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Require Import BinInt.
Require Import Lia.
Require Import ZBinary.
From bbv Require Import ZLib.
From compcert Require Import Integers Coqlib.
Require Import Coquplib.
Local Open Scope Z_scope.
Module PtrofsExtra.
Lemma mul_divs :
forall x y,
0 <= Ptrofs.signed y ->
0 < Ptrofs.signed x ->
Ptrofs.signed y mod Ptrofs.signed x = 0 ->
(Integers.Ptrofs.mul x (Integers.Ptrofs.divs y x)) = y.
Proof.
intros.
pose proof (Ptrofs.mods_divs_Euclid y x).
pose proof (Zquot.Zrem_Zmod_zero (Ptrofs.signed y) (Ptrofs.signed x)).
apply <- H3 in H1; try lia; clear H3.
unfold Ptrofs.mods in H2.
rewrite H1 in H2.
replace (Ptrofs.repr 0) with (Ptrofs.zero) in H2 by reflexivity.
rewrite Ptrofs.add_zero in H2.
rewrite Ptrofs.mul_commut.
congruence.
Qed.
Lemma Z_div_distr_add :
forall x y z,
x mod z = 0 ->
y mod z = 0 ->
z <> 0 ->
x / z + y / z = (x + y) / z.
Proof.
intros.
assert ((x + y) mod z = 0).
{ rewrite <- Z.add_mod_idemp_l; try assumption.
rewrite H. assumption. }
rewrite <- Z.mul_cancel_r with (p := z); try assumption.
rewrite Z.mul_add_distr_r.
repeat rewrite ZLib.div_mul_undo; lia.
Qed.
Lemma mul_unsigned :
forall x y,
Ptrofs.mul x y =
Ptrofs.repr (Ptrofs.unsigned x * Ptrofs.unsigned y).
Proof.
intros; unfold Ptrofs.mul.
apply Ptrofs.eqm_samerepr.
apply Ptrofs.eqm_mult; exists 0; lia.
Qed.
Lemma mul_repr :
forall x y,
Ptrofs.min_signed <= y <= Ptrofs.max_signed ->
Ptrofs.min_signed <= x <= Ptrofs.max_signed ->
Ptrofs.mul (Ptrofs.repr y) (Ptrofs.repr x) = Ptrofs.repr (x * y).
Proof.
intros; unfold Ptrofs.mul.
destruct (Z_ge_lt_dec x 0); destruct (Z_ge_lt_dec y 0).
- f_equal.
repeat rewrite Ptrofs.unsigned_repr_eq.
repeat rewrite Z.mod_small; simplify; lia.
- assert (Ptrofs.lt (Ptrofs.repr y) Ptrofs.zero = true).
{
unfold Ptrofs.lt.
rewrite Ptrofs.signed_repr; auto.
rewrite Ptrofs.signed_zero.
destruct (zlt y 0); try lia; auto.
}
rewrite Ptrofs.unsigned_signed with (n := Ptrofs.repr y).
rewrite H1.
rewrite Ptrofs.signed_repr; auto.
rewrite Ptrofs.unsigned_repr_eq.
rewrite Z.mod_small; simplify; try lia.
rewrite Z.mul_add_distr_r.
apply Ptrofs.eqm_samerepr.
exists x. simplify. lia.
- assert (Ptrofs.lt (Ptrofs.repr x) Ptrofs.zero = true).
{
unfold Ptrofs.lt.
rewrite Ptrofs.signed_repr; auto.
rewrite Ptrofs.signed_zero.
destruct (zlt x 0); try lia; auto.
}
rewrite Ptrofs.unsigned_signed with (n := Ptrofs.repr x).
rewrite H1.
rewrite Ptrofs.signed_repr; auto.
rewrite Ptrofs.unsigned_repr_eq.
rewrite Z.mod_small; simplify; try lia.
rewrite Z.mul_add_distr_l.
apply Ptrofs.eqm_samerepr.
exists y. simplify. lia.
- assert (Ptrofs.lt (Ptrofs.repr x) Ptrofs.zero = true).
{
unfold Ptrofs.lt.
rewrite Ptrofs.signed_repr; auto.
rewrite Ptrofs.signed_zero.
destruct (zlt x 0); try lia; auto.
}
assert (Ptrofs.lt (Ptrofs.repr y) Ptrofs.zero = true).
{
unfold Ptrofs.lt.
rewrite Ptrofs.signed_repr; auto.
rewrite Ptrofs.signed_zero.
destruct (zlt y 0); try lia; auto.
}
rewrite Ptrofs.unsigned_signed with (n := Ptrofs.repr x).
rewrite Ptrofs.unsigned_signed with (n := Ptrofs.repr y).
rewrite H2.
rewrite H1.
repeat rewrite Ptrofs.signed_repr; auto.
replace ((y + Ptrofs.modulus) * (x + Ptrofs.modulus)) with
(x * y + (x + y + Ptrofs.modulus) * Ptrofs.modulus) by lia.
apply Ptrofs.eqm_samerepr.
exists (x + y + Ptrofs.modulus). lia.
Qed.
End PtrofsExtra.
Module IntExtra.
Lemma mul_unsigned :
forall x y,
Int.mul x y =
Int.repr (Int.unsigned x * Int.unsigned y).
Proof.
intros; unfold Int.mul.
apply Int.eqm_samerepr.
apply Int.eqm_mult; exists 0; lia.
Qed.
Lemma mul_repr :
forall x y,
Int.min_signed <= y <= Int.max_signed ->
Int.min_signed <= x <= Int.max_signed ->
Int.mul (Int.repr y) (Int.repr x) = Int.repr (x * y).
Proof.
intros; unfold Int.mul.
destruct (Z_ge_lt_dec x 0); destruct (Z_ge_lt_dec y 0).
- f_equal.
repeat rewrite Int.unsigned_repr_eq.
repeat rewrite Z.mod_small; simplify; lia.
- assert (Int.lt (Int.repr y) Int.zero = true).
{
unfold Int.lt.
rewrite Int.signed_repr; auto.
rewrite Int.signed_zero.
destruct (zlt y 0); try lia; auto.
}
rewrite Int.unsigned_signed with (n := Int.repr y).
rewrite H1.
rewrite Int.signed_repr; auto.
rewrite Int.unsigned_repr_eq.
rewrite Z.mod_small; simplify; try lia.
rewrite Z.mul_add_distr_r.
apply Int.eqm_samerepr.
exists x. simplify. lia.
- assert (Int.lt (Int.repr x) Int.zero = true).
{
unfold Int.lt.
rewrite Int.signed_repr; auto.
rewrite Int.signed_zero.
destruct (zlt x 0); try lia; auto.
}
rewrite Int.unsigned_signed with (n := Int.repr x).
rewrite H1.
rewrite Int.signed_repr; auto.
rewrite Int.unsigned_repr_eq.
rewrite Z.mod_small; simplify; try lia.
rewrite Z.mul_add_distr_l.
apply Int.eqm_samerepr.
exists y. simplify. lia.
- assert (Int.lt (Int.repr x) Int.zero = true).
{
unfold Int.lt.
rewrite Int.signed_repr; auto.
rewrite Int.signed_zero.
destruct (zlt x 0); try lia; auto.
}
assert (Int.lt (Int.repr y) Int.zero = true).
{
unfold Int.lt.
rewrite Int.signed_repr; auto.
rewrite Int.signed_zero.
destruct (zlt y 0); try lia; auto.
}
rewrite Int.unsigned_signed with (n := Int.repr x).
rewrite Int.unsigned_signed with (n := Int.repr y).
rewrite H2.
rewrite H1.
repeat rewrite Int.signed_repr; auto.
replace ((y + Int.modulus) * (x + Int.modulus)) with
(x * y + (x + y + Int.modulus) * Int.modulus) by lia.
apply Int.eqm_samerepr.
exists (x + y + Int.modulus). lia.
Qed.
End IntExtra.
Lemma mul_of_int :
forall x y,
0 <= x < Integers.Ptrofs.modulus ->
Integers.Ptrofs.mul (Integers.Ptrofs.repr x) (Integers.Ptrofs.of_int y) =
Integers.Ptrofs.of_int (Integers.Int.mul (Integers.Int.repr x) y).
Proof.
intros.
pose proof (Integers.Ptrofs.agree32_of_int eq_refl y) as A.
pose proof (Integers.Ptrofs.agree32_to_int eq_refl (Integers.Ptrofs.repr x)) as B.
exploit Integers.Ptrofs.agree32_mul; [> reflexivity | exact B | exact A | intro C].
unfold Integers.Ptrofs.to_int in C.
unfold Integers.Ptrofs.of_int in C.
rewrite Integers.Ptrofs.unsigned_repr_eq in C.
rewrite Z.mod_small in C; auto.
symmetry.
apply Integers.Ptrofs.agree32_of_int_eq; auto.
Qed.
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