend/SelectDivproof.v +++ b/backend/SelectDivproof.v @@ -12,7 +12,7 @@ (** Correctness of instruction selection for integer division *) -Require Import Zquot Coqlib. +Require Import Zquot Coqlib Zbits. Require Import AST Integers Floats Values Memory Globalenvs Events. Require Import Cminor Op CminorSel. Require Import SelectOp SelectOpproof SplitLong SplitLongproof SelectLong SelectLongproof SelectDiv. @@ -378,7 +378,7 @@ Qed. Remark int64_shr'_div_two_p: forall x y, Int64.shr' x y = Int64.repr (Int64.signed x / two_p (Int.unsigned y)). Proof. - intros; unfold Int64.shr'. rewrite Int64.Zshiftr_div_two_p; auto. generalize (Int.unsigned_range y); omega. + intros; unfold Int64.shr'. rewrite Zshiftr_div_two_p; auto. generalize (Int.unsigned_range y); omega. Qed. Lemma divls_mul_shift_gen: @@ -453,7 +453,7 @@ Qed. Remark int64_shru'_div_two_p: forall x y, Int64.shru' x y = Int64.repr (Int64.unsigned x / two_p (Int.unsigned y)). Proof. - intros; unfold Int64.shru'. rewrite Int64.Zshiftr_div_two_p; auto. generalize (Int.unsigned_range y); omega. + intros; unfold Int64.shru'. rewrite Zshiftr_div_two_p; auto. generalize (Int.unsigned_range y); omega. Qed. Theorem divlu_mul_shift: diff --git a/backend/ValueDomain.v b/backend/ValueDomain.v index 47b87bfb..3ba2a35b 100644 --- a/backend/ValueDomain.v +++ b/backend/ValueDomain.v @@ -11,7 +11,7 @@ (* *********************************************************************) Require Import FunInd. -Require Import Zwf Coqlib Maps Integers Floats Lattice. +Require Import Zwf Coqlib Maps Zbits Integers Floats Lattice. Require Import Compopts AST. Require Import Values Memory Globalenvs Events. Require Import Registers RTL. @@ -1670,7 +1670,7 @@ Proof. assert (UNS: forall i j, j <> Int.zero -> is_uns (usize j) (Int.modu i j)). { intros. apply is_uns_mon with (usize (Int.modu i j)); auto with va. - unfold usize, Int.size. apply Int.Zsize_monotone. + unfold usize, Int.size. apply Zsize_monotone. generalize (Int.unsigned_range_2 j); intros RANGE. assert (Int.unsigned j <> 0). { red; intros; elim H. rewrite <- (Int.repr_unsigned j). rewrite H0. auto. } diff --git a/common/Memdata.v b/common/Memdata.v index c53f0e5d..7144d72c 100644 --- a/common/Memdata.v +++ b/common/Memdata.v @@ -17,6 +17,7 @@ (** In-memory representation of values. *) Require Import Coqlib. +Require Import Zbits. Require Archi. Require Import AST. Require Import Integers. @@ -272,7 +273,7 @@ Qed. Lemma bytes_of_int_mod: forall n x y, - Int.eqmod (two_p (Z.of_nat n * 8)) x y -> + eqmod (two_p (Z.of_nat n * 8)) x y -> bytes_of_int n x = bytes_of_int n y. Proof. induction n. @@ -284,7 +285,7 @@ Proof. intro EQM. simpl; decEq. apply Byte.eqm_samerepr. red. - eapply Int.eqmod_divides; eauto. apply Z.divide_factor_r. + eapply eqmod_divides; eauto. apply Z.divide_factor_r. apply IHn. destruct EQM as [k EQ]. exists k. rewrite EQ. rewrite <- Z_div_plus_full_l. decEq. change (two_p 8) with 256. ring. omega. @@ -292,7 +293,7 @@ Qed. Lemma encode_int_8_mod: forall x y, - Int.eqmod (two_p 8) x y -> + eqmod (two_p 8) x y -> encode_int 1%nat x = encode_int 1%nat y. Proof. intros. unfold encode_int. decEq. apply bytes_of_int_mod. auto. @@ -300,7 +301,7 @@ Qed. Lemma encode_int_16_mod: forall x y, - Int.eqmod (two_p 16) x y -> + eqmod (two_p 16) x y -> encode_int 2%nat x = encode_int 2%nat y. Proof. intros. unfold encode_int. decEq. apply bytes_of_int_mod. auto. diff --git a/lib/Floats.v b/lib/Floats.v index f93505fc..21d09a5e 100644 --- a/lib/Floats.v +++ b/lib/Floats.v @@ -16,8 +16,7 @@ (** Formalization of floating-point numbers, using the Flocq library. *) -Require Import Coqlib. -Require Import Integers. +Require Import Coqlib Zbits Integers. (*From Flocq*) Require Import Binary Bits Core. Require Import Fappli_IEEE_extra. @@ -1356,9 +1355,9 @@ Proof. rewrite Int64.testbit_repr by auto. f_equal. f_equal. unfold Int64.and. change (Int64.unsigned (Int64.repr 2047)) with 2047. change 2047 with (Z.ones 11). rewrite ! Z.land_ones by omega. - rewrite Int64.unsigned_repr. apply Int64.eqmod_mod_eq. + rewrite Int64.unsigned_repr. apply eqmod_mod_eq. apply Z.lt_gt. apply (Zpower_gt_0 radix2); omega. - apply Int64.eqmod_divides with (2^64). apply Int64.eqm_signed_unsigned. + apply eqmod_divides with (2^64). apply Int64.eqm_signed_unsigned. exists (2^(64-11)); auto. exploit (Z_mod_lt (Int64.unsigned n) (2^11)). compute; auto. assert (2^11 < Int64.max_unsigned) by (compute; auto). omega. diff --git a/lib/Integers.v b/lib/Integers.v index 64263b97..8f528079 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 *) @@ -91,6 +91,8 @@ Proof. 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 @@ -101,17 +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 @@ -119,51 +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. @@ -173,22 +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 @@ -323,54 +257,11 @@ 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). @@ -503,101 +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 (Z.div_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. @@ -643,6 +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. @@ -1304,298 +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: @@ -1894,7 +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. @@ -2013,7 +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 -> @@ -2626,56 +2144,7 @@ Proof. 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. - -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. -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. -Qed. +(** ** Properties of [is_power2]. *) Lemma is_power2_rng: forall n logn, @@ -2683,7 +2152,7 @@ Lemma is_power2_rng: 0 <= unsigned logn < zwordsize. Proof. intros n logn. unfold is_power2. - generalize (Z_one_bits_range (unsigned n)). + generalize (Z_one_bits_range wordsize (unsigned n)). destruct (Z_one_bits wordsize (unsigned n) 0). intros; discriminate. destruct l. @@ -2708,14 +2177,14 @@ Lemma is_power2_correct: 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)). + generalize (Z_one_bits_powerserie wordsize (unsigned n) (unsigned_range n)). + generalize (Z_one_bits_range wordsize (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. + auto. omega. elim (H z); change (Z.of_nat wordsize) with zwordsize; intros. generalize wordsize_max_unsigned; omega. auto with coqlib. intros; discriminate. @@ -2733,28 +2202,6 @@ 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). @@ -2768,19 +2215,6 @@ 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))). @@ -2840,21 +2274,6 @@ 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)). @@ -2897,43 +2316,6 @@ 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 -> @@ -2955,21 +2337,6 @@ 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. @@ -3048,17 +2415,6 @@ 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 -> @@ -3149,51 +2505,6 @@ 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. @@ -3203,35 +2514,6 @@ 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). @@ -3533,7 +2815,7 @@ 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. @@ -3541,7 +2823,8 @@ 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. @@ -3741,8 +3024,7 @@ 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 @@ -3768,94 +3050,10 @@ 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. @@ -4321,7 +3519,7 @@ 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. @@ -4380,7 +3578,7 @@ 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. } diff --git a/lib/Zbits.v b/lib/Zbits.v new file mode 100644 index 00000000..74f66b6e --- /dev/null +++ b/lib/Zbits.v @@ -0,0 +1,945 @@ +(* *********************************************************************) +(* *) +(* The Compcert verified compiler *) +(* *) +(* Xavier Leroy, Collège de France and Inria Paris *) +(* *) +(* Copyright Institut National de Recherche en Informatique et en *) +(* Automatique. All rights reserved. This file is distributed *) +(* under the terms of the GNU General Public License as published by *) +(* the Free Software Foundation, either version 2 of the License, or *) +(* (at your option) any later version. This file is also distributed *) +(* under the terms of the INRIA Non-Commercial License Agreement. *) +(* *) +(* *********************************************************************) + +(** Additional operations and proofs about binary integers, + on top of the ZArith standard library. *) + +Require Import Psatz Zquot. +Require Import Coqlib. + +(** ** 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 (Z.div_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. + +(** ** Fast normalization modulo [2^n] *) + +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_two_p (x: Z) (n: nat) : Z := + match x with + | Z0 => 0 + | Zpos p => P_mod_two_p p n + | Zneg p => let r := P_mod_two_p p n in if zeq r 0 then 0 else two_power_nat n - 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_two_p_range: + forall n x, 0 <= Z_mod_two_p x n < two_power_nat n. +Proof. + intros; unfold Z_mod_two_p. generalize (two_power_nat_pos n); intros. + destruct x. + - intuition. + - apply P_mod_two_p_range. + - set (r := P_mod_two_p p n). + assert (0 <= r < two_power_nat n) by apply P_mod_two_p_range. + destruct (zeq r 0). + + intuition. + + Psatz.lia. +Qed. + +Lemma Z_mod_two_p_eq: + forall n x, Z_mod_two_p x n = x mod (two_power_nat n). +Proof. + intros. unfold Z_mod_two_p. generalize (two_power_nat_pos n); intros. + destruct x. + - rewrite Zmod_0_l. auto. + - apply P_mod_two_p_eq. + - generalize (P_mod_two_p_range n p) (P_mod_two_p_eq n p). intros A B. + exploit (Z_div_mod_eq (Zpos p) (two_power_nat n)); auto. intros C. + set (q := Zpos p / two_power_nat n) in *. + set (r := P_mod_two_p p n) 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. + +(** ** Bit-level operations and properties *) + +(** Shift [x] left by one and insert [b] as the low bit of the result. *) + +Definition Zshiftin (b: bool) (x: Z) : Z := + if b then Z.succ_double x else Z.double x. + +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. + +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. + +(** ** Bit-wise decomposition ([Z.testbit]) *) + +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 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 equal_same_bits: + forall x y, + (forall i, 0 <= i -> Z.testbit x i = Z.testbit y i) -> + x = y. +Proof Z.bits_inj'. + +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 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. + +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. + +Corollary Ztestbit_neg_two_p: + forall n i, 0 <= n -> 0 <= i -> + Z.testbit (- (two_p n)) i = if zlt i n then false else true. +Proof. + intros. + replace (- two_p n) with (- (two_p n - 1) - 1) by omega. + rewrite Z_one_complement by auto. + rewrite Ztestbit_two_p_m1 by auto. + destruct (zlt i n); auto. +Qed. + +Lemma Z_add_is_or: + forall i, 0 <= i -> + forall x y, + (forall j, 0 <= j <= i -> Z.testbit x j && Z.testbit y j = false) -> + Z.testbit (x + y) i = Z.testbit x i || Z.testbit y i. +Proof. + intros i0 POS0. pattern i0. apply Zlt_0_ind; auto. + intros i IND POS x y EXCL. + rewrite (Zdecomp x) in *. rewrite (Zdecomp y) in *. + transitivity (Z.testbit (Zshiftin (Z.odd x || Z.odd y) (Z.div2 x + Z.div2 y)) i). + - f_equal. rewrite !Zshiftin_spec. + exploit (EXCL 0). omega. rewrite !Ztestbit_shiftin_base. intros. +Opaque Z.mul. + destruct (Z.odd x); destruct (Z.odd y); simpl in *; discriminate || ring. + - rewrite !Ztestbit_shiftin; auto. + destruct (zeq i 0). + + auto. + + apply IND. omega. intros. + exploit (EXCL (Z.succ j)). omega. + rewrite !Ztestbit_shiftin_succ. auto. + omega. omega. +Qed. + +(** ** Zero and sign extensions *) + +(** 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. + +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 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. + +(** [Zzero_ext n x] is [x modulo 2^n] *) + +Lemma Zzero_ext_mod: + forall n x, 0 <= n -> Zzero_ext n x = x mod (two_p n). +Proof. + intros. apply equal_same_bits; intros. + rewrite Zzero_ext_spec, Ztestbit_mod_two_p by auto. auto. +Qed. + +(** [Zzero_ext n x] is the unique integer congruent to [x] modulo [2^n] in the range [0...2^n-1]. *) + +Lemma Zzero_ext_range: + forall n x, 0 <= n -> 0 <= Zzero_ext n x < two_p n. +Proof. + intros. rewrite Zzero_ext_mod; auto. apply Z_mod_lt. apply two_p_gt_ZERO. omega. +Qed. + +Lemma eqmod_Zzero_ext: + forall n x, 0 <= n -> eqmod (two_p n) (Zzero_ext n x) x. +Proof. + intros. rewrite Zzero_ext_mod; auto. apply eqmod_sym. apply eqmod_mod. + apply two_p_gt_ZERO. omega. +Qed. + +(** Relation between [Zsign_ext n x] and (Zzero_ext n x] *) + +Lemma Zsign_ext_zero_ext: + forall n, 0 < n -> forall x, + Zsign_ext n x = Zzero_ext n x - (if Z.testbit x (n - 1) then two_p n else 0). +Proof. + intros. apply equal_same_bits; intros. + rewrite Zsign_ext_spec by auto. + destruct (Z.testbit x (n - 1)) eqn:SIGNBIT. +- set (n' := - two_p n). + replace (Zzero_ext n x - two_p n) with (Zzero_ext n x + n') by (unfold n'; omega). + rewrite Z_add_is_or; auto. + rewrite Zzero_ext_spec by auto. unfold n'; rewrite Ztestbit_neg_two_p by omega. + destruct (zlt i n). rewrite orb_false_r; auto. auto. + intros. rewrite Zzero_ext_spec by omega. unfold n'; rewrite Ztestbit_neg_two_p by omega. + destruct (zlt j n); auto using andb_false_r. +- replace (Zzero_ext n x - 0) with (Zzero_ext n x) by omega. + rewrite Zzero_ext_spec by auto. + destruct (zlt i n); auto. +Qed. + +(** [Zsign_ext n x] is the unique integer congruent to [x] modulo [2^n] + in the range [-2^(n-1)...2^(n-1) - 1]. *) + +Lemma Zsign_ext_range: + forall n x, 0 < n -> -two_p (n-1) <= Zsign_ext n x < two_p (n-1). +Proof. + intros. + assert (A: 0 <= Zzero_ext n x < two_p n) by (apply Zzero_ext_range; omega). + assert (B: Z.testbit (Zzero_ext n x) (n - 1) = + if zlt (Zzero_ext n x) (two_p (n - 1)) then false else true). + { set (N := Z.to_nat (n - 1)). + generalize (Zsign_bit N (Zzero_ext n x)). + rewrite ! two_power_nat_two_p. + rewrite inj_S. unfold N; rewrite Z2Nat.id by omega. + intros X; apply X. replace (Z.succ (n - 1)) with n by omega. exact A. + } + assert (C: two_p n = 2 * two_p (n - 1)). + { rewrite <- two_p_S by omega. f_equal; omega. } + rewrite Zzero_ext_spec, zlt_true in B by omega. + rewrite Zsign_ext_zero_ext by auto. rewrite B. + destruct (zlt (Zzero_ext n x) (two_p (n - 1))); omega. +Qed. + +Lemma eqmod_Zsign_ext: + forall n x, 0 < n -> + eqmod (two_p n) (Zsign_ext n x) x. +Proof. + intros. rewrite Zsign_ext_zero_ext by auto. + apply eqmod_trans with (x - 0). + apply eqmod_sub. + apply eqmod_Zzero_ext; omega. + exists (if Z.testbit x (n - 1) then 1 else 0). destruct (Z.testbit x (n - 1)); ring. + apply eqmod_refl2; omega. +Qed. + +(** ** 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. + +Fixpoint powerserie (l: list Z): Z := + match l with + | nil => 0 + | x :: xs => two_p x + powerserie xs + end. + +Lemma Z_one_bits_powerserie: + forall n x, 0 <= x < two_power_nat n -> x = powerserie (Z_one_bits n 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 n x i, In i (Z_one_bits n x 0) -> 0 <= i < Z.of_nat n. +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 n x 0 i H0). 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. + +(** ** Relation between bitwise operations and multiplications / divisions by powers of 2 *) + +(** 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. + +(** Right shifts and 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. + +(** ** 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. + +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. + +(** ** Size of integers, in bits. *) + +Definition Zsize (x: Z) : Z := + match x with + | Zpos p => Zpos (Pos.size p) + | _ => 0 + end. + +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. diff --git a/powerpc/Asmgenproof1.v b/powerpc/Asmgenproof1.v index c18757b2..afbba882 100644 --- a/powerpc/Asmgenproof1.v +++ b/powerpc/Asmgenproof1.v @@ -16,6 +16,7 @@ Require Import Coqlib. Require Import Errors. Require Import Maps. Require Import AST. +Require Import Zbits. Require Import Integers. Require Import Floats. Require Import Values. @@ -80,13 +81,13 @@ Proof. unfold Int.modu, Int.zero. decEq. change 0 with (0 mod 65536). change (Int.unsigned (Int.repr 65536)) with 65536. - apply Int.eqmod_mod_eq. omega. - unfold x, low_s. eapply Int.eqmod_trans. - apply Int.eqmod_divides with Int.modulus. + apply eqmod_mod_eq. omega. + unfold x, low_s. eapply eqmod_trans. + apply eqmod_divides with Int.modulus. unfold Int.sub. apply Int.eqm_unsigned_repr_l. apply Int.eqm_refl. exists 65536. compute; auto. replace 0 with (Int.unsigned n - Int.unsigned n) by omega. - apply Int.eqmod_sub. apply Int.eqmod_refl. apply Int.eqmod_sign_ext'. + apply eqmod_sub. apply eqmod_refl. apply Int.eqmod_sign_ext'. compute; auto. rewrite H0 in H. rewrite Int.add_zero in H. rewrite <- H. unfold x. rewrite Int.sub_add_opp. rewrite Int.add_assoc. diff --git a/powerpc/SelectLongproof.v b/powerpc/SelectLongproof.v index a214d131..f16c967e 100644 --- a/powerpc/SelectLongproof.v +++ b/powerpc/SelectLongproof.v @@ -12,7 +12,7 @@ (** Correctness of instruction selection for 64-bit integer operations *) -Require Import String Coqlib Maps Integers Floats Errors. +Require Import String Coqlib Maps Zbits Integers Floats Errors. Require Archi. Require Import AST Values Memory Globalenvs Events. Require Import Cminor Op CminorSel. @@ -221,11 +221,11 @@ Proof. change (Int64.unsigned Int64.iwordsize) with 64. f_equal. rewrite Int.unsigned_repr. - apply Int.eqmod_mod_eq. omega. - apply Int.eqmod_trans with a. - apply Int.eqmod_divides with Int.modulus. apply Int.eqm_sym. apply Int.eqm_unsigned_repr. + apply eqmod_mod_eq. omega. + apply eqmod_trans with a. + apply eqmod_divides with Int.modulus. apply Int.eqm_sym. apply Int.eqm_unsigned_repr. exists (two_p (32-6)); auto. - apply Int.eqmod_divides with Int64.modulus. apply Int64.eqm_unsigned_repr. + apply eqmod_divides with Int64.modulus. apply Int64.eqm_unsigned_repr. exists (two_p (64-6)); auto. assert (0 <= Int.unsigned (Int.repr a) mod 64 < 64) by (apply Z_mod_lt; omega). assert (64 < Int.max_unsigned) by (compute; auto). diff --git a/riscV/Asmgenproof1.v b/riscV/Asmgenproof1.v index 7f070c12..98d5bd33 100644 --- a/riscV/Asmgenproof1.v +++ b/riscV/Asmgenproof1.v @@ -16,7 +16,7 @@ (* *********************************************************************) Require Import Coqlib Errors Maps. -Require Import AST Integers Floats Values Memory Globalenvs. +Require Import AST Zbits Integers Floats Values Memory Globalenvs. Require Import Op Locations Mach Conventions. Require Import Asm Asmgen Asmgenproof0. @@ -33,16 +33,16 @@ Proof. predSpec Int.eq Int.eq_spec n lo. - auto. - set (m := Int.sub n lo). - assert (A: Int.eqmod (two_p 12) (Int.unsigned lo) (Int.unsigned n)) by (apply Int.eqmod_sign_ext'; compute; auto). - assert (B: Int.eqmod (two_p 12) (Int.unsigned n - Int.unsigned lo) 0). + assert (A: eqmod (two_p 12) (Int.unsigned lo) (Int.unsigned n)) by (apply Int.eqmod_sign_ext'; compute; auto). + assert (B: eqmod (two_p 12) (Int.unsigned n - Int.unsigned lo) 0). { replace 0 with (Int.unsigned n - Int.unsigned n) by omega. - auto using Int.eqmod_sub, Int.eqmod_refl. } - assert (C: Int.eqmod (two_p 12) (Int.unsigned m) 0). - { apply Int.eqmod_trans with (Int.unsigned n - Int.unsigned lo); auto. - apply Int.eqmod_divides with Int.modulus. apply Int.eqm_sym; apply Int.eqm_unsigned_repr. + auto using eqmod_sub, eqmod_refl. } + assert (C: eqmod (two_p 12) (Int.unsigned m) 0). + { apply eqmod_trans with (Int.unsigned n - Int.unsigned lo); auto. + apply eqmod_divides with Int.modulus. apply Int.eqm_sym; apply Int.eqm_unsigned_repr. exists (two_p (32-12)); auto. } assert (D: Int.modu m (Int.repr 4096) = Int.zero). - { apply Int.eqmod_mod_eq in C. unfold Int.modu. + { apply eqmod_mod_eq in C. unfold Int.modu. change (Int.unsigned (Int.repr 4096)) with (two_p 12). rewrite C. reflexivity. apply two_p_gt_ZERO; omega. } diff --git a/riscV/SelectOpproof.v b/riscV/SelectOpproof.v index 90f077db..d12bd8af 100644 --- a/riscV/SelectOpproof.v +++ b/riscV/SelectOpproof.v @@ -20,6 +20,7 @@ Require Import Coqlib. Require Import Maps. Require Import AST. +Require Import Zbits. Require Import Integers. Require Import Floats. Require Import Values. @@ -372,7 +373,7 @@ Proof. change (Int.ltu (Int.repr 32) Int64.iwordsize') with true; simpl. apply Val.lessdef_same. f_equal. transitivity (Int.repr (Z.shiftr (Int.signed i * Int.signed i0) 32)). - unfold Int.mulhs; f_equal. rewrite Int.Zshiftr_div_two_p by omega. reflexivity. + unfold Int.mulhs; f_equal. rewrite Zshiftr_div_two_p by omega. reflexivity. apply Int.same_bits_eq; intros n N. change Int.zwordsize with 32 in *. assert (N1: 0 <= n < 64) by omega. @@ -400,7 +401,7 @@ Proof. change (Int.ltu (Int.repr 32) Int64.iwordsize') with true; simpl. apply Val.lessdef_same. f_equal. transitivity (Int.repr (Z.shiftr (Int.unsigned i * Int.unsigned i0) 32)). - unfold Int.mulhu; f_equal. rewrite Int.Zshiftr_div_two_p by omega. reflexivity. + unfold Int.mulhu; f_equal. rewrite Zshiftr_div_two_p by omega. reflexivity. apply Int.same_bits_eq; intros n N. change Int.zwordsize with 32 in *. assert (N1: 0 <= n < 64) by omega. |