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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 Vericertlib.
Local Open Scope Z_scope.
Module PtrofsExtra.
Ltac ptrofs_mod_match m :=
match goal with
| [ H : ?x = 0 |- context[?x] ] => rewrite H
| [ _ : _ |- context[_ - 0] ] => rewrite Z.sub_0_r
| [ _ : _ |- context[0 + _] ] => rewrite Z.add_0_l
| [ _ : _ |- context[_ + 0] ] => rewrite Z.add_0_r
| [ _ : _ |- context[0 * _] ] => rewrite Z.mul_0_l
| [ _ : _ |- context[_ * 0] ] => rewrite Z.mul_0_r
| [ _ : _ |- context[?m mod ?m] ] =>
rewrite Z_mod_same_full with (a := m)
| [ _ : _ |- context[0 mod _] ] =>
rewrite Z.mod_0_l
| [ _ : _ |- context[(_ mod ?m) mod ?m] ] =>
rewrite Zmod_mod
| [ _ : _ |- context[(_ mod Ptrofs.modulus) mod m ] ] =>
rewrite <- Zmod_div_mod;
try (crush; lia || assumption)
| [ _ : _ |- context[Ptrofs.modulus mod m] ] =>
rewrite Zdivide_mod with (a := Ptrofs.modulus);
try (lia || assumption)
| [ _ : _ |- context[Ptrofs.signed ?a mod Ptrofs.modulus] ] =>
rewrite Z.mod_small with (a := Ptrofs.signed a) (b := Ptrofs.modulus)
| [ _ : _ |- context[(?x - ?y) mod ?m] ] =>
rewrite Zminus_mod with (a := x) (b := y) (n := m)
| [ _ : _ |- context[((?x + ?y) mod ?m) + _] ] =>
rewrite Zplus_mod with (a := x) (b := y) (n := m)
| [ _ : _ |- context[(?x + ?y) mod ?m] ] =>
rewrite Zplus_mod with (a := x) (b := y) (n := m)
| [ _ : _ |- context[(?x * ?y) mod ?m] ] =>
rewrite Zmult_mod with (a := x) (b := y) (n := m)
end.
Ltac ptrofs_mod_tac m :=
repeat (ptrofs_mod_match m); lia.
Lemma signed_mod_unsigned_mod :
forall x m,
0 < m ->
Ptrofs.modulus mod m = 0 ->
Ptrofs.signed x mod m = 0 ->
Ptrofs.unsigned x mod m = 0.
Proof.
intros.
repeat match goal with
| [ _ : _ |- context[if ?x then _ else _] ] => destruct x
| [ _ : _ |- context[_ mod Ptrofs.modulus mod m] ] =>
rewrite <- Zmod_div_mod; try lia; try assumption
| [ _ : _ |- context[Ptrofs.unsigned _] ] => rewrite Ptrofs.unsigned_signed
end; try crush; ptrofs_mod_tac m.
Qed.
Lemma of_int_mod :
forall x m,
Int.unsigned x mod m = 0 ->
Ptrofs.unsigned (Ptrofs.of_int x) mod m = 0.
Proof.
intros.
unfold Ptrofs.of_int.
rewrite Ptrofs.unsigned_repr; crush;
apply Int.unsigned_range_2.
Qed.
Lemma mul_mod :
forall x y m,
0 < m ->
(m | Ptrofs.modulus) ->
Ptrofs.unsigned x mod m = 0 ->
Ptrofs.unsigned y mod m = 0 ->
(Ptrofs.signed (Ptrofs.mul x y)) mod m = 0.
Proof.
intros. unfold Ptrofs.mul.
rewrite Ptrofs.signed_repr_eq.
repeat match goal with
| [ _ : _ |- context[if ?x then _ else _] ] => destruct x
| [ _ : _ |- context[_ mod Ptrofs.modulus mod m] ] =>
rewrite <- Zmod_div_mod; try lia; try assumption
end; try(crush; lia); ptrofs_mod_tac m.
Qed.
Lemma add_mod :
forall x y m,
0 < m ->
(m | Ptrofs.modulus) ->
Ptrofs.unsigned x mod m = 0 ->
Ptrofs.unsigned y mod m = 0 ->
(Ptrofs.unsigned (Ptrofs.add x y)) mod m = 0.
Proof.
intros. unfold Ptrofs.add.
rewrite Ptrofs.unsigned_repr_eq.
repeat match goal with
| [ _ : _ |- context[if ?x then _ else _] ] => destruct x
| [ _ : _ |- context[_ mod Ptrofs.modulus mod m] ] =>
rewrite <- Zmod_div_mod; try lia; try assumption
end; try (crush; lia); ptrofs_mod_tac m.
Qed.
Lemma mul_divu :
forall x y,
0 < Ptrofs.unsigned x ->
Ptrofs.unsigned y mod Ptrofs.unsigned x = 0 ->
(Integers.Ptrofs.mul x (Integers.Ptrofs.divu y x)) = y.
Proof.
intros.
assert (x <> Ptrofs.zero).
{ intro.
rewrite H1 in H.
replace (Ptrofs.unsigned Ptrofs.zero) with 0 in H by reflexivity.
lia. }
exploit (Ptrofs.modu_divu_Euclid y x); auto; intros.
unfold Ptrofs.modu in H2. rewrite H0 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 divu_unsigned :
forall x y,
0 < Ptrofs.unsigned y ->
Ptrofs.unsigned x <= Ptrofs.max_unsigned ->
Ptrofs.unsigned (Ptrofs.divu x y) = Ptrofs.unsigned x / Ptrofs.unsigned y.
Proof.
intros.
unfold Ptrofs.divu.
rewrite Ptrofs.unsigned_repr; auto.
split.
apply Z.div_pos; auto.
apply Ptrofs.unsigned_range.
apply Z.div_le_upper_bound; auto.
eapply Z.le_trans.
exact H0.
rewrite Z.mul_comm.
apply Z.le_mul_diag_r; crush.
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 pos_signed_unsigned :
forall x,
0 <= Ptrofs.signed x ->
Ptrofs.signed x = Ptrofs.unsigned x.
Proof.
intros.
rewrite Ptrofs.unsigned_signed.
destruct (Ptrofs.lt x Ptrofs.zero) eqn:EQ.
unfold Ptrofs.lt in EQ.
destruct (zlt (Ptrofs.signed x) (Ptrofs.signed Ptrofs.zero)); try discriminate.
replace (Ptrofs.signed (Ptrofs.zero)) with 0 in l by reflexivity. lia.
reflexivity.
Qed.
End PtrofsExtra.
Ltac ptrofs :=
repeat match goal with
| [ |- context[Ptrofs.add (Ptrofs.zero) _] ] => setoid_rewrite Ptrofs.add_zero_l
| [ H : context[Ptrofs.add (Ptrofs.zero) _] |- _ ] => setoid_rewrite Ptrofs.add_zero_l in H
| [ |- context[Ptrofs.repr 0] ] => replace (Ptrofs.repr 0) with Ptrofs.zero by reflexivity
| [ H : context[Ptrofs.repr 0] |- _ ] =>
replace (Ptrofs.repr 0) with Ptrofs.zero in H by reflexivity
| [ H: context[Ptrofs.unsigned (Ptrofs.repr (Ptrofs.unsigned _))] |- _ ] =>
setoid_rewrite Ptrofs.unsigned_repr in H; [>| apply Ptrofs.unsigned_range_2]
| [ |- context[Ptrofs.unsigned (Ptrofs.repr (Ptrofs.unsigned _))] ] =>
rewrite Ptrofs.unsigned_repr; [>| apply Ptrofs.unsigned_range_2]
| [ |- context[0 <= Ptrofs.unsigned _] ] => apply Ptrofs.unsigned_range_2
end.
Module IntExtra.
Import Int.
Ltac int_mod_match m :=
match goal with
| [ H : ?x = 0 |- context[?x] ] => rewrite H
| [ _ : _ |- context[_ - 0] ] => rewrite Z.sub_0_r
| [ _ : _ |- context[0 + _] ] => rewrite Z.add_0_l
| [ _ : _ |- context[_ + 0] ] => rewrite Z.add_0_r
| [ _ : _ |- context[0 * _] ] => rewrite Z.mul_0_l
| [ _ : _ |- context[_ * 0] ] => rewrite Z.mul_0_r
| [ _ : _ |- context[?m mod ?m] ] =>
rewrite Z_mod_same_full with (a := m)
| [ _ : _ |- context[0 mod _] ] =>
rewrite Z.mod_0_l
| [ _ : _ |- context[(_ mod ?m) mod ?m] ] =>
rewrite Zmod_mod
| [ _ : _ |- context[(_ mod Int.modulus) mod m ] ] =>
rewrite <- Zmod_div_mod;
try (crush; lia || assumption)
| [ _ : _ |- context[Int.modulus mod m] ] =>
rewrite Zdivide_mod with (a := Int.modulus);
try (lia || assumption)
| [ _ : _ |- context[Int.signed ?a mod Int.modulus] ] =>
rewrite Z.mod_small with (a := Int.signed a) (b := Int.modulus)
| [ _ : _ |- context[(?x - ?y) mod ?m] ] =>
rewrite Zminus_mod with (a := x) (b := y) (n := m)
| [ _ : _ |- context[((?x + ?y) mod ?m) + _] ] =>
rewrite Zplus_mod with (a := x) (b := y) (n := m)
| [ _ : _ |- context[(?x + ?y) mod ?m] ] =>
rewrite Zplus_mod with (a := x) (b := y) (n := m)
| [ _ : _ |- context[(?x * ?y) mod ?m] ] =>
rewrite Zmult_mod with (a := x) (b := y) (n := m)
end.
Ltac int_mod_tac m :=
repeat (int_mod_match m); lia.
Lemma mul_mod1 :
forall x y m,
0 < m ->
(m | Int.modulus) ->
Int.unsigned x mod m = 0 ->
(Int.unsigned (Int.mul x y)) mod m = 0.
Proof.
intros. unfold Int.mul.
rewrite Int.unsigned_repr_eq.
repeat match goal with
| [ _ : _ |- context[if ?x then _ else _] ] => destruct x
| [ _ : _ |- context[_ mod Int.modulus mod m] ] =>
rewrite <- Zmod_div_mod; try lia; try assumption
end; try (crush; lia); int_mod_tac m.
Qed.
Lemma mul_mod2 :
forall x y m,
0 < m ->
(m | Int.modulus) ->
Int.unsigned y mod m = 0 ->
(Int.unsigned (Int.mul x y)) mod m = 0.
Proof.
intros. unfold Int.mul.
rewrite Int.unsigned_repr_eq.
repeat match goal with
| [ _ : _ |- context[if ?x then _ else _] ] => destruct x
| [ _ : _ |- context[_ mod Int.modulus mod m] ] =>
rewrite <- Zmod_div_mod; try lia; try assumption
end; try (crush; lia); int_mod_tac m.
Qed.
Lemma add_mod :
forall x y m,
0 < m ->
(m | Int.modulus) ->
Int.unsigned x mod m = 0 ->
Int.unsigned y mod m = 0 ->
(Int.unsigned (Int.add x y)) mod m = 0.
Proof.
intros. unfold Int.add.
rewrite Int.unsigned_repr_eq.
repeat match goal with
| [ _ : _ |- context[if ?x then _ else _] ] => destruct x
| [ _ : _ |- context[_ mod Int.modulus mod m] ] =>
rewrite <- Zmod_div_mod; try lia; try assumption
end; try (crush; lia); int_mod_tac m.
Qed.
Definition ofbytes (a b c d : byte) : int :=
or (shl (repr (Byte.unsigned a)) (repr (3 * Byte.zwordsize)))
(or (shl (repr (Byte.unsigned b)) (repr (2 * Byte.zwordsize)))
(or (shl (repr (Byte.unsigned c)) (repr Byte.zwordsize))
(repr (Byte.unsigned d)))).
Definition byte1 (n: int) : byte := Byte.repr (unsigned n).
Definition byte2 (n: int) : byte := Byte.repr (unsigned (shru n (repr Byte.zwordsize))).
Definition byte3 (n: int) : byte := Byte.repr (unsigned (shru n (repr (2 * Byte.zwordsize)))).
Definition byte4 (n: int) : byte := Byte.repr (unsigned (shru n (repr (3 * Byte.zwordsize)))).
Lemma bits_byte1:
forall n i, 0 <= i < Byte.zwordsize -> Byte.testbit (byte1 n) i = testbit n i.
Proof.
intros. unfold byte1. rewrite Byte.testbit_repr; auto.
Qed.
Lemma bits_byte2:
forall n i, 0 <= i < Byte.zwordsize -> Byte.testbit (byte2 n) i = testbit n (i + Byte.zwordsize).
Proof.
intros. unfold byte2. rewrite Byte.testbit_repr; auto.
assert (zwordsize = 4 * Byte.zwordsize) by reflexivity.
fold (testbit (shru n (repr Byte.zwordsize)) i). rewrite bits_shru.
change (unsigned (repr Byte.zwordsize)) with Byte.zwordsize.
apply zlt_true. omega. omega.
Qed.
Lemma bits_byte3:
forall n i, 0 <= i < Byte.zwordsize -> Byte.testbit (byte3 n) i = testbit n (i + (2 * Byte.zwordsize)).
Proof.
intros. unfold byte3. rewrite Byte.testbit_repr; auto.
assert (zwordsize = 4 * Byte.zwordsize) by reflexivity.
fold (testbit (shru n (repr (2 * Byte.zwordsize))) i). rewrite bits_shru.
change (unsigned (repr (2 * Byte.zwordsize))) with (2 * Byte.zwordsize).
apply zlt_true. omega. omega.
Qed.
Lemma bits_byte4:
forall n i, 0 <= i < Byte.zwordsize -> Byte.testbit (byte4 n) i = testbit n (i + (3 * Byte.zwordsize)).
Proof.
intros. unfold byte4. rewrite Byte.testbit_repr; auto.
assert (zwordsize = 4 * Byte.zwordsize) by reflexivity.
fold (testbit (shru n (repr (3 * Byte.zwordsize))) i). rewrite bits_shru.
change (unsigned (repr (3 * Byte.zwordsize))) with (3 * Byte.zwordsize).
apply zlt_true. omega. omega.
Qed.
Lemma bits_ofwords:
forall b4 b3 b2 b1 i, 0 <= i < zwordsize ->
testbit (ofbytes b4 b3 b2 b1) i =
if zlt i Byte.zwordsize
then Byte.testbit b1 i
else (if zlt i (2 * Byte.zwordsize)
then Byte.testbit b2 (i - Byte.zwordsize)
else (if zlt i (3 * Byte.zwordsize)
then Byte.testbit b2 (i - 2 * Byte.zwordsize)
else Byte.testbit b2 (i - 3 * Byte.zwordsize))).
Proof.
intros. unfold ofbytes. repeat (rewrite bits_or; auto). repeat (rewrite bits_shl; auto).
change (unsigned (repr Byte.zwordsize)) with Byte.zwordsize.
change (unsigned (repr (2 * Byte.zwordsize))) with (2 * Byte.zwordsize).
change (unsigned (repr (3 * Byte.zwordsize))) with (3 * Byte.zwordsize).
assert (zwordsize = 4 * Byte.zwordsize) by reflexivity.
destruct (zlt i Byte.zwordsize).
rewrite testbit_repr; auto.
Abort.
End IntExtra.
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