Library CompOptCert.Integers
Formalizations of machine integers modulo 2N.
Inductive comparison : Type :=
| Ceq : comparison
| Cne : comparison
| Clt : comparison
| Cle : comparison
| Cgt : comparison
| Cge : comparison.
Definition negate_comparison (c: comparison): comparison :=
match c with
| Ceq ⇒ Cne
| Cne ⇒ Ceq
| Clt ⇒ Cge
| Cle ⇒ Cgt
| Cgt ⇒ Cle
| Cge ⇒ Clt
end.
Definition swap_comparison (c: comparison): comparison :=
match c with
| Ceq ⇒ Ceq
| Cne ⇒ Cne
| Clt ⇒ Cgt
| Cle ⇒ Cge
| Cgt ⇒ Clt
| Cge ⇒ Cle
end.
Module Type WORDSIZE.
Parameter wordsize: nat.
Axiom wordsize_not_zero: wordsize ≠ 0%nat.
End WORDSIZE.
Local Unset Elimination Schemes.
Local Unset Case Analysis Schemes.
Module Make(WS: WORDSIZE).
Definition wordsize: nat := WS.wordsize.
Definition zwordsize: Z := Z.of_nat wordsize.
Definition modulus : Z := two_power_nat wordsize.
Definition half_modulus : Z := modulus / 2.
Definition max_unsigned : Z := modulus - 1.
Definition max_signed : Z := half_modulus - 1.
Definition min_signed : Z := - half_modulus.
Remark wordsize_pos: zwordsize > 0.
Proof.
unfold zwordsize, wordsize. generalize WS.wordsize_not_zero. lia.
Qed.
Remark modulus_power: modulus = two_p zwordsize.
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; lia.
Qed.
Remark modulus_pos: modulus > 0.
Proof.
generalize modulus_gt_one; lia.
Qed.
Global Hint Resolve modulus_pos: ints.
Representation of machine integers
Fast normalization modulo 2^wordsize
Definition Z_mod_modulus (x: Z) : Z :=
match x with
| Z0 ⇒ 0
| Zpos p ⇒ P_mod_two_p p wordsize
| Zneg p ⇒ let r := P_mod_two_p p wordsize in if zeq r 0 then 0 else modulus - r
end.
Lemma Z_mod_modulus_range:
∀ x, 0 ≤ Z_mod_modulus x < modulus.
Proof (Z_mod_two_p_range wordsize).
Lemma Z_mod_modulus_range':
∀ x, -1 < Z_mod_modulus x < modulus.
Proof.
intros. generalize (Z_mod_modulus_range x); intuition.
Qed.
Lemma Z_mod_modulus_eq:
∀ x, Z_mod_modulus x = x mod modulus.
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
respectively.
Definition unsigned (n: int) : Z := intval n.
Definition signed (n: int) : Z :=
let x := unsigned n in
if zlt x half_modulus then x else x - modulus.
Conversely, repr takes a Coq integer and returns the corresponding
machine integer. The argument is treated modulo modulus.
Definition repr (x: Z) : int :=
mkint (Z_mod_modulus x) (Z_mod_modulus_range' x).
Definition zero := repr 0.
Definition one := repr 1.
Definition mone := repr (-1).
Definition iwordsize := repr zwordsize.
Lemma mkint_eq:
∀ x y Px Py, x = y → mkint x Px = mkint y Py.
Proof.
intros. subst y.
assert (∀ (n m: Z) (P1 P2: n < m), P1 = P2).
{
unfold Z.lt; intros.
apply eq_proofs_unicity.
intros c1 c2. destruct c1; destruct c2; (left; reflexivity) || (right; congruence).
}
destruct Px as [Px1 Px2]. destruct Py as [Py1 Py2].
rewrite (H _ _ Px1 Py1).
rewrite (H _ _ Px2 Py2).
reflexivity.
Qed.
Lemma eq_dec: ∀ (x y: int), {x = y} + {x ≠ y}.
Proof.
intros. destruct x; destruct y. destruct (zeq intval0 intval1).
left. apply mkint_eq. auto.
right. red; intro. injection H. exact n.
Defined.
Definition eq (x y: int) : bool :=
if zeq (unsigned x) (unsigned y) then true else false.
Definition lt (x y: int) : bool :=
if zlt (signed x) (signed y) then true else false.
Definition ltu (x y: int) : bool :=
if zlt (unsigned x) (unsigned y) then true else false.
Definition neg (x: int) : int := repr (- unsigned x).
Definition add (x y: int) : int :=
repr (unsigned x + unsigned y).
Definition sub (x y: int) : int :=
repr (unsigned x - unsigned y).
Definition mul (x y: int) : int :=
repr (unsigned x × unsigned y).
Definition divs (x y: int) : int :=
repr (Z.quot (signed x) (signed y)).
Definition mods (x y: int) : int :=
repr (Z.rem (signed x) (signed y)).
Definition divu (x y: int) : int :=
repr (unsigned x / unsigned y).
Definition modu (x y: int) : int :=
repr ((unsigned x) mod (unsigned y)).
Bitwise boolean operations.
Definition and (x y: int): int := repr (Z.land (unsigned x) (unsigned y)).
Definition or (x y: int): int := repr (Z.lor (unsigned x) (unsigned y)).
Definition xor (x y: int) : int := repr (Z.lxor (unsigned x) (unsigned y)).
Definition not (x: int) : int := xor x mone.
Shifts and rotates.
Definition shl (x y: int): int := repr (Z.shiftl (unsigned x) (unsigned y)).
Definition shru (x y: int): int := repr (Z.shiftr (unsigned x) (unsigned y)).
Definition shr (x y: int): int := repr (Z.shiftr (signed x) (unsigned y)).
Definition rol (x y: int) : int :=
let n := (unsigned y) mod zwordsize in
repr (Z.lor (Z.shiftl (unsigned x) n) (Z.shiftr (unsigned x) (zwordsize - n))).
Definition ror (x y: int) : int :=
let n := (unsigned y) mod zwordsize in
repr (Z.lor (Z.shiftr (unsigned x) n) (Z.shiftl (unsigned x) (zwordsize - n))).
Definition rolm (x a m: int): int := and (rol x a) m.
Viewed as signed divisions by powers of two, shrx rounds towards
zero, while shr rounds towards minus infinity.
High half of full multiply.
Definition mulhu (x y: int): int := repr ((unsigned x × unsigned y) / modulus).
Definition mulhs (x y: int): int := repr ((signed x × signed y) / modulus).
Condition flags
Definition negative (x: int): int :=
if lt x zero then one else zero.
Definition add_carry (x y cin: int): int :=
if zlt (unsigned x + unsigned y + unsigned cin) modulus then zero else one.
Definition add_overflow (x y cin: int): int :=
let s := signed x + signed y + signed cin in
if zle min_signed s && zle s max_signed then zero else one.
Definition sub_borrow (x y bin: int): int :=
if zlt (unsigned x - unsigned y - unsigned bin) 0 then one else zero.
Definition sub_overflow (x y bin: int): int :=
let s := signed x - signed y - signed bin in
if zle min_signed s && zle s max_signed then zero else one.
Definition shr_carry (x y: int) : int :=
if lt x zero && negb (eq (and x (sub (shl one y) one)) zero)
then one else zero.
Zero and sign extensions
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.
Recognition of powers of two.
Definition is_power2 (x: int) : option int :=
match Z_is_power2 (unsigned x) with
| Some i ⇒ Some (repr i)
| None ⇒ None
end.
Comparisons.
Definition cmp (c: comparison) (x y: int) : bool :=
match c with
| Ceq ⇒ eq x y
| Cne ⇒ negb (eq x y)
| Clt ⇒ lt x y
| Cle ⇒ negb (lt y x)
| Cgt ⇒ lt y x
| Cge ⇒ negb (lt x y)
end.
Definition cmpu (c: comparison) (x y: int) : bool :=
match c with
| Ceq ⇒ eq x y
| Cne ⇒ negb (eq x y)
| Clt ⇒ ltu x y
| Cle ⇒ negb (ltu y x)
| Cgt ⇒ ltu y x
| Cge ⇒ negb (ltu x y)
end.
Definition is_false (x: int) : Prop := x = zero.
Definition is_true (x: int) : Prop := x ≠ zero.
Definition notbool (x: int) : int := if eq x zero then one else zero.
x86-style extended division and modulus
Definition divmodu2 (nhi nlo: int) (d: int) : option (int × int) :=
if eq_dec d zero then None else
(let (q, r) := Z.div_eucl (unsigned nhi × modulus + unsigned nlo) (unsigned d) in
if zle q max_unsigned then Some(repr q, repr r) else None).
Definition divmods2 (nhi nlo: int) (d: int) : option (int × int) :=
if eq_dec d zero then None else
(let (q, r) := Z.quotrem (signed nhi × modulus + unsigned nlo) (signed d) in
if zle min_signed q && zle q max_signed then Some(repr q, repr r) else None).
Remark half_modulus_power:
half_modulus = two_p (zwordsize - 1).
Proof.
unfold half_modulus. rewrite modulus_power.
set (ws1 := zwordsize - 1).
replace (zwordsize) with (Z.succ ws1).
rewrite two_p_S. rewrite Z.mul_comm. apply Z_div_mult. lia.
unfold ws1. generalize wordsize_pos; lia.
unfold ws1. lia.
Qed.
Remark half_modulus_modulus: modulus = 2 × half_modulus.
Proof.
rewrite half_modulus_power. rewrite modulus_power.
rewrite <- two_p_S. apply f_equal. lia.
generalize wordsize_pos; lia.
Qed.
Relative positions, from greatest to smallest:
max_unsigned
max_signed
2*wordsize-1
wordsize
0
min_signed
Remark half_modulus_pos: half_modulus > 0.
Proof.
rewrite half_modulus_power. apply two_p_gt_ZERO. generalize wordsize_pos; lia.
Qed.
Remark min_signed_neg: min_signed < 0.
Proof.
unfold min_signed. generalize half_modulus_pos. lia.
Qed.
Remark max_signed_pos: max_signed ≥ 0.
Proof.
unfold max_signed. generalize half_modulus_pos. lia.
Qed.
Remark wordsize_max_unsigned: zwordsize ≤ max_unsigned.
Proof.
assert (zwordsize < modulus).
rewrite modulus_power. apply two_p_strict.
generalize wordsize_pos. lia.
unfold max_unsigned. lia.
Qed.
Remark two_wordsize_max_unsigned: 2 × zwordsize - 1 ≤ max_unsigned.
Proof.
assert (2 × zwordsize - 1 < modulus).
rewrite modulus_power. apply two_p_strict_2. generalize wordsize_pos; lia.
unfold max_unsigned; lia.
Qed.
Remark max_signed_unsigned: max_signed < max_unsigned.
Proof.
unfold max_signed, max_unsigned. rewrite half_modulus_modulus.
generalize half_modulus_pos. lia.
Qed.
Lemma unsigned_repr_eq:
∀ x, unsigned (repr x) = Z.modulo x modulus.
Proof.
intros. simpl. apply Z_mod_modulus_eq.
Qed.
Lemma signed_repr_eq:
∀ x, signed (repr x) = if zlt (Z.modulo x modulus) half_modulus then Z.modulo x modulus else Z.modulo x modulus - modulus.
Proof.
intros. unfold signed. rewrite unsigned_repr_eq. auto.
Qed.
Definition eqm := eqmod modulus.
Lemma eqm_refl: ∀ x, eqm x x.
Proof (eqmod_refl modulus).
Global Hint Resolve eqm_refl: ints.
Lemma eqm_refl2:
∀ x y, x = y → eqm x y.
Proof (eqmod_refl2 modulus).
Global Hint Resolve eqm_refl2: ints.
Lemma eqm_sym: ∀ x y, eqm x y → eqm y x.
Proof (eqmod_sym modulus).
Global Hint Resolve eqm_sym: ints.
Lemma eqm_trans: ∀ x y z, eqm x y → eqm y z → eqm x z.
Proof (eqmod_trans modulus).
Global Hint Resolve eqm_trans: ints.
Lemma eqm_small_eq:
∀ x y, eqm x y → 0 ≤ x < modulus → 0 ≤ y < modulus → x = y.
Proof (eqmod_small_eq modulus).
Global Hint Resolve eqm_small_eq: ints.
Lemma eqm_add:
∀ a b c d, eqm a b → eqm c d → eqm (a + c) (b + d).
Proof (eqmod_add modulus).
Global Hint Resolve eqm_add: ints.
Lemma eqm_neg:
∀ x y, eqm x y → eqm (-x) (-y).
Proof (eqmod_neg modulus).
Global Hint Resolve eqm_neg: ints.
Lemma eqm_sub:
∀ a b c d, eqm a b → eqm c d → eqm (a - c) (b - d).
Proof (eqmod_sub modulus).
Global Hint Resolve eqm_sub: ints.
Lemma eqm_mult:
∀ a b c d, eqm a c → eqm b d → eqm (a × b) (c × d).
Proof (eqmod_mult modulus).
Global Hint Resolve eqm_mult: ints.
Lemma eqm_same_bits:
∀ x y,
(∀ i, 0 ≤ i < zwordsize → Z.testbit x i = Z.testbit y i) →
eqm x y.
Proof (eqmod_same_bits wordsize).
Lemma same_bits_eqm:
∀ x y i,
eqm x y →
0 ≤ i < zwordsize →
Z.testbit x i = Z.testbit y i.
Proof (same_bits_eqmod wordsize).
Lemma eqm_samerepr: ∀ x y, eqm x y → repr x = repr y.
Proof.
intros. unfold repr. apply mkint_eq.
rewrite !Z_mod_modulus_eq. apply eqmod_mod_eq. auto with ints. exact H.
Qed.
Lemma eqm_unsigned_repr:
∀ z, eqm z (unsigned (repr z)).
Proof.
unfold eqm; intros. rewrite unsigned_repr_eq. apply eqmod_mod. auto with ints.
Qed.
Global Hint Resolve eqm_unsigned_repr: ints.
Lemma eqm_unsigned_repr_l:
∀ a b, eqm a b → eqm (unsigned (repr a)) b.
Proof.
intros. apply eqm_trans with a.
apply eqm_sym. apply eqm_unsigned_repr. auto.
Qed.
Global Hint Resolve eqm_unsigned_repr_l: ints.
Lemma eqm_unsigned_repr_r:
∀ a b, eqm a b → eqm a (unsigned (repr b)).
Proof.
intros. apply eqm_trans with b. auto.
apply eqm_unsigned_repr.
Qed.
Global Hint Resolve eqm_unsigned_repr_r: ints.
Lemma eqm_signed_unsigned:
∀ x, eqm (signed x) (unsigned x).
Proof.
intros; red. unfold signed. set (y := unsigned x).
case (zlt y half_modulus); intro.
apply eqmod_refl. red; ∃ (-1); ring.
Qed.
Theorem unsigned_range:
∀ i, 0 ≤ unsigned i < modulus.
Proof.
destruct i. simpl. lia.
Qed.
Global Hint Resolve unsigned_range: ints.
Theorem unsigned_range_2:
∀ i, 0 ≤ unsigned i ≤ max_unsigned.
Proof.
intro; unfold max_unsigned.
generalize (unsigned_range i). lia.
Qed.
Global Hint Resolve unsigned_range_2: ints.
Theorem signed_range:
∀ i, min_signed ≤ signed i ≤ max_signed.
Proof.
intros. unfold signed.
generalize (unsigned_range i). set (n := unsigned i). intros.
case (zlt n half_modulus); intro.
unfold max_signed. generalize min_signed_neg. lia.
unfold min_signed, max_signed.
rewrite half_modulus_modulus in ×. lia.
Qed.
Theorem repr_unsigned:
∀ i, repr (unsigned i) = i.
Proof.
destruct i; simpl. unfold repr. apply mkint_eq.
rewrite Z_mod_modulus_eq. apply Z.mod_small; lia.
Qed.
Global Hint Resolve repr_unsigned: ints.
Lemma repr_signed:
∀ i, repr (signed i) = i.
Proof.
intros. transitivity (repr (unsigned i)).
apply eqm_samerepr. apply eqm_signed_unsigned. auto with ints.
Qed.
Global Hint Resolve repr_signed: ints.
Opaque repr.
Lemma eqm_repr_eq: ∀ x y, eqm x (unsigned y) → repr x = y.
Proof.
intros. rewrite <- (repr_unsigned y). apply eqm_samerepr; auto.
Qed.
Theorem unsigned_repr:
∀ z, 0 ≤ z ≤ max_unsigned → unsigned (repr z) = z.
Proof.
intros. rewrite unsigned_repr_eq.
apply Z.mod_small. unfold max_unsigned in H. lia.
Qed.
Global Hint Resolve unsigned_repr: ints.
Theorem signed_repr:
∀ z, min_signed ≤ z ≤ max_signed → signed (repr z) = z.
Proof.
intros. unfold signed. destruct (zle 0 z).
replace (unsigned (repr z)) with z.
rewrite zlt_true. auto. unfold max_signed in H. lia.
symmetry. apply unsigned_repr. generalize max_signed_unsigned. lia.
pose (z' := z + modulus).
replace (repr z) with (repr z').
replace (unsigned (repr z')) with z'.
rewrite zlt_false. unfold z'. lia.
unfold z'. unfold min_signed in H.
rewrite half_modulus_modulus. lia.
symmetry. apply unsigned_repr.
unfold z', max_unsigned. unfold min_signed, max_signed in H.
rewrite half_modulus_modulus. lia.
apply eqm_samerepr. unfold z'; red. ∃ 1. lia.
Qed.
Theorem signed_eq_unsigned:
∀ x, unsigned x ≤ max_signed → signed x = unsigned x.
Proof.
intros. unfold signed. destruct (zlt (unsigned x) half_modulus).
auto. unfold max_signed in H. extlia.
Qed.
Theorem signed_positive:
∀ x, signed x ≥ 0 ↔ unsigned x ≤ max_signed.
Proof.
intros. unfold signed, max_signed.
generalize (unsigned_range x) half_modulus_modulus half_modulus_pos; intros.
destruct (zlt (unsigned x) half_modulus); lia.
Qed.
Theorem unsigned_zero: unsigned zero = 0.
Proof.
unfold zero; rewrite unsigned_repr_eq. apply Zmod_0_l.
Qed.
Theorem unsigned_one: unsigned one = 1.
Proof.
unfold one; rewrite unsigned_repr_eq. apply Z.mod_small. split. lia.
unfold modulus. replace wordsize with (S(Init.Nat.pred wordsize)).
rewrite two_power_nat_S. generalize (two_power_nat_pos (Init.Nat.pred wordsize)).
lia.
generalize wordsize_pos. unfold zwordsize. lia.
Qed.
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 Z.mod_small.
generalize modulus_pos. lia. lia.
Qed.
Theorem signed_zero: signed zero = 0.
Proof.
unfold signed. rewrite unsigned_zero. apply zlt_true. generalize half_modulus_pos; lia.
Qed.
Theorem signed_one: zwordsize > 1 → signed one = 1.
Proof.
intros. unfold signed. rewrite unsigned_one. apply zlt_true.
change 1 with (two_p 0). rewrite half_modulus_power. apply two_p_monotone_strict. lia.
Qed.
Theorem signed_mone: signed mone = -1.
Proof.
unfold signed. rewrite unsigned_mone.
rewrite zlt_false. lia.
rewrite half_modulus_modulus. generalize half_modulus_pos. lia.
Qed.
Theorem one_not_zero: one ≠ zero.
Proof.
assert (unsigned one ≠ unsigned zero).
rewrite unsigned_one; rewrite unsigned_zero; congruence.
congruence.
Qed.
Theorem unsigned_repr_wordsize:
unsigned iwordsize = zwordsize.
Proof.
unfold iwordsize; rewrite unsigned_repr_eq. apply Z.mod_small.
generalize wordsize_pos wordsize_max_unsigned; unfold max_unsigned; lia.
Qed.
Theorem eq_sym:
∀ x y, eq x y = eq y x.
Proof.
intros; unfold eq. case (zeq (unsigned x) (unsigned y)); intro.
rewrite e. rewrite zeq_true. auto.
rewrite zeq_false. auto. auto.
Qed.
Theorem eq_spec: ∀ (x y: int), if eq x y then x = y else x ≠ y.
Proof.
intros; unfold eq. case (eq_dec x y); intro.
subst y. rewrite zeq_true. auto.
rewrite zeq_false. auto.
destruct x; destruct y.
simpl. red; intro. elim n. apply mkint_eq. auto.
Qed.
Theorem eq_true: ∀ x, eq x x = true.
Proof.
intros. generalize (eq_spec x x); case (eq x x); intros; congruence.
Qed.
Theorem eq_false: ∀ x y, x ≠ y → eq x y = false.
Proof.
intros. generalize (eq_spec x y); case (eq x y); intros; congruence.
Qed.
Theorem same_if_eq: ∀ x y, eq x y = true → x = y.
Proof.
intros. generalize (eq_spec x y); rewrite H; auto.
Qed.
Theorem eq_signed:
∀ x y, eq x y = if zeq (signed x) (signed y) then true else false.
Proof.
intros. predSpec eq eq_spec x y.
subst x. rewrite zeq_true; auto.
destruct (zeq (signed x) (signed y)); auto.
elim H. rewrite <- (repr_signed x). rewrite <- (repr_signed y). congruence.
Qed.
Theorem add_unsigned: ∀ x y, add x y = repr (unsigned x + unsigned y).
Proof. intros; reflexivity.
Qed.
Theorem add_signed: ∀ x y, add x y = repr (signed x + signed y).
Proof.
intros. rewrite add_unsigned. apply eqm_samerepr.
apply eqm_add; apply eqm_sym; apply eqm_signed_unsigned.
Qed.
Theorem add_commut: ∀ x y, add x y = add y x.
Proof. intros; unfold add. decEq. lia. Qed.
Theorem add_zero: ∀ x, add x zero = x.
Proof.
intros. unfold add. rewrite unsigned_zero.
rewrite Z.add_0_r. apply repr_unsigned.
Qed.
Theorem add_zero_l: ∀ x, add zero x = x.
Proof.
intros. rewrite add_commut. apply add_zero.
Qed.
Theorem add_assoc: ∀ x y z, add (add x y) z = add x (add y z).
Proof.
intros; unfold add.
set (x' := unsigned x).
set (y' := unsigned y).
set (z' := unsigned z).
apply eqm_samerepr.
apply eqm_trans with ((x' + y') + z').
auto with ints.
rewrite <- Z.add_assoc. auto with ints.
Qed.
Theorem add_permut: ∀ x y z, add x (add y z) = add y (add x z).
Proof.
intros. rewrite (add_commut y z). rewrite <- add_assoc. apply add_commut.
Qed.
Theorem add_neg_zero: ∀ x, add x (neg x) = zero.
Proof.
intros; unfold add, neg, zero. apply eqm_samerepr.
replace 0 with (unsigned x + (- (unsigned x))).
auto with ints. lia.
Qed.
Theorem unsigned_add_carry:
∀ x y,
unsigned (add x y) = unsigned x + unsigned y - unsigned (add_carry x y zero) × modulus.
Proof.
intros.
unfold add, add_carry. rewrite unsigned_zero. rewrite Z.add_0_r.
rewrite unsigned_repr_eq.
generalize (unsigned_range x) (unsigned_range y). intros.
destruct (zlt (unsigned x + unsigned y) modulus).
rewrite unsigned_zero. apply Zmod_unique with 0. lia. lia.
rewrite unsigned_one. apply Zmod_unique with 1. lia. lia.
Qed.
Corollary unsigned_add_either:
∀ x y,
unsigned (add x y) = unsigned x + unsigned y
∨ unsigned (add x y) = unsigned x + unsigned y - modulus.
Proof.
intros. rewrite unsigned_add_carry. unfold add_carry.
rewrite unsigned_zero. rewrite Z.add_0_r.
destruct (zlt (unsigned x + unsigned y) modulus).
rewrite unsigned_zero. left; lia.
rewrite unsigned_one. right; lia.
Qed.
Theorem neg_repr: ∀ z, neg (repr z) = repr (-z).
Proof.
intros; unfold neg. apply eqm_samerepr. auto with ints.
Qed.
Theorem neg_zero: neg zero = zero.
Proof.
unfold neg. rewrite unsigned_zero. auto.
Qed.
Theorem neg_involutive: ∀ x, neg (neg x) = x.
Proof.
intros; unfold neg.
apply eqm_repr_eq. eapply eqm_trans. apply eqm_neg.
apply eqm_unsigned_repr_l. apply eqm_refl. apply eqm_refl2. lia.
Qed.
Theorem neg_add_distr: ∀ x y, neg(add x y) = add (neg x) (neg y).
Proof.
intros; unfold neg, add. apply eqm_samerepr.
apply eqm_trans with (- (unsigned x + unsigned y)).
auto with ints.
replace (- (unsigned x + unsigned y))
with ((- unsigned x) + (- unsigned y)).
auto with ints. lia.
Qed.
Theorem sub_zero_l: ∀ x, sub x zero = x.
Proof.
intros; unfold sub. rewrite unsigned_zero.
replace (unsigned x - 0) with (unsigned x) by lia. apply repr_unsigned.
Qed.
Theorem sub_zero_r: ∀ x, sub zero x = neg x.
Proof.
intros; unfold sub, neg. rewrite unsigned_zero. auto.
Qed.
Theorem sub_add_opp: ∀ x y, sub x y = add x (neg y).
Proof.
intros; unfold sub, add, neg. apply eqm_samerepr.
apply eqm_add; auto with ints.
Qed.
Theorem sub_idem: ∀ x, sub x x = zero.
Proof.
intros; unfold sub. unfold zero. decEq. lia.
Qed.
Theorem sub_add_l: ∀ x y z, sub (add x y) z = add (sub x z) y.
Proof.
intros. repeat rewrite sub_add_opp.
repeat rewrite add_assoc. decEq. apply add_commut.
Qed.
Theorem sub_add_r: ∀ x y z, sub x (add y z) = add (sub x z) (neg y).
Proof.
intros. repeat rewrite sub_add_opp.
rewrite neg_add_distr. rewrite add_permut. apply add_commut.
Qed.
Theorem sub_shifted:
∀ x y z,
sub (add x z) (add y z) = sub x y.
Proof.
intros. rewrite sub_add_opp. rewrite neg_add_distr.
rewrite add_assoc.
rewrite (add_commut (neg y) (neg z)).
rewrite <- (add_assoc z). rewrite add_neg_zero.
rewrite (add_commut zero). rewrite add_zero.
symmetry. apply sub_add_opp.
Qed.
Theorem sub_signed:
∀ x y, sub x y = repr (signed x - signed y).
Proof.
intros. unfold sub. apply eqm_samerepr.
apply eqm_sub; apply eqm_sym; apply eqm_signed_unsigned.
Qed.
Theorem unsigned_sub_borrow:
∀ x y,
unsigned (sub x y) = unsigned x - unsigned y + unsigned (sub_borrow x y zero) × modulus.
Proof.
intros.
unfold sub, sub_borrow. rewrite unsigned_zero. rewrite Z.sub_0_r.
rewrite unsigned_repr_eq.
generalize (unsigned_range x) (unsigned_range y). intros.
destruct (zlt (unsigned x - unsigned y) 0).
rewrite unsigned_one. apply Zmod_unique with (-1). lia. lia.
rewrite unsigned_zero. apply Zmod_unique with 0. lia. lia.
Qed.
Theorem mul_commut: ∀ x y, mul x y = mul y x.
Proof.
intros; unfold mul. decEq. ring.
Qed.
Theorem mul_zero: ∀ x, mul x zero = zero.
Proof.
intros; unfold mul. rewrite unsigned_zero.
unfold zero. decEq. ring.
Qed.
Theorem mul_one: ∀ x, mul x one = x.
Proof.
intros; unfold mul. rewrite unsigned_one.
transitivity (repr (unsigned x)). decEq. ring.
apply repr_unsigned.
Qed.
Theorem mul_mone: ∀ x, mul x mone = neg x.
Proof.
intros; unfold mul, neg. rewrite unsigned_mone.
apply eqm_samerepr.
replace (-unsigned x) with (0 - unsigned x) by lia.
replace (unsigned x × (modulus - 1)) with (unsigned x × modulus - unsigned x) by ring.
apply eqm_sub. ∃ (unsigned x). lia. apply eqm_refl.
Qed.
Theorem mul_assoc: ∀ x y z, mul (mul x y) z = mul x (mul y z).
Proof.
intros; unfold mul.
set (x' := unsigned x).
set (y' := unsigned y).
set (z' := unsigned z).
apply eqm_samerepr. apply eqm_trans with ((x' × y') × z').
auto with ints.
rewrite <- Z.mul_assoc. auto with ints.
Qed.
Theorem mul_add_distr_l:
∀ x y z, mul (add x y) z = add (mul x z) (mul y z).
Proof.
intros; unfold mul, add.
apply eqm_samerepr.
set (x' := unsigned x).
set (y' := unsigned y).
set (z' := unsigned z).
apply eqm_trans with ((x' + y') × z').
auto with ints.
replace ((x' + y') × z') with (x' × z' + y' × z').
auto with ints.
ring.
Qed.
Theorem mul_add_distr_r:
∀ x y z, mul x (add y z) = add (mul x y) (mul x z).
Proof.
intros. rewrite mul_commut. rewrite mul_add_distr_l.
decEq; apply mul_commut.
Qed.
Theorem neg_mul_distr_l:
∀ x y, neg(mul x y) = mul (neg x) y.
Proof.
intros. unfold mul, neg.
set (x' := unsigned x). set (y' := unsigned y).
apply eqm_samerepr. apply eqm_trans with (- (x' × y')).
auto with ints.
replace (- (x' × y')) with ((-x') × y') by ring.
auto with ints.
Qed.
Theorem neg_mul_distr_r:
∀ x y, neg(mul x y) = mul x (neg y).
Proof.
intros. rewrite (mul_commut x y). rewrite (mul_commut x (neg y)).
apply neg_mul_distr_l.
Qed.
Theorem mul_signed:
∀ x y, mul x y = repr (signed x × signed y).
Proof.
intros; unfold mul. apply eqm_samerepr.
apply eqm_mult; apply eqm_sym; apply eqm_signed_unsigned.
Qed.
Lemma modu_divu_Euclid:
∀ x y, y ≠ zero → x = add (mul (divu x y) y) (modu x y).
Proof.
intros. unfold add, mul, divu, modu.
transitivity (repr (unsigned x)). auto with ints.
apply eqm_samerepr.
set (x' := unsigned x). set (y' := unsigned y).
apply eqm_trans with ((x' / y') × y' + x' mod y').
apply eqm_refl2. rewrite Z.mul_comm. apply Z_div_mod_eq.
generalize (unsigned_range y); intro.
assert (unsigned y ≠ 0). red; intro.
elim H. rewrite <- (repr_unsigned y). unfold zero. congruence.
unfold y'. lia.
auto with ints.
Qed.
Theorem modu_divu:
∀ x y, y ≠ zero → modu x y = sub x (mul (divu x y) y).
Proof.
intros.
assert (∀ a b c, a = add b c → c = sub a b).
intros. subst a. rewrite sub_add_l. rewrite sub_idem.
rewrite add_commut. rewrite add_zero. auto.
apply H0. apply modu_divu_Euclid. auto.
Qed.
Lemma mods_divs_Euclid:
∀ x y, x = add (mul (divs x y) y) (mods x y).
Proof.
intros. unfold add, mul, divs, mods.
transitivity (repr (signed x)). auto with ints.
apply eqm_samerepr.
set (x' := signed x). set (y' := signed y).
apply eqm_trans with ((Z.quot x' y') × y' + Z.rem x' y').
apply eqm_refl2. rewrite Z.mul_comm. apply Z.quot_rem'.
apply eqm_add; auto with ints.
apply eqm_unsigned_repr_r. apply eqm_mult; auto with ints.
unfold y'. apply eqm_signed_unsigned.
Qed.
Theorem mods_divs:
∀ x y, mods x y = sub x (mul (divs x y) y).
Proof.
intros.
assert (∀ a b c, a = add b c → c = sub a b).
intros. subst a. rewrite sub_add_l. rewrite sub_idem.
rewrite add_commut. rewrite add_zero. auto.
apply H. apply mods_divs_Euclid.
Qed.
Theorem divu_one:
∀ x, divu x one = x.
Proof.
unfold divu; intros. rewrite unsigned_one. rewrite Zdiv_1_r. apply repr_unsigned.
Qed.
Theorem divs_one:
∀ x, zwordsize > 1 → divs x one = x.
Proof.
unfold divs; intros. rewrite signed_one. rewrite Z.quot_1_r. apply repr_signed. auto.
Qed.
Theorem modu_one:
∀ x, modu x one = zero.
Proof.
intros. rewrite modu_divu. rewrite divu_one. rewrite mul_one. apply sub_idem.
apply one_not_zero.
Qed.
Theorem divs_mone:
∀ x, divs x mone = neg x.
Proof.
unfold divs, neg; intros.
rewrite signed_mone.
replace (Z.quot (signed x) (-1)) with (- (signed x)).
apply eqm_samerepr. apply eqm_neg. apply eqm_signed_unsigned.
set (x' := signed x).
set (one := 1).
change (-1) with (- one). rewrite Zquot_opp_r.
assert (Z.quot x' one = x').
symmetry. apply Zquot_unique_full with 0. red.
change (Z.abs one) with 1.
destruct (zle 0 x'). left. lia. right. lia.
unfold one; ring.
congruence.
Qed.
Theorem mods_mone:
∀ x, mods x mone = zero.
Proof.
intros. rewrite mods_divs. rewrite divs_mone.
rewrite <- neg_mul_distr_l. rewrite mul_mone. rewrite neg_involutive. apply sub_idem.
Qed.
Theorem divmodu2_divu_modu:
∀ n d,
d ≠ zero → divmodu2 zero n d = Some (divu n d, modu n d).
Proof.
unfold divmodu2, divu, modu; intros.
rewrite dec_eq_false by auto.
set (N := unsigned zero × modulus + unsigned n).
assert (E1: unsigned n = N) by (unfold N; rewrite unsigned_zero; ring). rewrite ! E1.
set (D := unsigned d).
set (Q := N / D); set (R := N mod D).
assert (E2: Z.div_eucl N D = (Q, R)).
{ unfold Q, R, Z.div, Z.modulo. destruct (Z.div_eucl N D); auto. }
rewrite E2. rewrite zle_true. auto.
assert (unsigned d ≠ 0).
{ red; intros. elim H. rewrite <- (repr_unsigned d). rewrite H0; auto. }
assert (0 < D).
{ unfold D. generalize (unsigned_range d); intros. lia. }
assert (0 ≤ Q ≤ max_unsigned).
{ unfold Q. apply Zdiv_interval_2.
rewrite <- E1; apply unsigned_range_2.
lia. unfold max_unsigned; generalize modulus_pos; lia. lia. }
lia.
Qed.
Lemma unsigned_signed:
∀ n, unsigned n = if lt n zero then signed n + modulus else signed n.
Proof.
intros. unfold lt. rewrite signed_zero. unfold signed.
generalize (unsigned_range n). rewrite half_modulus_modulus. intros.
destruct (zlt (unsigned n) half_modulus).
- rewrite zlt_false by lia. auto.
- rewrite zlt_true by lia. ring.
Qed.
Theorem divmods2_divs_mods:
∀ n d,
d ≠ zero → n ≠ repr min_signed ∨ d ≠ mone →
divmods2 (if lt n zero then mone else zero) n d = Some (divs n d, mods n d).
Proof.
unfold divmods2, divs, mods; intros.
rewrite dec_eq_false by auto.
set (N := signed (if lt n zero then mone else zero) × modulus + unsigned n).
set (D := signed d).
assert (D ≠ 0).
{ unfold D; red; intros. elim H. rewrite <- (repr_signed d). rewrite H1; auto. }
assert (N = signed n).
{ unfold N. rewrite unsigned_signed. destruct (lt n zero).
rewrite signed_mone. ring.
rewrite signed_zero. ring. }
set (Q := Z.quot N D); set (R := Z.rem N D).
assert (E2: Z.quotrem N D = (Q, R)).
{ unfold Q, R, Z.quot, Z.rem. destruct (Z.quotrem N D); auto. }
rewrite E2.
assert (min_signed ≤ N ≤ max_signed) by (rewrite H2; apply signed_range).
assert (min_signed ≤ Q ≤ max_signed).
{ unfold Q. destruct (zeq D 1); [ | destruct (zeq D (-1))].
-
rewrite e. rewrite Z.quot_1_r; auto.
-
rewrite e. change (-1) with (Z.opp 1). rewrite Z.quot_opp_r by lia.
rewrite Z.quot_1_r.
assert (N ≠ min_signed).
{ red; intros; destruct H0.
+ elim H0. rewrite <- (repr_signed n). rewrite <- H2. rewrite H4. auto.
+ elim H0. rewrite <- (repr_signed d). unfold D in e; rewrite e; auto. }
unfold min_signed, max_signed in ×. lia.
-
assert (Z.abs (Z.quot N D) < half_modulus).
{ rewrite <- Z.quot_abs by lia. apply Zquot_lt_upper_bound.
extlia. extlia.
apply Z.le_lt_trans with (half_modulus × 1).
rewrite Z.mul_1_r. unfold min_signed, max_signed in H3; extlia.
apply Zmult_lt_compat_l. generalize half_modulus_pos; lia. extlia. }
rewrite Z.abs_lt in H4.
unfold min_signed, max_signed; lia.
}
unfold proj_sumbool; rewrite ! zle_true by lia; simpl.
unfold Q, R; rewrite H2; auto.
Qed.
Definition testbit (x: int) (i: Z) : bool := Z.testbit (unsigned x) i.
Lemma testbit_repr:
∀ x i,
0 ≤ i < zwordsize →
testbit (repr x) i = Z.testbit x i.
Proof.
intros. unfold testbit. apply same_bits_eqm; auto with ints.
Qed.
Lemma same_bits_eq:
∀ x y,
(∀ i, 0 ≤ i < zwordsize → testbit x i = testbit y i) →
x = y.
Proof.
intros. rewrite <- (repr_unsigned x). rewrite <- (repr_unsigned y).
apply eqm_samerepr. apply eqm_same_bits. auto.
Qed.
Lemma bits_above:
∀ x i, i ≥ zwordsize → testbit x i = false.
Proof.
intros. apply Ztestbit_above with wordsize; auto. apply unsigned_range.
Qed.
Lemma bits_below:
∀ x i, i < 0 → testbit x i = false.
Proof.
intros. apply Z.testbit_neg_r; auto.
Qed.
Lemma bits_zero:
∀ i, testbit zero i = false.
Proof.
intros. unfold testbit. rewrite unsigned_zero. apply Ztestbit_0.
Qed.
Remark bits_one: ∀ n, testbit one n = zeq n 0.
Proof.
unfold testbit; intros. rewrite unsigned_one. apply Ztestbit_1.
Qed.
Lemma bits_mone:
∀ i, 0 ≤ i < zwordsize → testbit mone i = true.
Proof.
intros. unfold mone. rewrite testbit_repr; auto. apply Ztestbit_m1. lia.
Qed.
Hint Rewrite bits_zero bits_mone : ints.
Ltac bit_solve :=
intros; apply same_bits_eq; intros; autorewrite with ints; auto with bool.
Lemma sign_bit_of_unsigned:
∀ x, testbit x (zwordsize - 1) = if zlt (unsigned x) half_modulus then false else true.
Proof.
intros. unfold testbit.
set (ws1 := Init.Nat.pred wordsize).
assert (zwordsize - 1 = Z.of_nat ws1).
unfold zwordsize, ws1, wordsize.
destruct WS.wordsize as [] eqn:E.
elim WS.wordsize_not_zero; auto.
rewrite Nat2Z.inj_succ. simpl. lia.
assert (half_modulus = two_power_nat ws1).
rewrite two_power_nat_two_p. rewrite <- H. apply half_modulus_power.
rewrite H; rewrite H0.
apply Zsign_bit. rewrite two_power_nat_S. rewrite <- H0.
rewrite <- half_modulus_modulus. apply unsigned_range.
Qed.
Lemma bits_signed:
∀ x i, 0 ≤ i →
Z.testbit (signed x) i = testbit x (if zlt i zwordsize then i else zwordsize - 1).
Proof.
intros.
destruct (zlt i zwordsize).
- apply same_bits_eqm. apply eqm_signed_unsigned. lia.
- unfold signed. rewrite sign_bit_of_unsigned. destruct (zlt (unsigned x) half_modulus).
+ apply Ztestbit_above with wordsize. apply unsigned_range. auto.
+ apply Ztestbit_above_neg with wordsize.
fold modulus. generalize (unsigned_range x). lia. auto.
Qed.
Lemma bits_le:
∀ x y,
(∀ i, 0 ≤ i < zwordsize → testbit x i = true → testbit y i = true) →
unsigned x ≤ unsigned y.
Proof.
intros. apply Ztestbit_le. generalize (unsigned_range y); lia.
intros. fold (testbit y i). destruct (zlt i zwordsize).
apply H. lia. auto.
fold (testbit x i) in H1. rewrite bits_above in H1; auto. congruence.
Qed.
Lemma bits_and:
∀ x y i, 0 ≤ i < zwordsize →
testbit (and x y) i = testbit x i && testbit y i.
Proof.
intros. unfold and. rewrite testbit_repr; auto. rewrite Z.land_spec; intuition.
Qed.
Lemma bits_or:
∀ x y i, 0 ≤ i < zwordsize →
testbit (or x y) i = testbit x i || testbit y i.
Proof.
intros. unfold or. rewrite testbit_repr; auto. rewrite Z.lor_spec; intuition.
Qed.
Lemma bits_xor:
∀ x y i, 0 ≤ i < zwordsize →
testbit (xor x y) i = xorb (testbit x i) (testbit y i).
Proof.
intros. unfold xor. rewrite testbit_repr; auto. rewrite Z.lxor_spec; intuition.
Qed.
Lemma bits_not:
∀ x i, 0 ≤ i < zwordsize →
testbit (not x) i = negb (testbit x i).
Proof.
intros. unfold not. rewrite bits_xor; auto. rewrite bits_mone; auto.
Qed.
Hint Rewrite bits_and bits_or bits_xor bits_not: ints.
Theorem and_commut: ∀ x y, and x y = and y x.
Proof.
bit_solve.
Qed.
Theorem and_assoc: ∀ x y z, and (and x y) z = and x (and y z).
Proof.
bit_solve.
Qed.
Theorem and_zero: ∀ x, and x zero = zero.
Proof.
bit_solve. apply andb_b_false.
Qed.
Corollary and_zero_l: ∀ x, and zero x = zero.
Proof.
intros. rewrite and_commut. apply and_zero.
Qed.
Theorem and_mone: ∀ x, and x mone = x.
Proof.
bit_solve. apply andb_b_true.
Qed.
Corollary and_mone_l: ∀ x, and mone x = x.
Proof.
intros. rewrite and_commut. apply and_mone.
Qed.
Theorem and_idem: ∀ x, and x x = x.
Proof.
bit_solve. destruct (testbit x i); auto.
Qed.
Theorem or_commut: ∀ x y, or x y = or y x.
Proof.
bit_solve.
Qed.
Theorem or_assoc: ∀ x y z, or (or x y) z = or x (or y z).
Proof.
bit_solve.
Qed.
Theorem or_zero: ∀ x, or x zero = x.
Proof.
bit_solve.
Qed.
Corollary or_zero_l: ∀ x, or zero x = x.
Proof.
intros. rewrite or_commut. apply or_zero.
Qed.
Theorem or_mone: ∀ x, or x mone = mone.
Proof.
bit_solve.
Qed.
Theorem or_idem: ∀ x, or x x = x.
Proof.
bit_solve. destruct (testbit x i); auto.
Qed.
Theorem and_or_distrib:
∀ x y z,
and x (or y z) = or (and x y) (and x z).
Proof.
bit_solve. apply demorgan1.
Qed.
Corollary and_or_distrib_l:
∀ x y z,
and (or x y) z = or (and x z) (and y z).
Proof.
intros. rewrite (and_commut (or x y)). rewrite and_or_distrib. f_equal; apply and_commut.
Qed.
Theorem or_and_distrib:
∀ x y z,
or x (and y z) = and (or x y) (or x z).
Proof.
bit_solve. apply orb_andb_distrib_r.
Qed.
Corollary or_and_distrib_l:
∀ x y z,
or (and x y) z = and (or x z) (or y z).
Proof.
intros. rewrite (or_commut (and x y)). rewrite or_and_distrib. f_equal; apply or_commut.
Qed.
Theorem and_or_absorb: ∀ x y, and x (or x y) = x.
Proof.
bit_solve.
assert (∀ a b, a && (a || b) = a) by destr_bool.
auto.
Qed.
Theorem or_and_absorb: ∀ x y, or x (and x y) = x.
Proof.
bit_solve.
assert (∀ a b, a || (a && b) = a) by destr_bool.
auto.
Qed.
Theorem xor_commut: ∀ x y, xor x y = xor y x.
Proof.
bit_solve. apply xorb_comm.
Qed.
Theorem xor_assoc: ∀ x y z, xor (xor x y) z = xor x (xor y z).
Proof.
bit_solve. apply xorb_assoc.
Qed.
Theorem xor_zero: ∀ x, xor x zero = x.
Proof.
bit_solve. apply xorb_false.
Qed.
Corollary xor_zero_l: ∀ x, xor zero x = x.
Proof.
intros. rewrite xor_commut. apply xor_zero.
Qed.
Theorem xor_idem: ∀ x, xor x x = zero.
Proof.
bit_solve. apply xorb_nilpotent.
Qed.
Theorem xor_zero_one: xor zero one = one.
Proof. rewrite xor_commut. apply xor_zero. Qed.
Theorem xor_one_one: xor one one = zero.
Proof. apply xor_idem. Qed.
Theorem xor_zero_equal: ∀ x y, xor x y = zero → x = y.
Proof.
intros. apply same_bits_eq; intros.
assert (xorb (testbit x i) (testbit y i) = false).
rewrite <- bits_xor; auto. rewrite H. apply bits_zero.
destruct (testbit x i); destruct (testbit y i); reflexivity || discriminate.
Qed.
Theorem xor_is_zero: ∀ x y, eq (xor x y) zero = eq x y.
Proof.
intros. predSpec eq eq_spec (xor x y) zero.
- apply xor_zero_equal in H. subst y. rewrite eq_true; auto.
- predSpec eq eq_spec x y.
+ elim H; subst y; apply xor_idem.
+ auto.
Qed.
Theorem and_xor_distrib:
∀ x y z,
and x (xor y z) = xor (and x y) (and x z).
Proof.
bit_solve.
assert (∀ a b c, a && (xorb b c) = xorb (a && b) (a && c)) by destr_bool.
auto.
Qed.
Theorem and_le:
∀ x y, unsigned (and x y) ≤ unsigned x.
Proof.
intros. apply bits_le; intros.
rewrite bits_and in H0; auto. rewrite andb_true_iff in H0. tauto.
Qed.
Theorem or_le:
∀ x y, unsigned x ≤ unsigned (or x y).
Proof.
intros. apply bits_le; intros.
rewrite bits_or; auto. rewrite H0; auto.
Qed.
Theorem not_involutive:
∀ (x: int), not (not x) = x.
Proof.
intros. unfold not. rewrite xor_assoc. rewrite xor_idem. apply xor_zero.
Qed.
Theorem not_zero:
not zero = mone.
Proof.
unfold not. rewrite xor_commut. apply xor_zero.
Qed.
Theorem not_mone:
not mone = zero.
Proof.
rewrite <- (not_involutive zero). symmetry. decEq. apply not_zero.
Qed.
Theorem not_or_and_not:
∀ x y, not (or x y) = and (not x) (not y).
Proof.
bit_solve. apply negb_orb.
Qed.
Theorem not_and_or_not:
∀ x y, not (and x y) = or (not x) (not y).
Proof.
bit_solve. apply negb_andb.
Qed.
Theorem and_not_self:
∀ x, and x (not x) = zero.
Proof.
bit_solve.
Qed.
Theorem or_not_self:
∀ x, or x (not x) = mone.
Proof.
bit_solve.
Qed.
Theorem xor_not_self:
∀ x, xor x (not x) = mone.
Proof.
bit_solve. destruct (testbit x i); auto.
Qed.
Lemma unsigned_not:
∀ x, unsigned (not x) = max_unsigned - unsigned x.
Proof.
intros. transitivity (unsigned (repr(-unsigned x - 1))).
f_equal. bit_solve. rewrite testbit_repr; auto. symmetry. apply Z_one_complement. lia.
rewrite unsigned_repr_eq. apply Zmod_unique with (-1).
unfold max_unsigned. lia.
generalize (unsigned_range x). unfold max_unsigned. lia.
Qed.
Theorem not_neg:
∀ x, not x = add (neg x) mone.
Proof.
bit_solve.
rewrite <- (repr_unsigned x) at 1. unfold add.
rewrite !testbit_repr; auto.
transitivity (Z.testbit (-unsigned x - 1) i).
symmetry. apply Z_one_complement. lia.
apply same_bits_eqm; auto.
replace (-unsigned x - 1) with (-unsigned x + (-1)) by lia.
apply eqm_add.
unfold neg. apply eqm_unsigned_repr.
rewrite unsigned_mone. ∃ (-1). ring.
Qed.
Theorem neg_not:
∀ x, neg x = add (not x) one.
Proof.
intros. rewrite not_neg. rewrite add_assoc.
replace (add mone one) with zero. rewrite add_zero. auto.
apply eqm_samerepr. rewrite unsigned_mone. rewrite unsigned_one.
∃ (-1). ring.
Qed.
Theorem sub_add_not:
∀ x y, sub x y = add (add x (not y)) one.
Proof.
intros. rewrite sub_add_opp. rewrite neg_not.
rewrite ! add_assoc. auto.
Qed.
Theorem sub_add_not_3:
∀ x y b,
b = zero ∨ b = one →
sub (sub x y) b = add (add x (not y)) (xor b one).
Proof.
intros. rewrite ! sub_add_not. rewrite ! add_assoc. f_equal. f_equal.
rewrite <- neg_not. rewrite <- sub_add_opp. destruct H; subst b.
rewrite xor_zero_l. rewrite sub_zero_l. auto.
rewrite xor_idem. rewrite sub_idem. auto.
Qed.
Theorem sub_borrow_add_carry:
∀ x y b,
b = zero ∨ b = one →
sub_borrow x y b = xor (add_carry x (not y) (xor b one)) one.
Proof.
intros. unfold sub_borrow, add_carry. rewrite unsigned_not.
replace (unsigned (xor b one)) with (1 - unsigned b).
destruct (zlt (unsigned x - unsigned y - unsigned b)).
rewrite zlt_true. rewrite xor_zero_l; auto.
unfold max_unsigned; lia.
rewrite zlt_false. rewrite xor_idem; auto.
unfold max_unsigned; lia.
destruct H; subst b.
rewrite xor_zero_l. rewrite unsigned_one, unsigned_zero; auto.
rewrite xor_idem. rewrite unsigned_one, unsigned_zero; auto.
Qed.
Connections between add and bitwise logical operations.
Lemma Z_add_is_or:
∀ i, 0 ≤ i →
∀ x y,
(∀ 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). lia. 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. lia. intros.
exploit (EXCL (Z.succ j)). lia.
rewrite !Ztestbit_shiftin_succ. auto.
lia. lia.
Qed.
Theorem add_is_or:
∀ x y,
and x y = zero →
add x y = or x y.
Proof.
bit_solve. unfold add. rewrite testbit_repr; auto.
apply Z_add_is_or. lia.
intros.
assert (testbit (and x y) j = testbit zero j) by congruence.
autorewrite with ints in H2. assumption. lia.
Qed.
Theorem xor_is_or:
∀ x y, and x y = zero → xor x y = or x y.
Proof.
bit_solve.
assert (testbit (and x y) i = testbit zero i) by congruence.
autorewrite with ints in H1; auto.
destruct (testbit x i); destruct (testbit y i); simpl in *; congruence.
Qed.
Theorem add_is_xor:
∀ x y,
and x y = zero →
add x y = xor x y.
Proof.
intros. rewrite xor_is_or; auto. apply add_is_or; auto.
Qed.
Theorem add_and:
∀ x y z,
and y z = zero →
add (and x y) (and x z) = and x (or y z).
Proof.
intros. rewrite add_is_or.
rewrite and_or_distrib; auto.
rewrite (and_commut x y).
rewrite and_assoc.
repeat rewrite <- (and_assoc x).
rewrite (and_commut (and x x)).
rewrite <- and_assoc.
rewrite H. rewrite and_commut. apply and_zero.
Qed.
Lemma bits_shl:
∀ x y i,
0 ≤ i < zwordsize →
testbit (shl x y) i =
if zlt i (unsigned y) then false else testbit x (i - unsigned y).
Proof.
intros. unfold shl. rewrite testbit_repr; auto.
destruct (zlt i (unsigned y)).
apply Z.shiftl_spec_low. auto.
apply Z.shiftl_spec_high. lia. lia.
Qed.
Lemma bits_shru:
∀ x y i,
0 ≤ i < zwordsize →
testbit (shru x y) i =
if zlt (i + unsigned y) zwordsize then testbit x (i + unsigned y) else false.
Proof.
intros. unfold shru. rewrite testbit_repr; auto.
rewrite Z.shiftr_spec. fold (testbit x (i + unsigned y)).
destruct (zlt (i + unsigned y) zwordsize).
auto.
apply bits_above; auto.
lia.
Qed.
Lemma bits_shr:
∀ x y i,
0 ≤ i < zwordsize →
testbit (shr x y) i =
testbit x (if zlt (i + unsigned y) zwordsize then i + unsigned y else zwordsize - 1).
Proof.
intros. unfold shr. rewrite testbit_repr; auto.
rewrite Z.shiftr_spec. apply bits_signed.
generalize (unsigned_range y); lia.
lia.
Qed.
Hint Rewrite bits_shl bits_shru bits_shr: ints.
Theorem shl_zero: ∀ x, shl x zero = x.
Proof.
bit_solve. rewrite unsigned_zero. rewrite zlt_false. f_equal; lia. lia.
Qed.
Lemma bitwise_binop_shl:
∀ f f' x y n,
(∀ x y i, 0 ≤ i < zwordsize → testbit (f x y) i = f' (testbit x i) (testbit y i)) →
f' false false = false →
f (shl x n) (shl y n) = shl (f x y) n.
Proof.
intros. apply same_bits_eq; intros.
rewrite H; auto. rewrite !bits_shl; auto.
destruct (zlt i (unsigned n)); auto.
rewrite H; auto. generalize (unsigned_range n); lia.
Qed.
Theorem and_shl:
∀ x y n,
and (shl x n) (shl y n) = shl (and x y) n.
Proof.
intros. apply bitwise_binop_shl with andb. exact bits_and. auto.
Qed.
Theorem or_shl:
∀ x y n,
or (shl x n) (shl y n) = shl (or x y) n.
Proof.
intros. apply bitwise_binop_shl with orb. exact bits_or. auto.
Qed.
Theorem xor_shl:
∀ x y n,
xor (shl x n) (shl y n) = shl (xor x y) n.
Proof.
intros. apply bitwise_binop_shl with xorb. exact bits_xor. auto.
Qed.
Lemma ltu_inv:
∀ x y, ltu x y = true → 0 ≤ unsigned x < unsigned y.
Proof.
unfold ltu; intros. destruct (zlt (unsigned x) (unsigned y)).
split; auto. generalize (unsigned_range x); lia.
discriminate.
Qed.
Lemma ltu_iwordsize_inv:
∀ x, ltu x iwordsize = true → 0 ≤ unsigned x < zwordsize.
Proof.
intros. generalize (ltu_inv _ _ H). rewrite unsigned_repr_wordsize. auto.
Qed.
Theorem shl_shl:
∀ x y z,
ltu y iwordsize = true →
ltu z iwordsize = true →
ltu (add y z) iwordsize = true →
shl (shl x y) z = shl x (add y z).
Proof.
intros.
generalize (ltu_iwordsize_inv _ H) (ltu_iwordsize_inv _ H0); intros.
assert (unsigned (add y z) = unsigned y + unsigned z).
unfold add. apply unsigned_repr.
generalize two_wordsize_max_unsigned; lia.
apply same_bits_eq; intros.
rewrite bits_shl; auto.
destruct (zlt i (unsigned z)).
- rewrite bits_shl; auto. rewrite zlt_true. auto. lia.
- rewrite bits_shl. destruct (zlt (i - unsigned z) (unsigned y)).
+ rewrite bits_shl; auto. rewrite zlt_true. auto. lia.
+ rewrite bits_shl; auto. rewrite zlt_false. f_equal. lia. lia.
+ lia.
Qed.
Theorem sub_ltu:
∀ x y,
ltu x y = true →
0 ≤ unsigned y - unsigned x ≤ unsigned y.
Proof.
intros.
generalize (ltu_inv x y H). intros .
split. lia. lia.
Qed.
Theorem shru_zero: ∀ x, shru x zero = x.
Proof.
bit_solve. rewrite unsigned_zero. rewrite zlt_true. f_equal; lia. lia.
Qed.
Lemma bitwise_binop_shru:
∀ f f' x y n,
(∀ x y i, 0 ≤ i < zwordsize → testbit (f x y) i = f' (testbit x i) (testbit y i)) →
f' false false = false →
f (shru x n) (shru y n) = shru (f x y) n.
Proof.
intros. apply same_bits_eq; intros.
rewrite H; auto. rewrite !bits_shru; auto.
destruct (zlt (i + unsigned n) zwordsize); auto.
rewrite H; auto. generalize (unsigned_range n); lia.
Qed.
Theorem and_shru:
∀ x y n,
and (shru x n) (shru y n) = shru (and x y) n.
Proof.
intros. apply bitwise_binop_shru with andb; auto. exact bits_and.
Qed.
Theorem or_shru:
∀ x y n,
or (shru x n) (shru y n) = shru (or x y) n.
Proof.
intros. apply bitwise_binop_shru with orb; auto. exact bits_or.
Qed.
Theorem xor_shru:
∀ x y n,
xor (shru x n) (shru y n) = shru (xor x y) n.
Proof.
intros. apply bitwise_binop_shru with xorb; auto. exact bits_xor.
Qed.
Theorem shru_shru:
∀ x y z,
ltu y iwordsize = true →
ltu z iwordsize = true →
ltu (add y z) iwordsize = true →
shru (shru x y) z = shru x (add y z).
Proof.
intros.
generalize (ltu_iwordsize_inv _ H) (ltu_iwordsize_inv _ H0); intros.
assert (unsigned (add y z) = unsigned y + unsigned z).
unfold add. apply unsigned_repr.
generalize two_wordsize_max_unsigned; lia.
apply same_bits_eq; intros.
rewrite bits_shru; auto.
destruct (zlt (i + unsigned z) zwordsize).
- rewrite bits_shru. destruct (zlt (i + unsigned z + unsigned y) zwordsize).
+ rewrite bits_shru; auto. rewrite zlt_true. f_equal. lia. lia.
+ rewrite bits_shru; auto. rewrite zlt_false. auto. lia.
+ lia.
- rewrite bits_shru; auto. rewrite zlt_false. auto. lia.
Qed.
Theorem shr_zero: ∀ x, shr x zero = x.
Proof.
bit_solve. rewrite unsigned_zero. rewrite zlt_true. f_equal; lia. lia.
Qed.
Lemma bitwise_binop_shr:
∀ f f' x y n,
(∀ x y i, 0 ≤ i < zwordsize → testbit (f x y) i = f' (testbit x i) (testbit y i)) →
f (shr x n) (shr y n) = shr (f x y) n.
Proof.
intros. apply same_bits_eq; intros.
rewrite H; auto. rewrite !bits_shr; auto.
rewrite H; auto.
destruct (zlt (i + unsigned n) zwordsize).
generalize (unsigned_range n); lia.
lia.
Qed.
Theorem and_shr:
∀ x y n,
and (shr x n) (shr y n) = shr (and x y) n.
Proof.
intros. apply bitwise_binop_shr with andb. exact bits_and.
Qed.
Theorem or_shr:
∀ x y n,
or (shr x n) (shr y n) = shr (or x y) n.
Proof.
intros. apply bitwise_binop_shr with orb. exact bits_or.
Qed.
Theorem xor_shr:
∀ x y n,
xor (shr x n) (shr y n) = shr (xor x y) n.
Proof.
intros. apply bitwise_binop_shr with xorb. exact bits_xor.
Qed.
Theorem shr_shr:
∀ x y z,
ltu y iwordsize = true →
ltu z iwordsize = true →
ltu (add y z) iwordsize = true →
shr (shr x y) z = shr x (add y z).
Proof.
intros.
generalize (ltu_iwordsize_inv _ H) (ltu_iwordsize_inv _ H0); intros.
assert (unsigned (add y z) = unsigned y + unsigned z).
unfold add. apply unsigned_repr.
generalize two_wordsize_max_unsigned; lia.
apply same_bits_eq; intros.
rewrite !bits_shr; auto. f_equal.
destruct (zlt (i + unsigned z) zwordsize).
rewrite H4. replace (i + (unsigned y + unsigned z)) with (i + unsigned z + unsigned y) by lia. auto.
rewrite (zlt_false _ (i + unsigned (add y z))).
destruct (zlt (zwordsize - 1 + unsigned y) zwordsize); lia.
lia.
destruct (zlt (i + unsigned z) zwordsize); lia.
Qed.
Theorem and_shr_shru:
∀ x y z,
and (shr x z) (shru y z) = shru (and x y) z.
Proof.
intros. apply same_bits_eq; intros.
rewrite bits_and; auto. rewrite bits_shr; auto. rewrite !bits_shru; auto.
destruct (zlt (i + unsigned z) zwordsize).
- rewrite bits_and; auto. generalize (unsigned_range z); lia.
- apply andb_false_r.
Qed.
Theorem shr_and_shru_and:
∀ x y z,
shru (shl z y) y = z →
and (shr x y) z = and (shru x y) z.
Proof.
intros.
rewrite <- H.
rewrite and_shru. rewrite and_shr_shru. auto.
Qed.
Theorem shru_lt_zero:
∀ x,
shru x (repr (zwordsize - 1)) = if lt x zero then one else zero.
Proof.
intros. apply same_bits_eq; intros.
rewrite bits_shru; auto.
rewrite unsigned_repr.
destruct (zeq i 0).
subst i. rewrite Z.add_0_l. rewrite zlt_true.
rewrite sign_bit_of_unsigned.
unfold lt. rewrite signed_zero. unfold signed.
destruct (zlt (unsigned x) half_modulus).
rewrite zlt_false. auto. generalize (unsigned_range x); lia.
rewrite zlt_true. unfold one; rewrite testbit_repr; auto.
generalize (unsigned_range x); lia.
lia.
rewrite zlt_false.
unfold testbit. rewrite Ztestbit_eq. rewrite zeq_false.
destruct (lt x zero).
rewrite unsigned_one. simpl Z.div2. rewrite Z.testbit_0_l; auto.
rewrite unsigned_zero. simpl Z.div2. rewrite Z.testbit_0_l; auto.
auto. lia. lia.
generalize wordsize_max_unsigned; lia.
Qed.
Theorem shr_lt_zero:
∀ x,
shr x (repr (zwordsize - 1)) = if lt x zero then mone else zero.
Proof.
intros. apply same_bits_eq; intros.
rewrite bits_shr; auto.
rewrite unsigned_repr.
transitivity (testbit x (zwordsize - 1)).
f_equal. destruct (zlt (i + (zwordsize - 1)) zwordsize); lia.
rewrite sign_bit_of_unsigned.
unfold lt. rewrite signed_zero. unfold signed.
destruct (zlt (unsigned x) half_modulus).
rewrite zlt_false. rewrite bits_zero; auto. generalize (unsigned_range x); lia.
rewrite zlt_true. rewrite bits_mone; auto. generalize (unsigned_range x); lia.
generalize wordsize_max_unsigned; lia.
Qed.
Lemma bits_rol:
∀ x y i,
0 ≤ i < zwordsize →
testbit (rol x y) i = testbit x ((i - unsigned y) mod zwordsize).
Proof.
intros. unfold rol.
exploit (Z_div_mod_eq (unsigned y) zwordsize). apply wordsize_pos.
set (j := unsigned y mod zwordsize). set (k := unsigned y / zwordsize).
intros EQ.
exploit (Z_mod_lt (unsigned y) zwordsize). apply wordsize_pos.
fold j. intros RANGE.
rewrite testbit_repr; auto.
rewrite Z.lor_spec. rewrite Z.shiftr_spec. 2: lia.
destruct (zlt i j).
- rewrite Z.shiftl_spec_low; auto. simpl.
unfold testbit. f_equal.
symmetry. apply Zmod_unique with (-k - 1).
rewrite EQ. ring.
lia.
- rewrite Z.shiftl_spec_high.
fold (testbit x (i + (zwordsize - j))).
rewrite bits_above. rewrite orb_false_r.
fold (testbit x (i - j)).
f_equal. symmetry. apply Zmod_unique with (-k).
rewrite EQ. ring.
lia. lia. lia. lia.
Qed.
Lemma bits_ror:
∀ x y i,
0 ≤ i < zwordsize →
testbit (ror x y) i = testbit x ((i + unsigned y) mod zwordsize).
Proof.
intros. unfold ror.
exploit (Z_div_mod_eq (unsigned y) zwordsize). apply wordsize_pos.
set (j := unsigned y mod zwordsize). set (k := unsigned y / zwordsize).
intros EQ.
exploit (Z_mod_lt (unsigned y) zwordsize). apply wordsize_pos.
fold j. intros RANGE.
rewrite testbit_repr; auto.
rewrite Z.lor_spec. rewrite Z.shiftr_spec. 2: lia.
destruct (zlt (i + j) zwordsize).
- rewrite Z.shiftl_spec_low; auto. rewrite orb_false_r.
unfold testbit. f_equal.
symmetry. apply Zmod_unique with k.
rewrite EQ. ring.
lia. lia.
- rewrite Z.shiftl_spec_high.
fold (testbit x (i + j)).
rewrite bits_above. simpl.
unfold testbit. f_equal.
symmetry. apply Zmod_unique with (k + 1).
rewrite EQ. ring.
lia. lia. lia. lia.
Qed.
Hint Rewrite bits_rol bits_ror: ints.
Theorem shl_rolm:
∀ x n,
ltu n iwordsize = true →
shl x n = rolm x n (shl mone n).
Proof.
intros. generalize (ltu_inv _ _ H). rewrite unsigned_repr_wordsize; intros.
unfold rolm. apply same_bits_eq; intros.
rewrite bits_and; auto. rewrite !bits_shl; auto. rewrite bits_rol; auto.
destruct (zlt i (unsigned n)).
- rewrite andb_false_r; auto.
- generalize (unsigned_range n); intros.
rewrite bits_mone. rewrite andb_true_r. f_equal.
symmetry. apply Z.mod_small. lia.
lia.
Qed.
Theorem shru_rolm:
∀ x n,
ltu n iwordsize = true →
shru x n = rolm x (sub iwordsize n) (shru mone n).
Proof.
intros. generalize (ltu_inv _ _ H). rewrite unsigned_repr_wordsize; intros.
unfold rolm. apply same_bits_eq; intros.
rewrite bits_and; auto. rewrite !bits_shru; auto. rewrite bits_rol; auto.
destruct (zlt (i + unsigned n) zwordsize).
- generalize (unsigned_range n); intros.
rewrite bits_mone. rewrite andb_true_r. f_equal.
unfold sub. rewrite unsigned_repr. rewrite unsigned_repr_wordsize.
symmetry. apply Zmod_unique with (-1). ring. lia.
rewrite unsigned_repr_wordsize. generalize wordsize_max_unsigned. lia.
lia.
- rewrite andb_false_r; auto.
Qed.
Theorem rol_zero:
∀ x,
rol x zero = x.
Proof.
bit_solve. f_equal. rewrite unsigned_zero. rewrite Z.sub_0_r.
apply Z.mod_small; auto.
Qed.
Lemma bitwise_binop_rol:
∀ f f' x y n,
(∀ x y i, 0 ≤ i < zwordsize → testbit (f x y) i = f' (testbit x i) (testbit y i)) →
rol (f x y) n = f (rol x n) (rol y n).
Proof.
intros. apply same_bits_eq; intros.
rewrite H; auto. rewrite !bits_rol; auto. rewrite H; auto.
apply Z_mod_lt. apply wordsize_pos.
Qed.
Theorem rol_and:
∀ x y n,
rol (and x y) n = and (rol x n) (rol y n).
Proof.
intros. apply bitwise_binop_rol with andb. exact bits_and.
Qed.
Theorem rol_or:
∀ x y n,
rol (or x y) n = or (rol x n) (rol y n).
Proof.
intros. apply bitwise_binop_rol with orb. exact bits_or.
Qed.
Theorem rol_xor:
∀ x y n,
rol (xor x y) n = xor (rol x n) (rol y n).
Proof.
intros. apply bitwise_binop_rol with xorb. exact bits_xor.
Qed.
Theorem rol_rol:
∀ x n m,
Z.divide zwordsize modulus →
rol (rol x n) m = rol x (modu (add n m) iwordsize).
Proof.
bit_solve. f_equal. apply eqmod_mod_eq. apply wordsize_pos.
set (M := unsigned m); set (N := unsigned n).
apply eqmod_trans with (i - M - N).
apply eqmod_sub.
apply eqmod_sym. apply eqmod_mod. apply wordsize_pos.
apply eqmod_refl.
replace (i - M - N) with (i - (M + N)) by lia.
apply eqmod_sub.
apply eqmod_refl.
apply eqmod_trans with (Z.modulo (unsigned n + unsigned m) zwordsize).
replace (M + N) with (N + M) by lia. apply eqmod_mod. apply wordsize_pos.
unfold modu, add. fold M; fold N. rewrite unsigned_repr_wordsize.
assert (∀ a, eqmod zwordsize a (unsigned (repr a))).
intros. eapply eqmod_divides. apply eqm_unsigned_repr. assumption.
eapply eqmod_trans. 2: apply H1.
apply eqmod_refl2. apply eqmod_mod_eq. apply wordsize_pos. auto.
apply Z_mod_lt. apply wordsize_pos.
Qed.
Theorem rolm_zero:
∀ x m,
rolm x zero m = and x m.
Proof.
intros. unfold rolm. rewrite rol_zero. auto.
Qed.
Theorem rolm_rolm:
∀ x n1 m1 n2 m2,
Z.divide zwordsize modulus →
rolm (rolm x n1 m1) n2 m2 =
rolm x (modu (add n1 n2) iwordsize)
(and (rol m1 n2) m2).
Proof.
intros.
unfold rolm. rewrite rol_and. rewrite and_assoc.
rewrite rol_rol. reflexivity. auto.
Qed.
Theorem or_rolm:
∀ x n m1 m2,
or (rolm x n m1) (rolm x n m2) = rolm x n (or m1 m2).
Proof.
intros; unfold rolm. symmetry. apply and_or_distrib.
Qed.
Theorem ror_rol:
∀ x y,
ltu y iwordsize = true →
ror x y = rol x (sub iwordsize y).
Proof.
intros.
generalize (ltu_iwordsize_inv _ H); intros.
apply same_bits_eq; intros.
rewrite bits_ror; auto. rewrite bits_rol; auto. f_equal.
unfold sub. rewrite unsigned_repr. rewrite unsigned_repr_wordsize.
apply eqmod_mod_eq. apply wordsize_pos. ∃ 1. ring.
rewrite unsigned_repr_wordsize.
generalize wordsize_pos; generalize wordsize_max_unsigned; lia.
Qed.
Theorem ror_rol_neg:
∀ x y, (zwordsize | modulus) → ror x y = rol x (neg y).
Proof.
intros. apply same_bits_eq; intros.
rewrite bits_ror by auto. rewrite bits_rol by auto.
f_equal. apply eqmod_mod_eq. lia.
apply eqmod_trans with (i - (- unsigned y)).
apply eqmod_refl2; lia.
apply eqmod_sub. apply eqmod_refl.
apply eqmod_divides with modulus.
apply eqm_unsigned_repr. auto.
Qed.
Theorem or_ror:
∀ x y z,
ltu y iwordsize = true →
ltu z iwordsize = true →
add y z = iwordsize →
ror x z = or (shl x y) (shru x z).
Proof.
intros.
generalize (ltu_iwordsize_inv _ H) (ltu_iwordsize_inv _ H0); intros.
unfold ror, or, shl, shru. apply same_bits_eq; intros.
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 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.
lia.
generalize two_wordsize_max_unsigned; lia.
- apply eqm_unsigned_repr_r. apply eqm_refl2. f_equal.
apply Z.mod_small; auto.
Qed.
Properties of is_power2.
Remark is_power2_inv:
∀ n logn,
is_power2 n = Some logn →
Z_is_power2 (unsigned n) = Some (unsigned logn) ∧ 0 ≤ unsigned logn < zwordsize.
Proof.
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; lia.
rewrite <- modulus_power. apply unsigned_range.
auto. }
rewrite unsigned_repr; auto. generalize wordsize_max_unsigned; lia.
Qed.
Lemma is_power2_rng:
∀ n logn,
is_power2 n = Some logn →
0 ≤ unsigned logn < zwordsize.
Proof.
intros. apply (is_power2_inv n logn); auto.
Qed.
Theorem is_power2_range:
∀ n logn,
is_power2 n = Some logn → ltu logn iwordsize = true.
Proof.
intros. unfold ltu. rewrite unsigned_repr_wordsize.
apply zlt_true. generalize (is_power2_rng _ _ H). tauto.
Qed.
Lemma is_power2_correct:
∀ n logn,
is_power2 n = Some logn →
unsigned n = two_p (unsigned logn).
Proof.
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:
∀ n,
0 ≤ n < zwordsize →
0 ≤ two_p n ≤ max_unsigned.
Proof.
intros. split.
assert (two_p n > 0). apply two_p_gt_ZERO. lia. lia.
generalize (two_p_monotone_strict _ _ H).
unfold zwordsize; rewrite <- two_power_nat_two_p.
unfold max_unsigned, modulus. lia.
Qed.
Lemma is_power2_two_p:
∀ n, 0 ≤ n < zwordsize →
is_power2 (repr (two_p n)) = Some (repr n).
Proof.
intros. unfold is_power2. rewrite unsigned_repr.
rewrite Z_is_power2_complete by lia; auto.
apply two_p_range. auto.
Qed.
Relation between bitwise operations and multiplications / divisions by powers of 2
Lemma shl_mul_two_p:
∀ x y,
shl x y = mul x (repr (two_p (unsigned y))).
Proof.
intros. unfold shl, mul. apply eqm_samerepr.
rewrite Zshiftl_mul_two_p. auto with ints.
generalize (unsigned_range y); lia.
Qed.
Theorem shl_mul:
∀ x y,
shl x y = mul x (shl one y).
Proof.
intros.
assert (shl one y = repr (two_p (unsigned y))).
{
rewrite shl_mul_two_p. rewrite mul_commut. rewrite mul_one. auto.
}
rewrite H. apply shl_mul_two_p.
Qed.
Theorem mul_pow2:
∀ x n logn,
is_power2 n = Some logn →
mul x n = shl x logn.
Proof.
intros. generalize (is_power2_correct n logn H); intro.
rewrite shl_mul_two_p. rewrite <- H0. rewrite repr_unsigned.
auto.
Qed.
Theorem shifted_or_is_add:
∀ x y n,
0 ≤ n < zwordsize →
unsigned y < two_p n →
or (shl x (repr n)) y = repr(unsigned x × two_p n + unsigned y).
Proof.
intros. rewrite <- add_is_or.
- unfold add. apply eqm_samerepr. apply eqm_add; auto with ints.
rewrite shl_mul_two_p. unfold mul. apply eqm_unsigned_repr_l.
apply eqm_mult; auto with ints. apply eqm_unsigned_repr_l.
apply eqm_refl2. rewrite unsigned_repr. auto.
generalize wordsize_max_unsigned; lia.
- bit_solve.
rewrite unsigned_repr.
destruct (zlt i n).
+ auto.
+ replace (testbit y i) with false. apply andb_false_r.
symmetry. unfold testbit.
assert (EQ: Z.of_nat (Z.to_nat n) = n) by (apply Z2Nat.id; lia).
apply Ztestbit_above with (Z.to_nat n).
rewrite <- EQ in H0. rewrite <- two_power_nat_two_p in H0.
generalize (unsigned_range y); lia.
rewrite EQ; auto.
+ generalize wordsize_max_unsigned; lia.
Qed.
Unsigned right shifts and unsigned divisions by powers of 2.
Lemma shru_div_two_p:
∀ x y,
shru x y = repr (unsigned x / two_p (unsigned y)).
Proof.
intros. unfold shru.
rewrite Zshiftr_div_two_p. auto.
generalize (unsigned_range y); lia.
Qed.
Theorem divu_pow2:
∀ x n logn,
is_power2 n = Some logn →
divu x n = shru x logn.
Proof.
intros. generalize (is_power2_correct n logn H). intro.
symmetry. unfold divu. rewrite H0. apply shru_div_two_p.
Qed.
Signed right shifts and signed divisions by powers of 2.
Lemma shr_div_two_p:
∀ x y,
shr x y = repr (signed x / two_p (unsigned y)).
Proof.
intros. unfold shr.
rewrite Zshiftr_div_two_p. auto.
generalize (unsigned_range y); lia.
Qed.
Theorem divs_pow2:
∀ x n logn,
is_power2 n = Some logn →
divs x n = shrx x logn.
Proof.
intros. generalize (is_power2_correct _ _ H); intro.
unfold shrx. rewrite shl_mul_two_p.
rewrite mul_commut. rewrite mul_one.
rewrite <- H0. rewrite repr_unsigned. auto.
Qed.
Unsigned modulus over 2^n is masking with 2^n-1.
Theorem modu_and:
∀ x n logn,
is_power2 n = Some logn →
modu x n = and x (sub n one).
Proof.
intros. generalize (is_power2_correct _ _ H); intro.
generalize (is_power2_rng _ _ H); intro.
apply same_bits_eq; intros.
rewrite bits_and; auto.
unfold sub. rewrite testbit_repr; auto.
rewrite H0. rewrite unsigned_one.
unfold modu. rewrite testbit_repr; auto. rewrite H0.
rewrite Ztestbit_mod_two_p. rewrite Ztestbit_two_p_m1.
destruct (zlt i (unsigned logn)).
rewrite andb_true_r; auto.
rewrite andb_false_r; auto.
tauto. tauto. tauto. tauto.
Qed.
Properties of shrx (signed division by a power of 2)
Theorem shrx_zero:
∀ x, zwordsize > 1 → shrx x zero = x.
Proof.
intros. unfold shrx. rewrite shl_zero. unfold divs. rewrite signed_one by auto.
rewrite Z.quot_1_r. apply repr_signed.
Qed.
Theorem shrx_shr:
∀ x y,
ltu y (repr (zwordsize - 1)) = true →
shrx x y = shr (if lt x zero then add x (sub (shl one y) one) else x) y.
Proof.
intros.
set (uy := unsigned y).
assert (0 ≤ uy < zwordsize - 1).
generalize (ltu_inv _ _ H). rewrite unsigned_repr. auto.
generalize wordsize_pos wordsize_max_unsigned; lia.
rewrite shr_div_two_p. unfold shrx. unfold divs.
assert (shl one y = repr (two_p uy)).
transitivity (mul one (repr (two_p uy))).
symmetry. apply mul_pow2. replace y with (repr uy).
apply is_power2_two_p. lia. apply repr_unsigned.
rewrite mul_commut. apply mul_one.
assert (two_p uy > 0). apply two_p_gt_ZERO. lia.
assert (two_p uy < half_modulus).
rewrite half_modulus_power.
apply two_p_monotone_strict. auto.
assert (two_p uy < modulus).
rewrite modulus_power. apply two_p_monotone_strict. lia.
assert (unsigned (shl one y) = two_p uy).
rewrite H1. apply unsigned_repr. unfold max_unsigned. lia.
assert (signed (shl one y) = two_p uy).
rewrite H1. apply signed_repr.
unfold max_signed. generalize min_signed_neg. lia.
rewrite H6.
rewrite Zquot_Zdiv; auto.
unfold lt. rewrite signed_zero.
destruct (zlt (signed x) 0); auto.
rewrite add_signed.
assert (signed (sub (shl one y) one) = two_p uy - 1).
unfold sub. rewrite H5. rewrite unsigned_one.
apply signed_repr.
generalize min_signed_neg. unfold max_signed. lia.
rewrite H7. rewrite signed_repr. f_equal. f_equal. lia.
generalize (signed_range x). intros.
assert (two_p uy - 1 ≤ max_signed). unfold max_signed. lia. lia.
Qed.
Theorem shrx_shr_2:
∀ x y,
ltu y (repr (zwordsize - 1)) = true →
shrx x y = shr (add x (shru (shr x (repr (zwordsize - 1))) (sub iwordsize y))) y.
Proof.
intros.
rewrite shrx_shr by auto. f_equal.
rewrite shr_lt_zero. destruct (lt x zero).
- set (uy := unsigned y).
generalize (unsigned_range y); fold uy; intros.
assert (0 ≤ uy < zwordsize - 1).
generalize (ltu_inv _ _ H). rewrite unsigned_repr. auto.
generalize wordsize_pos wordsize_max_unsigned; lia.
assert (two_p uy < modulus).
rewrite modulus_power. apply two_p_monotone_strict. lia.
f_equal. rewrite shl_mul_two_p. fold uy. rewrite mul_commut. rewrite mul_one.
unfold sub. rewrite unsigned_one. rewrite unsigned_repr.
rewrite unsigned_repr_wordsize. fold uy.
apply same_bits_eq; intros. rewrite bits_shru by auto.
rewrite testbit_repr by auto. rewrite Ztestbit_two_p_m1 by lia.
rewrite unsigned_repr by (generalize wordsize_max_unsigned; lia).
destruct (zlt i uy).
rewrite zlt_true by lia. rewrite bits_mone by lia. auto.
rewrite zlt_false by lia. auto.
assert (two_p uy > 0) by (apply two_p_gt_ZERO; lia). unfold max_unsigned; lia.
- replace (shru zero (sub iwordsize y)) with zero.
rewrite add_zero; auto.
bit_solve. destruct (zlt (i + unsigned (sub iwordsize y)) zwordsize); auto.
Qed.
Theorem shrx_carry:
∀ x y,
ltu y (repr (zwordsize - 1)) = true →
shrx x y = add (shr x y) (shr_carry x y).
Proof.
intros. rewrite shrx_shr; auto. unfold shr_carry.
unfold lt. set (sx := signed x). rewrite signed_zero.
destruct (zlt sx 0); simpl.
2: rewrite add_zero; auto.
set (uy := unsigned y).
assert (0 ≤ uy < zwordsize - 1).
generalize (ltu_inv _ _ H). rewrite unsigned_repr. auto.
generalize wordsize_pos wordsize_max_unsigned; lia.
assert (shl one y = repr (two_p uy)).
rewrite shl_mul_two_p. rewrite mul_commut. apply mul_one.
assert (and x (sub (shl one y) one) = modu x (repr (two_p uy))).
symmetry. rewrite H1. apply modu_and with (logn := y).
rewrite is_power2_two_p. unfold uy. rewrite repr_unsigned. auto.
lia.
rewrite H2. rewrite H1.
repeat rewrite shr_div_two_p. fold sx. fold uy.
assert (two_p uy > 0). apply two_p_gt_ZERO. lia.
assert (two_p uy < modulus).
rewrite modulus_power. apply two_p_monotone_strict. lia.
assert (two_p uy < half_modulus).
rewrite half_modulus_power.
apply two_p_monotone_strict. auto.
assert (two_p uy < modulus).
rewrite modulus_power. apply two_p_monotone_strict. lia.
assert (sub (repr (two_p uy)) one = repr (two_p uy - 1)).
unfold sub. apply eqm_samerepr. apply eqm_sub. apply eqm_sym; apply eqm_unsigned_repr.
rewrite unsigned_one. apply eqm_refl.
rewrite H7. rewrite add_signed. fold sx.
rewrite (signed_repr (two_p uy - 1)). rewrite signed_repr.
unfold modu. rewrite unsigned_repr.
unfold eq. rewrite unsigned_zero. rewrite unsigned_repr.
assert (unsigned x mod two_p uy = sx mod two_p uy).
apply eqmod_mod_eq; auto. apply eqmod_divides with modulus.
fold eqm. unfold sx. apply eqm_sym. apply eqm_signed_unsigned.
unfold modulus. rewrite two_power_nat_two_p.
∃ (two_p (zwordsize - uy)). rewrite <- two_p_is_exp.
f_equal. fold zwordsize; lia. lia. lia.
rewrite H8. rewrite Zdiv_shift; auto.
unfold add. apply eqm_samerepr. apply eqm_add.
apply eqm_unsigned_repr.
destruct (zeq (sx mod two_p uy) 0); simpl.
rewrite unsigned_zero. apply eqm_refl.
rewrite unsigned_one. apply eqm_refl.
generalize (Z_mod_lt (unsigned x) (two_p uy) H3). unfold max_unsigned. lia.
unfold max_unsigned; lia.
generalize (signed_range x). fold sx. intros. split. lia. unfold max_signed. lia.
generalize min_signed_neg. unfold max_signed. lia.
Qed.
Lemma shr_shru_positive:
∀ x y,
signed x ≥ 0 →
shr x y = shru x y.
Proof.
intros.
rewrite shr_div_two_p. rewrite shru_div_two_p.
rewrite signed_eq_unsigned. auto. apply signed_positive. auto.
Qed.
Lemma and_positive:
∀ x y, signed y ≥ 0 → signed (and x y) ≥ 0.
Proof.
intros.
assert (unsigned y < half_modulus). rewrite signed_positive in H. unfold max_signed in H; lia.
generalize (sign_bit_of_unsigned y). rewrite zlt_true; auto. intros A.
generalize (sign_bit_of_unsigned (and x y)). rewrite bits_and. rewrite A.
rewrite andb_false_r. unfold signed.
destruct (zlt (unsigned (and x y)) half_modulus).
intros. generalize (unsigned_range (and x y)); lia.
congruence.
generalize wordsize_pos; lia.
Qed.
Theorem shr_and_is_shru_and:
∀ x y z,
lt y zero = false → shr (and x y) z = shru (and x y) z.
Proof.
intros. apply shr_shru_positive. apply and_positive.
unfold lt in H. rewrite signed_zero in H. destruct (zlt (signed y) 0). congruence. auto.
Qed.
Lemma bits_zero_ext:
∀ n x i, 0 ≤ i →
testbit (zero_ext n x) i = if zlt i n then testbit x i else false.
Proof.
intros. unfold zero_ext. destruct (zlt i zwordsize).
rewrite testbit_repr; auto. rewrite Zzero_ext_spec. auto. auto.
rewrite !bits_above; auto. destruct (zlt i n); auto.
Qed.
Lemma bits_sign_ext:
∀ n x i, 0 ≤ i < zwordsize →
testbit (sign_ext n x) i = testbit x (if zlt i n then i else n - 1).
Proof.
intros. unfold sign_ext.
rewrite testbit_repr; auto. apply Zsign_ext_spec. lia.
Qed.
Hint Rewrite bits_zero_ext bits_sign_ext: ints.
Theorem zero_ext_above:
∀ n x, n ≥ zwordsize → zero_ext n x = x.
Proof.
intros. apply same_bits_eq; intros.
rewrite bits_zero_ext. apply zlt_true. lia. lia.
Qed.
Theorem zero_ext_below:
∀ n x, n ≤ 0 → zero_ext n x = zero.
Proof.
intros. bit_solve. destruct (zlt i n); auto. apply bits_below; lia. lia.
Qed.
Theorem sign_ext_above:
∀ n x, n ≥ zwordsize → sign_ext n x = x.
Proof.
intros. apply same_bits_eq; intros.
unfold sign_ext; rewrite testbit_repr; auto.
rewrite Zsign_ext_spec. rewrite zlt_true. auto. lia. lia.
Qed.
Theorem sign_ext_below:
∀ n x, n ≤ 0 → sign_ext n x = zero.
Proof.
intros. bit_solve. apply bits_below. destruct (zlt i n); lia.
Qed.
Theorem zero_ext_and:
∀ n x, 0 ≤ n → zero_ext n x = and x (repr (two_p n - 1)).
Proof.
bit_solve. rewrite testbit_repr; auto. rewrite Ztestbit_two_p_m1; intuition.
destruct (zlt i n).
rewrite andb_true_r; auto.
rewrite andb_false_r; auto.
tauto.
Qed.
Theorem zero_ext_mod:
∀ n x, 0 ≤ n < zwordsize →
unsigned (zero_ext n x) = Z.modulo (unsigned x) (two_p n).
Proof.
intros. apply equal_same_bits. intros.
rewrite Ztestbit_mod_two_p; auto.
fold (testbit (zero_ext n x) i).
destruct (zlt i zwordsize).
rewrite bits_zero_ext; auto.
rewrite bits_above. rewrite zlt_false; auto. lia. lia.
lia.
Qed.
Theorem zero_ext_widen:
∀ x n n', 0 ≤ n ≤ n' →
zero_ext n' (zero_ext n x) = zero_ext n x.
Proof.
bit_solve. destruct (zlt i n).
apply zlt_true. lia.
destruct (zlt i n'); auto.
tauto. tauto.
Qed.
Theorem sign_ext_widen:
∀ x n n', 0 < n ≤ n' →
sign_ext n' (sign_ext n x) = sign_ext n x.
Proof.
intros. destruct (zlt n' zwordsize).
bit_solve. destruct (zlt i n').
auto.
rewrite (zlt_false _ i n).
destruct (zlt (n' - 1) n); f_equal; lia.
lia.
destruct (zlt i n'); lia.
apply sign_ext_above; auto.
Qed.
Theorem sign_zero_ext_widen:
∀ x n n', 0 ≤ n < n' →
sign_ext n' (zero_ext n x) = zero_ext n x.
Proof.
intros. destruct (zlt n' zwordsize).
bit_solve.
destruct (zlt i n').
auto.
rewrite !zlt_false. auto. lia. lia. lia.
destruct (zlt i n'); lia.
apply sign_ext_above; auto.
Qed.
Theorem zero_ext_narrow:
∀ x n n', 0 ≤ n ≤ n' →
zero_ext n (zero_ext n' x) = zero_ext n x.
Proof.
bit_solve. destruct (zlt i n).
apply zlt_true. lia.
auto.
lia. lia. lia.
Qed.
Theorem sign_ext_narrow:
∀ x n n', 0 < n ≤ n' →
sign_ext n (sign_ext n' x) = sign_ext n x.
Proof.
intros. destruct (zlt n zwordsize).
bit_solve. destruct (zlt i n); f_equal; apply zlt_true; lia.
destruct (zlt i n); lia.
rewrite (sign_ext_above n'). auto. lia.
Qed.
Theorem zero_sign_ext_narrow:
∀ x n n', 0 < n ≤ n' →
zero_ext n (sign_ext n' x) = zero_ext n x.
Proof.
intros. destruct (zlt n' zwordsize).
bit_solve.
destruct (zlt i n); auto.
rewrite zlt_true; auto. lia.
lia. lia.
rewrite sign_ext_above; auto.
Qed.
Theorem zero_ext_idem:
∀ n x, 0 ≤ n → zero_ext n (zero_ext n x) = zero_ext n x.
Proof.
intros. apply zero_ext_widen. lia.
Qed.
Theorem sign_ext_idem:
∀ n x, 0 < n → sign_ext n (sign_ext n x) = sign_ext n x.
Proof.
intros. apply sign_ext_widen. lia.
Qed.
Theorem sign_ext_zero_ext:
∀ n x, 0 < n → sign_ext n (zero_ext n x) = sign_ext n x.
Proof.
intros. destruct (zlt n zwordsize).
bit_solve.
destruct (zlt i n).
rewrite zlt_true; auto.
rewrite zlt_true; auto. lia.
destruct (zlt i n); lia.
rewrite zero_ext_above; auto.
Qed.
Theorem zero_ext_sign_ext:
∀ n x, 0 < n → zero_ext n (sign_ext n x) = zero_ext n x.
Proof.
intros. apply zero_sign_ext_narrow. lia.
Qed.
Theorem sign_ext_equal_if_zero_equal:
∀ n x y, 0 < n →
zero_ext n x = zero_ext n y →
sign_ext n x = sign_ext n y.
Proof.
intros. rewrite <- (sign_ext_zero_ext n x H).
rewrite <- (sign_ext_zero_ext n y H). congruence.
Qed.
Theorem shru_shl:
∀ x y z, ltu y iwordsize = true → ltu z iwordsize = true →
shru (shl x y) z =
if ltu z y then shl (zero_ext (zwordsize - unsigned y) x) (sub y z)
else zero_ext (zwordsize - unsigned z) (shru x (sub z y)).
Proof.
intros. apply ltu_iwordsize_inv in H; apply ltu_iwordsize_inv in H0.
unfold ltu. set (Y := unsigned y) in *; set (Z := unsigned z) in ×.
apply same_bits_eq; intros. rewrite bits_shru by auto. fold Z.
destruct (zlt Z Y).
- assert (A: unsigned (sub y z) = Y - Z).
{ apply unsigned_repr. generalize wordsize_max_unsigned; lia. }
symmetry; rewrite bits_shl, A by lia.
destruct (zlt (i + Z) zwordsize).
+ rewrite bits_shl by lia. fold Y.
destruct (zlt i (Y - Z)); [rewrite zlt_true by lia|rewrite zlt_false by lia]; auto.
rewrite bits_zero_ext by lia. rewrite zlt_true by lia. f_equal; lia.
+ rewrite bits_zero_ext by lia. rewrite ! zlt_false by lia. auto.
- assert (A: unsigned (sub z y) = Z - Y).
{ apply unsigned_repr. generalize wordsize_max_unsigned; lia. }
rewrite bits_zero_ext, bits_shru, A by lia.
destruct (zlt (i + Z) zwordsize); [rewrite zlt_true by lia|rewrite zlt_false by lia]; auto.
rewrite bits_shl by lia. fold Y.
destruct (zlt (i + Z) Y).
+ rewrite zlt_false by lia. auto.
+ rewrite zlt_true by lia. f_equal; lia.
Qed.
Corollary zero_ext_shru_shl:
∀ n x,
0 < n < zwordsize →
let y := repr (zwordsize - n) in
zero_ext n x = shru (shl x y) y.
Proof.
intros.
assert (A: unsigned y = zwordsize - n).
{ unfold y. apply unsigned_repr. generalize wordsize_max_unsigned. lia. }
assert (B: ltu y iwordsize = true).
{ unfold ltu; rewrite A, unsigned_repr_wordsize. apply zlt_true; lia. }
rewrite shru_shl by auto. unfold ltu; rewrite zlt_false by lia.
rewrite sub_idem, shru_zero. f_equal. rewrite A; lia.
Qed.
Theorem shr_shl:
∀ x y z, ltu y iwordsize = true → ltu z iwordsize = true →
shr (shl x y) z =
if ltu z y then shl (sign_ext (zwordsize - unsigned y) x) (sub y z)
else sign_ext (zwordsize - unsigned z) (shr x (sub z y)).
Proof.
intros. apply ltu_iwordsize_inv in H; apply ltu_iwordsize_inv in H0.
unfold ltu. set (Y := unsigned y) in *; set (Z := unsigned z) in ×.
apply same_bits_eq; intros. rewrite bits_shr by auto. fold Z.
rewrite bits_shl by (destruct (zlt (i + Z) zwordsize); lia). fold Y.
destruct (zlt Z Y).
- assert (A: unsigned (sub y z) = Y - Z).
{ apply unsigned_repr. generalize wordsize_max_unsigned; lia. }
rewrite bits_shl, A by lia.
destruct (zlt i (Y - Z)).
+ apply zlt_true. destruct (zlt (i + Z) zwordsize); lia.
+ rewrite zlt_false by (destruct (zlt (i + Z) zwordsize); lia).
rewrite bits_sign_ext by lia. f_equal.
destruct (zlt (i + Z) zwordsize).
rewrite zlt_true by lia. lia.
rewrite zlt_false by lia. lia.
- assert (A: unsigned (sub z y) = Z - Y).
{ apply unsigned_repr. generalize wordsize_max_unsigned; lia. }
rewrite bits_sign_ext by lia.
rewrite bits_shr by (destruct (zlt i (zwordsize - Z)); lia).
rewrite A. rewrite zlt_false by (destruct (zlt (i + Z) zwordsize); lia).
f_equal. destruct (zlt i (zwordsize - Z)).
+ rewrite ! zlt_true by lia. lia.
+ rewrite ! zlt_false by lia. rewrite zlt_true by lia. lia.
Qed.
Corollary sign_ext_shr_shl:
∀ n x,
0 < n < zwordsize →
let y := repr (zwordsize - n) in
sign_ext n x = shr (shl x y) y.
Proof.
intros.
assert (A: unsigned y = zwordsize - n).
{ unfold y. apply unsigned_repr. generalize wordsize_max_unsigned. lia. }
assert (B: ltu y iwordsize = true).
{ unfold ltu; rewrite A, unsigned_repr_wordsize. apply zlt_true; lia. }
rewrite shr_shl by auto. unfold ltu; rewrite zlt_false by lia.
rewrite sub_idem, shr_zero. f_equal. rewrite A; lia.
Qed.
Lemma zero_ext_range:
∀ n x, 0 ≤ n < zwordsize → 0 ≤ unsigned (zero_ext n x) < two_p n.
Proof.
intros. rewrite zero_ext_mod; auto. apply Z_mod_lt. apply two_p_gt_ZERO. lia.
Qed.
Lemma eqmod_zero_ext:
∀ n x, 0 ≤ n < zwordsize → eqmod (two_p n) (unsigned (zero_ext n x)) (unsigned x).
Proof.
intros. rewrite zero_ext_mod; auto. apply eqmod_sym. apply eqmod_mod.
apply two_p_gt_ZERO. lia.
Qed.
Lemma sign_ext_range:
∀ n x, 0 < n < zwordsize → -two_p (n-1) ≤ signed (sign_ext n x) < two_p (n-1).
Proof.
intros. rewrite sign_ext_shr_shl; auto.
set (X := shl x (repr (zwordsize - n))).
assert (two_p (n - 1) > 0) by (apply two_p_gt_ZERO; lia).
assert (unsigned (repr (zwordsize - n)) = zwordsize - n).
apply unsigned_repr.
split. lia. generalize wordsize_max_unsigned; lia.
rewrite shr_div_two_p.
rewrite signed_repr.
rewrite H1.
apply Zdiv_interval_1.
lia. lia. apply two_p_gt_ZERO; lia.
replace (- two_p (n - 1) × two_p (zwordsize - n))
with (- (two_p (n - 1) × two_p (zwordsize - n))) by ring.
rewrite <- two_p_is_exp.
replace (n - 1 + (zwordsize - n)) with (zwordsize - 1) by lia.
rewrite <- half_modulus_power.
generalize (signed_range X). unfold min_signed, max_signed. lia.
lia. lia.
apply Zdiv_interval_2. apply signed_range.
generalize min_signed_neg; lia.
generalize max_signed_pos; lia.
rewrite H1. apply two_p_gt_ZERO. lia.
Qed.
Lemma eqmod_sign_ext':
∀ n x, 0 < n < zwordsize →
eqmod (two_p n) (unsigned (sign_ext n x)) (unsigned x).
Proof.
intros.
set (N := Z.to_nat n).
assert (Z.of_nat N = n) by (apply Z2Nat.id; lia).
rewrite <- H0. rewrite <- two_power_nat_two_p.
apply eqmod_same_bits; intros.
rewrite H0 in H1. rewrite H0.
fold (testbit (sign_ext n x) i). rewrite bits_sign_ext.
rewrite zlt_true. auto. lia. lia.
Qed.
Lemma eqmod_sign_ext:
∀ n x, 0 < n < zwordsize →
eqmod (two_p n) (signed (sign_ext n x)) (unsigned x).
Proof.
intros. apply eqmod_trans with (unsigned (sign_ext n x)).
apply eqmod_divides with modulus. apply eqm_signed_unsigned.
∃ (two_p (zwordsize - n)).
unfold modulus. rewrite two_power_nat_two_p. fold zwordsize.
rewrite <- two_p_is_exp. f_equal. lia. lia. lia.
apply eqmod_sign_ext'; auto.
Qed.
Combinations of shifts and zero/sign extensions
Lemma shl_zero_ext:
∀ n m x, 0 ≤ n →
shl (zero_ext n x) m = zero_ext (n + unsigned m) (shl x m).
Proof.
intros. apply same_bits_eq; intros.
rewrite bits_zero_ext, ! bits_shl by lia.
destruct (zlt i (unsigned m)).
- rewrite zlt_true by lia; auto.
- rewrite bits_zero_ext by lia.
destruct (zlt (i - unsigned m) n); [rewrite zlt_true by lia|rewrite zlt_false by lia]; auto.
Qed.
Lemma shl_sign_ext:
∀ n m x, 0 < n →
shl (sign_ext n x) m = sign_ext (n + unsigned m) (shl x m).
Proof.
intros. generalize (unsigned_range m); intros.
apply same_bits_eq; intros.
rewrite bits_sign_ext, ! bits_shl by lia.
destruct (zlt i (n + unsigned m)).
- rewrite bits_shl by auto. destruct (zlt i (unsigned m)); auto.
rewrite bits_sign_ext by lia. f_equal. apply zlt_true. lia.
- rewrite zlt_false by lia. rewrite bits_shl by lia. rewrite zlt_false by lia.
rewrite bits_sign_ext by lia. f_equal. rewrite zlt_false by lia. lia.
Qed.
Lemma shru_zero_ext:
∀ n m x, 0 ≤ n →
shru (zero_ext (n + unsigned m) x) m = zero_ext n (shru x m).
Proof.
intros. bit_solve.
- destruct (zlt (i + unsigned m) zwordsize).
× destruct (zlt i n); [rewrite zlt_true by lia|rewrite zlt_false by lia]; auto.
× destruct (zlt i n); auto.
- generalize (unsigned_range m); lia.
- lia.
Qed.
Lemma shru_zero_ext_0:
∀ n m x, n ≤ unsigned m →
shru (zero_ext n x) m = zero.
Proof.
intros. bit_solve.
- destruct (zlt (i + unsigned m) zwordsize); auto.
apply zlt_false. lia.
- generalize (unsigned_range m); lia.
Qed.
Lemma shr_sign_ext:
∀ n m x, 0 < n → n + unsigned m < zwordsize →
shr (sign_ext (n + unsigned m) x) m = sign_ext n (shr x m).
Proof.
intros. generalize (unsigned_range m); intros.
apply same_bits_eq; intros.
rewrite bits_sign_ext, bits_shr by auto.
rewrite bits_sign_ext, bits_shr.
- f_equal.
destruct (zlt i n), (zlt (i + unsigned m) zwordsize).
+ apply zlt_true; lia.
+ apply zlt_true; lia.
+ rewrite zlt_false by lia. rewrite zlt_true by lia. lia.
+ rewrite zlt_false by lia. rewrite zlt_true by lia. lia.
- destruct (zlt i n); lia.
- destruct (zlt (i + unsigned m) zwordsize); lia.
Qed.
Lemma zero_ext_shru_min:
∀ s x n, ltu n iwordsize = true →
zero_ext s (shru x n) = zero_ext (Z.min s (zwordsize - unsigned n)) (shru x n).
Proof.
intros. apply ltu_iwordsize_inv in H.
apply Z.min_case_strong; intros; auto.
bit_solve; try lia.
destruct (zlt i (zwordsize - unsigned n)).
rewrite zlt_true by lia. auto.
destruct (zlt i s); auto. rewrite zlt_false by lia; auto.
Qed.
Lemma sign_ext_shr_min:
∀ s x n, ltu n iwordsize = true →
sign_ext s (shr x n) = sign_ext (Z.min s (zwordsize - unsigned n)) (shr x n).
Proof.
intros. apply ltu_iwordsize_inv in H.
rewrite Z.min_comm.
destruct (Z.min_spec (zwordsize - unsigned n) s) as [[A B] | [A B]]; rewrite B; auto.
apply same_bits_eq; intros. rewrite ! bits_sign_ext by auto.
destruct (zlt i (zwordsize - unsigned n)).
rewrite zlt_true by lia. auto.
assert (C: testbit (shr x n) (zwordsize - unsigned n - 1) = testbit x (zwordsize - 1)).
{ rewrite bits_shr by lia. rewrite zlt_true by lia. f_equal; lia. }
rewrite C. destruct (zlt i s); rewrite bits_shr by lia.
rewrite zlt_false by lia. auto.
rewrite zlt_false by lia. auto.
Qed.
Lemma shl_zero_ext_min:
∀ s x n, ltu n iwordsize = true →
shl (zero_ext s x) n = shl (zero_ext (Z.min s (zwordsize - unsigned n)) x) n.
Proof.
intros. apply ltu_iwordsize_inv in H.
apply Z.min_case_strong; intros; auto.
apply same_bits_eq; intros. rewrite ! bits_shl by auto.
destruct (zlt i (unsigned n)); auto.
rewrite ! bits_zero_ext by lia.
destruct (zlt (i - unsigned n) s).
rewrite zlt_true by lia; auto.
rewrite zlt_false by lia; auto.
Qed.
Lemma shl_sign_ext_min:
∀ s x n, ltu n iwordsize = true →
shl (sign_ext s x) n = shl (sign_ext (Z.min s (zwordsize - unsigned n)) x) n.
Proof.
intros. apply ltu_iwordsize_inv in H.
rewrite Z.min_comm.
destruct (Z.min_spec (zwordsize - unsigned n) s) as [[A B] | [A B]]; rewrite B; auto.
apply same_bits_eq; intros. rewrite ! bits_shl by auto.
destruct (zlt i (unsigned n)); auto.
rewrite ! bits_sign_ext by lia. f_equal.
destruct (zlt (i - unsigned n) s).
rewrite zlt_true by lia; auto.
extlia.
Qed.
Properties of one_bits (decomposition in sum of powers of two)
Theorem one_bits_range:
∀ x i, In i (one_bits x) → ltu i iwordsize = true.
Proof.
assert (A: ∀ p, 0 ≤ p < zwordsize → ltu (repr p) iwordsize = true).
intros. unfold ltu, iwordsize. apply zlt_true.
repeat rewrite unsigned_repr. tauto.
generalize wordsize_max_unsigned; lia.
generalize wordsize_max_unsigned; lia.
unfold one_bits. intros.
destruct (list_in_map_inv _ _ _ H) as [i0 [EQ IN]].
subst i. apply A. apply Z_one_bits_range with (unsigned x); auto.
Qed.
Fixpoint int_of_one_bits (l: list int) : int :=
match l with
| nil ⇒ zero
| a :: b ⇒ add (shl one a) (int_of_one_bits b)
end.
Theorem one_bits_decomp:
∀ x, x = int_of_one_bits (one_bits x).
Proof.
intros.
transitivity (repr (powerserie (Z_one_bits wordsize (unsigned x) 0))).
transitivity (repr (unsigned x)).
auto with ints. decEq. apply Z_one_bits_powerserie.
auto with ints.
unfold one_bits.
generalize (Z_one_bits_range wordsize (unsigned x)).
generalize (Z_one_bits wordsize (unsigned x) 0).
induction l.
intros; reflexivity.
intros; simpl. rewrite <- IHl. unfold add. apply eqm_samerepr.
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 _ _)). change (Z.of_nat wordsize) with zwordsize.
generalize wordsize_max_unsigned. lia.
auto with ints.
intros; apply H; auto with coqlib.
Qed.
Theorem negate_cmp:
∀ c x y, cmp (negate_comparison c) x y = negb (cmp c x y).
Proof.
intros. destruct c; simpl; try rewrite negb_elim; auto.
Qed.
Theorem negate_cmpu:
∀ c x y, cmpu (negate_comparison c) x y = negb (cmpu c x y).
Proof.
intros. destruct c; simpl; try rewrite negb_elim; auto.
Qed.
Theorem swap_cmp:
∀ c x y, cmp (swap_comparison c) x y = cmp c y x.
Proof.
intros. destruct c; simpl; auto. apply eq_sym. decEq. apply eq_sym.
Qed.
Theorem swap_cmpu:
∀ c x y, cmpu (swap_comparison c) x y = cmpu c y x.
Proof.
intros. destruct c; simpl; auto. apply eq_sym. decEq. apply eq_sym.
Qed.
Lemma translate_eq:
∀ x y d,
eq (add x d) (add y d) = eq x y.
Proof.
intros. unfold eq. case (zeq (unsigned x) (unsigned y)); intro.
unfold add. rewrite e. apply zeq_true.
apply zeq_false. unfold add. red; intro. apply n.
apply eqm_small_eq; auto with ints.
replace (unsigned x) with ((unsigned x + unsigned d) - unsigned d).
replace (unsigned y) with ((unsigned y + unsigned d) - unsigned d).
apply eqm_sub. apply eqm_trans with (unsigned (repr (unsigned x + unsigned d))).
eauto with ints. apply eqm_trans with (unsigned (repr (unsigned y + unsigned d))).
eauto with ints. eauto with ints. eauto with ints.
lia. lia.
Qed.
Lemma translate_ltu:
∀ x y d,
0 ≤ unsigned x + unsigned d ≤ max_unsigned →
0 ≤ unsigned y + unsigned d ≤ max_unsigned →
ltu (add x d) (add y d) = ltu x y.
Proof.
intros. unfold add. unfold ltu.
repeat rewrite unsigned_repr; auto. case (zlt (unsigned x) (unsigned y)); intro.
apply zlt_true. lia.
apply zlt_false. lia.
Qed.
Theorem translate_cmpu:
∀ c x y d,
0 ≤ unsigned x + unsigned d ≤ max_unsigned →
0 ≤ unsigned y + unsigned d ≤ max_unsigned →
cmpu c (add x d) (add y d) = cmpu c x y.
Proof.
intros. unfold cmpu.
rewrite translate_eq. repeat rewrite translate_ltu; auto.
Qed.
Lemma translate_lt:
∀ x y d,
min_signed ≤ signed x + signed d ≤ max_signed →
min_signed ≤ signed y + signed d ≤ max_signed →
lt (add x d) (add y d) = lt x y.
Proof.
intros. repeat rewrite add_signed. unfold lt.
repeat rewrite signed_repr; auto. case (zlt (signed x) (signed y)); intro.
apply zlt_true. lia.
apply zlt_false. lia.
Qed.
Theorem translate_cmp:
∀ c x y d,
min_signed ≤ signed x + signed d ≤ max_signed →
min_signed ≤ signed y + signed d ≤ max_signed →
cmp c (add x d) (add y d) = cmp c x y.
Proof.
intros. unfold cmp.
rewrite translate_eq. repeat rewrite translate_lt; auto.
Qed.
Theorem notbool_isfalse_istrue:
∀ x, is_false x → is_true (notbool x).
Proof.
unfold is_false, is_true, notbool; intros; subst x.
rewrite eq_true. apply one_not_zero.
Qed.
Theorem notbool_istrue_isfalse:
∀ x, is_true x → is_false (notbool x).
Proof.
unfold is_false, is_true, notbool; intros.
generalize (eq_spec x zero). case (eq x zero); intro.
contradiction. auto.
Qed.
Theorem ltu_range_test:
∀ x y,
ltu x y = true → unsigned y ≤ max_signed →
0 ≤ signed x < unsigned y.
Proof.
intros.
unfold ltu in H. destruct (zlt (unsigned x) (unsigned y)); try discriminate.
rewrite signed_eq_unsigned.
generalize (unsigned_range x). lia. lia.
Qed.
Theorem lt_sub_overflow:
∀ x y,
xor (sub_overflow x y zero) (negative (sub x y)) = if lt x y then one else zero.
Proof.
intros. unfold negative, sub_overflow, lt. rewrite sub_signed.
rewrite signed_zero. rewrite Z.sub_0_r.
generalize (signed_range x) (signed_range y).
set (X := signed x); set (Y := signed y). intros RX RY.
unfold min_signed, max_signed in ×.
generalize half_modulus_pos half_modulus_modulus; intros HM MM.
destruct (zle 0 (X - Y)).
- unfold proj_sumbool at 1; rewrite zle_true at 1 by lia. simpl.
rewrite (zlt_false _ X) by lia.
destruct (zlt (X - Y) half_modulus).
+ unfold proj_sumbool; rewrite zle_true by lia.
rewrite signed_repr. rewrite zlt_false by lia. apply xor_idem.
unfold min_signed, max_signed; lia.
+ unfold proj_sumbool; rewrite zle_false by lia.
replace (signed (repr (X - Y))) with (X - Y - modulus).
rewrite zlt_true by lia. apply xor_idem.
rewrite signed_repr_eq. replace ((X - Y) mod modulus) with (X - Y).
rewrite zlt_false; auto.
symmetry. apply Zmod_unique with 0; lia.
- unfold proj_sumbool at 2. rewrite zle_true at 1 by lia. rewrite andb_true_r.
rewrite (zlt_true _ X) by lia.
destruct (zlt (X - Y) (-half_modulus)).
+ unfold proj_sumbool; rewrite zle_false by lia.
replace (signed (repr (X - Y))) with (X - Y + modulus).
rewrite zlt_false by lia. apply xor_zero.
rewrite signed_repr_eq. replace ((X - Y) mod modulus) with (X - Y + modulus).
rewrite zlt_true by lia; auto.
symmetry. apply Zmod_unique with (-1); lia.
+ unfold proj_sumbool; rewrite zle_true by lia.
rewrite signed_repr. rewrite zlt_true by lia. apply xor_zero_l.
unfold min_signed, max_signed; lia.
Qed.
Lemma signed_eq:
∀ x y, eq x y = zeq (signed x) (signed y).
Proof.
intros. unfold eq. unfold proj_sumbool.
destruct (zeq (unsigned x) (unsigned y));
destruct (zeq (signed x) (signed y)); auto.
elim n. unfold signed. rewrite e; auto.
elim n. apply eqm_small_eq; auto with ints.
eapply eqm_trans. apply eqm_sym. apply eqm_signed_unsigned.
rewrite e. apply eqm_signed_unsigned.
Qed.
Lemma not_lt:
∀ x y, negb (lt y x) = (lt x y || eq x y).
Proof.
intros. unfold lt. rewrite signed_eq. unfold proj_sumbool.
destruct (zlt (signed y) (signed x)).
rewrite zlt_false. rewrite zeq_false. auto. lia. lia.
destruct (zeq (signed x) (signed y)).
rewrite zlt_false. auto. lia.
rewrite zlt_true. auto. lia.
Qed.
Lemma lt_not:
∀ x y, lt y x = negb (lt x y) && negb (eq x y).
Proof.
intros. rewrite <- negb_orb. rewrite <- not_lt. rewrite negb_involutive. auto.
Qed.
Lemma not_ltu:
∀ x y, negb (ltu y x) = (ltu x y || eq x y).
Proof.
intros. unfold ltu, eq.
destruct (zlt (unsigned y) (unsigned x)).
rewrite zlt_false. rewrite zeq_false. auto. lia. lia.
destruct (zeq (unsigned x) (unsigned y)).
rewrite zlt_false. auto. lia.
rewrite zlt_true. auto. lia.
Qed.
Lemma ltu_not:
∀ x y, ltu y x = negb (ltu x y) && negb (eq x y).
Proof.
intros. rewrite <- negb_orb. rewrite <- not_ltu. rewrite negb_involutive. auto.
Qed.
Definition no_overlap (ofs1: int) (sz1: Z) (ofs2: int) (sz2: Z) : bool :=
let x1 := unsigned ofs1 in let x2 := unsigned ofs2 in
zlt (x1 + sz1) modulus && zlt (x2 + sz2) modulus
&& (zle (x1 + sz1) x2 || zle (x2 + sz2) x1).
Lemma no_overlap_sound:
∀ ofs1 sz1 ofs2 sz2 base,
sz1 > 0 → sz2 > 0 → no_overlap ofs1 sz1 ofs2 sz2 = true →
unsigned (add base ofs1) + sz1 ≤ unsigned (add base ofs2)
∨ unsigned (add base ofs2) + sz2 ≤ unsigned (add base ofs1).
Proof.
intros.
destruct (andb_prop _ _ H1). clear H1.
destruct (andb_prop _ _ H2). clear H2.
apply proj_sumbool_true in H1.
apply proj_sumbool_true in H4.
assert (unsigned ofs1 + sz1 ≤ unsigned ofs2 ∨ unsigned ofs2 + sz2 ≤ unsigned ofs1).
destruct (orb_prop _ _ H3). left. eapply proj_sumbool_true; eauto. right. eapply proj_sumbool_true; eauto.
clear H3.
generalize (unsigned_range ofs1) (unsigned_range ofs2). intros P Q.
generalize (unsigned_add_either base ofs1) (unsigned_add_either base ofs2).
intros [C|C] [D|D]; lia.
Qed.
Definition size (x: int) : Z := Zsize (unsigned x).
Theorem size_zero: size zero = 0.
Proof.
unfold size; rewrite unsigned_zero; auto.
Qed.
Theorem bits_size_1:
∀ x, x = zero ∨ testbit x (Z.pred (size x)) = true.
Proof.
intros. destruct (zeq (unsigned x) 0).
left. rewrite <- (repr_unsigned x). rewrite e; auto.
right. apply Ztestbit_size_1. generalize (unsigned_range x); lia.
Qed.
Theorem bits_size_2:
∀ x i, size x ≤ i → testbit x i = false.
Proof.
intros. apply Ztestbit_size_2. generalize (unsigned_range x); lia.
fold (size x); lia.
Qed.
Theorem size_range:
∀ x, 0 ≤ size x ≤ zwordsize.
Proof.
intros; split. apply Zsize_pos.
destruct (bits_size_1 x).
subst x; unfold size; rewrite unsigned_zero; simpl. generalize wordsize_pos; lia.
destruct (zle (size x) zwordsize); auto.
rewrite bits_above in H. congruence. lia.
Qed.
Theorem bits_size_3:
∀ x n,
0 ≤ n →
(∀ i, n ≤ i < zwordsize → testbit x i = false) →
size x ≤ n.
Proof.
intros. destruct (zle (size x) n). auto.
destruct (bits_size_1 x).
subst x. unfold size; rewrite unsigned_zero; assumption.
rewrite (H0 (Z.pred (size x))) in H1. congruence.
generalize (size_range x); lia.
Qed.
Theorem bits_size_4:
∀ x n,
0 ≤ n →
testbit x (Z.pred n) = true →
(∀ i, n ≤ i < zwordsize → testbit x i = false) →
size x = n.
Proof.
intros.
assert (size x ≤ n).
apply bits_size_3; auto.
destruct (zlt (size x) n).
rewrite bits_size_2 in H0. congruence. lia.
lia.
Qed.
Theorem size_interval_1:
∀ x, 0 ≤ unsigned x < two_p (size x).
Proof.
intros; apply Zsize_interval_1. generalize (unsigned_range x); lia.
Qed.
Theorem size_interval_2:
∀ x n, 0 ≤ n → 0 ≤ unsigned x < two_p n → n ≥ size x.
Proof.
intros. apply Zsize_interval_2; auto.
Qed.
Theorem size_and:
∀ a b, size (and a b) ≤ Z.min (size a) (size b).
Proof.
intros.
assert (0 ≤ Z.min (size a) (size b)).
generalize (size_range a) (size_range b). zify; lia.
apply bits_size_3. auto. intros.
rewrite bits_and by lia.
rewrite andb_false_iff.
generalize (bits_size_2 a i).
generalize (bits_size_2 b i).
zify; intuition.
Qed.
Corollary and_interval:
∀ a b, 0 ≤ unsigned (and a b) < two_p (Z.min (size a) (size b)).
Proof.
intros.
generalize (size_interval_1 (and a b)); intros.
assert (two_p (size (and a b)) ≤ two_p (Z.min (size a) (size b))).
apply two_p_monotone. split. generalize (size_range (and a b)); lia.
apply size_and.
lia.
Qed.
Theorem size_or:
∀ a b, size (or a b) = Z.max (size a) (size b).
Proof.
intros. generalize (size_range a) (size_range b); intros.
destruct (bits_size_1 a).
subst a. rewrite size_zero. rewrite or_zero_l. zify; lia.
destruct (bits_size_1 b).
subst b. rewrite size_zero. rewrite or_zero. zify; lia.
zify. destruct H3 as [[P Q] | [P Q]]; subst.
apply bits_size_4. tauto. rewrite bits_or. rewrite H2. apply orb_true_r.
lia.
intros. rewrite bits_or. rewrite !bits_size_2. auto. lia. lia. lia.
apply bits_size_4. tauto. rewrite bits_or. rewrite H1. apply orb_true_l.
destruct (zeq (size a) 0). unfold testbit in H1. rewrite Z.testbit_neg_r in H1.
congruence. lia. lia.
intros. rewrite bits_or. rewrite !bits_size_2. auto. lia. lia. lia.
Qed.
Corollary or_interval:
∀ a b, 0 ≤ unsigned (or a b) < two_p (Z.max (size a) (size b)).
Proof.
intros. rewrite <- size_or. apply size_interval_1.
Qed.
Theorem size_xor:
∀ a b, size (xor a b) ≤ Z.max (size a) (size b).
Proof.
intros.
assert (0 ≤ Z.max (size a) (size b)).
generalize (size_range a) (size_range b). zify; lia.
apply bits_size_3. auto. intros.
rewrite bits_xor. rewrite !bits_size_2. auto.
zify; lia.
zify; lia.
lia.
Qed.
Corollary xor_interval:
∀ a b, 0 ≤ unsigned (xor a b) < two_p (Z.max (size a) (size b)).
Proof.
intros.
generalize (size_interval_1 (xor a b)); intros.
assert (two_p (size (xor a b)) ≤ two_p (Z.max (size a) (size b))).
apply two_p_monotone. split. generalize (size_range (xor a b)); lia.
apply size_xor.
lia.
Qed.
End Make.
Module Wordsize_32.
Definition wordsize := 32%nat.
Remark wordsize_not_zero: wordsize ≠ 0%nat.
Proof. unfold wordsize; congruence. Qed.
End Wordsize_32.
Strategy opaque [Wordsize_32.wordsize].
Module Int := Make(Wordsize_32).
Strategy 0 [Wordsize_32.wordsize].
Notation int := Int.int.
Remark int_wordsize_divides_modulus:
Z.divide (Z.of_nat Int.wordsize) Int.modulus.
Proof.
∃ (two_p (32-5)); reflexivity.
Qed.
Module Wordsize_8.
Definition wordsize := 8%nat.
Remark wordsize_not_zero: wordsize ≠ 0%nat.
Proof. unfold wordsize; congruence. Qed.
End Wordsize_8.
Strategy opaque [Wordsize_8.wordsize].
Module Byte := Make(Wordsize_8).
Strategy 0 [Wordsize_8.wordsize].
Notation byte := Byte.int.
Module Wordsize_64.
Definition wordsize := 64%nat.
Remark wordsize_not_zero: wordsize ≠ 0%nat.
Proof. unfold wordsize; congruence. Qed.
End Wordsize_64.
Strategy opaque [Wordsize_64.wordsize].
Module Int64.
Include Make(Wordsize_64).
Shifts with amount given as a 32-bit integer
Definition iwordsize': Int.int := Int.repr zwordsize.
Definition shl' (x: int) (y: Int.int): int :=
repr (Z.shiftl (unsigned x) (Int.unsigned y)).
Definition shru' (x: int) (y: Int.int): int :=
repr (Z.shiftr (unsigned x) (Int.unsigned y)).
Definition shr' (x: int) (y: Int.int): int :=
repr (Z.shiftr (signed x) (Int.unsigned y)).
Definition rol' (x: int) (y: Int.int): int :=
rol x (repr (Int.unsigned y)).
Definition shrx' (x: int) (y: Int.int): int :=
divs x (shl' one y).
Definition shr_carry' (x: int) (y: Int.int): int :=
if lt x zero && negb (eq (and x (sub (shl' one y) one)) zero)
then one else zero.
Lemma bits_shl':
∀ x y i,
0 ≤ i < zwordsize →
testbit (shl' x y) i =
if zlt i (Int.unsigned y) then false else testbit x (i - Int.unsigned y).
Proof.
intros. unfold shl'. rewrite testbit_repr; auto.
destruct (zlt i (Int.unsigned y)).
apply Z.shiftl_spec_low. auto.
apply Z.shiftl_spec_high. lia. lia.
Qed.
Lemma bits_shru':
∀ x y i,
0 ≤ i < zwordsize →
testbit (shru' x y) i =
if zlt (i + Int.unsigned y) zwordsize then testbit x (i + Int.unsigned y) else false.
Proof.
intros. unfold shru'. rewrite testbit_repr; auto.
rewrite Z.shiftr_spec. fold (testbit x (i + Int.unsigned y)).
destruct (zlt (i + Int.unsigned y) zwordsize).
auto.
apply bits_above; auto.
lia.
Qed.
Lemma bits_shr':
∀ x y i,
0 ≤ i < zwordsize →
testbit (shr' x y) i =
testbit x (if zlt (i + Int.unsigned y) zwordsize then i + Int.unsigned y else zwordsize - 1).
Proof.
intros. unfold shr'. rewrite testbit_repr; auto.
rewrite Z.shiftr_spec. apply bits_signed.
generalize (Int.unsigned_range y); lia.
lia.
Qed.
Lemma shl'_mul_two_p:
∀ x y,
shl' x y = mul x (repr (two_p (Int.unsigned y))).
Proof.
intros. unfold shl', mul. apply eqm_samerepr.
rewrite Zshiftl_mul_two_p. apply eqm_mult. apply eqm_refl. apply eqm_unsigned_repr.
generalize (Int.unsigned_range y); lia.
Qed.
Lemma shl'_one_two_p:
∀ y, shl' one y = repr (two_p (Int.unsigned y)).
Proof.
intros. rewrite shl'_mul_two_p. rewrite mul_commut. rewrite mul_one. auto.
Qed.
Theorem shl'_mul:
∀ x y,
shl' x y = mul x (shl' one y).
Proof.
intros. rewrite shl'_one_two_p. apply shl'_mul_two_p.
Qed.
Theorem shl'_zero:
∀ x, shl' x Int.zero = x.
Proof.
intros. unfold shl'. rewrite Int.unsigned_zero. unfold Z.shiftl.
apply repr_unsigned.
Qed.
Theorem shru'_zero :
∀ x, shru' x Int.zero = x.
Proof.
intros. unfold shru'. rewrite Int.unsigned_zero. unfold Z.shiftr.
apply repr_unsigned.
Qed.
Theorem shr'_zero :
∀ x, shr' x Int.zero = x.
Proof.
intros. unfold shr'. rewrite Int.unsigned_zero. unfold Z.shiftr.
apply repr_signed.
Qed.
Theorem shrx'_zero:
∀ x, shrx' x Int.zero = x.
Proof.
intros. change (shrx' x Int.zero) with (shrx x zero). apply shrx_zero. compute; auto.
Qed.
Theorem shrx'_carry:
∀ x y,
Int.ltu y (Int.repr 63) = true →
shrx' x y = add (shr' x y) (shr_carry' x y).
Proof.
intros. apply Int.ltu_inv in H. change (Int.unsigned (Int.repr 63)) with 63 in H.
set (y1 := Int64.repr (Int.unsigned y)).
assert (U: unsigned y1 = Int.unsigned y).
{ apply unsigned_repr. assert (63 < max_unsigned) by reflexivity. lia. }
transitivity (shrx x y1).
- unfold shrx', shrx, shl', shl. rewrite U; auto.
- rewrite shrx_carry.
+ f_equal.
unfold shr, shr'. rewrite U; auto.
unfold shr_carry, shr_carry', shl, shl'. rewrite U; auto.
+ unfold ltu. apply zlt_true. rewrite U; tauto.
Qed.
Theorem shrx'_shr_2:
∀ x y,
Int.ltu y (Int.repr 63) = true →
shrx' x y = shr' (add x (shru' (shr' x (Int.repr 63)) (Int.sub (Int.repr 64) y))) y.
Proof.
intros.
set (z := repr (Int.unsigned y)).
apply Int.ltu_inv in H. change (Int.unsigned (Int.repr 63)) with 63 in H.
assert (N1: 63 < max_unsigned) by reflexivity.
assert (N2: 63 < Int.max_unsigned) by reflexivity.
assert (A: unsigned z = Int.unsigned y).
{ unfold z; apply unsigned_repr; lia. }
assert (B: unsigned (sub (repr 64) z) = Int.unsigned (Int.sub (Int.repr 64) y)).
{ unfold z. unfold sub, Int.sub.
change (unsigned (repr 64)) with 64.
change (Int.unsigned (Int.repr 64)) with 64.
rewrite (unsigned_repr (Int.unsigned y)) by lia.
rewrite unsigned_repr, Int.unsigned_repr by lia.
auto. }
unfold shrx', shr', shru', shl'.
rewrite <- A.
change (Int.unsigned (Int.repr 63)) with (unsigned (repr 63)).
rewrite <- B.
apply shrx_shr_2.
unfold ltu. apply zlt_true. change (unsigned z < 63). rewrite A; lia.
Qed.
Remark int_ltu_2_inv:
∀ y z,
Int.ltu y iwordsize' = true →
Int.ltu z iwordsize' = true →
Int.unsigned (Int.add y z) ≤ Int.unsigned iwordsize' →
let y' := repr (Int.unsigned y) in
let z' := repr (Int.unsigned z) in
Int.unsigned y = unsigned y'
∧ Int.unsigned z = unsigned z'
∧ ltu y' iwordsize = true
∧ ltu z' iwordsize = true
∧ Int.unsigned (Int.add y z) = unsigned (add y' z')
∧ add y' z' = repr (Int.unsigned (Int.add y z)).
Proof.
intros. apply Int.ltu_inv in H. apply Int.ltu_inv in H0.
change (Int.unsigned iwordsize') with 64 in ×.
assert (128 < max_unsigned) by reflexivity.
assert (128 < Int.max_unsigned) by reflexivity.
assert (Y: unsigned y' = Int.unsigned y) by (apply unsigned_repr; lia).
assert (Z: unsigned z' = Int.unsigned z) by (apply unsigned_repr; lia).
assert (P: Int.unsigned (Int.add y z) = unsigned (add y' z')).
{ unfold Int.add. rewrite Int.unsigned_repr by lia.
unfold add. rewrite unsigned_repr by lia. congruence. }
intuition auto.
apply zlt_true. rewrite Y; auto.
apply zlt_true. rewrite Z; auto.
rewrite P. rewrite repr_unsigned. auto.
Qed.
Theorem or_ror':
∀ x y z,
Int.ltu y iwordsize' = true →
Int.ltu z iwordsize' = true →
Int.add y z = iwordsize' →
ror x (repr (Int.unsigned z)) = or (shl' x y) (shru' x z).
Proof.
intros. destruct (int_ltu_2_inv y z) as (A & B & C & D & E & F); auto. rewrite H1; lia.
replace (shl' x y) with (shl x (repr (Int.unsigned y))).
replace (shru' x z) with (shru x (repr (Int.unsigned z))).
apply or_ror; auto. rewrite F, H1. reflexivity.
unfold shru, shru'; rewrite <- B; auto.
unfold shl, shl'; rewrite <- A; auto.
Qed.
Theorem shl'_shl':
∀ x y z,
Int.ltu y iwordsize' = true →
Int.ltu z iwordsize' = true →
Int.ltu (Int.add y z) iwordsize' = true →
shl' (shl' x y) z = shl' x (Int.add y z).
Proof.
intros. apply Int.ltu_inv in H1.
destruct (int_ltu_2_inv y z) as (A & B & C & D & E & F); auto. lia.
set (y' := repr (Int.unsigned y)) in ×.
set (z' := repr (Int.unsigned z)) in ×.
replace (shl' x y) with (shl x y').
replace (shl' (shl x y') z) with (shl (shl x y') z').
replace (shl' x (Int.add y z)) with (shl x (add y' z')).
apply shl_shl; auto. apply zlt_true. rewrite <- E.
change (unsigned iwordsize) with zwordsize. tauto.
unfold shl, shl'. rewrite E; auto.
unfold shl at 1, shl'. rewrite <- B; auto.
unfold shl, shl'; rewrite <- A; auto.
Qed.
Theorem shru'_shru':
∀ x y z,
Int.ltu y iwordsize' = true →
Int.ltu z iwordsize' = true →
Int.ltu (Int.add y z) iwordsize' = true →
shru' (shru' x y) z = shru' x (Int.add y z).
Proof.
intros. apply Int.ltu_inv in H1.
destruct (int_ltu_2_inv y z) as (A & B & C & D & E & F); auto. lia.
set (y' := repr (Int.unsigned y)) in ×.
set (z' := repr (Int.unsigned z)) in ×.
replace (shru' x y) with (shru x y').
replace (shru' (shru x y') z) with (shru (shru x y') z').
replace (shru' x (Int.add y z)) with (shru x (add y' z')).
apply shru_shru; auto. apply zlt_true. rewrite <- E.
change (unsigned iwordsize) with zwordsize. tauto.
unfold shru, shru'. rewrite E; auto.
unfold shru at 1, shru'. rewrite <- B; auto.
unfold shru, shru'; rewrite <- A; auto.
Qed.
Theorem shr'_shr':
∀ x y z,
Int.ltu y iwordsize' = true →
Int.ltu z iwordsize' = true →
Int.ltu (Int.add y z) iwordsize' = true →
shr' (shr' x y) z = shr' x (Int.add y z).
Proof.
intros. apply Int.ltu_inv in H1.
destruct (int_ltu_2_inv y z) as (A & B & C & D & E & F); auto. lia.
set (y' := repr (Int.unsigned y)) in ×.
set (z' := repr (Int.unsigned z)) in ×.
replace (shr' x y) with (shr x y').
replace (shr' (shr x y') z) with (shr (shr x y') z').
replace (shr' x (Int.add y z)) with (shr x (add y' z')).
apply shr_shr; auto. apply zlt_true. rewrite <- E.
change (unsigned iwordsize) with zwordsize. tauto.
unfold shr, shr'. rewrite E; auto.
unfold shr at 1, shr'. rewrite <- B; auto.
unfold shr, shr'; rewrite <- A; auto.
Qed.
Theorem shru'_shl':
∀ x y z, Int.ltu y iwordsize' = true → Int.ltu z iwordsize' = true →
shru' (shl' x y) z =
if Int.ltu z y then shl' (zero_ext (zwordsize - Int.unsigned y) x) (Int.sub y z)
else zero_ext (zwordsize - Int.unsigned z) (shru' x (Int.sub z y)).
Proof.
intros. apply Int.ltu_inv in H; apply Int.ltu_inv in H0.
change (Int.unsigned iwordsize') with zwordsize in ×.
unfold Int.ltu. set (Y := Int.unsigned y) in *; set (Z := Int.unsigned z) in ×.
apply same_bits_eq; intros. rewrite bits_shru' by auto. fold Z.
destruct (zlt Z Y).
- assert (A: Int.unsigned (Int.sub y z) = Y - Z).
{ apply Int.unsigned_repr. assert (zwordsize < Int.max_unsigned) by reflexivity. lia. }
symmetry; rewrite bits_shl', A by lia.
destruct (zlt (i + Z) zwordsize).
+ rewrite bits_shl' by lia. fold Y.
destruct (zlt i (Y - Z)); [rewrite zlt_true by lia|rewrite zlt_false by lia]; auto.
rewrite bits_zero_ext by lia. rewrite zlt_true by lia. f_equal; lia.
+ rewrite bits_zero_ext by lia. rewrite ! zlt_false by lia. auto.
- assert (A: Int.unsigned (Int.sub z y) = Z - Y).
{ apply Int.unsigned_repr. assert (zwordsize < Int.max_unsigned) by reflexivity. lia. }
rewrite bits_zero_ext, bits_shru', A by lia.
destruct (zlt (i + Z) zwordsize); [rewrite zlt_true by lia|rewrite zlt_false by lia]; auto.
rewrite bits_shl' by lia. fold Y.
destruct (zlt (i + Z) Y).
+ rewrite zlt_false by lia. auto.
+ rewrite zlt_true by lia. f_equal; lia.
Qed.
Theorem shr'_shl':
∀ x y z, Int.ltu y iwordsize' = true → Int.ltu z iwordsize' = true →
shr' (shl' x y) z =
if Int.ltu z y then shl' (sign_ext (zwordsize - Int.unsigned y) x) (Int.sub y z)
else sign_ext (zwordsize - Int.unsigned z) (shr' x (Int.sub z y)).
Proof.
intros. apply Int.ltu_inv in H; apply Int.ltu_inv in H0.
change (Int.unsigned iwordsize') with zwordsize in ×.
unfold Int.ltu. set (Y := Int.unsigned y) in *; set (Z := Int.unsigned z) in ×.
apply same_bits_eq; intros. rewrite bits_shr' by auto. fold Z.
rewrite bits_shl' by (destruct (zlt (i + Z) zwordsize); lia). fold Y.
destruct (zlt Z Y).
- assert (A: Int.unsigned (Int.sub y z) = Y - Z).
{ apply Int.unsigned_repr. assert (zwordsize < Int.max_unsigned) by reflexivity. lia. }
rewrite bits_shl', A by lia.
destruct (zlt i (Y - Z)).
+ apply zlt_true. destruct (zlt (i + Z) zwordsize); lia.
+ rewrite zlt_false by (destruct (zlt (i + Z) zwordsize); lia).
rewrite bits_sign_ext by lia. f_equal.
destruct (zlt (i + Z) zwordsize).
rewrite zlt_true by lia. lia.
rewrite zlt_false by lia. lia.
- assert (A: Int.unsigned (Int.sub z y) = Z - Y).
{ apply Int.unsigned_repr. assert (zwordsize < Int.max_unsigned) by reflexivity. lia. }
rewrite bits_sign_ext by lia.
rewrite bits_shr' by (destruct (zlt i (zwordsize - Z)); lia).
rewrite A. rewrite zlt_false by (destruct (zlt (i + Z) zwordsize); lia).
f_equal. destruct (zlt i (zwordsize - Z)).
+ rewrite ! zlt_true by lia. lia.
+ rewrite ! zlt_false by lia. rewrite zlt_true by lia. lia.
Qed.
Lemma shl'_zero_ext:
∀ n m x, 0 ≤ n →
shl' (zero_ext n x) m = zero_ext (n + Int.unsigned m) (shl' x m).
Proof.
intros. apply same_bits_eq; intros.
rewrite bits_zero_ext, ! bits_shl' by lia.
destruct (zlt i (Int.unsigned m)).
- rewrite zlt_true by lia; auto.
- rewrite bits_zero_ext by lia.
destruct (zlt (i - Int.unsigned m) n); [rewrite zlt_true by lia|rewrite zlt_false by lia]; auto.
Qed.
Lemma shl'_sign_ext:
∀ n m x, 0 < n →
shl' (sign_ext n x) m = sign_ext (n + Int.unsigned m) (shl' x m).
Proof.
intros. generalize (Int.unsigned_range m); intros.
apply same_bits_eq; intros.
rewrite bits_sign_ext, ! bits_shl' by lia.
destruct (zlt i (n + Int.unsigned m)).
- rewrite bits_shl' by auto. destruct (zlt i (Int.unsigned m)); auto.
rewrite bits_sign_ext by lia. f_equal. apply zlt_true. lia.
- rewrite zlt_false by lia. rewrite bits_shl' by lia. rewrite zlt_false by lia.
rewrite bits_sign_ext by lia. f_equal. rewrite zlt_false by lia. lia.
Qed.
Lemma shru'_zero_ext:
∀ n m x, 0 ≤ n →
shru' (zero_ext (n + Int.unsigned m) x) m = zero_ext n (shru' x m).
Proof.
intros. generalize (Int.unsigned_range m); intros.
bit_solve; [|lia]. rewrite bits_shru', bits_zero_ext, bits_shru' by lia.
destruct (zlt (i + Int.unsigned m) zwordsize).
× destruct (zlt i n); [rewrite zlt_true by lia|rewrite zlt_false by lia]; auto.
× destruct (zlt i n); auto.
Qed.
Lemma shru'_zero_ext_0:
∀ n m x, n ≤ Int.unsigned m →
shru' (zero_ext n x) m = zero.
Proof.
intros. generalize (Int.unsigned_range m); intros.
bit_solve. rewrite bits_shru', bits_zero_ext by lia.
destruct (zlt (i + Int.unsigned m) zwordsize); auto.
apply zlt_false. lia.
Qed.
Lemma shr'_sign_ext:
∀ n m x, 0 < n → n + Int.unsigned m < zwordsize →
shr' (sign_ext (n + Int.unsigned m) x) m = sign_ext n (shr' x m).
Proof.
intros. generalize (Int.unsigned_range m); intros.
apply same_bits_eq; intros.
rewrite bits_sign_ext, bits_shr' by auto.
rewrite bits_sign_ext, bits_shr'.
- f_equal.
destruct (zlt i n), (zlt (i + Int.unsigned m) zwordsize).
+ apply zlt_true; lia.
+ apply zlt_true; lia.
+ rewrite zlt_false by lia. rewrite zlt_true by lia. lia.
+ rewrite zlt_false by lia. rewrite zlt_true by lia. lia.
- destruct (zlt i n); lia.
- destruct (zlt (i + Int.unsigned m) zwordsize); lia.
Qed.
Lemma zero_ext_shru'_min:
∀ s x n, Int.ltu n iwordsize' = true →
zero_ext s (shru' x n) = zero_ext (Z.min s (zwordsize - Int.unsigned n)) (shru' x n).
Proof.
intros. apply Int.ltu_inv in H. change (Int.unsigned iwordsize') with zwordsize in H.
apply Z.min_case_strong; intros; auto.
bit_solve; try lia. rewrite ! bits_shru' by lia.
destruct (zlt i (zwordsize - Int.unsigned n)).
rewrite zlt_true by lia. auto.
destruct (zlt i s); auto. rewrite zlt_false by lia; auto.
Qed.
Lemma sign_ext_shr'_min:
∀ s x n, Int.ltu n iwordsize' = true →
sign_ext s (shr' x n) = sign_ext (Z.min s (zwordsize - Int.unsigned n)) (shr' x n).
Proof.
intros. apply Int.ltu_inv in H. change (Int.unsigned iwordsize') with zwordsize in H.
rewrite Z.min_comm.
destruct (Z.min_spec (zwordsize - Int.unsigned n) s) as [[A B] | [A B]]; rewrite B; auto.
apply same_bits_eq; intros. rewrite ! bits_sign_ext by auto.
destruct (zlt i (zwordsize - Int.unsigned n)).
rewrite zlt_true by lia. auto.
assert (C: testbit (shr' x n) (zwordsize - Int.unsigned n - 1) = testbit x (zwordsize - 1)).
{ rewrite bits_shr' by lia. rewrite zlt_true by lia. f_equal; lia. }
rewrite C. destruct (zlt i s); rewrite bits_shr' by lia.
rewrite zlt_false by lia. auto.
rewrite zlt_false by lia. auto.
Qed.
Lemma shl'_zero_ext_min:
∀ s x n, Int.ltu n iwordsize' = true →
shl' (zero_ext s x) n = shl' (zero_ext (Z.min s (zwordsize - Int.unsigned n)) x) n.
Proof.
intros. apply Int.ltu_inv in H. change (Int.unsigned iwordsize') with zwordsize in H.
apply Z.min_case_strong; intros; auto.
apply same_bits_eq; intros. rewrite ! bits_shl' by auto.
destruct (zlt i (Int.unsigned n)); auto.
rewrite ! bits_zero_ext by lia.
destruct (zlt (i - Int.unsigned n) s).
rewrite zlt_true by lia; auto.
rewrite zlt_false by lia; auto.
Qed.
Lemma shl'_sign_ext_min:
∀ s x n, Int.ltu n iwordsize' = true →
shl' (sign_ext s x) n = shl' (sign_ext (Z.min s (zwordsize - Int.unsigned n)) x) n.
Proof.
intros. apply Int.ltu_inv in H. change (Int.unsigned iwordsize') with zwordsize in H.
rewrite Z.min_comm.
destruct (Z.min_spec (zwordsize - Int.unsigned n) s) as [[A B] | [A B]]; rewrite B; auto.
apply same_bits_eq; intros. rewrite ! bits_shl' by auto.
destruct (zlt i (Int.unsigned n)); auto.
rewrite ! bits_sign_ext by lia. f_equal.
destruct (zlt (i - Int.unsigned n) s).
rewrite zlt_true by lia; auto.
extlia.
Qed.
Powers of two with exponents given as 32-bit ints
Definition one_bits' (x: int) : list Int.int :=
List.map Int.repr (Z_one_bits wordsize (unsigned x) 0).
Definition is_power2' (x: int) : option Int.int :=
match Z_one_bits wordsize (unsigned x) 0 with
| i :: nil ⇒ Some (Int.repr i)
| _ ⇒ None
end.
Theorem one_bits'_range:
∀ x i, In i (one_bits' x) → Int.ltu i iwordsize' = true.
Proof.
intros.
destruct (list_in_map_inv _ _ _ H) as [i0 [EQ IN]].
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. lia.
assert (zwordsize < Int.max_unsigned) by reflexivity. lia.
Qed.
Fixpoint int_of_one_bits' (l: list Int.int) : int :=
match l with
| nil ⇒ zero
| a :: b ⇒ add (shl' one a) (int_of_one_bits' b)
end.
Theorem one_bits'_decomp:
∀ x, x = int_of_one_bits' (one_bits' x).
Proof.
assert (REC: ∀ l,
(∀ i, In i l → 0 ≤ i < zwordsize) →
int_of_one_bits' (List.map Int.repr l) = repr (powerserie l)).
{ induction l; simpl; intros.
- auto.
- rewrite IHl by eauto. apply eqm_samerepr; apply eqm_add.
+ rewrite shl'_one_two_p. rewrite Int.unsigned_repr. apply eqm_sym; apply eqm_unsigned_repr.
exploit (H a). auto. assert (zwordsize < Int.max_unsigned) by reflexivity. lia.
+ apply eqm_sym; apply eqm_unsigned_repr.
}
intros. rewrite <- (repr_unsigned x) at 1. unfold one_bits'. rewrite REC.
rewrite <- Z_one_bits_powerserie. auto. apply unsigned_range.
apply Z_one_bits_range.
Qed.
Lemma is_power2'_rng:
∀ n logn,
is_power2' n = Some logn →
0 ≤ Int.unsigned logn < zwordsize.
Proof.
unfold is_power2'; intros n logn P2.
destruct (Z_one_bits wordsize (unsigned n) 0) as [ | i [ | ? ?]] eqn:B; inv P2.
assert (0 ≤ i < zwordsize).
{ apply Z_one_bits_range with (unsigned n). rewrite B; auto with coqlib. }
rewrite Int.unsigned_repr. auto.
assert (zwordsize < Int.max_unsigned) by reflexivity.
lia.
Qed.
Theorem is_power2'_range:
∀ n logn,
is_power2' n = Some logn → Int.ltu logn iwordsize' = true.
Proof.
intros. unfold Int.ltu. change (Int.unsigned iwordsize') with zwordsize.
apply zlt_true. generalize (is_power2'_rng _ _ H). tauto.
Qed.
Lemma is_power2'_correct:
∀ n logn,
is_power2' n = Some logn →
unsigned n = two_p (Int.unsigned logn).
Proof.
unfold is_power2'; intros.
destruct (Z_one_bits wordsize (unsigned n) 0) as [ | i [ | ? ?]] eqn:B; inv H.
rewrite (Z_one_bits_powerserie wordsize (unsigned n)) by (apply unsigned_range).
rewrite Int.unsigned_repr. rewrite B; simpl. lia.
assert (0 ≤ i < zwordsize).
{ apply Z_one_bits_range with (unsigned n). rewrite B; auto with coqlib. }
assert (zwordsize < Int.max_unsigned) by reflexivity.
lia.
Qed.
Theorem mul_pow2':
∀ x n logn,
is_power2' n = Some logn →
mul x n = shl' x logn.
Proof.
intros. rewrite shl'_mul. f_equal. rewrite shl'_one_two_p.
rewrite <- (repr_unsigned n). f_equal. apply is_power2'_correct; auto.
Qed.
Theorem divu_pow2':
∀ x n logn,
is_power2' n = Some logn →
divu x n = shru' x logn.
Proof.
intros. generalize (is_power2'_correct n logn H). intro.
symmetry. unfold divu. rewrite H0. unfold shru'. rewrite Zshiftr_div_two_p. auto.
eapply is_power2'_rng; eauto.
Qed.
Decomposing 64-bit ints as pairs of 32-bit ints
Definition loword (n: int) : Int.int := Int.repr (unsigned n).
Definition hiword (n: int) : Int.int := Int.repr (unsigned (shru n (repr Int.zwordsize))).
Definition ofwords (hi lo: Int.int) : int :=
or (shl (repr (Int.unsigned hi)) (repr Int.zwordsize)) (repr (Int.unsigned lo)).
Lemma bits_loword:
∀ n i, 0 ≤ i < Int.zwordsize → Int.testbit (loword n) i = testbit n i.
Proof.
intros. unfold loword. rewrite Int.testbit_repr; auto.
Qed.
Lemma bits_hiword:
∀ n i, 0 ≤ i < Int.zwordsize → Int.testbit (hiword n) i = testbit n (i + Int.zwordsize).
Proof.
intros. unfold hiword. rewrite Int.testbit_repr; auto.
assert (zwordsize = 2 × Int.zwordsize) by reflexivity.
fold (testbit (shru n (repr Int.zwordsize)) i). rewrite bits_shru.
change (unsigned (repr Int.zwordsize)) with Int.zwordsize.
apply zlt_true. lia. lia.
Qed.
Lemma bits_ofwords:
∀ hi lo i, 0 ≤ i < zwordsize →
testbit (ofwords hi lo) i =
if zlt i Int.zwordsize then Int.testbit lo i else Int.testbit hi (i - Int.zwordsize).
Proof.
intros. unfold ofwords. rewrite bits_or; auto. rewrite bits_shl; auto.
change (unsigned (repr Int.zwordsize)) with Int.zwordsize.
assert (zwordsize = 2 × Int.zwordsize) by reflexivity.
destruct (zlt i Int.zwordsize).
rewrite testbit_repr; auto.
rewrite !testbit_repr; auto.
fold (Int.testbit lo i). rewrite Int.bits_above. apply orb_false_r. auto.
lia.
Qed.
Lemma lo_ofwords:
∀ hi lo, loword (ofwords hi lo) = lo.
Proof.
intros. apply Int.same_bits_eq; intros.
rewrite bits_loword; auto. rewrite bits_ofwords. apply zlt_true. lia.
assert (zwordsize = 2 × Int.zwordsize) by reflexivity. lia.
Qed.
Lemma hi_ofwords:
∀ hi lo, hiword (ofwords hi lo) = hi.
Proof.
intros. apply Int.same_bits_eq; intros.
rewrite bits_hiword; auto. rewrite bits_ofwords.
rewrite zlt_false. f_equal. lia. lia.
assert (zwordsize = 2 × Int.zwordsize) by reflexivity. lia.
Qed.
Lemma ofwords_recompose:
∀ n, ofwords (hiword n) (loword n) = n.
Proof.
intros. apply same_bits_eq; intros. rewrite bits_ofwords; auto.
destruct (zlt i Int.zwordsize).
apply bits_loword. lia.
rewrite bits_hiword. f_equal. lia.
assert (zwordsize = 2 × Int.zwordsize) by reflexivity. lia.
Qed.
Lemma ofwords_add:
∀ lo hi, ofwords hi lo = repr (Int.unsigned hi × two_p 32 + Int.unsigned lo).
Proof.
intros. unfold ofwords. rewrite shifted_or_is_add.
apply eqm_samerepr. apply eqm_add. apply eqm_mult.
apply eqm_sym; apply eqm_unsigned_repr.
apply eqm_refl.
apply eqm_sym; apply eqm_unsigned_repr.
change Int.zwordsize with 32; change zwordsize with 64; lia.
rewrite unsigned_repr. generalize (Int.unsigned_range lo). intros [A B]. exact B.
assert (Int.max_unsigned < max_unsigned) by (compute; auto).
generalize (Int.unsigned_range_2 lo); lia.
Qed.
Lemma ofwords_add':
∀ lo hi, unsigned (ofwords hi lo) = Int.unsigned hi × two_p 32 + Int.unsigned lo.
Proof.
intros. rewrite ofwords_add. apply unsigned_repr.
generalize (Int.unsigned_range hi) (Int.unsigned_range lo).
change (two_p 32) with Int.modulus.
change Int.modulus with 4294967296.
change max_unsigned with 18446744073709551615.
lia.
Qed.
Remark eqm_mul_2p32:
∀ x y, Int.eqm x y → eqm (x × two_p 32) (y × two_p 32).
Proof.
intros. destruct H as [k EQ]. ∃ k. rewrite EQ.
change Int.modulus with (two_p 32).
change modulus with (two_p 32 × two_p 32).
ring.
Qed.
Lemma ofwords_add'':
∀ lo hi, signed (ofwords hi lo) = Int.signed hi × two_p 32 + Int.unsigned lo.
Proof.
intros. rewrite ofwords_add.
replace (repr (Int.unsigned hi × two_p 32 + Int.unsigned lo))
with (repr (Int.signed hi × two_p 32 + Int.unsigned lo)).
apply signed_repr.
generalize (Int.signed_range hi) (Int.unsigned_range lo).
change (two_p 32) with Int.modulus.
change min_signed with (Int.min_signed × Int.modulus).
change max_signed with (Int.max_signed × Int.modulus + Int.modulus - 1).
change Int.modulus with 4294967296.
lia.
apply eqm_samerepr. apply eqm_add. apply eqm_mul_2p32. apply Int.eqm_signed_unsigned. apply eqm_refl.
Qed.
Expressing 64-bit operations in terms of 32-bit operations
Lemma decompose_bitwise_binop:
∀ f f64 f32 xh xl yh yl,
(∀ x y i, 0 ≤ i < zwordsize → testbit (f64 x y) i = f (testbit x i) (testbit y i)) →
(∀ x y i, 0 ≤ i < Int.zwordsize → Int.testbit (f32 x y) i = f (Int.testbit x i) (Int.testbit y i)) →
f64 (ofwords xh xl) (ofwords yh yl) = ofwords (f32 xh yh) (f32 xl yl).
Proof.
intros. apply Int64.same_bits_eq; intros.
rewrite H by auto. rewrite ! bits_ofwords by auto.
assert (zwordsize = 2 × Int.zwordsize) by reflexivity.
destruct (zlt i Int.zwordsize); rewrite H0 by lia; auto.
Qed.
Lemma decompose_and:
∀ xh xl yh yl,
and (ofwords xh xl) (ofwords yh yl) = ofwords (Int.and xh yh) (Int.and xl yl).
Proof.
intros. apply decompose_bitwise_binop with andb.
apply bits_and. apply Int.bits_and.
Qed.
Lemma decompose_or:
∀ xh xl yh yl,
or (ofwords xh xl) (ofwords yh yl) = ofwords (Int.or xh yh) (Int.or xl yl).
Proof.
intros. apply decompose_bitwise_binop with orb.
apply bits_or. apply Int.bits_or.
Qed.
Lemma decompose_xor:
∀ xh xl yh yl,
xor (ofwords xh xl) (ofwords yh yl) = ofwords (Int.xor xh yh) (Int.xor xl yl).
Proof.
intros. apply decompose_bitwise_binop with xorb.
apply bits_xor. apply Int.bits_xor.
Qed.
Lemma decompose_not:
∀ xh xl,
not (ofwords xh xl) = ofwords (Int.not xh) (Int.not xl).
Proof.
intros. unfold not, Int.not. rewrite <- decompose_xor. f_equal.
apply (Int64.eq_spec mone (ofwords Int.mone Int.mone)).
Qed.
Lemma decompose_shl_1:
∀ xh xl y,
0 ≤ Int.unsigned y < Int.zwordsize →
shl' (ofwords xh xl) y =
ofwords (Int.or (Int.shl xh y) (Int.shru xl (Int.sub Int.iwordsize y)))
(Int.shl xl y).
Proof.
intros.
assert (Int.unsigned (Int.sub Int.iwordsize y) = Int.zwordsize - Int.unsigned y).
{ unfold Int.sub. rewrite Int.unsigned_repr. auto.
rewrite Int.unsigned_repr_wordsize. generalize Int.wordsize_max_unsigned; lia. }
assert (zwordsize = 2 × Int.zwordsize) by reflexivity.
apply Int64.same_bits_eq; intros.
rewrite bits_shl' by auto. symmetry. rewrite bits_ofwords by auto.
destruct (zlt i Int.zwordsize). rewrite Int.bits_shl by lia.
destruct (zlt i (Int.unsigned y)). auto.
rewrite bits_ofwords by lia. rewrite zlt_true by lia. auto.
rewrite zlt_false by lia. rewrite bits_ofwords by lia.
rewrite Int.bits_or by lia. rewrite Int.bits_shl by lia.
rewrite Int.bits_shru by lia. rewrite H0.
destruct (zlt (i - Int.unsigned y) (Int.zwordsize)).
rewrite zlt_true by lia. rewrite zlt_true by lia.
rewrite orb_false_l. f_equal. lia.
rewrite zlt_false by lia. rewrite zlt_false by lia.
rewrite orb_false_r. f_equal. lia.
Qed.
Lemma decompose_shl_2:
∀ xh xl y,
Int.zwordsize ≤ Int.unsigned y < zwordsize →
shl' (ofwords xh xl) y =
ofwords (Int.shl xl (Int.sub y Int.iwordsize)) Int.zero.
Proof.
intros.
assert (zwordsize = 2 × Int.zwordsize) by reflexivity.
assert (Int.unsigned (Int.sub y Int.iwordsize) = Int.unsigned y - Int.zwordsize).
{ unfold Int.sub. rewrite Int.unsigned_repr. auto.
rewrite Int.unsigned_repr_wordsize. generalize (Int.unsigned_range_2 y). lia. }
apply Int64.same_bits_eq; intros.
rewrite bits_shl' by auto. symmetry. rewrite bits_ofwords by auto.
destruct (zlt i Int.zwordsize). rewrite zlt_true by lia. apply Int.bits_zero.
rewrite Int.bits_shl by lia.
destruct (zlt i (Int.unsigned y)).
rewrite zlt_true by lia. auto.
rewrite zlt_false by lia.
rewrite bits_ofwords by lia. rewrite zlt_true by lia. f_equal. lia.
Qed.
Lemma decompose_shru_1:
∀ xh xl y,
0 ≤ Int.unsigned y < Int.zwordsize →
shru' (ofwords xh xl) y =
ofwords (Int.shru xh y)
(Int.or (Int.shru xl y) (Int.shl xh (Int.sub Int.iwordsize y))).
Proof.
intros.
assert (Int.unsigned (Int.sub Int.iwordsize y) = Int.zwordsize - Int.unsigned y).
{ unfold Int.sub. rewrite Int.unsigned_repr. auto.
rewrite Int.unsigned_repr_wordsize. generalize Int.wordsize_max_unsigned; lia. }
assert (zwordsize = 2 × Int.zwordsize) by reflexivity.
apply Int64.same_bits_eq; intros.
rewrite bits_shru' by auto. symmetry. rewrite bits_ofwords by auto.
destruct (zlt i Int.zwordsize).
rewrite zlt_true by lia.
rewrite bits_ofwords by lia.
rewrite Int.bits_or by lia. rewrite Int.bits_shl by lia.
rewrite Int.bits_shru by lia. rewrite H0.
destruct (zlt (i + Int.unsigned y) (Int.zwordsize)).
rewrite zlt_true by lia.
rewrite orb_false_r. auto.
rewrite zlt_false by lia.
rewrite orb_false_l. f_equal. lia.
rewrite Int.bits_shru by lia.
destruct (zlt (i + Int.unsigned y) zwordsize).
rewrite bits_ofwords by lia.
rewrite zlt_true by lia. rewrite zlt_false by lia. f_equal. lia.
rewrite zlt_false by lia. auto.
Qed.
Lemma decompose_shru_2:
∀ xh xl y,
Int.zwordsize ≤ Int.unsigned y < zwordsize →
shru' (ofwords xh xl) y =
ofwords Int.zero (Int.shru xh (Int.sub y Int.iwordsize)).
Proof.
intros.
assert (zwordsize = 2 × Int.zwordsize) by reflexivity.
assert (Int.unsigned (Int.sub y Int.iwordsize) = Int.unsigned y - Int.zwordsize).
{ unfold Int.sub. rewrite Int.unsigned_repr. auto.
rewrite Int.unsigned_repr_wordsize. generalize (Int.unsigned_range_2 y). lia. }
apply Int64.same_bits_eq; intros.
rewrite bits_shru' by auto. symmetry. rewrite bits_ofwords by auto.
destruct (zlt i Int.zwordsize).
rewrite Int.bits_shru by lia. rewrite H1.
destruct (zlt (i + Int.unsigned y) zwordsize).
rewrite zlt_true by lia. rewrite bits_ofwords by lia.
rewrite zlt_false by lia. f_equal; lia.
rewrite zlt_false by lia. auto.
rewrite zlt_false by lia. apply Int.bits_zero.
Qed.
Lemma decompose_shr_1:
∀ xh xl y,
0 ≤ Int.unsigned y < Int.zwordsize →
shr' (ofwords xh xl) y =
ofwords (Int.shr xh y)
(Int.or (Int.shru xl y) (Int.shl xh (Int.sub Int.iwordsize y))).
Proof.
intros.
assert (Int.unsigned (Int.sub Int.iwordsize y) = Int.zwordsize - Int.unsigned y).
{ unfold Int.sub. rewrite Int.unsigned_repr. auto.
rewrite Int.unsigned_repr_wordsize. generalize Int.wordsize_max_unsigned; lia. }
assert (zwordsize = 2 × Int.zwordsize) by reflexivity.
apply Int64.same_bits_eq; intros.
rewrite bits_shr' by auto. symmetry. rewrite bits_ofwords by auto.
destruct (zlt i Int.zwordsize).
rewrite zlt_true by lia.
rewrite bits_ofwords by lia.
rewrite Int.bits_or by lia. rewrite Int.bits_shl by lia.
rewrite Int.bits_shru by lia. rewrite H0.
destruct (zlt (i + Int.unsigned y) (Int.zwordsize)).
rewrite zlt_true by lia.
rewrite orb_false_r. auto.
rewrite zlt_false by lia.
rewrite orb_false_l. f_equal. lia.
rewrite Int.bits_shr by lia.
destruct (zlt (i + Int.unsigned y) zwordsize).
rewrite bits_ofwords by lia.
rewrite zlt_true by lia. rewrite zlt_false by lia. f_equal. lia.
rewrite zlt_false by lia. rewrite bits_ofwords by lia.
rewrite zlt_false by lia. f_equal.
Qed.
Lemma decompose_shr_2:
∀ xh xl y,
Int.zwordsize ≤ Int.unsigned y < zwordsize →
shr' (ofwords xh xl) y =
ofwords (Int.shr xh (Int.sub Int.iwordsize Int.one))
(Int.shr xh (Int.sub y Int.iwordsize)).
Proof.
intros.
assert (zwordsize = 2 × Int.zwordsize) by reflexivity.
assert (Int.unsigned (Int.sub y Int.iwordsize) = Int.unsigned y - Int.zwordsize).
{ unfold Int.sub. rewrite Int.unsigned_repr. auto.
rewrite Int.unsigned_repr_wordsize. generalize (Int.unsigned_range_2 y). lia. }
apply Int64.same_bits_eq; intros.
rewrite bits_shr' by auto. symmetry. rewrite bits_ofwords by auto.
destruct (zlt i Int.zwordsize).
rewrite Int.bits_shr by lia. rewrite H1.
destruct (zlt (i + Int.unsigned y) zwordsize).
rewrite zlt_true by lia. rewrite bits_ofwords by lia.
rewrite zlt_false by lia. f_equal; lia.
rewrite zlt_false by lia. rewrite bits_ofwords by lia.
rewrite zlt_false by lia. auto.
rewrite Int.bits_shr by lia.
change (Int.unsigned (Int.sub Int.iwordsize Int.one)) with (Int.zwordsize - 1).
destruct (zlt (i + Int.unsigned y) zwordsize);
rewrite bits_ofwords by lia.
symmetry. rewrite zlt_false by lia. f_equal.
destruct (zlt (i - Int.zwordsize + (Int.zwordsize - 1)) Int.zwordsize); lia.
symmetry. rewrite zlt_false by lia. f_equal.
destruct (zlt (i - Int.zwordsize + (Int.zwordsize - 1)) Int.zwordsize); lia.
Qed.
Lemma decompose_add:
∀ xh xl yh yl,
add (ofwords xh xl) (ofwords yh yl) =
ofwords (Int.add (Int.add xh yh) (Int.add_carry xl yl Int.zero))
(Int.add xl yl).
Proof.
intros. symmetry. rewrite ofwords_add. rewrite add_unsigned.
apply eqm_samerepr.
rewrite ! ofwords_add'. rewrite (Int.unsigned_add_carry xl yl).
set (cc := Int.add_carry xl yl Int.zero).
set (Xl := Int.unsigned xl); set (Xh := Int.unsigned xh);
set (Yl := Int.unsigned yl); set (Yh := Int.unsigned yh).
change Int.modulus with (two_p 32).
replace (Xh × two_p 32 + Xl + (Yh × two_p 32 + Yl))
with ((Xh + Yh) × two_p 32 + (Xl + Yl)) by ring.
replace (Int.unsigned (Int.add (Int.add xh yh) cc) × two_p 32 +
(Xl + Yl - Int.unsigned cc × two_p 32))
with ((Int.unsigned (Int.add (Int.add xh yh) cc) - Int.unsigned cc) × two_p 32
+ (Xl + Yl)) by ring.
apply eqm_add. 2: apply eqm_refl. apply eqm_mul_2p32.
replace (Xh + Yh) with ((Xh + Yh + Int.unsigned cc) - Int.unsigned cc) by ring.
apply Int.eqm_sub. 2: apply Int.eqm_refl.
apply Int.eqm_unsigned_repr_l. apply Int.eqm_add. 2: apply Int.eqm_refl.
apply Int.eqm_unsigned_repr_l. apply Int.eqm_refl.
Qed.
Lemma decompose_sub:
∀ xh xl yh yl,
sub (ofwords xh xl) (ofwords yh yl) =
ofwords (Int.sub (Int.sub xh yh) (Int.sub_borrow xl yl Int.zero))
(Int.sub xl yl).
Proof.
intros. symmetry. rewrite ofwords_add.
apply eqm_samerepr.
rewrite ! ofwords_add'. rewrite (Int.unsigned_sub_borrow xl yl).
set (bb := Int.sub_borrow xl yl Int.zero).
set (Xl := Int.unsigned xl); set (Xh := Int.unsigned xh);
set (Yl := Int.unsigned yl); set (Yh := Int.unsigned yh).
change Int.modulus with (two_p 32).
replace (Xh × two_p 32 + Xl - (Yh × two_p 32 + Yl))
with ((Xh - Yh) × two_p 32 + (Xl - Yl)) by ring.
replace (Int.unsigned (Int.sub (Int.sub xh yh) bb) × two_p 32 +
(Xl - Yl + Int.unsigned bb × two_p 32))
with ((Int.unsigned (Int.sub (Int.sub xh yh) bb) + Int.unsigned bb) × two_p 32
+ (Xl - Yl)) by ring.
apply eqm_add. 2: apply eqm_refl. apply eqm_mul_2p32.
replace (Xh - Yh) with ((Xh - Yh - Int.unsigned bb) + Int.unsigned bb) by ring.
apply Int.eqm_add. 2: apply Int.eqm_refl.
apply Int.eqm_unsigned_repr_l. apply Int.eqm_add. 2: apply Int.eqm_refl.
apply Int.eqm_unsigned_repr_l. apply Int.eqm_refl.
Qed.
Lemma decompose_sub':
∀ xh xl yh yl,
sub (ofwords xh xl) (ofwords yh yl) =
ofwords (Int.add (Int.add xh (Int.not yh)) (Int.add_carry xl (Int.not yl) Int.one))
(Int.sub xl yl).
Proof.
intros. rewrite decompose_sub. f_equal.
rewrite Int.sub_borrow_add_carry by auto.
rewrite Int.sub_add_not_3. rewrite Int.xor_assoc. rewrite Int.xor_idem.
rewrite Int.xor_zero. auto.
rewrite Int.xor_zero_l. unfold Int.add_carry.
destruct (zlt (Int.unsigned xl + Int.unsigned (Int.not yl) + Int.unsigned Int.one) Int.modulus);
compute; [right|left]; apply Int.mkint_eq; auto.
Qed.
Definition mul' (x y: Int.int) : int := repr (Int.unsigned x × Int.unsigned y).
Lemma mul'_mulhu:
∀ x y, mul' x y = ofwords (Int.mulhu x y) (Int.mul x y).
Proof.
intros.
rewrite ofwords_add. unfold mul', Int.mulhu, Int.mul.
set (p := Int.unsigned x × Int.unsigned y).
set (ph := p / Int.modulus). set (pl := p mod Int.modulus).
transitivity (repr (ph × Int.modulus + pl)).
- f_equal. rewrite Z.mul_comm. apply Z_div_mod_eq. apply Int.modulus_pos.
- apply eqm_samerepr. apply eqm_add. apply eqm_mul_2p32. auto with ints.
rewrite Int.unsigned_repr_eq. apply eqm_refl.
Qed.
Lemma decompose_mul:
∀ xh xl yh yl,
mul (ofwords xh xl) (ofwords yh yl) =
ofwords (Int.add (Int.add (hiword (mul' xl yl)) (Int.mul xl yh)) (Int.mul xh yl))
(loword (mul' xl yl)).
Proof.
intros.
set (pl := loword (mul' xl yl)); set (ph := hiword (mul' xl yl)).
assert (EQ0: unsigned (mul' xl yl) = Int.unsigned ph × two_p 32 + Int.unsigned pl).
{ rewrite <- (ofwords_recompose (mul' xl yl)). apply ofwords_add'. }
symmetry. rewrite ofwords_add. unfold mul. rewrite !ofwords_add'.
set (XL := Int.unsigned xl); set (XH := Int.unsigned xh);
set (YL := Int.unsigned yl); set (YH := Int.unsigned yh).
set (PH := Int.unsigned ph) in ×. set (PL := Int.unsigned pl) in ×.
transitivity (repr (((PH + XL × YH) + XH × YL) × two_p 32 + PL)).
apply eqm_samerepr. apply eqm_add. 2: apply eqm_refl.
apply eqm_mul_2p32.
rewrite Int.add_unsigned. apply Int.eqm_unsigned_repr_l. apply Int.eqm_add.
rewrite Int.add_unsigned. apply Int.eqm_unsigned_repr_l. apply Int.eqm_add.
apply Int.eqm_refl.
unfold Int.mul. apply Int.eqm_unsigned_repr_l. apply Int.eqm_refl.
unfold Int.mul. apply Int.eqm_unsigned_repr_l. apply Int.eqm_refl.
transitivity (repr (unsigned (mul' xl yl) + (XL × YH + XH × YL) × two_p 32)).
rewrite EQ0. f_equal. ring.
transitivity (repr ((XL × YL + (XL × YH + XH × YL) × two_p 32))).
apply eqm_samerepr. apply eqm_add. 2: apply eqm_refl.
unfold mul'. apply eqm_unsigned_repr_l. apply eqm_refl.
transitivity (repr (0 + (XL × YL + (XL × YH + XH × YL) × two_p 32))).
rewrite Z.add_0_l; auto.
transitivity (repr (XH × YH × (two_p 32 × two_p 32) + (XL × YL + (XL × YH + XH × YL) × two_p 32))).
apply eqm_samerepr. apply eqm_add. 2: apply eqm_refl.
change (two_p 32 × two_p 32) with modulus. ∃ (- XH × YH). ring.
f_equal. ring.
Qed.
Lemma decompose_mul_2:
∀ xh xl yh yl,
mul (ofwords xh xl) (ofwords yh yl) =
ofwords (Int.add (Int.add (Int.mulhu xl yl) (Int.mul xl yh)) (Int.mul xh yl))
(Int.mul xl yl).
Proof.
intros. rewrite decompose_mul. rewrite mul'_mulhu.
rewrite hi_ofwords, lo_ofwords. auto.
Qed.
Lemma decompose_ltu:
∀ xh xl yh yl,
ltu (ofwords xh xl) (ofwords yh yl) = if Int.eq xh yh then Int.ltu xl yl else Int.ltu xh yh.
Proof.
intros. unfold ltu. rewrite ! ofwords_add'. unfold Int.ltu, Int.eq.
destruct (zeq (Int.unsigned xh) (Int.unsigned yh)).
rewrite e. destruct (zlt (Int.unsigned xl) (Int.unsigned yl)).
apply zlt_true; lia.
apply zlt_false; lia.
change (two_p 32) with Int.modulus.
generalize (Int.unsigned_range xl) (Int.unsigned_range yl).
change Int.modulus with 4294967296. intros.
destruct (zlt (Int.unsigned xh) (Int.unsigned yh)).
apply zlt_true; lia.
apply zlt_false; lia.
Qed.
Lemma decompose_leu:
∀ xh xl yh yl,
negb (ltu (ofwords yh yl) (ofwords xh xl)) =
if Int.eq xh yh then negb (Int.ltu yl xl) else Int.ltu xh yh.
Proof.
intros. rewrite decompose_ltu. rewrite Int.eq_sym.
unfold Int.eq. destruct (zeq (Int.unsigned xh) (Int.unsigned yh)).
auto.
unfold Int.ltu. destruct (zlt (Int.unsigned xh) (Int.unsigned yh)).
rewrite zlt_false by lia; auto.
rewrite zlt_true by lia; auto.
Qed.
Lemma decompose_lt:
∀ xh xl yh yl,
lt (ofwords xh xl) (ofwords yh yl) = if Int.eq xh yh then Int.ltu xl yl else Int.lt xh yh.
Proof.
intros. unfold lt. rewrite ! ofwords_add''. rewrite Int.eq_signed.
destruct (zeq (Int.signed xh) (Int.signed yh)).
rewrite e. unfold Int.ltu. destruct (zlt (Int.unsigned xl) (Int.unsigned yl)).
apply zlt_true; lia.
apply zlt_false; lia.
change (two_p 32) with Int.modulus.
generalize (Int.unsigned_range xl) (Int.unsigned_range yl).
change Int.modulus with 4294967296. intros.
unfold Int.lt. destruct (zlt (Int.signed xh) (Int.signed yh)).
apply zlt_true; lia.
apply zlt_false; lia.
Qed.
Lemma decompose_le:
∀ xh xl yh yl,
negb (lt (ofwords yh yl) (ofwords xh xl)) =
if Int.eq xh yh then negb (Int.ltu yl xl) else Int.lt xh yh.
Proof.
intros. rewrite decompose_lt. rewrite Int.eq_sym.
rewrite Int.eq_signed. destruct (zeq (Int.signed xh) (Int.signed yh)).
auto.
unfold Int.lt. destruct (zlt (Int.signed xh) (Int.signed yh)).
rewrite zlt_false by lia; auto.
rewrite zlt_true by lia; auto.
Qed.
Utility proofs for mixed 32bit and 64bit arithmetic
Remark int_unsigned_range:
∀ x, 0 ≤ Int.unsigned x ≤ max_unsigned.
Proof.
intros.
unfold max_unsigned. unfold modulus.
generalize (Int.unsigned_range x).
unfold Int.modulus in ×.
change (wordsize) with 64%nat in ×.
change (Int.wordsize) with 32%nat in ×.
unfold two_power_nat. simpl.
lia.
Qed.
Remark int_unsigned_repr:
∀ x, unsigned (repr (Int.unsigned x)) = Int.unsigned x.
Proof.
intros. rewrite unsigned_repr. auto.
apply int_unsigned_range.
Qed.
Lemma int_sub_ltu:
∀ x y,
Int.ltu x y= true →
Int.unsigned (Int.sub y x) = unsigned (sub (repr (Int.unsigned y)) (repr (Int.unsigned x))).
Proof.
intros. generalize (Int.sub_ltu x y H). intros. unfold Int.sub. unfold sub.
rewrite Int.unsigned_repr. rewrite unsigned_repr.
rewrite unsigned_repr by apply int_unsigned_range. rewrite int_unsigned_repr. reflexivity.
rewrite unsigned_repr by apply int_unsigned_range.
rewrite int_unsigned_repr. generalize (int_unsigned_range y).
lia.
generalize (Int.sub_ltu x y H). intros.
generalize (Int.unsigned_range_2 y). intros. lia.
Qed.
End Int64.
Strategy 0 [Wordsize_64.wordsize].
Notation int64 := Int64.int.
Global Opaque Int.repr Int64.repr Byte.repr.