bedrock.lang.cpp.arith.builtins
(*
* Copyright (c) 2020 BedRock Systems, Inc.
* This software is distributed under the terms of the BedRock Open-Source License.
* See the LICENSE-BedRock file in the repository root for details.
*)
Require Import bedrock.prelude.base.
Require Import bedrock.lang.cpp.arith.types.
Require Import bedrock.lang.cpp.arith.operator.
#[local] Open Scope Z_scope.
(* see
* https://gcc.gnu.org/onlinedocs/gcc/Other-Builtins.html
*)
(* Returns one plus the index of the least significant 1-bit of x,
or if x is zero, returns zero. *)
Definition first_set (sz : bitsize) (n : Z) : Z :=
let n := trim (bitsize.bitsN sz) n in
if Z.eqb n 0 then 0
else
(fix get ls : Z :=
match ls with
| nil => 64
| l :: ls =>
if Z.testbit n l
then 1 + l
else get ls
end)%Z (List.map Z.of_nat (seq 0 (N.to_nat (bitsize.bitsN sz)))).
#[global] Arguments first_set : simpl never.
(* Returns the number of trailing 0-bits in x, starting at the least
significant bit position. If x is 0, the result is undefined. *)
Definition trailing_zeros (sz : bitsize) (n : Z) : Z :=
(fix get ls : Z :=
match ls with
| nil => 64
| l :: ls =>
if Z.testbit n l
then l
else get ls
end)%Z (List.map Z.of_nat (seq 0 (N.to_nat (bitsize.bitsN sz)))).
#[global] Arguments trailing_zeros : simpl never.
(* Returns the number of leading 0-bits in x, starting at the most significant
bit position. If x is 0, the result is undefined. *)
Definition leading_zeros (sz : bitsize) (l : Z) : Z :=
bitsize.bitsZ sz - Z.log2 (l mod (2^64)).
#[global] Arguments leading_zeros : simpl never.
Module Import churn_bits.
(* TODO: using bool_decide would simplify this reasoning. *)
Ltac churn_bits' :=
repeat match goal with
| |- context[(?l <=? ?r)%Z] =>
let Hnb := fresh "Hnb" in
let Hn := fresh "Hn" in
destruct (l <=? r)%Z eqn:Hnb;
set (Hn := Hnb);
[ apply Z.leb_le in Hn
| apply Z.leb_gt in Hn]
| |- context[(?l <? ?r)%Z] =>
let Hnb := fresh "Hnb" in
let Hn := fresh "Hn" in
destruct (l <? r)%Z eqn:Hnb;
set (Hn := Hnb);
[ apply Z.ltb_lt in Hn
| apply Z.ltb_ge in Hn]
end; rewrite ?andb_false_l ?andb_false_r ?andb_true_l ?andb_true_r
?orb_false_l ?orb_false_r ?orb_true_l ?orb_true_r ?Z.bits_0 //=;
try lia.
Ltac churn_bits :=
apply Z.bits_inj'=> n ?;
repeat (rewrite ?Z.lor_spec ?Z.shiftl_spec ?Z.land_spec ?Z.shiftr_spec; try lia);
rewrite !Z.testbit_ones; try lia;
churn_bits'.
End churn_bits.
Section BitsTheory.
Lemma log2_lt_pow2ge:
∀ a b : Z,
0 <= a →
Z.log2 a < b ->
a < 2 ^ b.
Proof.
intros.
assert (a=0 \/ 0 < a) as [Hc | Hc] by lia.
{
subst.
simpl in *.
apply Z.pow_pos_nonneg; lia.
}
{
apply Z.log2_lt_pow2; auto.
}
Qed.
Lemma ZlogLtPow2: forall (a w: Z),
0 < w ->
0 ≤ a < 2 ^ w → Z.log2 a < w.
Proof.
intros.
assert (a=0 \/ 0 < a) as Hc by lia.
destruct Hc as [Hc | Hc]; [subst; simpl; lia | ].
apply Z.log2_lt_pow2; lia.
Qed.
Lemma ZlorRange:
forall (a b: Z) (w : N),
0 <= a < (2 ^ Z.of_N w) ->
0 <= b < (2 ^ Z.of_N w) ->
0 <= Z.lor a b < (2 ^ Z.of_N w).
Proof.
intros.
assert (w=0 \/ 0 < w)%N as [Hc | Hc] by lia.
{
subst. simpl in *.
assert (a=0) by lia.
assert (b=0) by lia.
subst.
simpl.
compute. firstorder.
}
apply and_wlog_r; [apply Z.lor_nonneg; lia | ].
intros.
apply log2_lt_pow2ge; [assumption | ].
rewrite -> Z.log2_lor by lia.
apply ZlogLtPow2 in H; try lia.
apply ZlogLtPow2 in H0; lia.
Qed.
End BitsTheory.
Section Byte.
Definition _get_byte (x: Z) (n: nat): Z := Z.land (x ≫ (8 * n)) (Z.ones 8).
Definition _set_byte (x: Z) (n: nat): Z := (Z.land (Z.ones 8) x) ≪ (8 * n).
Section Theory.
Lemma _get_byte_0:
forall (idx: nat),
_get_byte 0 idx = 0.
Proof. intros; by rewrite /_get_byte Z.shiftr_0_l Z.land_0_l. Qed.
Lemma _set_byte_0:
forall (idx: nat),
_set_byte 0 idx = 0.
Proof. intros; by rewrite /_set_byte Z.shiftl_0_l. Qed.
#[local] Lemma Z_mul_of_nat_S (z : Z) (n : nat) : (z * S n)%Z = (z + z * n)%Z.
Proof. lia. Qed.
Lemma _get_byte_S_idx:
forall (v: Z) (idx: nat),
_get_byte v (S idx) = _get_byte (v ≫ 8) idx.
Proof.
intros; rewrite /_get_byte.
by rewrite Z.shiftr_shiftr ?Z_mul_of_nat_S; lia.
Qed.
Lemma _get_byte_lor:
forall (v v': Z) (idx: nat),
_get_byte (Z.lor v v') idx =
Z.lor (_get_byte v idx) (_get_byte v' idx).
Proof. intros *; rewrite /_get_byte; churn_bits. Qed.
Lemma _set_byte_S_idx:
forall (v: Z) (idx: nat),
_set_byte v (S idx) = ((_set_byte v idx) ≪ 8)%Z.
Proof.
intros; rewrite /_set_byte.
by rewrite Z.shiftl_shiftl ?Z_mul_of_nat_S ?(Z.add_comm 8); lia.
Qed.
Lemma _set_byte_lor:
forall (v v': Z) (idx: nat),
_set_byte (Z.lor v v') idx =
Z.lor (_set_byte v idx) (_set_byte v' idx).
Proof. intros *; rewrite /_set_byte; churn_bits. Qed.
Lemma _set_byte_shiftl_idx:
forall (idx idx': nat) shift v,
idx' <= idx ->
(shift = 8 * Z.of_nat idx')%Z ->
(_set_byte v idx ≫ shift)%Z = _set_byte v (idx - idx').
Proof.
intros * Hidx Hshift; rewrite /_set_byte; subst; churn_bits.
f_equal; lia.
Qed.
Lemma _get_byte_bound:
forall (v : Z) (idx : nat),
(0 <= _get_byte v idx < 256)%Z.
Proof.
intros *; rewrite /_get_byte Z.land_ones; try lia.
pose proof (Z.mod_pos_bound (v ≫ (8 * idx)) 256) as [? ?]; try lia.
now replace (2 ^ 8) with 256%Z by lia.
Qed.
Lemma _get_byte_nonneg:
forall (v: Z) (idx: nat),
(0 <= _get_byte v idx)%Z.
Proof. intros *; pose proof (_get_byte_bound v idx); lia. Qed.
Lemma _set_byte_bound:
forall (v : Z) (idx : nat),
(0 <= _set_byte v idx < 2 ^ (8 * (idx + 1)))%Z.
Proof.
intros; rewrite /_set_byte Z.land_comm Z.land_ones; try lia.
rewrite Z.shiftl_mul_pow2; try lia.
pose proof (Z.mod_pos_bound v 256) as [? ?]; try lia.
replace (2 ^ (8 * (idx + 1))) with ((2 ^ 8) * (2 ^ (8 * idx)))
by (rewrite Z.mul_add_distr_l Zpower_exp; lia).
replace (2 ^ 8) with 256%Z by lia.
split.
- apply Z.mul_nonneg_nonneg; auto.
apply Z.pow_nonneg; lia.
- apply Zmult_lt_compat_r; auto.
apply Z.pow_pos_nonneg; lia.
Qed.
Lemma _set_byte_nonneg:
forall (v: Z) (idx: nat),
(0 <= _set_byte v idx)%Z.
Proof. intros *; pose proof (_set_byte_bound v idx); lia. Qed.
Lemma _set_byte_testbit_low:
forall idx v n,
0 <= n ->
n < 8 * Z.of_nat idx ->
Z.testbit (_set_byte v idx) n = false.
Proof.
intros * Hnonneg Hn.
repeat (rewrite ?Z.lor_spec ?Z.shiftl_spec ?Z.land_spec ?Z.shiftr_spec; try lia).
rewrite !Z.testbit_ones; lia.
Qed.
Lemma _set_byte_testbit_high:
forall idx' idx v n,
(idx' < idx)%nat ->
8 * Z.of_nat idx <= n ->
Z.testbit (_set_byte v idx') n = false.
Proof.
intros * Hidx Hn.
repeat (rewrite ?Z.lor_spec ?Z.shiftl_spec ?Z.land_spec ?Z.shiftr_spec; try lia).
rewrite !Z.testbit_ones; lia.
Qed.
Lemma _set_byte_land_no_overlap:
forall (idx idx': nat) mask v,
idx <> idx' ->
(mask = Z.ones 8 ≪ (8 * Z.of_nat idx'))%Z ->
Z.land (_set_byte v idx) mask = 0.
Proof. intros * Hidx Hmask; rewrite /_set_byte; subst; churn_bits. Qed.
Lemma _set_byte_land_useless:
forall idx mask v,
(mask = Z.ones 8 ≪ (8 * Z.of_nat idx))%Z ->
Z.land (_set_byte v idx) mask =
_set_byte v idx.
Proof. intros * Hmask; rewrite /_set_byte; subst; churn_bits. Qed.
Lemma _set_byte_shiftr_big:
forall (idx: nat) (idx': Z) v,
(Z.of_nat idx < idx')%Z ->
(_set_byte v idx ≫ (8 * idx'))%Z = 0.
Proof. intros * Hidx; rewrite /_set_byte; churn_bits. Qed.
Lemma _get_0_set_0_eq:
forall v,
_get_byte v 0 = _set_byte v 0.
Proof. intros *; rewrite /_get_byte/_set_byte; churn_bits. Qed.
Lemma _set_get_0:
forall v idx,
_set_byte (_get_byte v 0) idx = _set_byte v idx.
Proof.
intros *; rewrite /_get_byte/_set_byte.
apply Z.bits_inj'=> n ?.
repeat (rewrite ?Z.lor_spec ?Z.land_spec; try lia).
repeat (rewrite ?Z.shiftl_spec; try lia).
repeat (rewrite ?Z.lor_spec ?Z.land_spec; try lia).
assert (n < 8 * idx \/ 8 * idx <= n) as [Hn | Hn] by lia.
- rewrite !Z.testbit_neg_r; [reflexivity | lia.. ].
- rewrite !Z.shiftr_spec; try lia.
rewrite !Z.testbit_ones; try lia.
churn_bits'.
Qed.
Lemma _get_set_byte_no_overlap:
forall (v: Z) (idx idx': nat),
idx <> idx' ->
_get_byte (_set_byte v idx) idx' = 0.
Proof. intros * Hidx; rewrite /_get_byte/_set_byte; churn_bits. Qed.
Lemma _get_set_byte_roundtrip:
forall (v: Z) (idx: nat),
_get_byte (_set_byte v idx) idx = _get_byte v 0.
Proof.
intros *; rewrite /_get_byte/_set_byte; churn_bits.
f_equal; lia.
Qed.
Lemma _set_get_byte_roundtrip:
forall (v: Z) (idx: nat),
_set_byte (_get_byte v idx) idx =
_get_byte v idx ≪ (8 * idx).
Proof.
intros *; rewrite /_get_byte/_set_byte.
apply Z.bits_inj' => n ?.
repeat (rewrite ?Z.lor_spec ?Z.land_spec; try lia).
repeat (rewrite ?Z.shiftl_spec; try lia).
repeat (rewrite ?Z.lor_spec ?Z.land_spec; try lia).
assert (n < 8 * idx \/ 8 * idx <= n) as [Hn | Hn] by lia.
- rewrite !Z.testbit_neg_r; [reflexivity | lia.. ].
- rewrite !Z.shiftr_spec; try lia.
rewrite !Z.testbit_ones; try lia.
churn_bits'.
Qed.
End Theory.
Section split.
Lemma split8:
forall z,
0 <= z < 2^bitsize.bitsZ bitsize.W8 ->
z = _set_byte (_get_byte z 0) 0.
Proof.
intros * Hbounds; rewrite _set_get_0.
rewrite /_set_byte Z.shiftl_0_r.
rewrite Z.land_comm Z.land_ones; try lia.
symmetry; now apply Z.mod_small.
Qed.
Lemma split16:
forall z,
0 <= z < 2^bitsize.bitsZ bitsize.W16 ->
z = Z.lor (_set_byte (_get_byte z 0) 0)
(_set_byte (_get_byte z 1) 1).
Proof.
intros * Hbounds; simpl in Hbounds.
apply Z.bits_inj'=> n ?.
assert (n < 8 * 1%nat \/ 8 * 1%nat <= n < 8 * 2%nat \/ 8 * 2%nat <= n)
as [Hn | [Hn | Hn]]
by lia; rewrite !Z.lor_spec;
[ rewrite (_set_byte_testbit_low 1); auto; try lia
| rewrite (_set_byte_testbit_high 0 1); auto; try lia
| rewrite (_set_byte_testbit_high 0 2); auto; try lia;
rewrite (_set_byte_testbit_high 1 2); auto; try lia;
apply Z.testbit_false; [by lia |];
rewrite Z.div_small; [apply Z.mod_0_l; lia |];
pose proof (Z.pow_le_mono_r 2 16 n ltac:(lia) ltac:(auto)); lia
]; rewrite ?orb_false_l ?orb_false_r;
repeat (rewrite ?Z.lor_spec ?Z.shiftl_spec ?Z.land_spec ?Z.shiftr_spec; try lia);
rewrite !Z.testbit_ones; try lia;
churn_bits'; now rewrite Z.sub_add.
Qed.
Lemma split32:
forall z,
0 <= z < 2^bitsize.bitsZ bitsize.W32 ->
z = Z.lor (_set_byte (_get_byte z 0) 0)
(Z.lor (_set_byte (_get_byte z 1) 1)
(Z.lor (_set_byte (_get_byte z 2) 2)
(_set_byte (_get_byte z 3) 3))).
Proof.
intros * Hbounds; simpl in Hbounds.
apply Z.bits_inj'=> n ?.
assert (n < 8 * 1%nat \/
8 * 1%nat <= n < 8 * 2%nat \/
8 * 2%nat <= n < 8 * 3%nat \/
8 * 3%nat <= n < 8 * 4%nat \/
8 * 4%nat <= n)
as [Hn | [Hn | [Hn | [Hn | Hn]]]]
by lia; rewrite !Z.lor_spec;
[ rewrite (_set_byte_testbit_low 1); auto; try lia;
rewrite (_set_byte_testbit_low 2); auto; try lia;
rewrite (_set_byte_testbit_low 3); auto; try lia
| rewrite (_set_byte_testbit_high 0 1); auto; try lia;
rewrite (_set_byte_testbit_low 2); auto; try lia;
rewrite (_set_byte_testbit_low 3); auto; try lia
| rewrite (_set_byte_testbit_high 0 2); auto; try lia;
rewrite (_set_byte_testbit_high 1 2); auto; try lia;
rewrite (_set_byte_testbit_low 3); auto; try lia
| rewrite (_set_byte_testbit_high 0 3); auto; try lia;
rewrite (_set_byte_testbit_high 1 3); auto; try lia;
rewrite (_set_byte_testbit_high 2 3); auto; try lia
| rewrite (_set_byte_testbit_high 0 4); auto; try lia;
rewrite (_set_byte_testbit_high 1 4); auto; try lia;
rewrite (_set_byte_testbit_high 2 4); auto; try lia;
rewrite (_set_byte_testbit_high 3 4); auto; try lia;
apply Z.testbit_false; [by lia |];
rewrite Z.div_small; [apply Z.mod_0_l; lia |];
pose proof (Z.pow_le_mono_r 2 32 n ltac:(lia) ltac:(auto)); lia
]; rewrite ?orb_false_l ?orb_false_r;
repeat (rewrite ?Z.lor_spec ?Z.shiftl_spec ?Z.land_spec ?Z.shiftr_spec; try lia);
rewrite !Z.testbit_ones; try lia;
churn_bits'; now rewrite Z.sub_add.
Qed.
Lemma split64:
forall z,
0 <= z < 2^bitsize.bitsZ bitsize.W64 ->
z = Z.lor (_set_byte (_get_byte z 0) 0)
(Z.lor (_set_byte (_get_byte z 1) 1)
(Z.lor (_set_byte (_get_byte z 2) 2)
(Z.lor (_set_byte (_get_byte z 3) 3)
(Z.lor (_set_byte (_get_byte z 4) 4)
(Z.lor (_set_byte (_get_byte z 5) 5)
(Z.lor (_set_byte (_get_byte z 6) 6)
(_set_byte (_get_byte z 7) 7))))))).
Proof.
intros * Hbounds; simpl in Hbounds.
apply Z.bits_inj'=> n ?.
assert (n < 8 * 1%nat \/
8 * 1%nat <= n < 8 * 2%nat \/
8 * 2%nat <= n < 8 * 3%nat \/
8 * 3%nat <= n < 8 * 4%nat \/
8 * 4%nat <= n < 8 * 5%nat \/
8 * 5%nat <= n < 8 * 6%nat \/
8 * 6%nat <= n < 8 * 7%nat \/
8 * 7%nat <= n < 8 * 8%nat \/
8 * 8%nat <= n)
as [Hn | [Hn | [Hn | [Hn | [Hn | [Hn | [Hn | [Hn | Hn]]]]]]]]
by lia; rewrite !Z.lor_spec;
[ rewrite (_set_byte_testbit_low 1); auto; try lia;
rewrite (_set_byte_testbit_low 2); auto; try lia;
rewrite (_set_byte_testbit_low 3); auto; try lia;
rewrite (_set_byte_testbit_low 4); auto; try lia;
rewrite (_set_byte_testbit_low 5); auto; try lia;
rewrite (_set_byte_testbit_low 6); auto; try lia;
rewrite (_set_byte_testbit_low 7); auto; try lia
| rewrite (_set_byte_testbit_high 0 1); auto; try lia;
rewrite (_set_byte_testbit_low 2); auto; try lia;
rewrite (_set_byte_testbit_low 3); auto; try lia;
rewrite (_set_byte_testbit_low 4); auto; try lia;
rewrite (_set_byte_testbit_low 5); auto; try lia;
rewrite (_set_byte_testbit_low 6); auto; try lia;
rewrite (_set_byte_testbit_low 7); auto; try lia
| rewrite (_set_byte_testbit_high 0 2); auto; try lia;
rewrite (_set_byte_testbit_high 1 2); auto; try lia;
rewrite (_set_byte_testbit_low 3); auto; try lia;
rewrite (_set_byte_testbit_low 4); auto; try lia;
rewrite (_set_byte_testbit_low 5); auto; try lia;
rewrite (_set_byte_testbit_low 6); auto; try lia;
rewrite (_set_byte_testbit_low 7); auto; try lia
| rewrite (_set_byte_testbit_high 0 3); auto; try lia;
rewrite (_set_byte_testbit_high 1 3); auto; try lia;
rewrite (_set_byte_testbit_high 2 3); auto; try lia;
rewrite (_set_byte_testbit_low 4); auto; try lia;
rewrite (_set_byte_testbit_low 5); auto; try lia;
rewrite (_set_byte_testbit_low 6); auto; try lia;
rewrite (_set_byte_testbit_low 7); auto; try lia
| rewrite (_set_byte_testbit_high 0 4); auto; try lia;
rewrite (_set_byte_testbit_high 1 4); auto; try lia;
rewrite (_set_byte_testbit_high 2 4); auto; try lia;
rewrite (_set_byte_testbit_high 3 4); auto; try lia;
rewrite (_set_byte_testbit_low 5); auto; try lia;
rewrite (_set_byte_testbit_low 6); auto; try lia;
rewrite (_set_byte_testbit_low 7); auto; try lia
| rewrite (_set_byte_testbit_high 0 5); auto; try lia;
rewrite (_set_byte_testbit_high 1 5); auto; try lia;
rewrite (_set_byte_testbit_high 2 5); auto; try lia;
rewrite (_set_byte_testbit_high 3 5); auto; try lia;
rewrite (_set_byte_testbit_high 4 5); auto; try lia;
rewrite (_set_byte_testbit_low 6); auto; try lia;
rewrite (_set_byte_testbit_low 7); auto; try lia
| rewrite (_set_byte_testbit_high 0 6); auto; try lia;
rewrite (_set_byte_testbit_high 1 6); auto; try lia;
rewrite (_set_byte_testbit_high 2 6); auto; try lia;
rewrite (_set_byte_testbit_high 3 6); auto; try lia;
rewrite (_set_byte_testbit_high 4 6); auto; try lia;
rewrite (_set_byte_testbit_high 5 6); auto; try lia;
rewrite (_set_byte_testbit_low 7); auto; try lia
| rewrite (_set_byte_testbit_high 0 7); auto; try lia;
rewrite (_set_byte_testbit_high 1 7); auto; try lia;
rewrite (_set_byte_testbit_high 2 7); auto; try lia;
rewrite (_set_byte_testbit_high 3 7); auto; try lia;
rewrite (_set_byte_testbit_high 4 7); auto; try lia;
rewrite (_set_byte_testbit_high 5 7); auto; try lia;
rewrite (_set_byte_testbit_high 6 7); auto; try lia
| rewrite (_set_byte_testbit_high 0 8); auto; try lia;
rewrite (_set_byte_testbit_high 1 8); auto; try lia;
rewrite (_set_byte_testbit_high 2 8); auto; try lia;
rewrite (_set_byte_testbit_high 3 8); auto; try lia;
rewrite (_set_byte_testbit_high 4 8); auto; try lia;
rewrite (_set_byte_testbit_high 5 8); auto; try lia;
rewrite (_set_byte_testbit_high 6 8); auto; try lia;
rewrite (_set_byte_testbit_high 7 8); auto; try lia;
apply Z.testbit_false; [by lia |];
rewrite Z.div_small; [apply Z.mod_0_l; lia |];
pose proof (Z.pow_le_mono_r 2 64 n ltac:(lia) ltac:(auto)); lia
]; rewrite ?orb_false_l ?orb_false_r;
repeat (rewrite ?Z.lor_spec ?Z.shiftl_spec ?Z.land_spec ?Z.shiftr_spec; try lia);
rewrite !Z.testbit_ones; try lia;
churn_bits'; now rewrite Z.sub_add.
Qed.
Lemma split128:
forall z,
0 <= z < 2^bitsize.bitsZ bitsize.W128 ->
z = Z.lor (_set_byte (_get_byte z 0) 0)
(Z.lor (_set_byte (_get_byte z 1) 1)
(Z.lor (_set_byte (_get_byte z 2) 2)
(Z.lor (_set_byte (_get_byte z 3) 3)
(Z.lor (_set_byte (_get_byte z 4) 4)
(Z.lor (_set_byte (_get_byte z 5) 5)
(Z.lor (_set_byte (_get_byte z 6) 6)
(Z.lor (_set_byte (_get_byte z 7) 7)
(Z.lor (_set_byte (_get_byte z 8) 8)
(Z.lor (_set_byte (_get_byte z 9) 9)
(Z.lor (_set_byte (_get_byte z 10) 10)
(Z.lor (_set_byte (_get_byte z 11) 11)
(Z.lor (_set_byte (_get_byte z 12) 12)
(Z.lor (_set_byte (_get_byte z 13) 13)
(Z.lor (_set_byte (_get_byte z 14) 14)
(_set_byte (_get_byte z 15) 15))))))))))))))).
Proof.
intros * Hbounds; simpl in Hbounds.
apply Z.bits_inj'=> n ?.
assert (n < 8 * 1%nat \/
8 * 1%nat <= n < 8 * 2%nat \/
8 * 2%nat <= n < 8 * 3%nat \/
8 * 3%nat <= n < 8 * 4%nat \/
8 * 4%nat <= n < 8 * 5%nat \/
8 * 5%nat <= n < 8 * 6%nat \/
8 * 6%nat <= n < 8 * 7%nat \/
8 * 7%nat <= n < 8 * 8%nat \/
8 * 8%nat <= n < 8 * 9%nat \/
8 * 9%nat <= n < 8 * 10%nat \/
8 * 10%nat <= n < 8 * 11%nat \/
8 * 11%nat <= n < 8 * 12%nat \/
8 * 12%nat <= n < 8 * 13%nat \/
8 * 13%nat <= n < 8 * 14%nat \/
8 * 14%nat <= n < 8 * 15%nat \/
8 * 15%nat <= n < 8 * 16%nat \/
8 * 16%nat <= n)
as [Hn | [Hn | [Hn | [Hn | [Hn | [Hn | [Hn | [Hn | [Hn | [Hn | [Hn | [Hn | [Hn | [Hn | [Hn | [Hn | Hn]]]]]]]]]]]]]]]]
by lia; rewrite !Z.lor_spec;
[ rewrite (_set_byte_testbit_low 1); auto; try lia;
rewrite (_set_byte_testbit_low 2); auto; try lia;
rewrite (_set_byte_testbit_low 3); auto; try lia;
rewrite (_set_byte_testbit_low 4); auto; try lia;
rewrite (_set_byte_testbit_low 5); auto; try lia;
rewrite (_set_byte_testbit_low 6); auto; try lia;
rewrite (_set_byte_testbit_low 7); auto; try lia;
rewrite (_set_byte_testbit_low 8); auto; try lia;
rewrite (_set_byte_testbit_low 9); auto; try lia;
rewrite (_set_byte_testbit_low 10); auto; try lia;
rewrite (_set_byte_testbit_low 11); auto; try lia;
rewrite (_set_byte_testbit_low 12); auto; try lia;
rewrite (_set_byte_testbit_low 13); auto; try lia;
rewrite (_set_byte_testbit_low 14); auto; try lia;
rewrite (_set_byte_testbit_low 15); auto; try lia
| rewrite (_set_byte_testbit_high 0 1); auto; try lia;
rewrite (_set_byte_testbit_low 2); auto; try lia;
rewrite (_set_byte_testbit_low 3); auto; try lia;
rewrite (_set_byte_testbit_low 4); auto; try lia;
rewrite (_set_byte_testbit_low 5); auto; try lia;
rewrite (_set_byte_testbit_low 6); auto; try lia;
rewrite (_set_byte_testbit_low 7); auto; try lia;
rewrite (_set_byte_testbit_low 8); auto; try lia;
rewrite (_set_byte_testbit_low 9); auto; try lia;
rewrite (_set_byte_testbit_low 10); auto; try lia;
rewrite (_set_byte_testbit_low 11); auto; try lia;
rewrite (_set_byte_testbit_low 12); auto; try lia;
rewrite (_set_byte_testbit_low 13); auto; try lia;
rewrite (_set_byte_testbit_low 14); auto; try lia;
rewrite (_set_byte_testbit_low 15); auto; try lia
| rewrite (_set_byte_testbit_high 0 2); auto; try lia;
rewrite (_set_byte_testbit_high 1 2); auto; try lia;
rewrite (_set_byte_testbit_low 3); auto; try lia;
rewrite (_set_byte_testbit_low 4); auto; try lia;
rewrite (_set_byte_testbit_low 5); auto; try lia;
rewrite (_set_byte_testbit_low 6); auto; try lia;
rewrite (_set_byte_testbit_low 7); auto; try lia;
rewrite (_set_byte_testbit_low 8); auto; try lia;
rewrite (_set_byte_testbit_low 9); auto; try lia;
rewrite (_set_byte_testbit_low 10); auto; try lia;
rewrite (_set_byte_testbit_low 11); auto; try lia;
rewrite (_set_byte_testbit_low 12); auto; try lia;
rewrite (_set_byte_testbit_low 13); auto; try lia;
rewrite (_set_byte_testbit_low 14); auto; try lia;
rewrite (_set_byte_testbit_low 15); auto; try lia
| rewrite (_set_byte_testbit_high 0 3); auto; try lia;
rewrite (_set_byte_testbit_high 1 3); auto; try lia;
rewrite (_set_byte_testbit_high 2 3); auto; try lia;
rewrite (_set_byte_testbit_low 4); auto; try lia;
rewrite (_set_byte_testbit_low 5); auto; try lia;
rewrite (_set_byte_testbit_low 6); auto; try lia;
rewrite (_set_byte_testbit_low 7); auto; try lia;
rewrite (_set_byte_testbit_low 8); auto; try lia;
rewrite (_set_byte_testbit_low 9); auto; try lia;
rewrite (_set_byte_testbit_low 10); auto; try lia;
rewrite (_set_byte_testbit_low 11); auto; try lia;
rewrite (_set_byte_testbit_low 12); auto; try lia;
rewrite (_set_byte_testbit_low 13); auto; try lia;
rewrite (_set_byte_testbit_low 14); auto; try lia;
rewrite (_set_byte_testbit_low 15); auto; try lia
| rewrite (_set_byte_testbit_high 0 4); auto; try lia;
rewrite (_set_byte_testbit_high 1 4); auto; try lia;
rewrite (_set_byte_testbit_high 2 4); auto; try lia;
rewrite (_set_byte_testbit_high 3 4); auto; try lia;
rewrite (_set_byte_testbit_low 5); auto; try lia;
rewrite (_set_byte_testbit_low 6); auto; try lia;
rewrite (_set_byte_testbit_low 7); auto; try lia;
rewrite (_set_byte_testbit_low 8); auto; try lia;
rewrite (_set_byte_testbit_low 9); auto; try lia;
rewrite (_set_byte_testbit_low 10); auto; try lia;
rewrite (_set_byte_testbit_low 11); auto; try lia;
rewrite (_set_byte_testbit_low 12); auto; try lia;
rewrite (_set_byte_testbit_low 13); auto; try lia;
rewrite (_set_byte_testbit_low 14); auto; try lia;
rewrite (_set_byte_testbit_low 15); auto; try lia
| rewrite (_set_byte_testbit_high 0 5); auto; try lia;
rewrite (_set_byte_testbit_high 1 5); auto; try lia;
rewrite (_set_byte_testbit_high 2 5); auto; try lia;
rewrite (_set_byte_testbit_high 3 5); auto; try lia;
rewrite (_set_byte_testbit_high 4 5); auto; try lia;
rewrite (_set_byte_testbit_low 6); auto; try lia;
rewrite (_set_byte_testbit_low 7); auto; try lia;
rewrite (_set_byte_testbit_low 8); auto; try lia;
rewrite (_set_byte_testbit_low 9); auto; try lia;
rewrite (_set_byte_testbit_low 10); auto; try lia;
rewrite (_set_byte_testbit_low 11); auto; try lia;
rewrite (_set_byte_testbit_low 12); auto; try lia;
rewrite (_set_byte_testbit_low 13); auto; try lia;
rewrite (_set_byte_testbit_low 14); auto; try lia;
rewrite (_set_byte_testbit_low 15); auto; try lia
| rewrite (_set_byte_testbit_high 0 6); auto; try lia;
rewrite (_set_byte_testbit_high 1 6); auto; try lia;
rewrite (_set_byte_testbit_high 2 6); auto; try lia;
rewrite (_set_byte_testbit_high 3 6); auto; try lia;
rewrite (_set_byte_testbit_high 4 6); auto; try lia;
rewrite (_set_byte_testbit_high 5 6); auto; try lia;
rewrite (_set_byte_testbit_low 7); auto; try lia;
rewrite (_set_byte_testbit_low 8); auto; try lia;
rewrite (_set_byte_testbit_low 9); auto; try lia;
rewrite (_set_byte_testbit_low 10); auto; try lia;
rewrite (_set_byte_testbit_low 11); auto; try lia;
rewrite (_set_byte_testbit_low 12); auto; try lia;
rewrite (_set_byte_testbit_low 13); auto; try lia;
rewrite (_set_byte_testbit_low 14); auto; try lia;
rewrite (_set_byte_testbit_low 15); auto; try lia
| rewrite (_set_byte_testbit_high 0 7); auto; try lia;
rewrite (_set_byte_testbit_high 1 7); auto; try lia;
rewrite (_set_byte_testbit_high 2 7); auto; try lia;
rewrite (_set_byte_testbit_high 3 7); auto; try lia;
rewrite (_set_byte_testbit_high 4 7); auto; try lia;
rewrite (_set_byte_testbit_high 5 7); auto; try lia;
rewrite (_set_byte_testbit_high 6 7); auto; try lia;
rewrite (_set_byte_testbit_low 8); auto; try lia;
rewrite (_set_byte_testbit_low 9); auto; try lia;
rewrite (_set_byte_testbit_low 10); auto; try lia;
rewrite (_set_byte_testbit_low 11); auto; try lia;
rewrite (_set_byte_testbit_low 12); auto; try lia;
rewrite (_set_byte_testbit_low 13); auto; try lia;
rewrite (_set_byte_testbit_low 14); auto; try lia;
rewrite (_set_byte_testbit_low 15); auto; try lia
| rewrite (_set_byte_testbit_high 0 8); auto; try lia;
rewrite (_set_byte_testbit_high 1 8); auto; try lia;
rewrite (_set_byte_testbit_high 2 8); auto; try lia;
rewrite (_set_byte_testbit_high 3 8); auto; try lia;
rewrite (_set_byte_testbit_high 4 8); auto; try lia;
rewrite (_set_byte_testbit_high 5 8); auto; try lia;
rewrite (_set_byte_testbit_high 6 8); auto; try lia;
rewrite (_set_byte_testbit_high 7 8); auto; try lia;
rewrite (_set_byte_testbit_low 9); auto; try lia;
rewrite (_set_byte_testbit_low 10); auto; try lia;
rewrite (_set_byte_testbit_low 11); auto; try lia;
rewrite (_set_byte_testbit_low 12); auto; try lia;
rewrite (_set_byte_testbit_low 13); auto; try lia;
rewrite (_set_byte_testbit_low 14); auto; try lia;
rewrite (_set_byte_testbit_low 15); auto; try lia
| rewrite (_set_byte_testbit_high 0 9); auto; try lia;
rewrite (_set_byte_testbit_high 1 9); auto; try lia;
rewrite (_set_byte_testbit_high 2 9); auto; try lia;
rewrite (_set_byte_testbit_high 3 9); auto; try lia;
rewrite (_set_byte_testbit_high 4 9); auto; try lia;
rewrite (_set_byte_testbit_high 5 9); auto; try lia;
rewrite (_set_byte_testbit_high 6 9); auto; try lia;
rewrite (_set_byte_testbit_high 7 9); auto; try lia;
rewrite (_set_byte_testbit_high 8 9); auto; try lia;
rewrite (_set_byte_testbit_low 10); auto; try lia;
rewrite (_set_byte_testbit_low 11); auto; try lia;
rewrite (_set_byte_testbit_low 12); auto; try lia;
rewrite (_set_byte_testbit_low 13); auto; try lia;
rewrite (_set_byte_testbit_low 14); auto; try lia;
rewrite (_set_byte_testbit_low 15); auto; try lia
| rewrite (_set_byte_testbit_high 0 10); auto; try lia;
rewrite (_set_byte_testbit_high 1 10); auto; try lia;
rewrite (_set_byte_testbit_high 2 10); auto; try lia;
rewrite (_set_byte_testbit_high 3 10); auto; try lia;
rewrite (_set_byte_testbit_high 4 10); auto; try lia;
rewrite (_set_byte_testbit_high 5 10); auto; try lia;
rewrite (_set_byte_testbit_high 6 10); auto; try lia;
rewrite (_set_byte_testbit_high 7 10); auto; try lia;
rewrite (_set_byte_testbit_high 8 10); auto; try lia;
rewrite (_set_byte_testbit_high 9 10); auto; try lia;
rewrite (_set_byte_testbit_low 11); auto; try lia;
rewrite (_set_byte_testbit_low 12); auto; try lia;
rewrite (_set_byte_testbit_low 13); auto; try lia;
rewrite (_set_byte_testbit_low 14); auto; try lia;
rewrite (_set_byte_testbit_low 15); auto; try lia
| rewrite (_set_byte_testbit_high 0 11); auto; try lia;
rewrite (_set_byte_testbit_high 1 11); auto; try lia;
rewrite (_set_byte_testbit_high 2 11); auto; try lia;
rewrite (_set_byte_testbit_high 3 11); auto; try lia;
rewrite (_set_byte_testbit_high 4 11); auto; try lia;
rewrite (_set_byte_testbit_high 5 11); auto; try lia;
rewrite (_set_byte_testbit_high 6 11); auto; try lia;
rewrite (_set_byte_testbit_high 7 11); auto; try lia;
rewrite (_set_byte_testbit_high 8 11); auto; try lia;
rewrite (_set_byte_testbit_high 9 11); auto; try lia;
rewrite (_set_byte_testbit_high 10 11); auto; try lia;
rewrite (_set_byte_testbit_low 12); auto; try lia;
rewrite (_set_byte_testbit_low 13); auto; try lia;
rewrite (_set_byte_testbit_low 14); auto; try lia;
rewrite (_set_byte_testbit_low 15); auto; try lia
| rewrite (_set_byte_testbit_high 0 12); auto; try lia;
rewrite (_set_byte_testbit_high 1 12); auto; try lia;
rewrite (_set_byte_testbit_high 2 12); auto; try lia;
rewrite (_set_byte_testbit_high 3 12); auto; try lia;
rewrite (_set_byte_testbit_high 4 12); auto; try lia;
rewrite (_set_byte_testbit_high 5 12); auto; try lia;
rewrite (_set_byte_testbit_high 6 12); auto; try lia;
rewrite (_set_byte_testbit_high 7 12); auto; try lia;
rewrite (_set_byte_testbit_high 8 12); auto; try lia;
rewrite (_set_byte_testbit_high 9 12); auto; try lia;
rewrite (_set_byte_testbit_high 10 12); auto; try lia;
rewrite (_set_byte_testbit_high 11 12); auto; try lia;
rewrite (_set_byte_testbit_low 13); auto; try lia;
rewrite (_set_byte_testbit_low 14); auto; try lia;
rewrite (_set_byte_testbit_low 15); auto; try lia
| rewrite (_set_byte_testbit_high 0 13); auto; try lia;
rewrite (_set_byte_testbit_high 1 13); auto; try lia;
rewrite (_set_byte_testbit_high 2 13); auto; try lia;
rewrite (_set_byte_testbit_high 3 13); auto; try lia;
rewrite (_set_byte_testbit_high 4 13); auto; try lia;
rewrite (_set_byte_testbit_high 5 13); auto; try lia;
rewrite (_set_byte_testbit_high 6 13); auto; try lia;
rewrite (_set_byte_testbit_high 7 13); auto; try lia;
rewrite (_set_byte_testbit_high 8 13); auto; try lia;
rewrite (_set_byte_testbit_high 9 13); auto; try lia;
rewrite (_set_byte_testbit_high 10 13); auto; try lia;
rewrite (_set_byte_testbit_high 11 13); auto; try lia;
rewrite (_set_byte_testbit_high 12 13); auto; try lia;
rewrite (_set_byte_testbit_low 14); auto; try lia;
rewrite (_set_byte_testbit_low 15); auto; try lia
| rewrite (_set_byte_testbit_high 0 14); auto; try lia;
rewrite (_set_byte_testbit_high 1 14); auto; try lia;
rewrite (_set_byte_testbit_high 2 14); auto; try lia;
rewrite (_set_byte_testbit_high 3 14); auto; try lia;
rewrite (_set_byte_testbit_high 4 14); auto; try lia;
rewrite (_set_byte_testbit_high 5 14); auto; try lia;
rewrite (_set_byte_testbit_high 6 14); auto; try lia;
rewrite (_set_byte_testbit_high 7 14); auto; try lia;
rewrite (_set_byte_testbit_high 8 14); auto; try lia;
rewrite (_set_byte_testbit_high 9 14); auto; try lia;
rewrite (_set_byte_testbit_high 10 14); auto; try lia;
rewrite (_set_byte_testbit_high 11 14); auto; try lia;
rewrite (_set_byte_testbit_high 12 14); auto; try lia;
rewrite (_set_byte_testbit_high 13 14); auto; try lia;
rewrite (_set_byte_testbit_low 15); auto; try lia
| rewrite (_set_byte_testbit_high 0 15); auto; try lia;
rewrite (_set_byte_testbit_high 1 15); auto; try lia;
rewrite (_set_byte_testbit_high 2 15); auto; try lia;
rewrite (_set_byte_testbit_high 3 15); auto; try lia;
rewrite (_set_byte_testbit_high 4 15); auto; try lia;
rewrite (_set_byte_testbit_high 5 15); auto; try lia;
rewrite (_set_byte_testbit_high 6 15); auto; try lia;
rewrite (_set_byte_testbit_high 7 15); auto; try lia;
rewrite (_set_byte_testbit_high 8 15); auto; try lia;
rewrite (_set_byte_testbit_high 9 15); auto; try lia;
rewrite (_set_byte_testbit_high 10 15); auto; try lia;
rewrite (_set_byte_testbit_high 11 15); auto; try lia;
rewrite (_set_byte_testbit_high 12 15); auto; try lia;
rewrite (_set_byte_testbit_high 13 15); auto; try lia;
rewrite (_set_byte_testbit_high 14 15); auto; try lia
| rewrite (_set_byte_testbit_high 0 16); auto; try lia;
rewrite (_set_byte_testbit_high 1 16); auto; try lia;
rewrite (_set_byte_testbit_high 2 16); auto; try lia;
rewrite (_set_byte_testbit_high 3 16); auto; try lia;
rewrite (_set_byte_testbit_high 4 16); auto; try lia;
rewrite (_set_byte_testbit_high 5 16); auto; try lia;
rewrite (_set_byte_testbit_high 6 16); auto; try lia;
rewrite (_set_byte_testbit_high 7 16); auto; try lia;
rewrite (_set_byte_testbit_high 8 16); auto; try lia;
rewrite (_set_byte_testbit_high 9 16); auto; try lia;
rewrite (_set_byte_testbit_high 10 16); auto; try lia;
rewrite (_set_byte_testbit_high 11 16); auto; try lia;
rewrite (_set_byte_testbit_high 12 16); auto; try lia;
rewrite (_set_byte_testbit_high 13 16); auto; try lia;
rewrite (_set_byte_testbit_high 14 16); auto; try lia;
rewrite (_set_byte_testbit_high 15 16); auto; try lia;
apply Z.testbit_false; [by lia |];
rewrite Z.div_small; [apply Z.mod_0_l; lia |];
pose proof (Z.pow_le_mono_r 2 128 n ltac:(lia) ltac:(auto)); lia
]; rewrite ?orb_false_l ?orb_false_r;
repeat (rewrite ?Z.lor_spec ?Z.shiftl_spec ?Z.land_spec ?Z.shiftr_spec; try lia);
rewrite !Z.testbit_ones; try lia;
churn_bits'; now rewrite Z.sub_add.
Qed.
End split.
End Byte.
#[global] Opaque _get_byte _set_byte.
Section Bswap.
#[local] Definition bswap_idxs (bytes: nat) :=
let idxs := seq 0 bytes in
zip idxs (rev idxs).
#[local] Definition bswap_ (bytes: nat) (n: Z): Z :=
fold_left (fun acc '(getIdx, setIdx) =>
Z.lor acc (_set_byte (_get_byte n getIdx) setIdx))
(bswap_idxs bytes)
0.
Definition bswap (sz : bitsize) : Z -> Z := bswap_ (bitsize.bytesNat sz).
Section test.
Local Definition bytes (ls : list Z) :=
fold_left (fun a b => a * 256 + b)%Z ls 0%Z.
Arguments bytes _%_Z.
Local Definition _bswap16_test : bswap bitsize.W16 (bytes (1::2::nil)%Z) = bytes (2::1::nil)%Z := eq_refl.
Local Definition _bswap32_test :
bswap bitsize.W32 (bytes (1::2::3::4::nil)%Z) = bytes (4::3::2::1::nil)%Z := eq_refl.
Local Definition _bswap64_test :
bswap bitsize.W64 (bytes (1::2::3::4::5::6::7::8::nil)%Z) = bytes (8::7::6::5::4::3::2::1::nil)%Z := eq_refl.
End test.
End Bswap.
Notation bswap8 := (bswap bitsize.W8) (only parsing).
Notation bswap16 := (bswap bitsize.W16) (only parsing).
Notation bswap32 := (bswap bitsize.W32) (only parsing).
Notation bswap64 := (bswap bitsize.W64) (only parsing).
Notation bswap128 := (bswap bitsize.W128) (only parsing).
#[global] Opaque bswap.
Section Bswap.
Section Theory.
Section bounded.
#[local] Transparent bswap.
Lemma bswap8_bounded:
forall v,
0 <= bswap8 v < 256.
Proof.
intros *; rewrite /bswap/bswap_/= Z.lor_0_l.
pose proof (_set_byte_bound (_get_byte v 0) 0).
lia.
Qed.
Lemma bswap16_bounded:
∀ v,
0 ≤ bswap16 v < 65536.
Proof.
intros *; rewrite /bswap/bswap_/= Z.lor_0_l.
pose proof (_set_byte_bound (_get_byte v 0) 1).
pose proof (_set_byte_bound (_get_byte v 1) 0).
repeat (try apply ZlorRange with (w:=16%N)); lia.
Qed.
Lemma bswap32_bounded:
forall v,
0 ≤ bswap32 v < 4294967296.
Proof.
intros *; rewrite /bswap/bswap_/= Z.lor_0_l.
pose proof (_set_byte_bound (_get_byte v 0) 3).
pose proof (_set_byte_bound (_get_byte v 1) 2).
pose proof (_set_byte_bound (_get_byte v 2) 1).
pose proof (_set_byte_bound (_get_byte v 3) 0).
repeat (try apply ZlorRange with (w:=32%N)); lia.
Qed.
Lemma bswap64_bounded:
forall v,
0 ≤ bswap64 v < 18446744073709551616.
Proof.
intros *; rewrite /bswap/bswap_/= Z.lor_0_l.
pose proof (_set_byte_bound (_get_byte v 0) 7).
pose proof (_set_byte_bound (_get_byte v 1) 6).
pose proof (_set_byte_bound (_get_byte v 2) 5).
pose proof (_set_byte_bound (_get_byte v 3) 4).
pose proof (_set_byte_bound (_get_byte v 4) 3).
pose proof (_set_byte_bound (_get_byte v 5) 2).
pose proof (_set_byte_bound (_get_byte v 6) 1).
pose proof (_set_byte_bound (_get_byte v 7) 0).
repeat (try apply ZlorRange with (w:=64%N)); lia.
Qed.
Lemma bswap128_bounded:
forall v,
0 ≤ bswap128 v < ltac:(let x := eval cbv in (2^128) in exact x).
Proof.
intros *; rewrite /bswap/bswap_/= Z.lor_0_l.
pose proof (_set_byte_bound (_get_byte v 0) 15).
pose proof (_set_byte_bound (_get_byte v 1) 14).
pose proof (_set_byte_bound (_get_byte v 2) 13).
pose proof (_set_byte_bound (_get_byte v 3) 12).
pose proof (_set_byte_bound (_get_byte v 4) 11).
pose proof (_set_byte_bound (_get_byte v 5) 10).
pose proof (_set_byte_bound (_get_byte v 6) 9).
pose proof (_set_byte_bound (_get_byte v 7) 8).
pose proof (_set_byte_bound (_get_byte v 8) 7).
pose proof (_set_byte_bound (_get_byte v 9) 6).
pose proof (_set_byte_bound (_get_byte v 10) 5).
pose proof (_set_byte_bound (_get_byte v 11) 4).
pose proof (_set_byte_bound (_get_byte v 12) 3).
pose proof (_set_byte_bound (_get_byte v 13) 2).
pose proof (_set_byte_bound (_get_byte v 14) 1).
pose proof (_set_byte_bound (_get_byte v 15) 0).
repeat (try apply ZlorRange with (w:=128%N)); lia.
Qed.
End bounded.
Lemma bswap_bounded:
forall sz v,
0 <= bswap sz v < 2^(bitsize.bitsZ sz).
Proof.
intros *; destruct sz;
eauto using
bswap8_bounded,
bswap16_bounded,
bswap32_bounded,
bswap64_bounded,
bswap128_bounded.
Qed.
End Theory.
End Bswap.
Section Bswap.
Section Theory.
Section useless_lor.
#[local] Transparent _get_byte _set_byte bswap.
Lemma bswap8_useless_lor:
forall v v' idx,
(idx >= 1)%nat ->
bswap8 (Z.lor v (_set_byte v' idx)) =
bswap8 v.
Proof.
move=> v v' idx Hidx.
rewrite /bswap8/bswap/bswap_/_get_byte/=; f_equal.
rewrite /_set_byte; churn_bits.
Qed.
Lemma bswap16_useless_lor:
forall v v' idx,
(idx >= 2)%nat ->
bswap16 (Z.lor v (_set_byte v' idx)) =
bswap16 v.
Proof.
move=> v v' idx Hidx.
rewrite /bswap16/bswap/bswap_/_get_byte/=.
do 1 (f_equal; [| f_equal; rewrite /_set_byte; churn_bits]).
f_equal; f_equal; rewrite /_set_byte; churn_bits.
Qed.
Lemma bswap32_useless_lor:
forall v v' idx,
(idx >= 4)%nat ->
bswap32 (Z.lor v (_set_byte v' idx)) =
bswap32 v.
Proof.
move=> v v' idx Hidx.
rewrite /bswap32/bswap/bswap_/_get_byte/=.
do 3 (f_equal; [| f_equal; rewrite /_set_byte; churn_bits]).
f_equal; f_equal; rewrite /_set_byte; churn_bits.
Qed.
Lemma bswap64_useless_lor:
forall v v' idx,
(idx >= 8)%nat ->
bswap64 (Z.lor v (_set_byte v' idx)) =
bswap64 v.
Proof.
move=> v v' idx Hidx.
rewrite /bswap64/bswap/bswap_/_get_byte/=.
do 7 (f_equal; [| f_equal; rewrite /_set_byte; churn_bits]).
f_equal; f_equal; rewrite /_set_byte; churn_bits.
Qed.
Lemma bswap128_useless_lor:
forall v v' idx,
(idx >= 16)%nat ->
bswap128 (Z.lor v (_set_byte v' idx)) =
bswap128 v.
Proof.
move=> v v' idx Hidx.
rewrite /bswap128/bswap/bswap_/_get_byte/=.
do 15 (f_equal; [| f_equal; rewrite /_set_byte; churn_bits]).
f_equal; f_equal; rewrite /_set_byte; churn_bits.
Qed.
End useless_lor.
Lemma bswap_useless_lor:
forall sz v v' idx,
(bitsize.bytesNat sz <= idx)%nat ->
bswap sz (Z.lor v (_set_byte v' idx)) =
bswap sz v.
Proof.
intros [] v v' idx Hidx; simpl in *;
eauto using
bswap8_useless_lor,
bswap16_useless_lor,
bswap32_useless_lor,
bswap64_useless_lor,
bswap128_useless_lor.
Qed.
Section _set_byte_reverse.
#[local] Transparent _get_byte _set_byte bswap.
Lemma bswap8_set_byte_reverse:
forall x,
bswap8 (_set_byte x 0) =
_set_byte x 0.
Proof.
intros *.
rewrite /bswap8/bswap/bswap_/=.
rewrite !_set_get_byte_roundtrip !Z.shiftl_0_r !Z.lor_0_l.
rewrite _get_set_byte_roundtrip; now apply _get_0_set_0_eq.
Qed.
Lemma bswap16_set_byte_reverse:
forall x1 x2,
bswap16 (Z.lor (_set_byte x1 0)
(_set_byte x2 1)) =
Z.lor (_set_byte x2 0)
(_set_byte x1 1).
Proof.
intros *.
rewrite /bswap16/bswap/bswap_/=.
rewrite {1}Z.lor_comm; f_equal;
try (rewrite !_get_byte_lor !_set_byte_lor
!_get_set_byte_roundtrip !_get_set_byte_no_overlap
?_set_byte_0 ?Z.lor_0_r ?Z.lor_0_l ?_set_get_0 //).
Qed.
Lemma bswap32_set_byte_reverse:
forall x1 x2 x3 x4,
bswap32 (Z.lor (_set_byte x1 0)
(Z.lor (_set_byte x2 1)
(Z.lor (_set_byte x3 2)
(_set_byte x4 3)))) =
Z.lor (_set_byte x4 0)
(Z.lor (_set_byte x3 1)
(Z.lor (_set_byte x2 2)
(_set_byte x1 3))).
Proof.
intros *.
rewrite /bswap32/bswap/bswap_/=.
rewrite {1}Z.lor_comm; f_equal;
try (rewrite !_get_byte_lor !_set_byte_lor
!_get_set_byte_roundtrip !_get_set_byte_no_overlap
?_set_byte_0 ?Z.lor_0_r ?Z.lor_0_l ?_set_get_0 //);
repeat (rewrite {1}Z.lor_comm; f_equal).
Qed.
Lemma bswap64_set_byte_reverse:
forall x1 x2 x3 x4 x5 x6 x7 x8,
bswap64 (Z.lor (_set_byte x1 0)
(Z.lor (_set_byte x2 1)
(Z.lor (_set_byte x3 2)
(Z.lor (_set_byte x4 3)
(Z.lor (_set_byte x5 4)
(Z.lor (_set_byte x6 5)
(Z.lor (_set_byte x7 6)
(_set_byte x8 7)))))))) =
Z.lor (_set_byte x8 0)
(Z.lor (_set_byte x7 1)
(Z.lor (_set_byte x6 2)
(Z.lor (_set_byte x5 3)
(Z.lor (_set_byte x4 4)
(Z.lor (_set_byte x3 5)
(Z.lor (_set_byte x2 6)
(_set_byte x1 7))))))).
Proof.
intros *.
rewrite /bswap64/bswap/bswap_/=.
repeat (rewrite {1}Z.lor_comm; f_equal;
[now (rewrite !_get_byte_lor !_set_byte_lor
!_get_set_byte_roundtrip !_get_set_byte_no_overlap
?_set_byte_0 ?Z.lor_0_r ?Z.lor_0_l ?_set_get_0 //) |]).
rewrite !_get_byte_lor !_set_byte_lor
!_get_set_byte_roundtrip !_get_set_byte_no_overlap
?_set_byte_0 ?Z.lor_0_r ?Z.lor_0_l ?_set_get_0 //.
Qed.
Lemma bswap128_set_byte_reverse:
forall x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16,
bswap128 (Z.lor (_set_byte x1 0)
(Z.lor (_set_byte x2 1)
(Z.lor (_set_byte x3 2)
(Z.lor (_set_byte x4 3)
(Z.lor (_set_byte x5 4)
(Z.lor (_set_byte x6 5)
(Z.lor (_set_byte x7 6)
(Z.lor (_set_byte x8 7)
(Z.lor (_set_byte x9 8)
(Z.lor (_set_byte x10 9)
(Z.lor (_set_byte x11 10)
(Z.lor (_set_byte x12 11)
(Z.lor (_set_byte x13 12)
(Z.lor (_set_byte x14 13)
(Z.lor (_set_byte x15 14)
(_set_byte x16 15)))))))))))))))) =
Z.lor (_set_byte x16 0)
(Z.lor (_set_byte x15 1)
(Z.lor (_set_byte x14 2)
(Z.lor (_set_byte x13 3)
(Z.lor (_set_byte x12 4)
(Z.lor (_set_byte x11 5)
(Z.lor (_set_byte x10 6)
(Z.lor (_set_byte x9 7)
(Z.lor (_set_byte x8 8)
(Z.lor (_set_byte x7 9)
(Z.lor (_set_byte x6 10)
(Z.lor (_set_byte x5 11)
(Z.lor (_set_byte x4 12)
(Z.lor (_set_byte x3 13)
(Z.lor (_set_byte x2 14)
(_set_byte x1 15))))))))))))))).
Proof.
intros *.
rewrite /bswap128/bswap/bswap_/=.
repeat (rewrite {1}Z.lor_comm; f_equal;
[now (rewrite !_get_byte_lor !_set_byte_lor
!_get_set_byte_roundtrip !_get_set_byte_no_overlap
?_set_byte_0 ?Z.lor_0_r ?Z.lor_0_l ?_set_get_0 //) |]).
rewrite !_get_byte_lor !_set_byte_lor
!_get_set_byte_roundtrip !_get_set_byte_no_overlap
?_set_byte_0 ?Z.lor_0_r ?Z.lor_0_l ?_set_get_0 //.
Qed.
End _set_byte_reverse.
Section larger.
#[local] Transparent _get_byte _set_byte bswap.
Lemma bswap8_larger z:
0 <= z ->
bswap8 z = bswap8 (Z.land z (Z.ones 8)).
Proof.
intros; rewrite /bswap8/bswap/bswap_/= ?Z.lor_0_l ?Z.lor_0_r.
f_equal; rewrite /_get_byte; churn_bits.
Qed.
Lemma bswap16_larger z:
0 <= z ->
bswap16 z = bswap16 (Z.land z (Z.ones 16)).
Proof.
intros; rewrite /bswap16/bswap/bswap_/= ?Z.lor_0_l ?Z.lor_0_r.
do 1 (f_equal; [| f_equal; rewrite /_get_byte; churn_bits]).
f_equal; rewrite /_get_byte; churn_bits.
Qed.
Lemma bswap32_larger z:
0 <= z ->
bswap32 z = bswap32 (Z.land z (Z.ones 32)).
Proof.
intros; rewrite /bswap32/bswap/bswap_/= ?Z.lor_0_l ?Z.lor_0_r.
do 3 (f_equal; [| f_equal; rewrite /_get_byte; churn_bits]).
f_equal; rewrite /_get_byte; churn_bits.
Qed.
Lemma bswap64_larger z:
0 <= z ->
bswap64 z = bswap64 (Z.land z (Z.ones 64)).
Proof.
intros; rewrite /bswap64/bswap/bswap_/= ?Z.lor_0_l ?Z.lor_0_r.
do 7 (f_equal; [| f_equal; rewrite /_get_byte; churn_bits]).
f_equal; rewrite /_get_byte; churn_bits.
Qed.
Lemma bswap128_larger z:
0 <= z ->
bswap128 z = bswap128 (Z.land z (Z.ones 128)).
Proof.
intros; rewrite /bswap128/bswap/bswap_/= ?Z.lor_0_l ?Z.lor_0_r.
do 15 (f_equal; [| f_equal; rewrite /_get_byte; churn_bits]).
f_equal; rewrite /_get_byte; churn_bits.
Qed.
End larger.
Lemma bswap_larger sz z:
0 <= z ->
bswap sz z = bswap sz (Z.land z (Z.ones (bitsize.bitsZ sz))).
Proof.
intros; destruct sz;
eauto using
bswap8_larger,
bswap16_larger,
bswap32_larger,
bswap64_larger,
bswap128_larger.
Qed.
Section involutive.
Lemma bswap8_involutive:
forall z,
0 <= z < 2^bitsize.bitsZ bitsize.W8 ->
bswap8 (bswap8 z) = z.
Proof.
intros * Hbounds.
rewrite (split8 z); auto.
now rewrite 2!bswap8_set_byte_reverse.
Qed.
Lemma bswap8_involutive_land:
forall z,
0 <= z < 2^bitsize.bitsZ bitsize.W8 ->
bswap8 (bswap8 z) = Z.land z (Z.ones (bitsize.bitsZ bitsize.W8)).
Proof.
intros * Hbounds; simpl in *.
rewrite bswap8_involutive; auto.
rewrite -> Z.land_ones by lia.
rewrite Z.mod_small; auto.
Qed.
Lemma bswap16_involutive:
forall z,
0 <= z < 2^bitsize.bitsZ bitsize.W16 ->
bswap16 (bswap16 z) = z.
Proof.
intros * Hbounds.
rewrite (split16 z); auto.
now rewrite 2!bswap16_set_byte_reverse.
Qed.
Lemma bswap16_involutive_land:
forall z,
0 <= z < 2^bitsize.bitsZ bitsize.W16 ->
bswap16 (bswap16 z) = Z.land z (Z.ones (bitsize.bitsZ bitsize.W16)).
Proof.
intros * Hbounds; simpl in *.
rewrite bswap16_involutive; auto.
rewrite -> Z.land_ones by lia.
rewrite Z.mod_small; auto.
Qed.
Lemma bswap32_involutive:
forall z,
0 <= z < 2^bitsize.bitsZ bitsize.W32 ->
bswap32 (bswap32 z) = z.
Proof.
intros * Hbounds.
rewrite (split32 z); auto.
now rewrite 2!bswap32_set_byte_reverse.
Qed.
Lemma bswap32_involutive_land:
forall z,
0 <= z < 2^bitsize.bitsZ bitsize.W32 ->
bswap32 (bswap32 z) = Z.land z (Z.ones (bitsize.bitsZ bitsize.W32)).
Proof.
intros * Hbounds; simpl in *.
rewrite bswap32_involutive; auto.
rewrite -> Z.land_ones by lia.
rewrite Z.mod_small; auto.
Qed.
Lemma bswap64_involutive:
forall z,
0 <= z < 2^bitsize.bitsZ bitsize.W64 ->
bswap64 (bswap64 z) = z.
Proof.
intros * Hbounds.
rewrite (split64 z); auto.
now rewrite 2!bswap64_set_byte_reverse.
Qed.
Lemma bswap64_involutive_land:
forall z,
0 <= z < 2^bitsize.bitsZ bitsize.W64 ->
bswap64 (bswap64 z) = Z.land z (Z.ones (bitsize.bitsZ bitsize.W64)).
Proof.
intros * Hbounds; simpl in *.
rewrite bswap64_involutive; auto.
rewrite -> Z.land_ones by lia.
rewrite Z.mod_small; auto.
Qed.
Lemma bswap128_involutive:
forall z,
0 <= z < 2^bitsize.bitsZ bitsize.W128 ->
bswap128 (bswap128 z) = z.
Proof.
intros * Hbounds.
rewrite (split128 z); auto.
now rewrite 2!bswap128_set_byte_reverse.
Qed.
Lemma bswap128_involutive_land:
forall z,
0 <= z < 2^bitsize.bitsZ bitsize.W128 ->
bswap128 (bswap128 z) = Z.land z (Z.ones (bitsize.bitsZ bitsize.W128)).
Proof.
intros * Hbounds; simpl in *.
rewrite bswap128_involutive; auto.
rewrite -> Z.land_ones by lia.
rewrite Z.mod_small; auto.
Qed.
End involutive.
Lemma bswap_involutive:
forall sz z,
0 <= z < 2^bitsize.bitsZ sz ->
bswap sz (bswap sz z) = z.
Proof.
intros * Hbounds; destruct sz;
eauto using
bswap8_involutive,
bswap16_involutive,
bswap32_involutive,
bswap64_involutive,
bswap128_involutive.
Qed.
Lemma bswap_involutive_land:
forall sz z,
0 <= z < 2^bitsize.bitsZ sz ->
bswap sz (bswap sz z) = Z.land z (Z.ones (bitsize.bitsZ sz)).
Proof.
intros * Hbounds; destruct sz;
eauto using
bswap8_involutive_land,
bswap16_involutive_land,
bswap32_involutive_land,
bswap64_involutive_land,
bswap128_involutive_land.
Qed.
End Theory.
End Bswap.
* Copyright (c) 2020 BedRock Systems, Inc.
* This software is distributed under the terms of the BedRock Open-Source License.
* See the LICENSE-BedRock file in the repository root for details.
*)
Require Import bedrock.prelude.base.
Require Import bedrock.lang.cpp.arith.types.
Require Import bedrock.lang.cpp.arith.operator.
#[local] Open Scope Z_scope.
(* see
* https://gcc.gnu.org/onlinedocs/gcc/Other-Builtins.html
*)
(* Returns one plus the index of the least significant 1-bit of x,
or if x is zero, returns zero. *)
Definition first_set (sz : bitsize) (n : Z) : Z :=
let n := trim (bitsize.bitsN sz) n in
if Z.eqb n 0 then 0
else
(fix get ls : Z :=
match ls with
| nil => 64
| l :: ls =>
if Z.testbit n l
then 1 + l
else get ls
end)%Z (List.map Z.of_nat (seq 0 (N.to_nat (bitsize.bitsN sz)))).
#[global] Arguments first_set : simpl never.
(* Returns the number of trailing 0-bits in x, starting at the least
significant bit position. If x is 0, the result is undefined. *)
Definition trailing_zeros (sz : bitsize) (n : Z) : Z :=
(fix get ls : Z :=
match ls with
| nil => 64
| l :: ls =>
if Z.testbit n l
then l
else get ls
end)%Z (List.map Z.of_nat (seq 0 (N.to_nat (bitsize.bitsN sz)))).
#[global] Arguments trailing_zeros : simpl never.
(* Returns the number of leading 0-bits in x, starting at the most significant
bit position. If x is 0, the result is undefined. *)
Definition leading_zeros (sz : bitsize) (l : Z) : Z :=
bitsize.bitsZ sz - Z.log2 (l mod (2^64)).
#[global] Arguments leading_zeros : simpl never.
Module Import churn_bits.
(* TODO: using bool_decide would simplify this reasoning. *)
Ltac churn_bits' :=
repeat match goal with
| |- context[(?l <=? ?r)%Z] =>
let Hnb := fresh "Hnb" in
let Hn := fresh "Hn" in
destruct (l <=? r)%Z eqn:Hnb;
set (Hn := Hnb);
[ apply Z.leb_le in Hn
| apply Z.leb_gt in Hn]
| |- context[(?l <? ?r)%Z] =>
let Hnb := fresh "Hnb" in
let Hn := fresh "Hn" in
destruct (l <? r)%Z eqn:Hnb;
set (Hn := Hnb);
[ apply Z.ltb_lt in Hn
| apply Z.ltb_ge in Hn]
end; rewrite ?andb_false_l ?andb_false_r ?andb_true_l ?andb_true_r
?orb_false_l ?orb_false_r ?orb_true_l ?orb_true_r ?Z.bits_0 //=;
try lia.
Ltac churn_bits :=
apply Z.bits_inj'=> n ?;
repeat (rewrite ?Z.lor_spec ?Z.shiftl_spec ?Z.land_spec ?Z.shiftr_spec; try lia);
rewrite !Z.testbit_ones; try lia;
churn_bits'.
End churn_bits.
Section BitsTheory.
Lemma log2_lt_pow2ge:
∀ a b : Z,
0 <= a →
Z.log2 a < b ->
a < 2 ^ b.
Proof.
intros.
assert (a=0 \/ 0 < a) as [Hc | Hc] by lia.
{
subst.
simpl in *.
apply Z.pow_pos_nonneg; lia.
}
{
apply Z.log2_lt_pow2; auto.
}
Qed.
Lemma ZlogLtPow2: forall (a w: Z),
0 < w ->
0 ≤ a < 2 ^ w → Z.log2 a < w.
Proof.
intros.
assert (a=0 \/ 0 < a) as Hc by lia.
destruct Hc as [Hc | Hc]; [subst; simpl; lia | ].
apply Z.log2_lt_pow2; lia.
Qed.
Lemma ZlorRange:
forall (a b: Z) (w : N),
0 <= a < (2 ^ Z.of_N w) ->
0 <= b < (2 ^ Z.of_N w) ->
0 <= Z.lor a b < (2 ^ Z.of_N w).
Proof.
intros.
assert (w=0 \/ 0 < w)%N as [Hc | Hc] by lia.
{
subst. simpl in *.
assert (a=0) by lia.
assert (b=0) by lia.
subst.
simpl.
compute. firstorder.
}
apply and_wlog_r; [apply Z.lor_nonneg; lia | ].
intros.
apply log2_lt_pow2ge; [assumption | ].
rewrite -> Z.log2_lor by lia.
apply ZlogLtPow2 in H; try lia.
apply ZlogLtPow2 in H0; lia.
Qed.
End BitsTheory.
Section Byte.
Definition _get_byte (x: Z) (n: nat): Z := Z.land (x ≫ (8 * n)) (Z.ones 8).
Definition _set_byte (x: Z) (n: nat): Z := (Z.land (Z.ones 8) x) ≪ (8 * n).
Section Theory.
Lemma _get_byte_0:
forall (idx: nat),
_get_byte 0 idx = 0.
Proof. intros; by rewrite /_get_byte Z.shiftr_0_l Z.land_0_l. Qed.
Lemma _set_byte_0:
forall (idx: nat),
_set_byte 0 idx = 0.
Proof. intros; by rewrite /_set_byte Z.shiftl_0_l. Qed.
#[local] Lemma Z_mul_of_nat_S (z : Z) (n : nat) : (z * S n)%Z = (z + z * n)%Z.
Proof. lia. Qed.
Lemma _get_byte_S_idx:
forall (v: Z) (idx: nat),
_get_byte v (S idx) = _get_byte (v ≫ 8) idx.
Proof.
intros; rewrite /_get_byte.
by rewrite Z.shiftr_shiftr ?Z_mul_of_nat_S; lia.
Qed.
Lemma _get_byte_lor:
forall (v v': Z) (idx: nat),
_get_byte (Z.lor v v') idx =
Z.lor (_get_byte v idx) (_get_byte v' idx).
Proof. intros *; rewrite /_get_byte; churn_bits. Qed.
Lemma _set_byte_S_idx:
forall (v: Z) (idx: nat),
_set_byte v (S idx) = ((_set_byte v idx) ≪ 8)%Z.
Proof.
intros; rewrite /_set_byte.
by rewrite Z.shiftl_shiftl ?Z_mul_of_nat_S ?(Z.add_comm 8); lia.
Qed.
Lemma _set_byte_lor:
forall (v v': Z) (idx: nat),
_set_byte (Z.lor v v') idx =
Z.lor (_set_byte v idx) (_set_byte v' idx).
Proof. intros *; rewrite /_set_byte; churn_bits. Qed.
Lemma _set_byte_shiftl_idx:
forall (idx idx': nat) shift v,
idx' <= idx ->
(shift = 8 * Z.of_nat idx')%Z ->
(_set_byte v idx ≫ shift)%Z = _set_byte v (idx - idx').
Proof.
intros * Hidx Hshift; rewrite /_set_byte; subst; churn_bits.
f_equal; lia.
Qed.
Lemma _get_byte_bound:
forall (v : Z) (idx : nat),
(0 <= _get_byte v idx < 256)%Z.
Proof.
intros *; rewrite /_get_byte Z.land_ones; try lia.
pose proof (Z.mod_pos_bound (v ≫ (8 * idx)) 256) as [? ?]; try lia.
now replace (2 ^ 8) with 256%Z by lia.
Qed.
Lemma _get_byte_nonneg:
forall (v: Z) (idx: nat),
(0 <= _get_byte v idx)%Z.
Proof. intros *; pose proof (_get_byte_bound v idx); lia. Qed.
Lemma _set_byte_bound:
forall (v : Z) (idx : nat),
(0 <= _set_byte v idx < 2 ^ (8 * (idx + 1)))%Z.
Proof.
intros; rewrite /_set_byte Z.land_comm Z.land_ones; try lia.
rewrite Z.shiftl_mul_pow2; try lia.
pose proof (Z.mod_pos_bound v 256) as [? ?]; try lia.
replace (2 ^ (8 * (idx + 1))) with ((2 ^ 8) * (2 ^ (8 * idx)))
by (rewrite Z.mul_add_distr_l Zpower_exp; lia).
replace (2 ^ 8) with 256%Z by lia.
split.
- apply Z.mul_nonneg_nonneg; auto.
apply Z.pow_nonneg; lia.
- apply Zmult_lt_compat_r; auto.
apply Z.pow_pos_nonneg; lia.
Qed.
Lemma _set_byte_nonneg:
forall (v: Z) (idx: nat),
(0 <= _set_byte v idx)%Z.
Proof. intros *; pose proof (_set_byte_bound v idx); lia. Qed.
Lemma _set_byte_testbit_low:
forall idx v n,
0 <= n ->
n < 8 * Z.of_nat idx ->
Z.testbit (_set_byte v idx) n = false.
Proof.
intros * Hnonneg Hn.
repeat (rewrite ?Z.lor_spec ?Z.shiftl_spec ?Z.land_spec ?Z.shiftr_spec; try lia).
rewrite !Z.testbit_ones; lia.
Qed.
Lemma _set_byte_testbit_high:
forall idx' idx v n,
(idx' < idx)%nat ->
8 * Z.of_nat idx <= n ->
Z.testbit (_set_byte v idx') n = false.
Proof.
intros * Hidx Hn.
repeat (rewrite ?Z.lor_spec ?Z.shiftl_spec ?Z.land_spec ?Z.shiftr_spec; try lia).
rewrite !Z.testbit_ones; lia.
Qed.
Lemma _set_byte_land_no_overlap:
forall (idx idx': nat) mask v,
idx <> idx' ->
(mask = Z.ones 8 ≪ (8 * Z.of_nat idx'))%Z ->
Z.land (_set_byte v idx) mask = 0.
Proof. intros * Hidx Hmask; rewrite /_set_byte; subst; churn_bits. Qed.
Lemma _set_byte_land_useless:
forall idx mask v,
(mask = Z.ones 8 ≪ (8 * Z.of_nat idx))%Z ->
Z.land (_set_byte v idx) mask =
_set_byte v idx.
Proof. intros * Hmask; rewrite /_set_byte; subst; churn_bits. Qed.
Lemma _set_byte_shiftr_big:
forall (idx: nat) (idx': Z) v,
(Z.of_nat idx < idx')%Z ->
(_set_byte v idx ≫ (8 * idx'))%Z = 0.
Proof. intros * Hidx; rewrite /_set_byte; churn_bits. Qed.
Lemma _get_0_set_0_eq:
forall v,
_get_byte v 0 = _set_byte v 0.
Proof. intros *; rewrite /_get_byte/_set_byte; churn_bits. Qed.
Lemma _set_get_0:
forall v idx,
_set_byte (_get_byte v 0) idx = _set_byte v idx.
Proof.
intros *; rewrite /_get_byte/_set_byte.
apply Z.bits_inj'=> n ?.
repeat (rewrite ?Z.lor_spec ?Z.land_spec; try lia).
repeat (rewrite ?Z.shiftl_spec; try lia).
repeat (rewrite ?Z.lor_spec ?Z.land_spec; try lia).
assert (n < 8 * idx \/ 8 * idx <= n) as [Hn | Hn] by lia.
- rewrite !Z.testbit_neg_r; [reflexivity | lia.. ].
- rewrite !Z.shiftr_spec; try lia.
rewrite !Z.testbit_ones; try lia.
churn_bits'.
Qed.
Lemma _get_set_byte_no_overlap:
forall (v: Z) (idx idx': nat),
idx <> idx' ->
_get_byte (_set_byte v idx) idx' = 0.
Proof. intros * Hidx; rewrite /_get_byte/_set_byte; churn_bits. Qed.
Lemma _get_set_byte_roundtrip:
forall (v: Z) (idx: nat),
_get_byte (_set_byte v idx) idx = _get_byte v 0.
Proof.
intros *; rewrite /_get_byte/_set_byte; churn_bits.
f_equal; lia.
Qed.
Lemma _set_get_byte_roundtrip:
forall (v: Z) (idx: nat),
_set_byte (_get_byte v idx) idx =
_get_byte v idx ≪ (8 * idx).
Proof.
intros *; rewrite /_get_byte/_set_byte.
apply Z.bits_inj' => n ?.
repeat (rewrite ?Z.lor_spec ?Z.land_spec; try lia).
repeat (rewrite ?Z.shiftl_spec; try lia).
repeat (rewrite ?Z.lor_spec ?Z.land_spec; try lia).
assert (n < 8 * idx \/ 8 * idx <= n) as [Hn | Hn] by lia.
- rewrite !Z.testbit_neg_r; [reflexivity | lia.. ].
- rewrite !Z.shiftr_spec; try lia.
rewrite !Z.testbit_ones; try lia.
churn_bits'.
Qed.
End Theory.
Section split.
Lemma split8:
forall z,
0 <= z < 2^bitsize.bitsZ bitsize.W8 ->
z = _set_byte (_get_byte z 0) 0.
Proof.
intros * Hbounds; rewrite _set_get_0.
rewrite /_set_byte Z.shiftl_0_r.
rewrite Z.land_comm Z.land_ones; try lia.
symmetry; now apply Z.mod_small.
Qed.
Lemma split16:
forall z,
0 <= z < 2^bitsize.bitsZ bitsize.W16 ->
z = Z.lor (_set_byte (_get_byte z 0) 0)
(_set_byte (_get_byte z 1) 1).
Proof.
intros * Hbounds; simpl in Hbounds.
apply Z.bits_inj'=> n ?.
assert (n < 8 * 1%nat \/ 8 * 1%nat <= n < 8 * 2%nat \/ 8 * 2%nat <= n)
as [Hn | [Hn | Hn]]
by lia; rewrite !Z.lor_spec;
[ rewrite (_set_byte_testbit_low 1); auto; try lia
| rewrite (_set_byte_testbit_high 0 1); auto; try lia
| rewrite (_set_byte_testbit_high 0 2); auto; try lia;
rewrite (_set_byte_testbit_high 1 2); auto; try lia;
apply Z.testbit_false; [by lia |];
rewrite Z.div_small; [apply Z.mod_0_l; lia |];
pose proof (Z.pow_le_mono_r 2 16 n ltac:(lia) ltac:(auto)); lia
]; rewrite ?orb_false_l ?orb_false_r;
repeat (rewrite ?Z.lor_spec ?Z.shiftl_spec ?Z.land_spec ?Z.shiftr_spec; try lia);
rewrite !Z.testbit_ones; try lia;
churn_bits'; now rewrite Z.sub_add.
Qed.
Lemma split32:
forall z,
0 <= z < 2^bitsize.bitsZ bitsize.W32 ->
z = Z.lor (_set_byte (_get_byte z 0) 0)
(Z.lor (_set_byte (_get_byte z 1) 1)
(Z.lor (_set_byte (_get_byte z 2) 2)
(_set_byte (_get_byte z 3) 3))).
Proof.
intros * Hbounds; simpl in Hbounds.
apply Z.bits_inj'=> n ?.
assert (n < 8 * 1%nat \/
8 * 1%nat <= n < 8 * 2%nat \/
8 * 2%nat <= n < 8 * 3%nat \/
8 * 3%nat <= n < 8 * 4%nat \/
8 * 4%nat <= n)
as [Hn | [Hn | [Hn | [Hn | Hn]]]]
by lia; rewrite !Z.lor_spec;
[ rewrite (_set_byte_testbit_low 1); auto; try lia;
rewrite (_set_byte_testbit_low 2); auto; try lia;
rewrite (_set_byte_testbit_low 3); auto; try lia
| rewrite (_set_byte_testbit_high 0 1); auto; try lia;
rewrite (_set_byte_testbit_low 2); auto; try lia;
rewrite (_set_byte_testbit_low 3); auto; try lia
| rewrite (_set_byte_testbit_high 0 2); auto; try lia;
rewrite (_set_byte_testbit_high 1 2); auto; try lia;
rewrite (_set_byte_testbit_low 3); auto; try lia
| rewrite (_set_byte_testbit_high 0 3); auto; try lia;
rewrite (_set_byte_testbit_high 1 3); auto; try lia;
rewrite (_set_byte_testbit_high 2 3); auto; try lia
| rewrite (_set_byte_testbit_high 0 4); auto; try lia;
rewrite (_set_byte_testbit_high 1 4); auto; try lia;
rewrite (_set_byte_testbit_high 2 4); auto; try lia;
rewrite (_set_byte_testbit_high 3 4); auto; try lia;
apply Z.testbit_false; [by lia |];
rewrite Z.div_small; [apply Z.mod_0_l; lia |];
pose proof (Z.pow_le_mono_r 2 32 n ltac:(lia) ltac:(auto)); lia
]; rewrite ?orb_false_l ?orb_false_r;
repeat (rewrite ?Z.lor_spec ?Z.shiftl_spec ?Z.land_spec ?Z.shiftr_spec; try lia);
rewrite !Z.testbit_ones; try lia;
churn_bits'; now rewrite Z.sub_add.
Qed.
Lemma split64:
forall z,
0 <= z < 2^bitsize.bitsZ bitsize.W64 ->
z = Z.lor (_set_byte (_get_byte z 0) 0)
(Z.lor (_set_byte (_get_byte z 1) 1)
(Z.lor (_set_byte (_get_byte z 2) 2)
(Z.lor (_set_byte (_get_byte z 3) 3)
(Z.lor (_set_byte (_get_byte z 4) 4)
(Z.lor (_set_byte (_get_byte z 5) 5)
(Z.lor (_set_byte (_get_byte z 6) 6)
(_set_byte (_get_byte z 7) 7))))))).
Proof.
intros * Hbounds; simpl in Hbounds.
apply Z.bits_inj'=> n ?.
assert (n < 8 * 1%nat \/
8 * 1%nat <= n < 8 * 2%nat \/
8 * 2%nat <= n < 8 * 3%nat \/
8 * 3%nat <= n < 8 * 4%nat \/
8 * 4%nat <= n < 8 * 5%nat \/
8 * 5%nat <= n < 8 * 6%nat \/
8 * 6%nat <= n < 8 * 7%nat \/
8 * 7%nat <= n < 8 * 8%nat \/
8 * 8%nat <= n)
as [Hn | [Hn | [Hn | [Hn | [Hn | [Hn | [Hn | [Hn | Hn]]]]]]]]
by lia; rewrite !Z.lor_spec;
[ rewrite (_set_byte_testbit_low 1); auto; try lia;
rewrite (_set_byte_testbit_low 2); auto; try lia;
rewrite (_set_byte_testbit_low 3); auto; try lia;
rewrite (_set_byte_testbit_low 4); auto; try lia;
rewrite (_set_byte_testbit_low 5); auto; try lia;
rewrite (_set_byte_testbit_low 6); auto; try lia;
rewrite (_set_byte_testbit_low 7); auto; try lia
| rewrite (_set_byte_testbit_high 0 1); auto; try lia;
rewrite (_set_byte_testbit_low 2); auto; try lia;
rewrite (_set_byte_testbit_low 3); auto; try lia;
rewrite (_set_byte_testbit_low 4); auto; try lia;
rewrite (_set_byte_testbit_low 5); auto; try lia;
rewrite (_set_byte_testbit_low 6); auto; try lia;
rewrite (_set_byte_testbit_low 7); auto; try lia
| rewrite (_set_byte_testbit_high 0 2); auto; try lia;
rewrite (_set_byte_testbit_high 1 2); auto; try lia;
rewrite (_set_byte_testbit_low 3); auto; try lia;
rewrite (_set_byte_testbit_low 4); auto; try lia;
rewrite (_set_byte_testbit_low 5); auto; try lia;
rewrite (_set_byte_testbit_low 6); auto; try lia;
rewrite (_set_byte_testbit_low 7); auto; try lia
| rewrite (_set_byte_testbit_high 0 3); auto; try lia;
rewrite (_set_byte_testbit_high 1 3); auto; try lia;
rewrite (_set_byte_testbit_high 2 3); auto; try lia;
rewrite (_set_byte_testbit_low 4); auto; try lia;
rewrite (_set_byte_testbit_low 5); auto; try lia;
rewrite (_set_byte_testbit_low 6); auto; try lia;
rewrite (_set_byte_testbit_low 7); auto; try lia
| rewrite (_set_byte_testbit_high 0 4); auto; try lia;
rewrite (_set_byte_testbit_high 1 4); auto; try lia;
rewrite (_set_byte_testbit_high 2 4); auto; try lia;
rewrite (_set_byte_testbit_high 3 4); auto; try lia;
rewrite (_set_byte_testbit_low 5); auto; try lia;
rewrite (_set_byte_testbit_low 6); auto; try lia;
rewrite (_set_byte_testbit_low 7); auto; try lia
| rewrite (_set_byte_testbit_high 0 5); auto; try lia;
rewrite (_set_byte_testbit_high 1 5); auto; try lia;
rewrite (_set_byte_testbit_high 2 5); auto; try lia;
rewrite (_set_byte_testbit_high 3 5); auto; try lia;
rewrite (_set_byte_testbit_high 4 5); auto; try lia;
rewrite (_set_byte_testbit_low 6); auto; try lia;
rewrite (_set_byte_testbit_low 7); auto; try lia
| rewrite (_set_byte_testbit_high 0 6); auto; try lia;
rewrite (_set_byte_testbit_high 1 6); auto; try lia;
rewrite (_set_byte_testbit_high 2 6); auto; try lia;
rewrite (_set_byte_testbit_high 3 6); auto; try lia;
rewrite (_set_byte_testbit_high 4 6); auto; try lia;
rewrite (_set_byte_testbit_high 5 6); auto; try lia;
rewrite (_set_byte_testbit_low 7); auto; try lia
| rewrite (_set_byte_testbit_high 0 7); auto; try lia;
rewrite (_set_byte_testbit_high 1 7); auto; try lia;
rewrite (_set_byte_testbit_high 2 7); auto; try lia;
rewrite (_set_byte_testbit_high 3 7); auto; try lia;
rewrite (_set_byte_testbit_high 4 7); auto; try lia;
rewrite (_set_byte_testbit_high 5 7); auto; try lia;
rewrite (_set_byte_testbit_high 6 7); auto; try lia
| rewrite (_set_byte_testbit_high 0 8); auto; try lia;
rewrite (_set_byte_testbit_high 1 8); auto; try lia;
rewrite (_set_byte_testbit_high 2 8); auto; try lia;
rewrite (_set_byte_testbit_high 3 8); auto; try lia;
rewrite (_set_byte_testbit_high 4 8); auto; try lia;
rewrite (_set_byte_testbit_high 5 8); auto; try lia;
rewrite (_set_byte_testbit_high 6 8); auto; try lia;
rewrite (_set_byte_testbit_high 7 8); auto; try lia;
apply Z.testbit_false; [by lia |];
rewrite Z.div_small; [apply Z.mod_0_l; lia |];
pose proof (Z.pow_le_mono_r 2 64 n ltac:(lia) ltac:(auto)); lia
]; rewrite ?orb_false_l ?orb_false_r;
repeat (rewrite ?Z.lor_spec ?Z.shiftl_spec ?Z.land_spec ?Z.shiftr_spec; try lia);
rewrite !Z.testbit_ones; try lia;
churn_bits'; now rewrite Z.sub_add.
Qed.
Lemma split128:
forall z,
0 <= z < 2^bitsize.bitsZ bitsize.W128 ->
z = Z.lor (_set_byte (_get_byte z 0) 0)
(Z.lor (_set_byte (_get_byte z 1) 1)
(Z.lor (_set_byte (_get_byte z 2) 2)
(Z.lor (_set_byte (_get_byte z 3) 3)
(Z.lor (_set_byte (_get_byte z 4) 4)
(Z.lor (_set_byte (_get_byte z 5) 5)
(Z.lor (_set_byte (_get_byte z 6) 6)
(Z.lor (_set_byte (_get_byte z 7) 7)
(Z.lor (_set_byte (_get_byte z 8) 8)
(Z.lor (_set_byte (_get_byte z 9) 9)
(Z.lor (_set_byte (_get_byte z 10) 10)
(Z.lor (_set_byte (_get_byte z 11) 11)
(Z.lor (_set_byte (_get_byte z 12) 12)
(Z.lor (_set_byte (_get_byte z 13) 13)
(Z.lor (_set_byte (_get_byte z 14) 14)
(_set_byte (_get_byte z 15) 15))))))))))))))).
Proof.
intros * Hbounds; simpl in Hbounds.
apply Z.bits_inj'=> n ?.
assert (n < 8 * 1%nat \/
8 * 1%nat <= n < 8 * 2%nat \/
8 * 2%nat <= n < 8 * 3%nat \/
8 * 3%nat <= n < 8 * 4%nat \/
8 * 4%nat <= n < 8 * 5%nat \/
8 * 5%nat <= n < 8 * 6%nat \/
8 * 6%nat <= n < 8 * 7%nat \/
8 * 7%nat <= n < 8 * 8%nat \/
8 * 8%nat <= n < 8 * 9%nat \/
8 * 9%nat <= n < 8 * 10%nat \/
8 * 10%nat <= n < 8 * 11%nat \/
8 * 11%nat <= n < 8 * 12%nat \/
8 * 12%nat <= n < 8 * 13%nat \/
8 * 13%nat <= n < 8 * 14%nat \/
8 * 14%nat <= n < 8 * 15%nat \/
8 * 15%nat <= n < 8 * 16%nat \/
8 * 16%nat <= n)
as [Hn | [Hn | [Hn | [Hn | [Hn | [Hn | [Hn | [Hn | [Hn | [Hn | [Hn | [Hn | [Hn | [Hn | [Hn | [Hn | Hn]]]]]]]]]]]]]]]]
by lia; rewrite !Z.lor_spec;
[ rewrite (_set_byte_testbit_low 1); auto; try lia;
rewrite (_set_byte_testbit_low 2); auto; try lia;
rewrite (_set_byte_testbit_low 3); auto; try lia;
rewrite (_set_byte_testbit_low 4); auto; try lia;
rewrite (_set_byte_testbit_low 5); auto; try lia;
rewrite (_set_byte_testbit_low 6); auto; try lia;
rewrite (_set_byte_testbit_low 7); auto; try lia;
rewrite (_set_byte_testbit_low 8); auto; try lia;
rewrite (_set_byte_testbit_low 9); auto; try lia;
rewrite (_set_byte_testbit_low 10); auto; try lia;
rewrite (_set_byte_testbit_low 11); auto; try lia;
rewrite (_set_byte_testbit_low 12); auto; try lia;
rewrite (_set_byte_testbit_low 13); auto; try lia;
rewrite (_set_byte_testbit_low 14); auto; try lia;
rewrite (_set_byte_testbit_low 15); auto; try lia
| rewrite (_set_byte_testbit_high 0 1); auto; try lia;
rewrite (_set_byte_testbit_low 2); auto; try lia;
rewrite (_set_byte_testbit_low 3); auto; try lia;
rewrite (_set_byte_testbit_low 4); auto; try lia;
rewrite (_set_byte_testbit_low 5); auto; try lia;
rewrite (_set_byte_testbit_low 6); auto; try lia;
rewrite (_set_byte_testbit_low 7); auto; try lia;
rewrite (_set_byte_testbit_low 8); auto; try lia;
rewrite (_set_byte_testbit_low 9); auto; try lia;
rewrite (_set_byte_testbit_low 10); auto; try lia;
rewrite (_set_byte_testbit_low 11); auto; try lia;
rewrite (_set_byte_testbit_low 12); auto; try lia;
rewrite (_set_byte_testbit_low 13); auto; try lia;
rewrite (_set_byte_testbit_low 14); auto; try lia;
rewrite (_set_byte_testbit_low 15); auto; try lia
| rewrite (_set_byte_testbit_high 0 2); auto; try lia;
rewrite (_set_byte_testbit_high 1 2); auto; try lia;
rewrite (_set_byte_testbit_low 3); auto; try lia;
rewrite (_set_byte_testbit_low 4); auto; try lia;
rewrite (_set_byte_testbit_low 5); auto; try lia;
rewrite (_set_byte_testbit_low 6); auto; try lia;
rewrite (_set_byte_testbit_low 7); auto; try lia;
rewrite (_set_byte_testbit_low 8); auto; try lia;
rewrite (_set_byte_testbit_low 9); auto; try lia;
rewrite (_set_byte_testbit_low 10); auto; try lia;
rewrite (_set_byte_testbit_low 11); auto; try lia;
rewrite (_set_byte_testbit_low 12); auto; try lia;
rewrite (_set_byte_testbit_low 13); auto; try lia;
rewrite (_set_byte_testbit_low 14); auto; try lia;
rewrite (_set_byte_testbit_low 15); auto; try lia
| rewrite (_set_byte_testbit_high 0 3); auto; try lia;
rewrite (_set_byte_testbit_high 1 3); auto; try lia;
rewrite (_set_byte_testbit_high 2 3); auto; try lia;
rewrite (_set_byte_testbit_low 4); auto; try lia;
rewrite (_set_byte_testbit_low 5); auto; try lia;
rewrite (_set_byte_testbit_low 6); auto; try lia;
rewrite (_set_byte_testbit_low 7); auto; try lia;
rewrite (_set_byte_testbit_low 8); auto; try lia;
rewrite (_set_byte_testbit_low 9); auto; try lia;
rewrite (_set_byte_testbit_low 10); auto; try lia;
rewrite (_set_byte_testbit_low 11); auto; try lia;
rewrite (_set_byte_testbit_low 12); auto; try lia;
rewrite (_set_byte_testbit_low 13); auto; try lia;
rewrite (_set_byte_testbit_low 14); auto; try lia;
rewrite (_set_byte_testbit_low 15); auto; try lia
| rewrite (_set_byte_testbit_high 0 4); auto; try lia;
rewrite (_set_byte_testbit_high 1 4); auto; try lia;
rewrite (_set_byte_testbit_high 2 4); auto; try lia;
rewrite (_set_byte_testbit_high 3 4); auto; try lia;
rewrite (_set_byte_testbit_low 5); auto; try lia;
rewrite (_set_byte_testbit_low 6); auto; try lia;
rewrite (_set_byte_testbit_low 7); auto; try lia;
rewrite (_set_byte_testbit_low 8); auto; try lia;
rewrite (_set_byte_testbit_low 9); auto; try lia;
rewrite (_set_byte_testbit_low 10); auto; try lia;
rewrite (_set_byte_testbit_low 11); auto; try lia;
rewrite (_set_byte_testbit_low 12); auto; try lia;
rewrite (_set_byte_testbit_low 13); auto; try lia;
rewrite (_set_byte_testbit_low 14); auto; try lia;
rewrite (_set_byte_testbit_low 15); auto; try lia
| rewrite (_set_byte_testbit_high 0 5); auto; try lia;
rewrite (_set_byte_testbit_high 1 5); auto; try lia;
rewrite (_set_byte_testbit_high 2 5); auto; try lia;
rewrite (_set_byte_testbit_high 3 5); auto; try lia;
rewrite (_set_byte_testbit_high 4 5); auto; try lia;
rewrite (_set_byte_testbit_low 6); auto; try lia;
rewrite (_set_byte_testbit_low 7); auto; try lia;
rewrite (_set_byte_testbit_low 8); auto; try lia;
rewrite (_set_byte_testbit_low 9); auto; try lia;
rewrite (_set_byte_testbit_low 10); auto; try lia;
rewrite (_set_byte_testbit_low 11); auto; try lia;
rewrite (_set_byte_testbit_low 12); auto; try lia;
rewrite (_set_byte_testbit_low 13); auto; try lia;
rewrite (_set_byte_testbit_low 14); auto; try lia;
rewrite (_set_byte_testbit_low 15); auto; try lia
| rewrite (_set_byte_testbit_high 0 6); auto; try lia;
rewrite (_set_byte_testbit_high 1 6); auto; try lia;
rewrite (_set_byte_testbit_high 2 6); auto; try lia;
rewrite (_set_byte_testbit_high 3 6); auto; try lia;
rewrite (_set_byte_testbit_high 4 6); auto; try lia;
rewrite (_set_byte_testbit_high 5 6); auto; try lia;
rewrite (_set_byte_testbit_low 7); auto; try lia;
rewrite (_set_byte_testbit_low 8); auto; try lia;
rewrite (_set_byte_testbit_low 9); auto; try lia;
rewrite (_set_byte_testbit_low 10); auto; try lia;
rewrite (_set_byte_testbit_low 11); auto; try lia;
rewrite (_set_byte_testbit_low 12); auto; try lia;
rewrite (_set_byte_testbit_low 13); auto; try lia;
rewrite (_set_byte_testbit_low 14); auto; try lia;
rewrite (_set_byte_testbit_low 15); auto; try lia
| rewrite (_set_byte_testbit_high 0 7); auto; try lia;
rewrite (_set_byte_testbit_high 1 7); auto; try lia;
rewrite (_set_byte_testbit_high 2 7); auto; try lia;
rewrite (_set_byte_testbit_high 3 7); auto; try lia;
rewrite (_set_byte_testbit_high 4 7); auto; try lia;
rewrite (_set_byte_testbit_high 5 7); auto; try lia;
rewrite (_set_byte_testbit_high 6 7); auto; try lia;
rewrite (_set_byte_testbit_low 8); auto; try lia;
rewrite (_set_byte_testbit_low 9); auto; try lia;
rewrite (_set_byte_testbit_low 10); auto; try lia;
rewrite (_set_byte_testbit_low 11); auto; try lia;
rewrite (_set_byte_testbit_low 12); auto; try lia;
rewrite (_set_byte_testbit_low 13); auto; try lia;
rewrite (_set_byte_testbit_low 14); auto; try lia;
rewrite (_set_byte_testbit_low 15); auto; try lia
| rewrite (_set_byte_testbit_high 0 8); auto; try lia;
rewrite (_set_byte_testbit_high 1 8); auto; try lia;
rewrite (_set_byte_testbit_high 2 8); auto; try lia;
rewrite (_set_byte_testbit_high 3 8); auto; try lia;
rewrite (_set_byte_testbit_high 4 8); auto; try lia;
rewrite (_set_byte_testbit_high 5 8); auto; try lia;
rewrite (_set_byte_testbit_high 6 8); auto; try lia;
rewrite (_set_byte_testbit_high 7 8); auto; try lia;
rewrite (_set_byte_testbit_low 9); auto; try lia;
rewrite (_set_byte_testbit_low 10); auto; try lia;
rewrite (_set_byte_testbit_low 11); auto; try lia;
rewrite (_set_byte_testbit_low 12); auto; try lia;
rewrite (_set_byte_testbit_low 13); auto; try lia;
rewrite (_set_byte_testbit_low 14); auto; try lia;
rewrite (_set_byte_testbit_low 15); auto; try lia
| rewrite (_set_byte_testbit_high 0 9); auto; try lia;
rewrite (_set_byte_testbit_high 1 9); auto; try lia;
rewrite (_set_byte_testbit_high 2 9); auto; try lia;
rewrite (_set_byte_testbit_high 3 9); auto; try lia;
rewrite (_set_byte_testbit_high 4 9); auto; try lia;
rewrite (_set_byte_testbit_high 5 9); auto; try lia;
rewrite (_set_byte_testbit_high 6 9); auto; try lia;
rewrite (_set_byte_testbit_high 7 9); auto; try lia;
rewrite (_set_byte_testbit_high 8 9); auto; try lia;
rewrite (_set_byte_testbit_low 10); auto; try lia;
rewrite (_set_byte_testbit_low 11); auto; try lia;
rewrite (_set_byte_testbit_low 12); auto; try lia;
rewrite (_set_byte_testbit_low 13); auto; try lia;
rewrite (_set_byte_testbit_low 14); auto; try lia;
rewrite (_set_byte_testbit_low 15); auto; try lia
| rewrite (_set_byte_testbit_high 0 10); auto; try lia;
rewrite (_set_byte_testbit_high 1 10); auto; try lia;
rewrite (_set_byte_testbit_high 2 10); auto; try lia;
rewrite (_set_byte_testbit_high 3 10); auto; try lia;
rewrite (_set_byte_testbit_high 4 10); auto; try lia;
rewrite (_set_byte_testbit_high 5 10); auto; try lia;
rewrite (_set_byte_testbit_high 6 10); auto; try lia;
rewrite (_set_byte_testbit_high 7 10); auto; try lia;
rewrite (_set_byte_testbit_high 8 10); auto; try lia;
rewrite (_set_byte_testbit_high 9 10); auto; try lia;
rewrite (_set_byte_testbit_low 11); auto; try lia;
rewrite (_set_byte_testbit_low 12); auto; try lia;
rewrite (_set_byte_testbit_low 13); auto; try lia;
rewrite (_set_byte_testbit_low 14); auto; try lia;
rewrite (_set_byte_testbit_low 15); auto; try lia
| rewrite (_set_byte_testbit_high 0 11); auto; try lia;
rewrite (_set_byte_testbit_high 1 11); auto; try lia;
rewrite (_set_byte_testbit_high 2 11); auto; try lia;
rewrite (_set_byte_testbit_high 3 11); auto; try lia;
rewrite (_set_byte_testbit_high 4 11); auto; try lia;
rewrite (_set_byte_testbit_high 5 11); auto; try lia;
rewrite (_set_byte_testbit_high 6 11); auto; try lia;
rewrite (_set_byte_testbit_high 7 11); auto; try lia;
rewrite (_set_byte_testbit_high 8 11); auto; try lia;
rewrite (_set_byte_testbit_high 9 11); auto; try lia;
rewrite (_set_byte_testbit_high 10 11); auto; try lia;
rewrite (_set_byte_testbit_low 12); auto; try lia;
rewrite (_set_byte_testbit_low 13); auto; try lia;
rewrite (_set_byte_testbit_low 14); auto; try lia;
rewrite (_set_byte_testbit_low 15); auto; try lia
| rewrite (_set_byte_testbit_high 0 12); auto; try lia;
rewrite (_set_byte_testbit_high 1 12); auto; try lia;
rewrite (_set_byte_testbit_high 2 12); auto; try lia;
rewrite (_set_byte_testbit_high 3 12); auto; try lia;
rewrite (_set_byte_testbit_high 4 12); auto; try lia;
rewrite (_set_byte_testbit_high 5 12); auto; try lia;
rewrite (_set_byte_testbit_high 6 12); auto; try lia;
rewrite (_set_byte_testbit_high 7 12); auto; try lia;
rewrite (_set_byte_testbit_high 8 12); auto; try lia;
rewrite (_set_byte_testbit_high 9 12); auto; try lia;
rewrite (_set_byte_testbit_high 10 12); auto; try lia;
rewrite (_set_byte_testbit_high 11 12); auto; try lia;
rewrite (_set_byte_testbit_low 13); auto; try lia;
rewrite (_set_byte_testbit_low 14); auto; try lia;
rewrite (_set_byte_testbit_low 15); auto; try lia
| rewrite (_set_byte_testbit_high 0 13); auto; try lia;
rewrite (_set_byte_testbit_high 1 13); auto; try lia;
rewrite (_set_byte_testbit_high 2 13); auto; try lia;
rewrite (_set_byte_testbit_high 3 13); auto; try lia;
rewrite (_set_byte_testbit_high 4 13); auto; try lia;
rewrite (_set_byte_testbit_high 5 13); auto; try lia;
rewrite (_set_byte_testbit_high 6 13); auto; try lia;
rewrite (_set_byte_testbit_high 7 13); auto; try lia;
rewrite (_set_byte_testbit_high 8 13); auto; try lia;
rewrite (_set_byte_testbit_high 9 13); auto; try lia;
rewrite (_set_byte_testbit_high 10 13); auto; try lia;
rewrite (_set_byte_testbit_high 11 13); auto; try lia;
rewrite (_set_byte_testbit_high 12 13); auto; try lia;
rewrite (_set_byte_testbit_low 14); auto; try lia;
rewrite (_set_byte_testbit_low 15); auto; try lia
| rewrite (_set_byte_testbit_high 0 14); auto; try lia;
rewrite (_set_byte_testbit_high 1 14); auto; try lia;
rewrite (_set_byte_testbit_high 2 14); auto; try lia;
rewrite (_set_byte_testbit_high 3 14); auto; try lia;
rewrite (_set_byte_testbit_high 4 14); auto; try lia;
rewrite (_set_byte_testbit_high 5 14); auto; try lia;
rewrite (_set_byte_testbit_high 6 14); auto; try lia;
rewrite (_set_byte_testbit_high 7 14); auto; try lia;
rewrite (_set_byte_testbit_high 8 14); auto; try lia;
rewrite (_set_byte_testbit_high 9 14); auto; try lia;
rewrite (_set_byte_testbit_high 10 14); auto; try lia;
rewrite (_set_byte_testbit_high 11 14); auto; try lia;
rewrite (_set_byte_testbit_high 12 14); auto; try lia;
rewrite (_set_byte_testbit_high 13 14); auto; try lia;
rewrite (_set_byte_testbit_low 15); auto; try lia
| rewrite (_set_byte_testbit_high 0 15); auto; try lia;
rewrite (_set_byte_testbit_high 1 15); auto; try lia;
rewrite (_set_byte_testbit_high 2 15); auto; try lia;
rewrite (_set_byte_testbit_high 3 15); auto; try lia;
rewrite (_set_byte_testbit_high 4 15); auto; try lia;
rewrite (_set_byte_testbit_high 5 15); auto; try lia;
rewrite (_set_byte_testbit_high 6 15); auto; try lia;
rewrite (_set_byte_testbit_high 7 15); auto; try lia;
rewrite (_set_byte_testbit_high 8 15); auto; try lia;
rewrite (_set_byte_testbit_high 9 15); auto; try lia;
rewrite (_set_byte_testbit_high 10 15); auto; try lia;
rewrite (_set_byte_testbit_high 11 15); auto; try lia;
rewrite (_set_byte_testbit_high 12 15); auto; try lia;
rewrite (_set_byte_testbit_high 13 15); auto; try lia;
rewrite (_set_byte_testbit_high 14 15); auto; try lia
| rewrite (_set_byte_testbit_high 0 16); auto; try lia;
rewrite (_set_byte_testbit_high 1 16); auto; try lia;
rewrite (_set_byte_testbit_high 2 16); auto; try lia;
rewrite (_set_byte_testbit_high 3 16); auto; try lia;
rewrite (_set_byte_testbit_high 4 16); auto; try lia;
rewrite (_set_byte_testbit_high 5 16); auto; try lia;
rewrite (_set_byte_testbit_high 6 16); auto; try lia;
rewrite (_set_byte_testbit_high 7 16); auto; try lia;
rewrite (_set_byte_testbit_high 8 16); auto; try lia;
rewrite (_set_byte_testbit_high 9 16); auto; try lia;
rewrite (_set_byte_testbit_high 10 16); auto; try lia;
rewrite (_set_byte_testbit_high 11 16); auto; try lia;
rewrite (_set_byte_testbit_high 12 16); auto; try lia;
rewrite (_set_byte_testbit_high 13 16); auto; try lia;
rewrite (_set_byte_testbit_high 14 16); auto; try lia;
rewrite (_set_byte_testbit_high 15 16); auto; try lia;
apply Z.testbit_false; [by lia |];
rewrite Z.div_small; [apply Z.mod_0_l; lia |];
pose proof (Z.pow_le_mono_r 2 128 n ltac:(lia) ltac:(auto)); lia
]; rewrite ?orb_false_l ?orb_false_r;
repeat (rewrite ?Z.lor_spec ?Z.shiftl_spec ?Z.land_spec ?Z.shiftr_spec; try lia);
rewrite !Z.testbit_ones; try lia;
churn_bits'; now rewrite Z.sub_add.
Qed.
End split.
End Byte.
#[global] Opaque _get_byte _set_byte.
Section Bswap.
#[local] Definition bswap_idxs (bytes: nat) :=
let idxs := seq 0 bytes in
zip idxs (rev idxs).
#[local] Definition bswap_ (bytes: nat) (n: Z): Z :=
fold_left (fun acc '(getIdx, setIdx) =>
Z.lor acc (_set_byte (_get_byte n getIdx) setIdx))
(bswap_idxs bytes)
0.
Definition bswap (sz : bitsize) : Z -> Z := bswap_ (bitsize.bytesNat sz).
Section test.
Local Definition bytes (ls : list Z) :=
fold_left (fun a b => a * 256 + b)%Z ls 0%Z.
Arguments bytes _%_Z.
Local Definition _bswap16_test : bswap bitsize.W16 (bytes (1::2::nil)%Z) = bytes (2::1::nil)%Z := eq_refl.
Local Definition _bswap32_test :
bswap bitsize.W32 (bytes (1::2::3::4::nil)%Z) = bytes (4::3::2::1::nil)%Z := eq_refl.
Local Definition _bswap64_test :
bswap bitsize.W64 (bytes (1::2::3::4::5::6::7::8::nil)%Z) = bytes (8::7::6::5::4::3::2::1::nil)%Z := eq_refl.
End test.
End Bswap.
Notation bswap8 := (bswap bitsize.W8) (only parsing).
Notation bswap16 := (bswap bitsize.W16) (only parsing).
Notation bswap32 := (bswap bitsize.W32) (only parsing).
Notation bswap64 := (bswap bitsize.W64) (only parsing).
Notation bswap128 := (bswap bitsize.W128) (only parsing).
#[global] Opaque bswap.
Section Bswap.
Section Theory.
Section bounded.
#[local] Transparent bswap.
Lemma bswap8_bounded:
forall v,
0 <= bswap8 v < 256.
Proof.
intros *; rewrite /bswap/bswap_/= Z.lor_0_l.
pose proof (_set_byte_bound (_get_byte v 0) 0).
lia.
Qed.
Lemma bswap16_bounded:
∀ v,
0 ≤ bswap16 v < 65536.
Proof.
intros *; rewrite /bswap/bswap_/= Z.lor_0_l.
pose proof (_set_byte_bound (_get_byte v 0) 1).
pose proof (_set_byte_bound (_get_byte v 1) 0).
repeat (try apply ZlorRange with (w:=16%N)); lia.
Qed.
Lemma bswap32_bounded:
forall v,
0 ≤ bswap32 v < 4294967296.
Proof.
intros *; rewrite /bswap/bswap_/= Z.lor_0_l.
pose proof (_set_byte_bound (_get_byte v 0) 3).
pose proof (_set_byte_bound (_get_byte v 1) 2).
pose proof (_set_byte_bound (_get_byte v 2) 1).
pose proof (_set_byte_bound (_get_byte v 3) 0).
repeat (try apply ZlorRange with (w:=32%N)); lia.
Qed.
Lemma bswap64_bounded:
forall v,
0 ≤ bswap64 v < 18446744073709551616.
Proof.
intros *; rewrite /bswap/bswap_/= Z.lor_0_l.
pose proof (_set_byte_bound (_get_byte v 0) 7).
pose proof (_set_byte_bound (_get_byte v 1) 6).
pose proof (_set_byte_bound (_get_byte v 2) 5).
pose proof (_set_byte_bound (_get_byte v 3) 4).
pose proof (_set_byte_bound (_get_byte v 4) 3).
pose proof (_set_byte_bound (_get_byte v 5) 2).
pose proof (_set_byte_bound (_get_byte v 6) 1).
pose proof (_set_byte_bound (_get_byte v 7) 0).
repeat (try apply ZlorRange with (w:=64%N)); lia.
Qed.
Lemma bswap128_bounded:
forall v,
0 ≤ bswap128 v < ltac:(let x := eval cbv in (2^128) in exact x).
Proof.
intros *; rewrite /bswap/bswap_/= Z.lor_0_l.
pose proof (_set_byte_bound (_get_byte v 0) 15).
pose proof (_set_byte_bound (_get_byte v 1) 14).
pose proof (_set_byte_bound (_get_byte v 2) 13).
pose proof (_set_byte_bound (_get_byte v 3) 12).
pose proof (_set_byte_bound (_get_byte v 4) 11).
pose proof (_set_byte_bound (_get_byte v 5) 10).
pose proof (_set_byte_bound (_get_byte v 6) 9).
pose proof (_set_byte_bound (_get_byte v 7) 8).
pose proof (_set_byte_bound (_get_byte v 8) 7).
pose proof (_set_byte_bound (_get_byte v 9) 6).
pose proof (_set_byte_bound (_get_byte v 10) 5).
pose proof (_set_byte_bound (_get_byte v 11) 4).
pose proof (_set_byte_bound (_get_byte v 12) 3).
pose proof (_set_byte_bound (_get_byte v 13) 2).
pose proof (_set_byte_bound (_get_byte v 14) 1).
pose proof (_set_byte_bound (_get_byte v 15) 0).
repeat (try apply ZlorRange with (w:=128%N)); lia.
Qed.
End bounded.
Lemma bswap_bounded:
forall sz v,
0 <= bswap sz v < 2^(bitsize.bitsZ sz).
Proof.
intros *; destruct sz;
eauto using
bswap8_bounded,
bswap16_bounded,
bswap32_bounded,
bswap64_bounded,
bswap128_bounded.
Qed.
End Theory.
End Bswap.
Section Bswap.
Section Theory.
Section useless_lor.
#[local] Transparent _get_byte _set_byte bswap.
Lemma bswap8_useless_lor:
forall v v' idx,
(idx >= 1)%nat ->
bswap8 (Z.lor v (_set_byte v' idx)) =
bswap8 v.
Proof.
move=> v v' idx Hidx.
rewrite /bswap8/bswap/bswap_/_get_byte/=; f_equal.
rewrite /_set_byte; churn_bits.
Qed.
Lemma bswap16_useless_lor:
forall v v' idx,
(idx >= 2)%nat ->
bswap16 (Z.lor v (_set_byte v' idx)) =
bswap16 v.
Proof.
move=> v v' idx Hidx.
rewrite /bswap16/bswap/bswap_/_get_byte/=.
do 1 (f_equal; [| f_equal; rewrite /_set_byte; churn_bits]).
f_equal; f_equal; rewrite /_set_byte; churn_bits.
Qed.
Lemma bswap32_useless_lor:
forall v v' idx,
(idx >= 4)%nat ->
bswap32 (Z.lor v (_set_byte v' idx)) =
bswap32 v.
Proof.
move=> v v' idx Hidx.
rewrite /bswap32/bswap/bswap_/_get_byte/=.
do 3 (f_equal; [| f_equal; rewrite /_set_byte; churn_bits]).
f_equal; f_equal; rewrite /_set_byte; churn_bits.
Qed.
Lemma bswap64_useless_lor:
forall v v' idx,
(idx >= 8)%nat ->
bswap64 (Z.lor v (_set_byte v' idx)) =
bswap64 v.
Proof.
move=> v v' idx Hidx.
rewrite /bswap64/bswap/bswap_/_get_byte/=.
do 7 (f_equal; [| f_equal; rewrite /_set_byte; churn_bits]).
f_equal; f_equal; rewrite /_set_byte; churn_bits.
Qed.
Lemma bswap128_useless_lor:
forall v v' idx,
(idx >= 16)%nat ->
bswap128 (Z.lor v (_set_byte v' idx)) =
bswap128 v.
Proof.
move=> v v' idx Hidx.
rewrite /bswap128/bswap/bswap_/_get_byte/=.
do 15 (f_equal; [| f_equal; rewrite /_set_byte; churn_bits]).
f_equal; f_equal; rewrite /_set_byte; churn_bits.
Qed.
End useless_lor.
Lemma bswap_useless_lor:
forall sz v v' idx,
(bitsize.bytesNat sz <= idx)%nat ->
bswap sz (Z.lor v (_set_byte v' idx)) =
bswap sz v.
Proof.
intros [] v v' idx Hidx; simpl in *;
eauto using
bswap8_useless_lor,
bswap16_useless_lor,
bswap32_useless_lor,
bswap64_useless_lor,
bswap128_useless_lor.
Qed.
Section _set_byte_reverse.
#[local] Transparent _get_byte _set_byte bswap.
Lemma bswap8_set_byte_reverse:
forall x,
bswap8 (_set_byte x 0) =
_set_byte x 0.
Proof.
intros *.
rewrite /bswap8/bswap/bswap_/=.
rewrite !_set_get_byte_roundtrip !Z.shiftl_0_r !Z.lor_0_l.
rewrite _get_set_byte_roundtrip; now apply _get_0_set_0_eq.
Qed.
Lemma bswap16_set_byte_reverse:
forall x1 x2,
bswap16 (Z.lor (_set_byte x1 0)
(_set_byte x2 1)) =
Z.lor (_set_byte x2 0)
(_set_byte x1 1).
Proof.
intros *.
rewrite /bswap16/bswap/bswap_/=.
rewrite {1}Z.lor_comm; f_equal;
try (rewrite !_get_byte_lor !_set_byte_lor
!_get_set_byte_roundtrip !_get_set_byte_no_overlap
?_set_byte_0 ?Z.lor_0_r ?Z.lor_0_l ?_set_get_0 //).
Qed.
Lemma bswap32_set_byte_reverse:
forall x1 x2 x3 x4,
bswap32 (Z.lor (_set_byte x1 0)
(Z.lor (_set_byte x2 1)
(Z.lor (_set_byte x3 2)
(_set_byte x4 3)))) =
Z.lor (_set_byte x4 0)
(Z.lor (_set_byte x3 1)
(Z.lor (_set_byte x2 2)
(_set_byte x1 3))).
Proof.
intros *.
rewrite /bswap32/bswap/bswap_/=.
rewrite {1}Z.lor_comm; f_equal;
try (rewrite !_get_byte_lor !_set_byte_lor
!_get_set_byte_roundtrip !_get_set_byte_no_overlap
?_set_byte_0 ?Z.lor_0_r ?Z.lor_0_l ?_set_get_0 //);
repeat (rewrite {1}Z.lor_comm; f_equal).
Qed.
Lemma bswap64_set_byte_reverse:
forall x1 x2 x3 x4 x5 x6 x7 x8,
bswap64 (Z.lor (_set_byte x1 0)
(Z.lor (_set_byte x2 1)
(Z.lor (_set_byte x3 2)
(Z.lor (_set_byte x4 3)
(Z.lor (_set_byte x5 4)
(Z.lor (_set_byte x6 5)
(Z.lor (_set_byte x7 6)
(_set_byte x8 7)))))))) =
Z.lor (_set_byte x8 0)
(Z.lor (_set_byte x7 1)
(Z.lor (_set_byte x6 2)
(Z.lor (_set_byte x5 3)
(Z.lor (_set_byte x4 4)
(Z.lor (_set_byte x3 5)
(Z.lor (_set_byte x2 6)
(_set_byte x1 7))))))).
Proof.
intros *.
rewrite /bswap64/bswap/bswap_/=.
repeat (rewrite {1}Z.lor_comm; f_equal;
[now (rewrite !_get_byte_lor !_set_byte_lor
!_get_set_byte_roundtrip !_get_set_byte_no_overlap
?_set_byte_0 ?Z.lor_0_r ?Z.lor_0_l ?_set_get_0 //) |]).
rewrite !_get_byte_lor !_set_byte_lor
!_get_set_byte_roundtrip !_get_set_byte_no_overlap
?_set_byte_0 ?Z.lor_0_r ?Z.lor_0_l ?_set_get_0 //.
Qed.
Lemma bswap128_set_byte_reverse:
forall x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16,
bswap128 (Z.lor (_set_byte x1 0)
(Z.lor (_set_byte x2 1)
(Z.lor (_set_byte x3 2)
(Z.lor (_set_byte x4 3)
(Z.lor (_set_byte x5 4)
(Z.lor (_set_byte x6 5)
(Z.lor (_set_byte x7 6)
(Z.lor (_set_byte x8 7)
(Z.lor (_set_byte x9 8)
(Z.lor (_set_byte x10 9)
(Z.lor (_set_byte x11 10)
(Z.lor (_set_byte x12 11)
(Z.lor (_set_byte x13 12)
(Z.lor (_set_byte x14 13)
(Z.lor (_set_byte x15 14)
(_set_byte x16 15)))))))))))))))) =
Z.lor (_set_byte x16 0)
(Z.lor (_set_byte x15 1)
(Z.lor (_set_byte x14 2)
(Z.lor (_set_byte x13 3)
(Z.lor (_set_byte x12 4)
(Z.lor (_set_byte x11 5)
(Z.lor (_set_byte x10 6)
(Z.lor (_set_byte x9 7)
(Z.lor (_set_byte x8 8)
(Z.lor (_set_byte x7 9)
(Z.lor (_set_byte x6 10)
(Z.lor (_set_byte x5 11)
(Z.lor (_set_byte x4 12)
(Z.lor (_set_byte x3 13)
(Z.lor (_set_byte x2 14)
(_set_byte x1 15))))))))))))))).
Proof.
intros *.
rewrite /bswap128/bswap/bswap_/=.
repeat (rewrite {1}Z.lor_comm; f_equal;
[now (rewrite !_get_byte_lor !_set_byte_lor
!_get_set_byte_roundtrip !_get_set_byte_no_overlap
?_set_byte_0 ?Z.lor_0_r ?Z.lor_0_l ?_set_get_0 //) |]).
rewrite !_get_byte_lor !_set_byte_lor
!_get_set_byte_roundtrip !_get_set_byte_no_overlap
?_set_byte_0 ?Z.lor_0_r ?Z.lor_0_l ?_set_get_0 //.
Qed.
End _set_byte_reverse.
Section larger.
#[local] Transparent _get_byte _set_byte bswap.
Lemma bswap8_larger z:
0 <= z ->
bswap8 z = bswap8 (Z.land z (Z.ones 8)).
Proof.
intros; rewrite /bswap8/bswap/bswap_/= ?Z.lor_0_l ?Z.lor_0_r.
f_equal; rewrite /_get_byte; churn_bits.
Qed.
Lemma bswap16_larger z:
0 <= z ->
bswap16 z = bswap16 (Z.land z (Z.ones 16)).
Proof.
intros; rewrite /bswap16/bswap/bswap_/= ?Z.lor_0_l ?Z.lor_0_r.
do 1 (f_equal; [| f_equal; rewrite /_get_byte; churn_bits]).
f_equal; rewrite /_get_byte; churn_bits.
Qed.
Lemma bswap32_larger z:
0 <= z ->
bswap32 z = bswap32 (Z.land z (Z.ones 32)).
Proof.
intros; rewrite /bswap32/bswap/bswap_/= ?Z.lor_0_l ?Z.lor_0_r.
do 3 (f_equal; [| f_equal; rewrite /_get_byte; churn_bits]).
f_equal; rewrite /_get_byte; churn_bits.
Qed.
Lemma bswap64_larger z:
0 <= z ->
bswap64 z = bswap64 (Z.land z (Z.ones 64)).
Proof.
intros; rewrite /bswap64/bswap/bswap_/= ?Z.lor_0_l ?Z.lor_0_r.
do 7 (f_equal; [| f_equal; rewrite /_get_byte; churn_bits]).
f_equal; rewrite /_get_byte; churn_bits.
Qed.
Lemma bswap128_larger z:
0 <= z ->
bswap128 z = bswap128 (Z.land z (Z.ones 128)).
Proof.
intros; rewrite /bswap128/bswap/bswap_/= ?Z.lor_0_l ?Z.lor_0_r.
do 15 (f_equal; [| f_equal; rewrite /_get_byte; churn_bits]).
f_equal; rewrite /_get_byte; churn_bits.
Qed.
End larger.
Lemma bswap_larger sz z:
0 <= z ->
bswap sz z = bswap sz (Z.land z (Z.ones (bitsize.bitsZ sz))).
Proof.
intros; destruct sz;
eauto using
bswap8_larger,
bswap16_larger,
bswap32_larger,
bswap64_larger,
bswap128_larger.
Qed.
Section involutive.
Lemma bswap8_involutive:
forall z,
0 <= z < 2^bitsize.bitsZ bitsize.W8 ->
bswap8 (bswap8 z) = z.
Proof.
intros * Hbounds.
rewrite (split8 z); auto.
now rewrite 2!bswap8_set_byte_reverse.
Qed.
Lemma bswap8_involutive_land:
forall z,
0 <= z < 2^bitsize.bitsZ bitsize.W8 ->
bswap8 (bswap8 z) = Z.land z (Z.ones (bitsize.bitsZ bitsize.W8)).
Proof.
intros * Hbounds; simpl in *.
rewrite bswap8_involutive; auto.
rewrite -> Z.land_ones by lia.
rewrite Z.mod_small; auto.
Qed.
Lemma bswap16_involutive:
forall z,
0 <= z < 2^bitsize.bitsZ bitsize.W16 ->
bswap16 (bswap16 z) = z.
Proof.
intros * Hbounds.
rewrite (split16 z); auto.
now rewrite 2!bswap16_set_byte_reverse.
Qed.
Lemma bswap16_involutive_land:
forall z,
0 <= z < 2^bitsize.bitsZ bitsize.W16 ->
bswap16 (bswap16 z) = Z.land z (Z.ones (bitsize.bitsZ bitsize.W16)).
Proof.
intros * Hbounds; simpl in *.
rewrite bswap16_involutive; auto.
rewrite -> Z.land_ones by lia.
rewrite Z.mod_small; auto.
Qed.
Lemma bswap32_involutive:
forall z,
0 <= z < 2^bitsize.bitsZ bitsize.W32 ->
bswap32 (bswap32 z) = z.
Proof.
intros * Hbounds.
rewrite (split32 z); auto.
now rewrite 2!bswap32_set_byte_reverse.
Qed.
Lemma bswap32_involutive_land:
forall z,
0 <= z < 2^bitsize.bitsZ bitsize.W32 ->
bswap32 (bswap32 z) = Z.land z (Z.ones (bitsize.bitsZ bitsize.W32)).
Proof.
intros * Hbounds; simpl in *.
rewrite bswap32_involutive; auto.
rewrite -> Z.land_ones by lia.
rewrite Z.mod_small; auto.
Qed.
Lemma bswap64_involutive:
forall z,
0 <= z < 2^bitsize.bitsZ bitsize.W64 ->
bswap64 (bswap64 z) = z.
Proof.
intros * Hbounds.
rewrite (split64 z); auto.
now rewrite 2!bswap64_set_byte_reverse.
Qed.
Lemma bswap64_involutive_land:
forall z,
0 <= z < 2^bitsize.bitsZ bitsize.W64 ->
bswap64 (bswap64 z) = Z.land z (Z.ones (bitsize.bitsZ bitsize.W64)).
Proof.
intros * Hbounds; simpl in *.
rewrite bswap64_involutive; auto.
rewrite -> Z.land_ones by lia.
rewrite Z.mod_small; auto.
Qed.
Lemma bswap128_involutive:
forall z,
0 <= z < 2^bitsize.bitsZ bitsize.W128 ->
bswap128 (bswap128 z) = z.
Proof.
intros * Hbounds.
rewrite (split128 z); auto.
now rewrite 2!bswap128_set_byte_reverse.
Qed.
Lemma bswap128_involutive_land:
forall z,
0 <= z < 2^bitsize.bitsZ bitsize.W128 ->
bswap128 (bswap128 z) = Z.land z (Z.ones (bitsize.bitsZ bitsize.W128)).
Proof.
intros * Hbounds; simpl in *.
rewrite bswap128_involutive; auto.
rewrite -> Z.land_ones by lia.
rewrite Z.mod_small; auto.
Qed.
End involutive.
Lemma bswap_involutive:
forall sz z,
0 <= z < 2^bitsize.bitsZ sz ->
bswap sz (bswap sz z) = z.
Proof.
intros * Hbounds; destruct sz;
eauto using
bswap8_involutive,
bswap16_involutive,
bswap32_involutive,
bswap64_involutive,
bswap128_involutive.
Qed.
Lemma bswap_involutive_land:
forall sz z,
0 <= z < 2^bitsize.bitsZ sz ->
bswap sz (bswap sz z) = Z.land z (Z.ones (bitsize.bitsZ sz)).
Proof.
intros * Hbounds; destruct sz;
eauto using
bswap8_involutive_land,
bswap16_involutive_land,
bswap32_involutive_land,
bswap64_involutive_land,
bswap128_involutive_land.
Qed.
End Theory.
End Bswap.