bedrock.lang.cpp.arith.z_to_bytes
(*
* 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.prelude.list_numbers.
Require Import bedrock.lang.cpp.arith.types.
Require Import bedrock.lang.cpp.arith.builtins.
Require Import bedrock.lang.cpp.arith.operator.
Import arith.builtins.churn_bits.
Section FromToBytes.
Section ExtraFacts.
Lemma repeat_cons_app:
forall (A: Type) (a: A) (cnt: nat),
(a :: repeat a cnt) = repeat a cnt ++ [a].
Proof.
induction cnt => //=.
by rewrite IHcnt.
Qed.
Lemma rev_repeat:
forall (A: Type) (a: A) (cnt: nat),
rev (repeat a cnt) = repeat a cnt.
Proof.
induction cnt => //=.
by rewrite IHcnt repeat_cons_app.
Qed.
Lemma Z_shiftr_small:
forall v e,
(0 <= e)%Z ->
(0 <= v)%Z ->
(v < 2 ^ e)%Z ->
(v ≫ e = 0)%Z.
Proof.
intros; rewrite Z.shiftr_div_pow2; try lia.
rewrite Z.div_small; lia.
Qed.
Lemma Z_pow2_trans_nat_l:
forall v (a b: nat),
(v < 2 ^ (8 * b))%Z ->
(v < 2 ^ (8 * (a + b)%nat))%Z.
Proof.
intros; destruct a.
- by replace (8 * (0%nat + b))%Z with (8 * b)%Z by lia.
- eapply Z.lt_trans; eauto; apply Z.pow_lt_mono_r; lia.
Qed.
Lemma Z_pow2_trans_nat_r:
forall v (a b: nat),
(v < 2 ^ (8 * a))%Z ->
(v < 2 ^ (8 * (a + b)%nat))%Z.
Proof.
intros; destruct b.
- by rewrite -plus_n_O.
- eapply Z.lt_trans; eauto; apply Z.pow_lt_mono_r; lia.
Qed.
Lemma Z_land_ldiff_no_overlap (mask offset v: Z) :
(0 < mask)%Z ->
(0 <= offset)%Z ->
(0 <= v)%Z ->
Z.land (mask ≪ offset) (Z.ldiff v (Z.ones offset)) = Z.land (mask ≪ offset) v.
Proof.
(* Intuition: the ldiff is going to remove the lowest
(idx+cnt) bytes, but the `255 ≪ (8 * (idx+cnt))`
mask doesn't overlap with any of those bits
so it is effectively a no-op.
*)
intros **.
apply Z.bits_inj' => n ?.
rewrite !Z.land_spec Z.ldiff_spec Z.shiftl_spec // Z.testbit_ones_nonneg; try lia.
destruct (n <? offset)%Z eqn:Hn => /=; rewrite ?andb_true_r //.
move: Hn => /Z.ltb_lt?.
rewrite !andb_false_r Z.testbit_neg_r //. lia.
Qed.
Lemma Z_land_ldiff_upper_byte (offset v: Z) :
(0 <= offset)%Z ->
(2^(8*offset) <= v)%Z ->
(v < 2^(8*Z.succ offset))%Z ->
Z.ldiff v (Z.ones (8 * offset)) = Z.land (255 ≪ (8 * offset)) v.
Proof.
(* Intuition: since `v < 2^(8*(idx+S cnt))`, we know
that there aren't going to be any bits
beyond the `255 ≪ (8 * (idx+cnt))` mask
which will be introduced by the change to ldiff.
*)
intros.
apply Z.bits_inj' => n ?.
rewrite !Z.land_spec Z.ldiff_spec Z.shiftl_spec // Z.testbit_ones_nonneg; try lia.
destruct (n <? 8*offset)%Z eqn:Hn => //=; rewrite ?andb_true_r ?andb_false_r //.
- apply Z.ltb_lt in Hn; rewrite Z.testbit_neg_r ?andb_false_l //=; lia.
- apply Z.ltb_ge in Hn.
replace (8 * Z.succ offset)%Z with (8 + (8 * offset))%Z in H1 by lia.
change (255)%Z with (Z.ones 8).
destruct (8 + (8 * offset) <? n)%Z eqn:Hn' => //=.
+ rewrite Z.bits_above_log2 ?andb_false_r //.
* apply Z.le_trans with (m := (2^(8*offset))%Z); try apply Z.pow_nonneg; lia.
* apply Z.log2_lt_pow2.
-- eapply Z.lt_le_trans; eauto; apply Z.pow_pos_nonneg; lia.
-- eapply Z.lt_trans; eauto.
apply Z.ltb_lt in Hn'.
apply Z.pow_lt_mono_r; try lia.
+ apply Z.ltb_ge in Hn'.
destruct (8 + (8 * offset) =? n)%Z eqn:Hn'' => //=.
* apply Z.eqb_eq in Hn''; subst.
rewrite Z.bits_above_log2 ?andb_false_l ?andb_false_r //.
-- apply Z.le_trans with (m := (2^(8*offset))%Z);
eauto; apply Z.pow_nonneg; lia.
-- apply Z.log2_lt_pow2; lia.
* apply Z.eqb_neq in Hn''.
assert (n < 8 + 8 * offset)%Z as Hn''' by lia.
rewrite Z.ones_spec_low ?andb_true_l //; lia.
Qed.
Lemma Z_ldiff_split:
forall (cnt idx: nat) (v: Z),
(0 <= v)%Z ->
(v < 2^(8*(idx+S cnt)))%Z ->
Z.ldiff v (Z.ones (8 * idx)) =
Z.lor (Z.land (Z.ones (8 * cnt) ≪ (8 * idx)) v)
(Z.land (Z.ones 8 ≪ (8 * (idx + cnt)%nat)) v).
Proof.
intros cnt idx v Hlower Hupper.
assert (v = 0 \/ 0 < v)%Z as [Hlower' | Hlower'] by lia;
[subst; by rewrite !Z.land_0_r Z.lor_diag Z.ldiff_0_l | clear Hlower].
apply Z.bits_inj'=> n ?.
rewrite Z.lor_spec Z.ldiff_spec !Z.land_spec !Z.shiftl_spec; try lia.
rewrite !Z.testbit_ones; try lia.
churn_bits'.
apply Z.bits_above_log2; try lia.
assert (8 * (idx+S cnt)%nat <= n)%Z by lia.
eapply Z.lt_le_trans; eauto.
apply Z.log2_lt_pow2; try lia.
Qed.
End ExtraFacts.
Section ToBytes_internal.
Definition _Z_to_bytes_unsigned_le' (idx: nat) (cnt: nat) (v: Z): list N :=
map (Z.to_N ∘ _get_byte v) $ seq idx cnt.
Definition _Z_to_bytes_unsigned_le (cnt: nat) (v: Z): list N :=
_Z_to_bytes_unsigned_le' 0 cnt v.
Definition _Z_to_bytes_le (cnt: nat) (sgn: signed) (v: Z): list N :=
_Z_to_bytes_unsigned_le
cnt (match sgn with
| Signed => to_unsigned_bits (8 * N.of_nat cnt) v
| Unsigned => v
end).
(* NOTE: This will be sealed once we finish the proofs for this section. *)
Definition _Z_to_bytes_def (cnt: nat) (endianness: endian) (sgn: signed) (v: Z): list N :=
let little := _Z_to_bytes_le cnt sgn v in
match endianness with
| Little => little
| Big => List.rev little
end.
End ToBytes_internal.
Section ToBytesFacts_internal.
#[local] Transparent _get_byte _set_byte.
Lemma _Z_to_bytes_unsigned_le'_length:
forall idx cnt v,
length (_Z_to_bytes_unsigned_le' idx cnt v) = cnt.
Proof.
rewrite /_Z_to_bytes_unsigned_le' => idx cnt v //=;
by rewrite length_map length_seq.
Qed.
Definition _Z_to_bytes_unsigned_le_length:
forall cnt v,
length (_Z_to_bytes_unsigned_le cnt v) = cnt.
Proof.
rewrite /_Z_to_bytes_unsigned_le => * //=;
by apply _Z_to_bytes_unsigned_le'_length.
Qed.
Lemma _Z_to_bytes_le_length:
forall cnt sgn v,
length (_Z_to_bytes_le cnt sgn v) = cnt.
Proof.
rewrite /_Z_to_bytes_le => * //=;
by apply _Z_to_bytes_unsigned_le_length.
Qed.
Lemma _Z_to_bytes_def_length:
forall cnt endianness sgn v,
length (_Z_to_bytes_def cnt endianness sgn v) = cnt.
Proof.
rewrite /_Z_to_bytes_def => cnt [ | ] sgn v //=;
try rewrite length_rev;
by apply _Z_to_bytes_le_length.
Qed.
Lemma _Z_to_bytes_unsigned_le'_0_bytes:
forall (idx: nat) (v: Z),
_Z_to_bytes_unsigned_le' idx 0 v = [].
Proof. done. Qed.
Lemma _Z_to_bytes_unsigned_le_0_bytes:
forall (v: Z),
_Z_to_bytes_unsigned_le 0 v = [].
Proof. apply (_Z_to_bytes_unsigned_le'_0_bytes 0). Qed.
Lemma _Z_to_bytes_le_0_bytes:
forall sgn (v: Z),
_Z_to_bytes_le 0 sgn v = [].
Proof.
move=> [ | ] v; rewrite /_Z_to_bytes_le;
[rewrite N.mul_0_r trim_0_l | ];
apply _Z_to_bytes_unsigned_le_0_bytes.
Qed.
Lemma _Z_to_bytes_def_0_bytes:
forall endianness sgn (v: Z),
_Z_to_bytes_def 0 endianness sgn v = [].
Proof.
move=> [ | ] sgn v;
by rewrite /_Z_to_bytes_def _Z_to_bytes_le_0_bytes //=.
Qed.
Lemma _Z_to_bytes_unsigned_le'_0_value:
forall (idx cnt: nat),
_Z_to_bytes_unsigned_le' idx cnt 0 = repeat 0%N cnt.
Proof.
induction cnt => //=.
rewrite /_Z_to_bytes_unsigned_le'.
rewrite seq_S map_app repeat_cons_app //=.
rewrite _get_byte_0; f_equal=> //=.
Qed.
Lemma _Z_to_bytes_unsigned_le_0_value:
forall (cnt: nat),
_Z_to_bytes_unsigned_le cnt 0 = repeat 0%N cnt.
Proof. apply _Z_to_bytes_unsigned_le'_0_value. Qed.
Lemma _Z_to_bytes_le_0_value:
forall (cnt: nat) sgn,
_Z_to_bytes_le cnt sgn 0 = repeat 0%N cnt.
Proof.
move=> cnt [ | ]; rewrite /_Z_to_bytes_le;
[rewrite trim_0_r | ];
apply _Z_to_bytes_unsigned_le_0_value.
Qed.
Lemma _Z_to_bytes_def_0_value:
forall sgn endianness (cnt: nat),
_Z_to_bytes_def cnt endianness sgn 0 = repeat 0%N cnt.
Proof.
move=> sgn [ | ] cnt;
rewrite /_Z_to_bytes_def
_Z_to_bytes_le_0_value //=;
by rewrite rev_repeat.
Qed.
Lemma _Z_to_bytes_unsigned_le'_S_cnt:
forall (idx cnt: nat) (v: Z),
_Z_to_bytes_unsigned_le' idx (S cnt) v =
_Z_to_bytes_unsigned_le' idx cnt v ++
_Z_to_bytes_unsigned_le' (idx+cnt) 1 v.
Proof.
intros; generalize dependent idx;
induction cnt; intros=> //=.
- by rewrite Nat.add_0_r.
- rewrite /_Z_to_bytes_unsigned_le'.
rewrite seq_S map_app //=.
Qed.
Lemma _Z_to_bytes_unsigned_le_S_cnt:
forall (cnt: nat) (v: Z),
_Z_to_bytes_unsigned_le (S cnt) v =
_Z_to_bytes_unsigned_le' 0 cnt v ++
_Z_to_bytes_unsigned_le' cnt 1 v.
Proof. apply _Z_to_bytes_unsigned_le'_S_cnt. Qed.
Lemma _Z_to_bytes_unsigned_le'_small:
forall (idx cnt: nat) (v: Z),
(0 <= v)%Z ->
(v < 2^(8*idx))%Z ->
_Z_to_bytes_unsigned_le' idx cnt v =
repeat 0%N cnt.
Proof.
intros; generalize dependent idx;
induction cnt ; intros=> //=.
rewrite _Z_to_bytes_unsigned_le'_S_cnt
/_Z_to_bytes_unsigned_le'
repeat_cons_app //=; f_equal.
- erewrite <- IHcnt; eauto.
- rewrite /_get_byte Z_shiftr_small; try lia;
[ rewrite Z.land_0_l
| apply Z_pow2_trans_nat_r]=> //=.
Qed.
Lemma _Z_to_bytes_unsigned_le'_shrink_cnt:
forall (idx cnt cnt': nat) (v: Z),
(0 <= v)%Z ->
(v < 2^(8*cnt))%Z ->
cnt < cnt' ->
_Z_to_bytes_unsigned_le' idx cnt' v =
_Z_to_bytes_unsigned_le' idx cnt v ++ repeat 0%N (cnt' - cnt).
Proof.
intros; generalize dependent idx; generalize dependent cnt;
induction cnt'; intros=> //=; [lia | ].
replace (S cnt' - cnt) with (S (cnt' - cnt)) by lia; simpl.
assert (cnt = cnt' \/ cnt < cnt') as [Hcnt | Hcnt] by lia.
- subst; rewrite _Z_to_bytes_unsigned_le'_S_cnt; f_equal.
rewrite Nat.sub_diag /_Z_to_bytes_unsigned_le' //=.
rewrite /_get_byte Z_shiftr_small; try lia;
[ rewrite Z.land_0_l
| apply Z_pow2_trans_nat_l]=> //=.
- rewrite repeat_cons_app app_assoc.
rewrite -IHcnt'; try lia.
rewrite _Z_to_bytes_unsigned_le'_S_cnt //=; f_equal.
rewrite /_Z_to_bytes_unsigned_le' //=.
rewrite /_get_byte Z_shiftr_small; try lia;
[ rewrite Z.land_0_l
| eapply Z.lt_trans; eauto; apply Z.pow_lt_mono_r; lia]=> //=.
Qed.
Lemma _Z_to_bytes_unsigned_le_shrink_cnt:
forall (cnt cnt': nat) (v: Z),
(0 <= v)%Z ->
(v < 2^(8*cnt))%Z ->
cnt < cnt' ->
_Z_to_bytes_unsigned_le cnt' v =
_Z_to_bytes_unsigned_le cnt v ++ repeat 0%N (cnt' - cnt).
Proof. apply _Z_to_bytes_unsigned_le'_shrink_cnt. Qed.
Lemma _Z_to_bytes_unsigned_le'_S_idx:
forall (idx cnt: nat) (v: Z),
(0 <= v)%Z ->
_Z_to_bytes_unsigned_le' (S idx) cnt v =
_Z_to_bytes_unsigned_le' idx cnt (v ≫ 8).
Proof.
intros idx cnt v Hlower; generalize dependent idx;
induction cnt; intros=> //=.
rewrite /_Z_to_bytes_unsigned_le' //=; f_equal.
- rewrite /_get_byte Z.shiftr_shiftr; try lia.
by replace (8 + 8 * idx)%Z with (8 * S idx)%Z by lia.
- fold (_Z_to_bytes_unsigned_le' (S (S idx)) cnt v).
fold (_Z_to_bytes_unsigned_le' (S idx) cnt (v ≫ 8)).
assert (v < 2^(8*cnt) \/ 2^(8*cnt) <= v)%Z as [Hv | Hv] by lia.
+ by apply IHcnt.
+ rewrite /_Z_to_bytes_unsigned_le' //=.
rewrite -seq_shift map_map.
generalize (seq (S idx) cnt); induction l=> //=; f_equal.
* rewrite /_get_byte Z.shiftr_shiftr; try lia.
by replace (8 + 8 * a)%Z with (8 * S a)%Z by lia.
* apply IHl.
Qed.
End ToBytesFacts_internal.
Section FromBytes_internal.
Definition _Z_from_bytes_unsigned_le' (idx: nat) (bytes: list N): Z :=
foldr (fun '(idx, byte) acc => Z.lor (_set_byte (Z.of_N byte) idx) acc)
0%Z (zip (seq idx (length bytes)) bytes).
Definition _Z_from_bytes_unsigned_le (bytes: list N): Z :=
_Z_from_bytes_unsigned_le' 0 bytes.
Definition _Z_from_bytes_le (sgn: signed) (bytes: list N): Z :=
let unsigned := _Z_from_bytes_unsigned_le bytes in
match sgn with
| Signed => to_signed_bits (8 * N.of_nat (length bytes)) unsigned
| Unsigned => unsigned
end.
(* NOTE: This will be sealed once we finish the proofs for this section. *)
Definition _Z_from_bytes_def (endianness: endian) (sgn: signed) (bytes: list N): Z :=
_Z_from_bytes_le
sgn match endianness with
| Little => bytes
| Big => List.rev bytes
end.
End FromBytes_internal.
Section FromBytesFacts_internal.
Lemma _Z_from_bytes_unsigned_le'_nil:
forall (idx: nat),
_Z_from_bytes_unsigned_le' idx [] = 0%Z.
Proof. rewrite /_Z_from_bytes_unsigned_le' //=. Qed.
Lemma _Z_from_bytes_unsigned_le_nil:
_Z_from_bytes_unsigned_le [] = 0%Z.
Proof. apply _Z_from_bytes_unsigned_le'_nil. Qed.
Lemma _Z_from_bytes_le_nil:
forall sgn,
_Z_from_bytes_le sgn [] = 0%Z.
Proof.
move=> [ | ];
rewrite /_Z_from_bytes_le
_Z_from_bytes_unsigned_le_nil //=.
Qed.
Lemma _Z_from_bytes_def_nil:
forall endianness sgn,
_Z_from_bytes_def endianness sgn [] = 0%Z.
Proof.
move=> [ | ] sgn;
by rewrite /_Z_from_bytes_def /rev
_Z_from_bytes_le_nil.
Qed.
Lemma _Z_from_bytes_unsigned_le'_cons:
forall (idx: nat) (byte: N) (bytes: list N),
_Z_from_bytes_unsigned_le' idx (byte :: bytes) =
Z.lor (_Z_from_bytes_unsigned_le' idx [byte])
(_Z_from_bytes_unsigned_le' (S idx) bytes).
Proof.
intros; generalize dependent idx; generalize dependent byte;
induction bytes => //=; intros.
- rewrite _Z_from_bytes_unsigned_le'_nil Z.lor_0_r; lia.
- rewrite /_Z_from_bytes_unsigned_le' //=.
by rewrite Z.lor_0_r.
Qed.
Lemma _Z_from_bytes_unsigned_le_cons:
forall (byte: N) (bytes: list N),
_Z_from_bytes_unsigned_le (byte :: bytes) =
Z.lor (_Z_from_bytes_unsigned_le' 0 [byte])
(_Z_from_bytes_unsigned_le' 1 bytes).
Proof. apply _Z_from_bytes_unsigned_le'_cons. Qed.
Lemma _Z_from_bytes_unsigned_le'_app:
forall (idx: nat) (bs1 bs2: list N),
_Z_from_bytes_unsigned_le' idx (bs1 ++ bs2) =
Z.lor (_Z_from_bytes_unsigned_le' idx bs1)
(_Z_from_bytes_unsigned_le' (idx + length bs1) bs2).
Proof.
intros; generalize dependent idx; generalize dependent bs2;
induction bs1 => //=; intros.
- repeat rewrite _Z_from_bytes_unsigned_le'_nil.
replace (idx + 0) with idx by lia.
by rewrite Z.lor_0_l.
- rewrite _Z_from_bytes_unsigned_le'_cons.
rewrite (_Z_from_bytes_unsigned_le'_cons idx a bs1).
rewrite IHbs1.
rewrite Nat.add_succ_r.
by rewrite Z.lor_assoc.
Qed.
Lemma _Z_from_bytes_unsigned_le_app:
forall (bs1 bs2: list N),
_Z_from_bytes_unsigned_le (bs1 ++ bs2) =
Z.lor (_Z_from_bytes_unsigned_le' 0 bs1)
(_Z_from_bytes_unsigned_le' (length bs1) bs2).
Proof. apply _Z_from_bytes_unsigned_le'_app. Qed.
Lemma _Z_from_bytes_unsigned_le'_0s:
forall (idx cnt: nat),
_Z_from_bytes_unsigned_le' idx (repeat 0%N cnt) = 0%Z.
Proof.
intros; generalize dependent idx.
induction cnt => idx //=.
rewrite _Z_from_bytes_unsigned_le'_cons.
rewrite IHcnt.
rewrite /_Z_from_bytes_unsigned_le' //=.
by rewrite _set_byte_0 !Z.lor_0_r.
Qed.
Lemma _Z_from_bytes_unsigned_le_0s:
forall (cnt: nat),
_Z_from_bytes_unsigned_le (repeat 0%N cnt) = 0%Z.
Proof. apply _Z_from_bytes_unsigned_le'_0s. Qed.
Lemma _Z_from_bytes_le_0s:
forall sgn (cnt: nat),
_Z_from_bytes_le sgn (repeat 0%N cnt) = 0%Z.
Proof.
move=> [ | ] cnt; rewrite /_Z_from_bytes_le.
- rewrite repeat_length _Z_from_bytes_unsigned_le_0s.
assert (8 * N.of_nat cnt = 0 \/ 0 < 8 * N.of_nat cnt)%N as [Hcnt | Hcnt] by lia.
+ rewrite /to_signed_bits bool_decide_eq_true_2; by lia.
+ rewrite to_signed_bits_id; intuition; [by reflexivity | ].
apply Z.pow_pos_nonneg; lia.
- apply _Z_from_bytes_unsigned_le_0s.
Qed.
Lemma _Z_from_bytes_def_0s:
forall endianness sgn (cnt: nat),
_Z_from_bytes_def endianness sgn (repeat 0%N cnt) = 0%Z.
Proof.
move=> [ | ] sgn cnt; rewrite /_Z_from_bytes_def ?rev_repeat;
apply _Z_from_bytes_le_0s.
Qed.
Lemma _Z_from_bytes_unsigned_le'_S_idx:
forall (idx: nat) (bytes: list N),
(_Z_from_bytes_unsigned_le' (S idx) (bytes) =
_Z_from_bytes_unsigned_le' idx bytes ≪ 8)%Z.
Proof.
intros idx bytes; generalize dependent idx;
induction bytes; intros=> //=.
rewrite /_Z_from_bytes_unsigned_le' //=.
fold (_Z_from_bytes_unsigned_le' (S (S idx)) bytes).
fold (_Z_from_bytes_unsigned_le' (S idx) bytes).
rewrite Z.shiftl_lor.
by rewrite IHbytes _set_byte_S_idx.
Qed.
#[local] Lemma length_1_inv:
forall {A} (l : list A),
length l = 1%nat ->
exists x0,
l = [x0].
Proof. intros; do 2 (try destruct l)=> //; do 1 eexists; eauto. Qed.
#[local] Lemma length_2_inv:
forall {A} (l : list A),
length l = 2%nat ->
exists x0 x1,
l = [x0; x1].
Proof. intros; do 3 (try destruct l)=> //; do 2 eexists; eauto. Qed.
#[local] Lemma length_4_inv:
forall {A} (l : list A),
length l = 4%nat ->
exists x0 x1 x2 x3,
l = [x0; x1; x2; x3].
Proof. intros; do 5 (try destruct l)=> //; do 4 eexists; eauto. Qed.
#[local] Lemma length_8_inv:
forall {A} (l : list A),
length l = 8%nat ->
exists x0 x1 x2 x3 x4 x5 x6 x7,
l = [x0; x1; x2; x3; x4; x5; x6; x7].
Proof. intros; do 9 (try destruct l)=> //; do 8 eexists; eauto. Qed.
#[local] Lemma length_16_inv:
forall {A} (l : list A),
length l = 16%nat ->
exists x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15,
l = [x0; x1; x2; x3; x4; x5; x6; x7; x8; x9; x10; x11; x12; x13; x14; x15].
Proof. intros; do 17 (try destruct l)=> //; do 16 eexists; eauto. Qed.
Lemma _Z_from_bytes_unsigned_le_bswap:
forall bsz sz (bytes: list N) v,
Datatypes.length bytes = sz ->
bitsize.bytesNat bsz = sz ->
_Z_from_bytes_unsigned_le bytes = v ->
_Z_from_bytes_unsigned_le (rev bytes) = bswap bsz v.
Proof.
rewrite /_Z_from_bytes_unsigned_le/_Z_from_bytes_unsigned_le';
move=> bsz sz bytes v Hlen Hsz Hdecodes; destruct bsz;
simpl in *; rewrite -Hsz in Hlen; subst;
[ apply length_1_inv in Hlen as [? Hbytes]
| apply length_2_inv in Hlen as [? [? Hbytes]]
| apply length_4_inv in Hlen as [? [? [? [? Hbytes]]]]
| apply length_8_inv in Hlen as [? [? [? [? [? [? [? [? Hbytes]]]]]]]]
| apply length_16_inv in Hlen as [? [? [? [? [? [? [? [?
[? [? [? [? [? [? [? [? Hbytes]]]]]]]]]]]]]]]]];
rewrite Hbytes //= !Z.lor_0_r.
- now rewrite bswap8_set_byte_reverse.
- now rewrite bswap16_set_byte_reverse.
- now rewrite bswap32_set_byte_reverse.
- now rewrite bswap64_set_byte_reverse.
- now rewrite bswap128_set_byte_reverse.
Qed.
Lemma _Z_from_bytes_unsigned_le'_singleton idx a :
_Z_from_bytes_unsigned_le' idx [a] = (_set_byte (Z.of_N a) idx).
Proof.
rewrite/_Z_from_bytes_unsigned_le'.
cbn.
by rewrite Z.lor_0_r.
Qed.
Lemma _Z_from_bytes_unsigned_le_singleton a :
_Z_from_bytes_unsigned_le [a] = (_get_byte (Z.of_N a) 0).
Proof.
by rewrite /_Z_from_bytes_unsigned_le _Z_from_bytes_unsigned_le'_singleton _get_0_set_0_eq.
Qed.
Lemma _Z_from_bytes_le_singleton sgn a :
_Z_from_bytes_le sgn [a] =
match sgn with
| Signed => to_signed_bits 8 (_get_byte a 0)
| Unsigned => _get_byte a 0
end.
Proof.
rewrite /_Z_from_bytes_le _Z_from_bytes_unsigned_le_singleton.
by rewrite /= N.mul_1_r.
Qed.
Lemma _Z_from_bytes_def_singleton endianness sgn a :
_Z_from_bytes_def endianness sgn [a] =
match sgn with
| Signed => to_signed_bits 8 (_get_byte a 0)
| Unsigned => _get_byte a 0
end.
Proof.
rewrite /_Z_from_bytes_def /=.
by destruct endianness; rewrite _Z_from_bytes_le_singleton.
Qed.
Lemma _Z_from_bytes_unsigned_le'_bounds bytes idx b :
(256 ^ (length bytes + Z.of_nat idx)%Z <= b)%Z ->
(0 ≤ _Z_from_bytes_unsigned_le' idx bytes < b)%Z.
Proof.
move: b idx.
elim bytes.
{
move => ? ?.
rewrite Z.add_0_l.
move => ?.
split; [by []|].
cbn.
eapply Z.lt_le_trans.
2: { eassumption. }
apply Z.pow_pos_nonneg; [ by[] | ].
exact: Zle_0_nat.
}
move => a l IH b idx /= H.
rewrite _Z_from_bytes_unsigned_le'_cons.
rewrite _Z_from_bytes_unsigned_le'_S_idx.
assert (
(0
≤ Z.lor (_Z_from_bytes_unsigned_le' idx [a]) (_Z_from_bytes_unsigned_le' idx l ≪ 8) <
2 ^ (Z.of_N $ N.of_nat (8 * (S (length l) + idx))))%Z
) as H1.
{
apply ZlorRange.
{
rewrite _Z_from_bytes_unsigned_le'_singleton.
move: (_set_byte_bound (Z.of_N a) idx) => [L0 L1].
split; first assumption.
eapply Z.lt_le_trans; first eassumption.
apply Z.pow_le_mono_r; first done.
lia.
}
move: (IH (2 ^ Z.of_N (N.of_nat (8 * (S (length l) + idx))))%Z (S idx)).
rewrite _Z_from_bytes_unsigned_le'_S_idx.
apply.
rewrite nat_N_Z.
rewrite Nat2Z.inj_mul.
rewrite Z.pow_mul_r //.
apply: Z.pow_le_mono.
1: done.
lia.
}
split; first tauto.
eapply Z.lt_le_trans.
{
move: H1 => [_ max].
eassumption.
}
rewrite nat_N_Z.
rewrite Nat2Z.inj_mul.
rewrite Z.pow_mul_r //.
cbn.
eapply Z.le_trans.
2: eassumption.
apply: Z.pow_le_mono.
1: done.
lia.
Qed.
Lemma _Z_from_bytes_unsigned_le'_bound bytes idx b :
(256 ^ (length bytes + Z.of_nat idx)%Z - 1 <= b)%Z ->
(0 ≤ _Z_from_bytes_unsigned_le' idx bytes ≤ b)%Z.
Proof.
pose proof (_Z_from_bytes_unsigned_le'_bounds bytes idx
(256 ^ (length bytes + Z.of_nat idx)%Z)%Z
ltac:(lia))
as Hbound'.
lia.
Qed.
Lemma _Z_from_bytes_unsigned_le_bounds bytes b :
(256 ^ length bytes <= b)%Z ->
(0 <= _Z_from_bytes_unsigned_le bytes < b)%Z.
Proof.
move => H.
apply _Z_from_bytes_unsigned_le'_bounds.
by rewrite Z.add_0_r.
Qed.
Lemma _Z_from_bytes_unsigned_le_bound bytes b :
(256 ^ length bytes - 1 <= b)%Z ->
(0 <= _Z_from_bytes_unsigned_le bytes <= b)%Z.
Proof.
pose proof (_Z_from_bytes_unsigned_le_bounds bytes
(256 ^ (length bytes))
ltac:(lia))
as Hbound'.
lia.
Qed.
End FromBytesFacts_internal.
Section FromToFacts_internal.
#[local] Transparent _get_byte _set_byte.
Lemma _Z_from_to_bytes_unsigned_le'_overflow:
forall (idx cnt: nat) (v: Z),
(2^(8*(idx+cnt)) <= v)%Z ->
_Z_from_bytes_unsigned_le' idx (_Z_to_bytes_unsigned_le' idx cnt v) =
Z.land ((Z.ones (8*cnt)) ≪ (8*idx)) v.
Proof.
intros idx cnt v Hlower;
generalize dependent idx;
generalize dependent v;
induction cnt; intros=> //=.
- rewrite /Z.ones Z.mul_0_r Z.shiftl_0_r
_Z_to_bytes_unsigned_le'_0_bytes
_Z_from_bytes_unsigned_le'_nil.
by rewrite Z.shiftl_0_l Z.land_0_l.
- rewrite _Z_to_bytes_unsigned_le'_S_cnt
_Z_from_bytes_unsigned_le'_app
_Z_to_bytes_unsigned_le'_length.
rewrite [_Z_to_bytes_unsigned_le' _ 1 _]/_Z_to_bytes_unsigned_le' //=.
rewrite [_Z_from_bytes_unsigned_le' _ [_]]/_Z_from_bytes_unsigned_le' //=.
rewrite Z.lor_0_r.
rewrite Z2N.id; [ | apply _get_byte_nonneg].
rewrite _set_get_byte_roundtrip/_get_byte.
assert (2^(8*(idx+cnt)) <= v)%Z as Hlower'. {
etrans; last eapply Hlower;
apply Z.pow_le_mono_r; lia.
}
specialize (IHcnt v idx Hlower'); rewrite IHcnt //=.
apply Z.bits_inj'=> n ?.
repeat (rewrite ?Z.lor_spec ?Z.land_spec; try lia).
rewrite Z.shiftl_land -Z.ldiff_ones_r; try lia.
repeat (rewrite ?Z.lor_spec ?Z.land_spec ?Z.shiftl_spec ?Z.ldiff_spec; try lia).
rewrite !Z.testbit_ones; try lia.
churn_bits'.
Qed.
Lemma _Z_from_to_bytes_unsigned_le'_roundtrip:
forall (idx cnt: nat) (v: Z),
(0 <= v)%Z ->
(v < 2^(8*(idx+cnt)))%Z ->
_Z_from_bytes_unsigned_le' idx (_Z_to_bytes_unsigned_le' idx cnt v) =
(v - v `mod` 2^(8*idx))%Z.
Proof.
intros idx cnt v Hlower Hupper;
generalize dependent idx;
generalize dependent v;
induction cnt; intros=> //=.
- rewrite Z.add_0_r in Hupper.
rewrite _Z_to_bytes_unsigned_le'_0_bytes
_Z_from_bytes_unsigned_le'_nil.
rewrite Zmod_small; try intuition; lia.
- assert (v < 2^(8*(idx+cnt)) \/ 2^(8*(idx+cnt)) <= v)%Z as [Hv | Hv] by lia.
+ rewrite _Z_to_bytes_unsigned_le'_S_cnt
_Z_from_bytes_unsigned_le'_app
_Z_to_bytes_unsigned_le'_length
IHcnt; try lia.
rewrite _Z_to_bytes_unsigned_le'_small; try lia.
1: by rewrite _Z_from_bytes_unsigned_le'_0s Z.lor_0_r.
+ clear IHcnt.
rewrite Zmod_eq_full.
2: pose proof (Z.pow_pos_nonneg
2 (8 * idx)
ltac:(lia) ltac:(lia)); lia.
rewrite -Z.shiftr_div_pow2; try lia.
rewrite -Z.shiftl_mul_pow2; try lia.
rewrite -Z.ldiff_ones_r; try lia.
rewrite _Z_to_bytes_unsigned_le'_S_cnt
_Z_from_bytes_unsigned_le'_app
_Z_to_bytes_unsigned_le'_length.
rewrite Z.sub_sub_distr Z.sub_diag Z.add_0_l.
rewrite [_Z_to_bytes_unsigned_le' _ 1 _]/_Z_to_bytes_unsigned_le' //=.
rewrite [_Z_from_bytes_unsigned_le' _ [_]]/_Z_from_bytes_unsigned_le' //=.
rewrite Z.lor_0_r.
rewrite Z2N.id; [ | apply _get_byte_nonneg].
rewrite _set_get_byte_roundtrip/_get_byte Z.shiftl_land.
rewrite _Z_from_to_bytes_unsigned_le'_overflow; try lia.
erewrite Z_ldiff_split; eauto; f_equal.
rewrite -Z.ldiff_ones_r; try lia.
rewrite Z.land_comm.
apply Z.bits_inj'=> n ?.
repeat (rewrite ?Z.lor_spec ?Z.land_spec ?Z.shiftl_spec ?Z.ldiff_spec; try lia).
rewrite !Z.testbit_ones; try lia.
churn_bits'.
Qed.
Lemma _Z_from_to_bytes_unsigned_le_roundtrip:
forall (cnt: nat) (v: Z),
(0 <= v)%Z ->
(v < 2^(8*cnt))%Z ->
_Z_from_bytes_unsigned_le (_Z_to_bytes_unsigned_le cnt v) = v.
Proof.
intros cnt v Hlower Hupper.
pose proof (_Z_from_to_bytes_unsigned_le'_roundtrip 0 cnt v Hlower) as Hpf.
rewrite Z.add_0_l in Hpf.
specialize (Hpf Hupper).
rewrite /_Z_from_bytes_unsigned_le
/_Z_to_bytes_unsigned_le Hpf.
rewrite Z.mul_0_r Z.pow_0_r Zmod_1_r; lia.
Qed.
Lemma _get_set_later_bytes_irrelevant idx start lst :
(idx < start)%nat ->
_get_byte
(foldr
(λ '(idx0, byte) (acc : Z),
(_set_byte (Z.of_N byte) idx0 `lor` acc)%Z) 0%Z
(zip (seq start (length lst)) lst)) idx = 0%Z.
Proof.
move: idx start.
induction lst.
{
move => ? ? ?.
by rewrite _get_byte_0.
}
move => ? ? ?.
cbn.
rewrite _get_byte_lor.
rewrite _get_set_byte_no_overlap.
2: lia.
rewrite Z.lor_0_l.
apply: IHlst.
lia.
Qed.
Lemma _get_byte_0_small_id (z : Z) :
_get_byte z 0 = Z.modulo z 256.
Proof. by rewrite/_get_byte /Z.of_nat Z.mul_0_r Z.shiftr_0_r Z.land_ones. Qed.
Lemma _Z_to_from_bytes_unsigned_le'_roundtrip (bytes : list N) (idx cnt : nat) :
Forall (λ b : N, (b < 256)%N) bytes →
cnt = length bytes →
_Z_to_bytes_unsigned_le' idx cnt (_Z_from_bytes_unsigned_le' idx bytes) = bytes.
Proof.
move: idx bytes.
induction cnt.
{ by move => ? []. }
move => idx [].
{ done. }
move => hd tl smallH.
apply Forall_cons_1 in smallH.
move: smallH => [small_hdH small_tlH].
move => [] cnt_eq.
rewrite/_Z_to_bytes_unsigned_le'.
cbn.
f_equal.
{
rewrite _get_byte_lor _get_set_later_bytes_irrelevant.
2: lia.
rewrite Z.lor_0_r _get_set_byte_roundtrip _get_byte_0_small_id Z2N.inj_mod.
2: lia.
2: lia.
by rewrite N2Z.id N.mod_small.
}
move: (IHcnt (S idx) tl small_tlH cnt_eq).
rewrite /_Z_to_bytes_unsigned_le' /_Z_from_bytes_unsigned_le'.
move => IH.
rewrite -{3}IH.
apply: map_ext_Forall.
rewrite Forall_seq.
move => ? ?.
cbn.
f_equal.
rewrite _get_byte_lor _get_set_byte_no_overlap.
2: lia.
by rewrite Z.lor_0_l.
Qed.
Lemma _Z_to_from_bytes_unsigned_le_roundtrip (bytes : list N) (cnt : nat) :
Forall (fun b => b < 256)%N bytes ->
cnt = length bytes ->
(_Z_to_bytes_unsigned_le cnt $ _Z_from_bytes_unsigned_le bytes) = bytes.
Proof.
move => ? ?.
by rewrite/_Z_to_bytes_unsigned_le/_Z_from_bytes_unsigned_le
_Z_to_from_bytes_unsigned_le'_roundtrip.
Qed.
Lemma _Z_from_to_bytes_le_roundtrip:
forall (cnt: nat) (sgn: signed) (v: Z),
match sgn with
| Signed => -2^((8*cnt)-1) <= v /\ v <= 2^((8*cnt)-1) - 1
| Unsigned => 0 <= v /\ v < 2^(8*cnt)
end%Z ->
_Z_from_bytes_le sgn (_Z_to_bytes_le cnt sgn v) = v.
Proof.
destruct sgn; intros v [Hlower Hupper];
[ | by apply _Z_from_to_bytes_unsigned_le_roundtrip].
rewrite /_Z_from_bytes_le /_Z_to_bytes_le _Z_to_bytes_unsigned_le_length.
assert (v < 0 \/ 0 <= v)%Z as [Hv | Hv] by lia.
- rewrite _Z_from_to_bytes_unsigned_le_roundtrip.
+ apply to_signed_unsigned_bits_roundtrip; intuition;
replace (Z.of_N (8 * N.of_nat cnt) - 1)%Z
with (8 * cnt - 1)%Z; lia.
+ rewrite /trim; apply Z.mod_pos;
apply Z.pow_pos_nonneg; lia.
+ rewrite /trim.
replace (Z.of_N (8 * N.of_nat cnt))
with (8 * cnt)%Z by lia.
by pose proof (Z_mod_lt v (2^(8*cnt))%Z
ltac:(apply Z.lt_gt; apply Z.pow_pos_nonneg; lia))
as [? ?].
- rewrite /trim Zmod_small; intuition; try lia.
+ rewrite _Z_from_to_bytes_unsigned_le_roundtrip; try lia.
* apply to_signed_bits_id; intuition.
eapply Z.le_lt_trans; eauto.
replace (Z.of_N (8 * N.of_nat cnt) - 1)%Z
with (8 * cnt - 1)%Z; lia.
* eapply Z.le_lt_trans; eauto.
match goal with
| |- (_ < ?r)%Z => replace r with (r - 0)%Z by lia
end.
apply Z.sub_lt_le_mono; try apply Z.pow_lt_mono_r; lia.
+ eapply Z.le_lt_trans; eauto.
match goal with
| |- (_ < ?r)%Z => replace r with (r - 0)%Z by lia
end.
apply Z.sub_lt_le_mono; try apply Z.pow_lt_mono_r; lia.
Qed.
Lemma _Z_from_unsigned_to_signed_bytes_le:
forall (cnt: nat) (v: Z),
(-2^((8*cnt)-1) <= v)%Z ->
(v <= 2^((8*cnt)-1) - 1)%Z ->
_Z_from_bytes_le Unsigned (_Z_to_bytes_le cnt Signed v) =
to_unsigned_bits (8*N.of_nat cnt) v.
Proof.
move=> cnt v Hlower Hupper.
rewrite /trim /_Z_from_bytes_le /_Z_to_bytes_le.
rewrite _Z_from_to_bytes_unsigned_le_roundtrip /trim //.
- apply Z_mod_lt; rewrite Z.gt_lt_iff;
apply Z.pow_pos_nonneg; lia.
- replace (Z.of_N (8 * N.of_nat cnt)) with (8 * cnt)%Z by lia;
apply Z_mod_lt; rewrite Z.gt_lt_iff;
apply Z.pow_pos_nonneg; lia.
Qed.
Lemma _Z_from_signed_to_unsigned_bytes_le:
forall (cnt: nat) (v: Z),
(0 <= v)%Z ->
(v < 2^(8*cnt))%Z ->
_Z_from_bytes_le Signed (_Z_to_bytes_le cnt Unsigned v) =
to_signed_bits (8*N.of_nat cnt) v.
Proof.
move=> cnt v Hlower Hupper.
rewrite /_Z_from_bytes_le /_Z_to_bytes_le
_Z_to_bytes_unsigned_le_length
_Z_from_to_bytes_unsigned_le_roundtrip //.
Qed.
Lemma _Z_from_to_bytes_def_roundtrip:
forall (cnt: nat) (endianness: endian) (sgn: signed) (v: Z),
match sgn with
| Signed => -2^((8*cnt)-1) <= v /\ v <= 2^((8*cnt)-1) - 1
| Unsigned => 0 <= v /\ v < 2^(8*cnt)
end%Z ->
_Z_from_bytes_def endianness sgn (_Z_to_bytes_def cnt endianness sgn v) = v.
Proof.
rewrite /_Z_from_bytes_def /_Z_to_bytes_def;
intros cnt [ | ] sgn v H;
try rewrite rev_involutive;
by apply _Z_from_to_bytes_le_roundtrip.
Qed.
Lemma _Z_to_from_bytes_le_round_trip sgn cnt bytes :
Forall (λ b : N, (b < 256)%N) bytes ->
cnt = length bytes ->
_Z_to_bytes_le cnt sgn (_Z_from_bytes_le sgn bytes) = bytes.
Proof.
move => small cntH.
move: sgn => [].
2: {
cbn.
rewrite/_Z_to_bytes_le.
by apply: _Z_to_from_bytes_unsigned_le_roundtrip.
}
rewrite/_Z_to_bytes_le.
cbn.
rewrite !cntH.
rewrite trim_to_signed_bits_agree.
rewrite to_unsigned_bits_id.
{ by apply: _Z_to_from_bytes_unsigned_le_roundtrip. }
apply: _Z_from_bytes_unsigned_le_bounds.
rewrite -(Z.pow_mul_r 2 8).
2: done.
2: lia.
apply: Z.eq_le_incl.
f_equal.
rewrite -nat_N_Z.
case (N.of_nat (length bytes)).
{ lia. }
move => ?.
lia.
Qed.
Lemma _Z_to_from_bytes_def_roundtrip bytes sgn endianness cnt :
Forall (λ b : N, (b < 256)%N) bytes →
cnt = length bytes →
_Z_to_bytes_def cnt endianness sgn (_Z_from_bytes_def endianness sgn bytes) = bytes.
Proof.
move => small cntH.
rewrite/_Z_from_bytes_def/_Z_to_bytes_def.
move: endianness => [].
{ exact: _Z_to_from_bytes_le_round_trip. }
rewrite _Z_to_from_bytes_le_round_trip.
{ exact: rev_involutive. }
{ exact: Forall_rev. }
by rewrite length_rev.
Qed.
Lemma _Z_from_unsigned_to_signed_bytes_def:
forall (cnt: nat) (endianness: endian) (v: Z),
(-2^((8*cnt)-1) <= v)%Z ->
(v <= 2^((8*cnt)-1) - 1)%Z ->
_Z_from_bytes_def endianness Unsigned (_Z_to_bytes_def cnt endianness Signed v) =
to_unsigned_bits (8*N.of_nat cnt) v.
Proof.
rewrite /_Z_from_bytes_def /_Z_to_bytes_def;
intros cnt [ | ] v Hlower Hupper;
try rewrite rev_involutive;
by apply _Z_from_unsigned_to_signed_bytes_le.
Qed.
Lemma _Z_from_signed_to_unsigned_bytes_def:
forall (cnt: nat) (endianness: endian) (v: Z),
(0 <= v)%Z ->
(v < 2^(8*cnt))%Z ->
_Z_from_bytes_def endianness Signed (_Z_to_bytes_def cnt endianness Unsigned v) =
to_signed_bits (8*N.of_nat cnt) v.
Proof.
rewrite /_Z_from_bytes_def /_Z_to_bytes_def;
intros cnt [ | ] v Hlower Hupper;
try rewrite rev_involutive;
by apply _Z_from_signed_to_unsigned_bytes_le.
Qed.
Lemma _Z_from_bytes_Unsigned_bound_internal (endianness : endian) :
forall lst,
(0 <= _Z_from_bytes_def endianness Unsigned (rev lst) < 256 ^ (length $ rev lst))%Z.
Proof.
rewrite/_Z_from_bytes_def.
rewrite/_Z_from_bytes_le.
move => lst.
apply _Z_from_bytes_unsigned_le_bounds.
case_match; by rewrite !length_rev.
Qed.
End FromToFacts_internal.
Section ToBytes_external.
Definition _Z_to_bytes_aux : seal (_Z_to_bytes_def). Proof. by eexists. Qed.
Definition _Z_to_bytes := _Z_to_bytes_aux.(unseal).
Definition _Z_to_bytes_eq: _Z_to_bytes = _ := _Z_to_bytes_aux.(seal_eq).
End ToBytes_external.
Section ToBytesFacts_external.
Lemma _Z_to_bytes_length:
forall (cnt: nat) endianness sgn v,
length (_Z_to_bytes cnt endianness sgn v) = cnt.
Proof. move=> *; rewrite _Z_to_bytes_eq; apply _Z_to_bytes_def_length. Qed.
Lemma _Z_to_bytes_lengthN:
forall (cnt: nat) endianness sgn v,
lengthN (_Z_to_bytes cnt endianness sgn v) = N.of_nat cnt.
Proof.
intros.
by rewrite/lengthN _Z_to_bytes_length.
Qed.
Lemma _Z_to_bytes_0_bytes:
forall sgn endianness v,
_Z_to_bytes 0 endianness sgn v = [].
Proof. move=> *; rewrite _Z_to_bytes_eq; apply _Z_to_bytes_def_0_bytes. Qed.
Lemma _Z_to_bytes_0_value:
forall (cnt: nat) endianness sgn,
_Z_to_bytes cnt endianness sgn 0%Z = repeat 0%N cnt.
Proof. move=> *; rewrite _Z_to_bytes_eq; apply _Z_to_bytes_def_0_value. Qed.
Lemma _Z_to_bytes_unsigned_le_range cnt v :
Forall (λ v : N, (v < 256)%N) (_Z_to_bytes_unsigned_le cnt v).
Proof.
rewrite/_Z_to_bytes_unsigned_le/_Z_to_bytes_unsigned_le'.
rewrite Forall_map.
apply: Forall_true.
move => x.
move: (_get_byte_bound v x).
cbn.
lia.
Qed.
Lemma _Z_to_bytes_le_range z cnt sign :
Forall (λ v : N, v < 256)%N (_Z_to_bytes_le cnt sign z).
Proof.
rewrite/_Z_to_bytes_le.
move: sign => []; exact: _Z_to_bytes_unsigned_le_range.
Qed.
Lemma _Z_to_bytes_range z cnt endianness sign:
Forall (λ v : N, v < 256)%N (_Z_to_bytes cnt endianness sign z).
Proof.
rewrite _Z_to_bytes_eq/_Z_to_bytes_def.
move: endianness => [].
{ exact: _Z_to_bytes_le_range. }
apply: Forall_rev.
exact: _Z_to_bytes_le_range.
Qed.
End ToBytesFacts_external.
Section FromBytes_external.
Definition _Z_from_bytes_aux: seal _Z_from_bytes_def. Proof. by eexists. Qed.
Definition _Z_from_bytes := _Z_from_bytes_aux.(unseal).
Definition _Z_from_bytes_eq: @_Z_from_bytes = _ := _Z_from_bytes_aux.(seal_eq).
End FromBytes_external.
Section FromBytesFacts_external.
Lemma _Z_from_bytes_nil:
forall endianness sgn,
_Z_from_bytes endianness sgn [] = 0%Z.
Proof. move=> *; rewrite _Z_from_bytes_eq; apply _Z_from_bytes_def_nil. Qed.
Lemma _Z_from_bytes_0s:
forall endianness sgn (cnt: nat),
_Z_from_bytes endianness sgn (repeat 0%N cnt) = 0%Z.
Proof. move=> *; rewrite _Z_from_bytes_eq; apply _Z_from_bytes_def_0s. Qed.
Lemma _Z_from_bytes_Unsigned_bound (endianness : endian) :
forall lst,
(0 <= _Z_from_bytes endianness Unsigned lst < 256 ^ (length lst))%Z.
Proof.
move => lst.
rewrite -[lst]rev_involutive.
rewrite _Z_from_bytes_eq.
exact: _Z_from_bytes_Unsigned_bound_internal.
Qed.
Lemma _Z_from_bytes_Unsigned_boundN (endianness : endian) :
forall lst,
(0 <= _Z_from_bytes endianness Unsigned lst < 256 ^ (lengthN lst))%Z.
Proof.
move=> *; rewrite/lengthN length_lengthN Nat2N.id nat_N_Z.
exact: _Z_from_bytes_Unsigned_bound.
Qed.
Lemma _Z_from_bytes_singleton endianness sgn a :
_Z_from_bytes endianness sgn [a] =
match sgn with
| Signed => to_signed_bits 8 (_get_byte a 0)
| Unsigned => _get_byte a 0
end.
Proof.
rewrite _Z_from_bytes_eq.
exact: _Z_from_bytes_def_singleton.
Qed.
End FromBytesFacts_external.
Section FromToFacts_external.
Lemma _Z_from_to_bytes_roundtrip:
forall (cnt: nat) (endianness: endian) (sgn: signed) (v: Z),
match sgn with
| Signed => -2^((8*cnt)-1) <= v /\ v <= 2^((8*cnt)-1) - 1
| Unsigned => 0 <= v /\ v < 2^(8*cnt)
end%Z ->
_Z_from_bytes endianness sgn (_Z_to_bytes cnt endianness sgn v) = v.
Proof.
move=> *; rewrite _Z_from_bytes_eq _Z_to_bytes_eq;
by apply _Z_from_to_bytes_def_roundtrip.
Qed.
Lemma _Z_to_from_bytes_roundtrip bytes sgn endianness cnt :
Forall (λ b : N, (b < 256)%N) bytes →
cnt = length bytes →
_Z_to_bytes cnt endianness sgn (_Z_from_bytes endianness sgn bytes) = bytes.
Proof.
move=> *; rewrite _Z_from_bytes_eq _Z_to_bytes_eq;
by apply _Z_to_from_bytes_def_roundtrip.
Qed.
Lemma _Z_from_unsigned_to_signed_bytes:
forall (cnt: nat) (endianness: endian) (v: Z),
(-2^((8*cnt)-1) <= v)%Z ->
(v <= 2^((8*cnt)-1) - 1)%Z ->
_Z_from_bytes endianness Unsigned (_Z_to_bytes cnt endianness Signed v) =
to_unsigned_bits (8*N.of_nat cnt) v.
Proof.
move=> *; rewrite _Z_from_bytes_eq _Z_to_bytes_eq;
by apply _Z_from_unsigned_to_signed_bytes_def.
Qed.
Lemma _Z_from_signed_to_unsigned_bytes:
forall (cnt: nat) (endianness: endian) (v: Z),
(0 <= v)%Z ->
(v < 2^(8*cnt))%Z ->
_Z_from_bytes endianness Signed (_Z_to_bytes cnt endianness Unsigned v) =
to_signed_bits (8*N.of_nat cnt) v.
Proof.
move=> *; rewrite _Z_from_bytes_eq _Z_to_bytes_eq;
by apply _Z_from_signed_to_unsigned_bytes_def.
Qed.
End FromToFacts_external.
End FromToBytes.
* 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.prelude.list_numbers.
Require Import bedrock.lang.cpp.arith.types.
Require Import bedrock.lang.cpp.arith.builtins.
Require Import bedrock.lang.cpp.arith.operator.
Import arith.builtins.churn_bits.
Section FromToBytes.
Section ExtraFacts.
Lemma repeat_cons_app:
forall (A: Type) (a: A) (cnt: nat),
(a :: repeat a cnt) = repeat a cnt ++ [a].
Proof.
induction cnt => //=.
by rewrite IHcnt.
Qed.
Lemma rev_repeat:
forall (A: Type) (a: A) (cnt: nat),
rev (repeat a cnt) = repeat a cnt.
Proof.
induction cnt => //=.
by rewrite IHcnt repeat_cons_app.
Qed.
Lemma Z_shiftr_small:
forall v e,
(0 <= e)%Z ->
(0 <= v)%Z ->
(v < 2 ^ e)%Z ->
(v ≫ e = 0)%Z.
Proof.
intros; rewrite Z.shiftr_div_pow2; try lia.
rewrite Z.div_small; lia.
Qed.
Lemma Z_pow2_trans_nat_l:
forall v (a b: nat),
(v < 2 ^ (8 * b))%Z ->
(v < 2 ^ (8 * (a + b)%nat))%Z.
Proof.
intros; destruct a.
- by replace (8 * (0%nat + b))%Z with (8 * b)%Z by lia.
- eapply Z.lt_trans; eauto; apply Z.pow_lt_mono_r; lia.
Qed.
Lemma Z_pow2_trans_nat_r:
forall v (a b: nat),
(v < 2 ^ (8 * a))%Z ->
(v < 2 ^ (8 * (a + b)%nat))%Z.
Proof.
intros; destruct b.
- by rewrite -plus_n_O.
- eapply Z.lt_trans; eauto; apply Z.pow_lt_mono_r; lia.
Qed.
Lemma Z_land_ldiff_no_overlap (mask offset v: Z) :
(0 < mask)%Z ->
(0 <= offset)%Z ->
(0 <= v)%Z ->
Z.land (mask ≪ offset) (Z.ldiff v (Z.ones offset)) = Z.land (mask ≪ offset) v.
Proof.
(* Intuition: the ldiff is going to remove the lowest
(idx+cnt) bytes, but the `255 ≪ (8 * (idx+cnt))`
mask doesn't overlap with any of those bits
so it is effectively a no-op.
*)
intros **.
apply Z.bits_inj' => n ?.
rewrite !Z.land_spec Z.ldiff_spec Z.shiftl_spec // Z.testbit_ones_nonneg; try lia.
destruct (n <? offset)%Z eqn:Hn => /=; rewrite ?andb_true_r //.
move: Hn => /Z.ltb_lt?.
rewrite !andb_false_r Z.testbit_neg_r //. lia.
Qed.
Lemma Z_land_ldiff_upper_byte (offset v: Z) :
(0 <= offset)%Z ->
(2^(8*offset) <= v)%Z ->
(v < 2^(8*Z.succ offset))%Z ->
Z.ldiff v (Z.ones (8 * offset)) = Z.land (255 ≪ (8 * offset)) v.
Proof.
(* Intuition: since `v < 2^(8*(idx+S cnt))`, we know
that there aren't going to be any bits
beyond the `255 ≪ (8 * (idx+cnt))` mask
which will be introduced by the change to ldiff.
*)
intros.
apply Z.bits_inj' => n ?.
rewrite !Z.land_spec Z.ldiff_spec Z.shiftl_spec // Z.testbit_ones_nonneg; try lia.
destruct (n <? 8*offset)%Z eqn:Hn => //=; rewrite ?andb_true_r ?andb_false_r //.
- apply Z.ltb_lt in Hn; rewrite Z.testbit_neg_r ?andb_false_l //=; lia.
- apply Z.ltb_ge in Hn.
replace (8 * Z.succ offset)%Z with (8 + (8 * offset))%Z in H1 by lia.
change (255)%Z with (Z.ones 8).
destruct (8 + (8 * offset) <? n)%Z eqn:Hn' => //=.
+ rewrite Z.bits_above_log2 ?andb_false_r //.
* apply Z.le_trans with (m := (2^(8*offset))%Z); try apply Z.pow_nonneg; lia.
* apply Z.log2_lt_pow2.
-- eapply Z.lt_le_trans; eauto; apply Z.pow_pos_nonneg; lia.
-- eapply Z.lt_trans; eauto.
apply Z.ltb_lt in Hn'.
apply Z.pow_lt_mono_r; try lia.
+ apply Z.ltb_ge in Hn'.
destruct (8 + (8 * offset) =? n)%Z eqn:Hn'' => //=.
* apply Z.eqb_eq in Hn''; subst.
rewrite Z.bits_above_log2 ?andb_false_l ?andb_false_r //.
-- apply Z.le_trans with (m := (2^(8*offset))%Z);
eauto; apply Z.pow_nonneg; lia.
-- apply Z.log2_lt_pow2; lia.
* apply Z.eqb_neq in Hn''.
assert (n < 8 + 8 * offset)%Z as Hn''' by lia.
rewrite Z.ones_spec_low ?andb_true_l //; lia.
Qed.
Lemma Z_ldiff_split:
forall (cnt idx: nat) (v: Z),
(0 <= v)%Z ->
(v < 2^(8*(idx+S cnt)))%Z ->
Z.ldiff v (Z.ones (8 * idx)) =
Z.lor (Z.land (Z.ones (8 * cnt) ≪ (8 * idx)) v)
(Z.land (Z.ones 8 ≪ (8 * (idx + cnt)%nat)) v).
Proof.
intros cnt idx v Hlower Hupper.
assert (v = 0 \/ 0 < v)%Z as [Hlower' | Hlower'] by lia;
[subst; by rewrite !Z.land_0_r Z.lor_diag Z.ldiff_0_l | clear Hlower].
apply Z.bits_inj'=> n ?.
rewrite Z.lor_spec Z.ldiff_spec !Z.land_spec !Z.shiftl_spec; try lia.
rewrite !Z.testbit_ones; try lia.
churn_bits'.
apply Z.bits_above_log2; try lia.
assert (8 * (idx+S cnt)%nat <= n)%Z by lia.
eapply Z.lt_le_trans; eauto.
apply Z.log2_lt_pow2; try lia.
Qed.
End ExtraFacts.
Section ToBytes_internal.
Definition _Z_to_bytes_unsigned_le' (idx: nat) (cnt: nat) (v: Z): list N :=
map (Z.to_N ∘ _get_byte v) $ seq idx cnt.
Definition _Z_to_bytes_unsigned_le (cnt: nat) (v: Z): list N :=
_Z_to_bytes_unsigned_le' 0 cnt v.
Definition _Z_to_bytes_le (cnt: nat) (sgn: signed) (v: Z): list N :=
_Z_to_bytes_unsigned_le
cnt (match sgn with
| Signed => to_unsigned_bits (8 * N.of_nat cnt) v
| Unsigned => v
end).
(* NOTE: This will be sealed once we finish the proofs for this section. *)
Definition _Z_to_bytes_def (cnt: nat) (endianness: endian) (sgn: signed) (v: Z): list N :=
let little := _Z_to_bytes_le cnt sgn v in
match endianness with
| Little => little
| Big => List.rev little
end.
End ToBytes_internal.
Section ToBytesFacts_internal.
#[local] Transparent _get_byte _set_byte.
Lemma _Z_to_bytes_unsigned_le'_length:
forall idx cnt v,
length (_Z_to_bytes_unsigned_le' idx cnt v) = cnt.
Proof.
rewrite /_Z_to_bytes_unsigned_le' => idx cnt v //=;
by rewrite length_map length_seq.
Qed.
Definition _Z_to_bytes_unsigned_le_length:
forall cnt v,
length (_Z_to_bytes_unsigned_le cnt v) = cnt.
Proof.
rewrite /_Z_to_bytes_unsigned_le => * //=;
by apply _Z_to_bytes_unsigned_le'_length.
Qed.
Lemma _Z_to_bytes_le_length:
forall cnt sgn v,
length (_Z_to_bytes_le cnt sgn v) = cnt.
Proof.
rewrite /_Z_to_bytes_le => * //=;
by apply _Z_to_bytes_unsigned_le_length.
Qed.
Lemma _Z_to_bytes_def_length:
forall cnt endianness sgn v,
length (_Z_to_bytes_def cnt endianness sgn v) = cnt.
Proof.
rewrite /_Z_to_bytes_def => cnt [ | ] sgn v //=;
try rewrite length_rev;
by apply _Z_to_bytes_le_length.
Qed.
Lemma _Z_to_bytes_unsigned_le'_0_bytes:
forall (idx: nat) (v: Z),
_Z_to_bytes_unsigned_le' idx 0 v = [].
Proof. done. Qed.
Lemma _Z_to_bytes_unsigned_le_0_bytes:
forall (v: Z),
_Z_to_bytes_unsigned_le 0 v = [].
Proof. apply (_Z_to_bytes_unsigned_le'_0_bytes 0). Qed.
Lemma _Z_to_bytes_le_0_bytes:
forall sgn (v: Z),
_Z_to_bytes_le 0 sgn v = [].
Proof.
move=> [ | ] v; rewrite /_Z_to_bytes_le;
[rewrite N.mul_0_r trim_0_l | ];
apply _Z_to_bytes_unsigned_le_0_bytes.
Qed.
Lemma _Z_to_bytes_def_0_bytes:
forall endianness sgn (v: Z),
_Z_to_bytes_def 0 endianness sgn v = [].
Proof.
move=> [ | ] sgn v;
by rewrite /_Z_to_bytes_def _Z_to_bytes_le_0_bytes //=.
Qed.
Lemma _Z_to_bytes_unsigned_le'_0_value:
forall (idx cnt: nat),
_Z_to_bytes_unsigned_le' idx cnt 0 = repeat 0%N cnt.
Proof.
induction cnt => //=.
rewrite /_Z_to_bytes_unsigned_le'.
rewrite seq_S map_app repeat_cons_app //=.
rewrite _get_byte_0; f_equal=> //=.
Qed.
Lemma _Z_to_bytes_unsigned_le_0_value:
forall (cnt: nat),
_Z_to_bytes_unsigned_le cnt 0 = repeat 0%N cnt.
Proof. apply _Z_to_bytes_unsigned_le'_0_value. Qed.
Lemma _Z_to_bytes_le_0_value:
forall (cnt: nat) sgn,
_Z_to_bytes_le cnt sgn 0 = repeat 0%N cnt.
Proof.
move=> cnt [ | ]; rewrite /_Z_to_bytes_le;
[rewrite trim_0_r | ];
apply _Z_to_bytes_unsigned_le_0_value.
Qed.
Lemma _Z_to_bytes_def_0_value:
forall sgn endianness (cnt: nat),
_Z_to_bytes_def cnt endianness sgn 0 = repeat 0%N cnt.
Proof.
move=> sgn [ | ] cnt;
rewrite /_Z_to_bytes_def
_Z_to_bytes_le_0_value //=;
by rewrite rev_repeat.
Qed.
Lemma _Z_to_bytes_unsigned_le'_S_cnt:
forall (idx cnt: nat) (v: Z),
_Z_to_bytes_unsigned_le' idx (S cnt) v =
_Z_to_bytes_unsigned_le' idx cnt v ++
_Z_to_bytes_unsigned_le' (idx+cnt) 1 v.
Proof.
intros; generalize dependent idx;
induction cnt; intros=> //=.
- by rewrite Nat.add_0_r.
- rewrite /_Z_to_bytes_unsigned_le'.
rewrite seq_S map_app //=.
Qed.
Lemma _Z_to_bytes_unsigned_le_S_cnt:
forall (cnt: nat) (v: Z),
_Z_to_bytes_unsigned_le (S cnt) v =
_Z_to_bytes_unsigned_le' 0 cnt v ++
_Z_to_bytes_unsigned_le' cnt 1 v.
Proof. apply _Z_to_bytes_unsigned_le'_S_cnt. Qed.
Lemma _Z_to_bytes_unsigned_le'_small:
forall (idx cnt: nat) (v: Z),
(0 <= v)%Z ->
(v < 2^(8*idx))%Z ->
_Z_to_bytes_unsigned_le' idx cnt v =
repeat 0%N cnt.
Proof.
intros; generalize dependent idx;
induction cnt ; intros=> //=.
rewrite _Z_to_bytes_unsigned_le'_S_cnt
/_Z_to_bytes_unsigned_le'
repeat_cons_app //=; f_equal.
- erewrite <- IHcnt; eauto.
- rewrite /_get_byte Z_shiftr_small; try lia;
[ rewrite Z.land_0_l
| apply Z_pow2_trans_nat_r]=> //=.
Qed.
Lemma _Z_to_bytes_unsigned_le'_shrink_cnt:
forall (idx cnt cnt': nat) (v: Z),
(0 <= v)%Z ->
(v < 2^(8*cnt))%Z ->
cnt < cnt' ->
_Z_to_bytes_unsigned_le' idx cnt' v =
_Z_to_bytes_unsigned_le' idx cnt v ++ repeat 0%N (cnt' - cnt).
Proof.
intros; generalize dependent idx; generalize dependent cnt;
induction cnt'; intros=> //=; [lia | ].
replace (S cnt' - cnt) with (S (cnt' - cnt)) by lia; simpl.
assert (cnt = cnt' \/ cnt < cnt') as [Hcnt | Hcnt] by lia.
- subst; rewrite _Z_to_bytes_unsigned_le'_S_cnt; f_equal.
rewrite Nat.sub_diag /_Z_to_bytes_unsigned_le' //=.
rewrite /_get_byte Z_shiftr_small; try lia;
[ rewrite Z.land_0_l
| apply Z_pow2_trans_nat_l]=> //=.
- rewrite repeat_cons_app app_assoc.
rewrite -IHcnt'; try lia.
rewrite _Z_to_bytes_unsigned_le'_S_cnt //=; f_equal.
rewrite /_Z_to_bytes_unsigned_le' //=.
rewrite /_get_byte Z_shiftr_small; try lia;
[ rewrite Z.land_0_l
| eapply Z.lt_trans; eauto; apply Z.pow_lt_mono_r; lia]=> //=.
Qed.
Lemma _Z_to_bytes_unsigned_le_shrink_cnt:
forall (cnt cnt': nat) (v: Z),
(0 <= v)%Z ->
(v < 2^(8*cnt))%Z ->
cnt < cnt' ->
_Z_to_bytes_unsigned_le cnt' v =
_Z_to_bytes_unsigned_le cnt v ++ repeat 0%N (cnt' - cnt).
Proof. apply _Z_to_bytes_unsigned_le'_shrink_cnt. Qed.
Lemma _Z_to_bytes_unsigned_le'_S_idx:
forall (idx cnt: nat) (v: Z),
(0 <= v)%Z ->
_Z_to_bytes_unsigned_le' (S idx) cnt v =
_Z_to_bytes_unsigned_le' idx cnt (v ≫ 8).
Proof.
intros idx cnt v Hlower; generalize dependent idx;
induction cnt; intros=> //=.
rewrite /_Z_to_bytes_unsigned_le' //=; f_equal.
- rewrite /_get_byte Z.shiftr_shiftr; try lia.
by replace (8 + 8 * idx)%Z with (8 * S idx)%Z by lia.
- fold (_Z_to_bytes_unsigned_le' (S (S idx)) cnt v).
fold (_Z_to_bytes_unsigned_le' (S idx) cnt (v ≫ 8)).
assert (v < 2^(8*cnt) \/ 2^(8*cnt) <= v)%Z as [Hv | Hv] by lia.
+ by apply IHcnt.
+ rewrite /_Z_to_bytes_unsigned_le' //=.
rewrite -seq_shift map_map.
generalize (seq (S idx) cnt); induction l=> //=; f_equal.
* rewrite /_get_byte Z.shiftr_shiftr; try lia.
by replace (8 + 8 * a)%Z with (8 * S a)%Z by lia.
* apply IHl.
Qed.
End ToBytesFacts_internal.
Section FromBytes_internal.
Definition _Z_from_bytes_unsigned_le' (idx: nat) (bytes: list N): Z :=
foldr (fun '(idx, byte) acc => Z.lor (_set_byte (Z.of_N byte) idx) acc)
0%Z (zip (seq idx (length bytes)) bytes).
Definition _Z_from_bytes_unsigned_le (bytes: list N): Z :=
_Z_from_bytes_unsigned_le' 0 bytes.
Definition _Z_from_bytes_le (sgn: signed) (bytes: list N): Z :=
let unsigned := _Z_from_bytes_unsigned_le bytes in
match sgn with
| Signed => to_signed_bits (8 * N.of_nat (length bytes)) unsigned
| Unsigned => unsigned
end.
(* NOTE: This will be sealed once we finish the proofs for this section. *)
Definition _Z_from_bytes_def (endianness: endian) (sgn: signed) (bytes: list N): Z :=
_Z_from_bytes_le
sgn match endianness with
| Little => bytes
| Big => List.rev bytes
end.
End FromBytes_internal.
Section FromBytesFacts_internal.
Lemma _Z_from_bytes_unsigned_le'_nil:
forall (idx: nat),
_Z_from_bytes_unsigned_le' idx [] = 0%Z.
Proof. rewrite /_Z_from_bytes_unsigned_le' //=. Qed.
Lemma _Z_from_bytes_unsigned_le_nil:
_Z_from_bytes_unsigned_le [] = 0%Z.
Proof. apply _Z_from_bytes_unsigned_le'_nil. Qed.
Lemma _Z_from_bytes_le_nil:
forall sgn,
_Z_from_bytes_le sgn [] = 0%Z.
Proof.
move=> [ | ];
rewrite /_Z_from_bytes_le
_Z_from_bytes_unsigned_le_nil //=.
Qed.
Lemma _Z_from_bytes_def_nil:
forall endianness sgn,
_Z_from_bytes_def endianness sgn [] = 0%Z.
Proof.
move=> [ | ] sgn;
by rewrite /_Z_from_bytes_def /rev
_Z_from_bytes_le_nil.
Qed.
Lemma _Z_from_bytes_unsigned_le'_cons:
forall (idx: nat) (byte: N) (bytes: list N),
_Z_from_bytes_unsigned_le' idx (byte :: bytes) =
Z.lor (_Z_from_bytes_unsigned_le' idx [byte])
(_Z_from_bytes_unsigned_le' (S idx) bytes).
Proof.
intros; generalize dependent idx; generalize dependent byte;
induction bytes => //=; intros.
- rewrite _Z_from_bytes_unsigned_le'_nil Z.lor_0_r; lia.
- rewrite /_Z_from_bytes_unsigned_le' //=.
by rewrite Z.lor_0_r.
Qed.
Lemma _Z_from_bytes_unsigned_le_cons:
forall (byte: N) (bytes: list N),
_Z_from_bytes_unsigned_le (byte :: bytes) =
Z.lor (_Z_from_bytes_unsigned_le' 0 [byte])
(_Z_from_bytes_unsigned_le' 1 bytes).
Proof. apply _Z_from_bytes_unsigned_le'_cons. Qed.
Lemma _Z_from_bytes_unsigned_le'_app:
forall (idx: nat) (bs1 bs2: list N),
_Z_from_bytes_unsigned_le' idx (bs1 ++ bs2) =
Z.lor (_Z_from_bytes_unsigned_le' idx bs1)
(_Z_from_bytes_unsigned_le' (idx + length bs1) bs2).
Proof.
intros; generalize dependent idx; generalize dependent bs2;
induction bs1 => //=; intros.
- repeat rewrite _Z_from_bytes_unsigned_le'_nil.
replace (idx + 0) with idx by lia.
by rewrite Z.lor_0_l.
- rewrite _Z_from_bytes_unsigned_le'_cons.
rewrite (_Z_from_bytes_unsigned_le'_cons idx a bs1).
rewrite IHbs1.
rewrite Nat.add_succ_r.
by rewrite Z.lor_assoc.
Qed.
Lemma _Z_from_bytes_unsigned_le_app:
forall (bs1 bs2: list N),
_Z_from_bytes_unsigned_le (bs1 ++ bs2) =
Z.lor (_Z_from_bytes_unsigned_le' 0 bs1)
(_Z_from_bytes_unsigned_le' (length bs1) bs2).
Proof. apply _Z_from_bytes_unsigned_le'_app. Qed.
Lemma _Z_from_bytes_unsigned_le'_0s:
forall (idx cnt: nat),
_Z_from_bytes_unsigned_le' idx (repeat 0%N cnt) = 0%Z.
Proof.
intros; generalize dependent idx.
induction cnt => idx //=.
rewrite _Z_from_bytes_unsigned_le'_cons.
rewrite IHcnt.
rewrite /_Z_from_bytes_unsigned_le' //=.
by rewrite _set_byte_0 !Z.lor_0_r.
Qed.
Lemma _Z_from_bytes_unsigned_le_0s:
forall (cnt: nat),
_Z_from_bytes_unsigned_le (repeat 0%N cnt) = 0%Z.
Proof. apply _Z_from_bytes_unsigned_le'_0s. Qed.
Lemma _Z_from_bytes_le_0s:
forall sgn (cnt: nat),
_Z_from_bytes_le sgn (repeat 0%N cnt) = 0%Z.
Proof.
move=> [ | ] cnt; rewrite /_Z_from_bytes_le.
- rewrite repeat_length _Z_from_bytes_unsigned_le_0s.
assert (8 * N.of_nat cnt = 0 \/ 0 < 8 * N.of_nat cnt)%N as [Hcnt | Hcnt] by lia.
+ rewrite /to_signed_bits bool_decide_eq_true_2; by lia.
+ rewrite to_signed_bits_id; intuition; [by reflexivity | ].
apply Z.pow_pos_nonneg; lia.
- apply _Z_from_bytes_unsigned_le_0s.
Qed.
Lemma _Z_from_bytes_def_0s:
forall endianness sgn (cnt: nat),
_Z_from_bytes_def endianness sgn (repeat 0%N cnt) = 0%Z.
Proof.
move=> [ | ] sgn cnt; rewrite /_Z_from_bytes_def ?rev_repeat;
apply _Z_from_bytes_le_0s.
Qed.
Lemma _Z_from_bytes_unsigned_le'_S_idx:
forall (idx: nat) (bytes: list N),
(_Z_from_bytes_unsigned_le' (S idx) (bytes) =
_Z_from_bytes_unsigned_le' idx bytes ≪ 8)%Z.
Proof.
intros idx bytes; generalize dependent idx;
induction bytes; intros=> //=.
rewrite /_Z_from_bytes_unsigned_le' //=.
fold (_Z_from_bytes_unsigned_le' (S (S idx)) bytes).
fold (_Z_from_bytes_unsigned_le' (S idx) bytes).
rewrite Z.shiftl_lor.
by rewrite IHbytes _set_byte_S_idx.
Qed.
#[local] Lemma length_1_inv:
forall {A} (l : list A),
length l = 1%nat ->
exists x0,
l = [x0].
Proof. intros; do 2 (try destruct l)=> //; do 1 eexists; eauto. Qed.
#[local] Lemma length_2_inv:
forall {A} (l : list A),
length l = 2%nat ->
exists x0 x1,
l = [x0; x1].
Proof. intros; do 3 (try destruct l)=> //; do 2 eexists; eauto. Qed.
#[local] Lemma length_4_inv:
forall {A} (l : list A),
length l = 4%nat ->
exists x0 x1 x2 x3,
l = [x0; x1; x2; x3].
Proof. intros; do 5 (try destruct l)=> //; do 4 eexists; eauto. Qed.
#[local] Lemma length_8_inv:
forall {A} (l : list A),
length l = 8%nat ->
exists x0 x1 x2 x3 x4 x5 x6 x7,
l = [x0; x1; x2; x3; x4; x5; x6; x7].
Proof. intros; do 9 (try destruct l)=> //; do 8 eexists; eauto. Qed.
#[local] Lemma length_16_inv:
forall {A} (l : list A),
length l = 16%nat ->
exists x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15,
l = [x0; x1; x2; x3; x4; x5; x6; x7; x8; x9; x10; x11; x12; x13; x14; x15].
Proof. intros; do 17 (try destruct l)=> //; do 16 eexists; eauto. Qed.
Lemma _Z_from_bytes_unsigned_le_bswap:
forall bsz sz (bytes: list N) v,
Datatypes.length bytes = sz ->
bitsize.bytesNat bsz = sz ->
_Z_from_bytes_unsigned_le bytes = v ->
_Z_from_bytes_unsigned_le (rev bytes) = bswap bsz v.
Proof.
rewrite /_Z_from_bytes_unsigned_le/_Z_from_bytes_unsigned_le';
move=> bsz sz bytes v Hlen Hsz Hdecodes; destruct bsz;
simpl in *; rewrite -Hsz in Hlen; subst;
[ apply length_1_inv in Hlen as [? Hbytes]
| apply length_2_inv in Hlen as [? [? Hbytes]]
| apply length_4_inv in Hlen as [? [? [? [? Hbytes]]]]
| apply length_8_inv in Hlen as [? [? [? [? [? [? [? [? Hbytes]]]]]]]]
| apply length_16_inv in Hlen as [? [? [? [? [? [? [? [?
[? [? [? [? [? [? [? [? Hbytes]]]]]]]]]]]]]]]]];
rewrite Hbytes //= !Z.lor_0_r.
- now rewrite bswap8_set_byte_reverse.
- now rewrite bswap16_set_byte_reverse.
- now rewrite bswap32_set_byte_reverse.
- now rewrite bswap64_set_byte_reverse.
- now rewrite bswap128_set_byte_reverse.
Qed.
Lemma _Z_from_bytes_unsigned_le'_singleton idx a :
_Z_from_bytes_unsigned_le' idx [a] = (_set_byte (Z.of_N a) idx).
Proof.
rewrite/_Z_from_bytes_unsigned_le'.
cbn.
by rewrite Z.lor_0_r.
Qed.
Lemma _Z_from_bytes_unsigned_le_singleton a :
_Z_from_bytes_unsigned_le [a] = (_get_byte (Z.of_N a) 0).
Proof.
by rewrite /_Z_from_bytes_unsigned_le _Z_from_bytes_unsigned_le'_singleton _get_0_set_0_eq.
Qed.
Lemma _Z_from_bytes_le_singleton sgn a :
_Z_from_bytes_le sgn [a] =
match sgn with
| Signed => to_signed_bits 8 (_get_byte a 0)
| Unsigned => _get_byte a 0
end.
Proof.
rewrite /_Z_from_bytes_le _Z_from_bytes_unsigned_le_singleton.
by rewrite /= N.mul_1_r.
Qed.
Lemma _Z_from_bytes_def_singleton endianness sgn a :
_Z_from_bytes_def endianness sgn [a] =
match sgn with
| Signed => to_signed_bits 8 (_get_byte a 0)
| Unsigned => _get_byte a 0
end.
Proof.
rewrite /_Z_from_bytes_def /=.
by destruct endianness; rewrite _Z_from_bytes_le_singleton.
Qed.
Lemma _Z_from_bytes_unsigned_le'_bounds bytes idx b :
(256 ^ (length bytes + Z.of_nat idx)%Z <= b)%Z ->
(0 ≤ _Z_from_bytes_unsigned_le' idx bytes < b)%Z.
Proof.
move: b idx.
elim bytes.
{
move => ? ?.
rewrite Z.add_0_l.
move => ?.
split; [by []|].
cbn.
eapply Z.lt_le_trans.
2: { eassumption. }
apply Z.pow_pos_nonneg; [ by[] | ].
exact: Zle_0_nat.
}
move => a l IH b idx /= H.
rewrite _Z_from_bytes_unsigned_le'_cons.
rewrite _Z_from_bytes_unsigned_le'_S_idx.
assert (
(0
≤ Z.lor (_Z_from_bytes_unsigned_le' idx [a]) (_Z_from_bytes_unsigned_le' idx l ≪ 8) <
2 ^ (Z.of_N $ N.of_nat (8 * (S (length l) + idx))))%Z
) as H1.
{
apply ZlorRange.
{
rewrite _Z_from_bytes_unsigned_le'_singleton.
move: (_set_byte_bound (Z.of_N a) idx) => [L0 L1].
split; first assumption.
eapply Z.lt_le_trans; first eassumption.
apply Z.pow_le_mono_r; first done.
lia.
}
move: (IH (2 ^ Z.of_N (N.of_nat (8 * (S (length l) + idx))))%Z (S idx)).
rewrite _Z_from_bytes_unsigned_le'_S_idx.
apply.
rewrite nat_N_Z.
rewrite Nat2Z.inj_mul.
rewrite Z.pow_mul_r //.
apply: Z.pow_le_mono.
1: done.
lia.
}
split; first tauto.
eapply Z.lt_le_trans.
{
move: H1 => [_ max].
eassumption.
}
rewrite nat_N_Z.
rewrite Nat2Z.inj_mul.
rewrite Z.pow_mul_r //.
cbn.
eapply Z.le_trans.
2: eassumption.
apply: Z.pow_le_mono.
1: done.
lia.
Qed.
Lemma _Z_from_bytes_unsigned_le'_bound bytes idx b :
(256 ^ (length bytes + Z.of_nat idx)%Z - 1 <= b)%Z ->
(0 ≤ _Z_from_bytes_unsigned_le' idx bytes ≤ b)%Z.
Proof.
pose proof (_Z_from_bytes_unsigned_le'_bounds bytes idx
(256 ^ (length bytes + Z.of_nat idx)%Z)%Z
ltac:(lia))
as Hbound'.
lia.
Qed.
Lemma _Z_from_bytes_unsigned_le_bounds bytes b :
(256 ^ length bytes <= b)%Z ->
(0 <= _Z_from_bytes_unsigned_le bytes < b)%Z.
Proof.
move => H.
apply _Z_from_bytes_unsigned_le'_bounds.
by rewrite Z.add_0_r.
Qed.
Lemma _Z_from_bytes_unsigned_le_bound bytes b :
(256 ^ length bytes - 1 <= b)%Z ->
(0 <= _Z_from_bytes_unsigned_le bytes <= b)%Z.
Proof.
pose proof (_Z_from_bytes_unsigned_le_bounds bytes
(256 ^ (length bytes))
ltac:(lia))
as Hbound'.
lia.
Qed.
End FromBytesFacts_internal.
Section FromToFacts_internal.
#[local] Transparent _get_byte _set_byte.
Lemma _Z_from_to_bytes_unsigned_le'_overflow:
forall (idx cnt: nat) (v: Z),
(2^(8*(idx+cnt)) <= v)%Z ->
_Z_from_bytes_unsigned_le' idx (_Z_to_bytes_unsigned_le' idx cnt v) =
Z.land ((Z.ones (8*cnt)) ≪ (8*idx)) v.
Proof.
intros idx cnt v Hlower;
generalize dependent idx;
generalize dependent v;
induction cnt; intros=> //=.
- rewrite /Z.ones Z.mul_0_r Z.shiftl_0_r
_Z_to_bytes_unsigned_le'_0_bytes
_Z_from_bytes_unsigned_le'_nil.
by rewrite Z.shiftl_0_l Z.land_0_l.
- rewrite _Z_to_bytes_unsigned_le'_S_cnt
_Z_from_bytes_unsigned_le'_app
_Z_to_bytes_unsigned_le'_length.
rewrite [_Z_to_bytes_unsigned_le' _ 1 _]/_Z_to_bytes_unsigned_le' //=.
rewrite [_Z_from_bytes_unsigned_le' _ [_]]/_Z_from_bytes_unsigned_le' //=.
rewrite Z.lor_0_r.
rewrite Z2N.id; [ | apply _get_byte_nonneg].
rewrite _set_get_byte_roundtrip/_get_byte.
assert (2^(8*(idx+cnt)) <= v)%Z as Hlower'. {
etrans; last eapply Hlower;
apply Z.pow_le_mono_r; lia.
}
specialize (IHcnt v idx Hlower'); rewrite IHcnt //=.
apply Z.bits_inj'=> n ?.
repeat (rewrite ?Z.lor_spec ?Z.land_spec; try lia).
rewrite Z.shiftl_land -Z.ldiff_ones_r; try lia.
repeat (rewrite ?Z.lor_spec ?Z.land_spec ?Z.shiftl_spec ?Z.ldiff_spec; try lia).
rewrite !Z.testbit_ones; try lia.
churn_bits'.
Qed.
Lemma _Z_from_to_bytes_unsigned_le'_roundtrip:
forall (idx cnt: nat) (v: Z),
(0 <= v)%Z ->
(v < 2^(8*(idx+cnt)))%Z ->
_Z_from_bytes_unsigned_le' idx (_Z_to_bytes_unsigned_le' idx cnt v) =
(v - v `mod` 2^(8*idx))%Z.
Proof.
intros idx cnt v Hlower Hupper;
generalize dependent idx;
generalize dependent v;
induction cnt; intros=> //=.
- rewrite Z.add_0_r in Hupper.
rewrite _Z_to_bytes_unsigned_le'_0_bytes
_Z_from_bytes_unsigned_le'_nil.
rewrite Zmod_small; try intuition; lia.
- assert (v < 2^(8*(idx+cnt)) \/ 2^(8*(idx+cnt)) <= v)%Z as [Hv | Hv] by lia.
+ rewrite _Z_to_bytes_unsigned_le'_S_cnt
_Z_from_bytes_unsigned_le'_app
_Z_to_bytes_unsigned_le'_length
IHcnt; try lia.
rewrite _Z_to_bytes_unsigned_le'_small; try lia.
1: by rewrite _Z_from_bytes_unsigned_le'_0s Z.lor_0_r.
+ clear IHcnt.
rewrite Zmod_eq_full.
2: pose proof (Z.pow_pos_nonneg
2 (8 * idx)
ltac:(lia) ltac:(lia)); lia.
rewrite -Z.shiftr_div_pow2; try lia.
rewrite -Z.shiftl_mul_pow2; try lia.
rewrite -Z.ldiff_ones_r; try lia.
rewrite _Z_to_bytes_unsigned_le'_S_cnt
_Z_from_bytes_unsigned_le'_app
_Z_to_bytes_unsigned_le'_length.
rewrite Z.sub_sub_distr Z.sub_diag Z.add_0_l.
rewrite [_Z_to_bytes_unsigned_le' _ 1 _]/_Z_to_bytes_unsigned_le' //=.
rewrite [_Z_from_bytes_unsigned_le' _ [_]]/_Z_from_bytes_unsigned_le' //=.
rewrite Z.lor_0_r.
rewrite Z2N.id; [ | apply _get_byte_nonneg].
rewrite _set_get_byte_roundtrip/_get_byte Z.shiftl_land.
rewrite _Z_from_to_bytes_unsigned_le'_overflow; try lia.
erewrite Z_ldiff_split; eauto; f_equal.
rewrite -Z.ldiff_ones_r; try lia.
rewrite Z.land_comm.
apply Z.bits_inj'=> n ?.
repeat (rewrite ?Z.lor_spec ?Z.land_spec ?Z.shiftl_spec ?Z.ldiff_spec; try lia).
rewrite !Z.testbit_ones; try lia.
churn_bits'.
Qed.
Lemma _Z_from_to_bytes_unsigned_le_roundtrip:
forall (cnt: nat) (v: Z),
(0 <= v)%Z ->
(v < 2^(8*cnt))%Z ->
_Z_from_bytes_unsigned_le (_Z_to_bytes_unsigned_le cnt v) = v.
Proof.
intros cnt v Hlower Hupper.
pose proof (_Z_from_to_bytes_unsigned_le'_roundtrip 0 cnt v Hlower) as Hpf.
rewrite Z.add_0_l in Hpf.
specialize (Hpf Hupper).
rewrite /_Z_from_bytes_unsigned_le
/_Z_to_bytes_unsigned_le Hpf.
rewrite Z.mul_0_r Z.pow_0_r Zmod_1_r; lia.
Qed.
Lemma _get_set_later_bytes_irrelevant idx start lst :
(idx < start)%nat ->
_get_byte
(foldr
(λ '(idx0, byte) (acc : Z),
(_set_byte (Z.of_N byte) idx0 `lor` acc)%Z) 0%Z
(zip (seq start (length lst)) lst)) idx = 0%Z.
Proof.
move: idx start.
induction lst.
{
move => ? ? ?.
by rewrite _get_byte_0.
}
move => ? ? ?.
cbn.
rewrite _get_byte_lor.
rewrite _get_set_byte_no_overlap.
2: lia.
rewrite Z.lor_0_l.
apply: IHlst.
lia.
Qed.
Lemma _get_byte_0_small_id (z : Z) :
_get_byte z 0 = Z.modulo z 256.
Proof. by rewrite/_get_byte /Z.of_nat Z.mul_0_r Z.shiftr_0_r Z.land_ones. Qed.
Lemma _Z_to_from_bytes_unsigned_le'_roundtrip (bytes : list N) (idx cnt : nat) :
Forall (λ b : N, (b < 256)%N) bytes →
cnt = length bytes →
_Z_to_bytes_unsigned_le' idx cnt (_Z_from_bytes_unsigned_le' idx bytes) = bytes.
Proof.
move: idx bytes.
induction cnt.
{ by move => ? []. }
move => idx [].
{ done. }
move => hd tl smallH.
apply Forall_cons_1 in smallH.
move: smallH => [small_hdH small_tlH].
move => [] cnt_eq.
rewrite/_Z_to_bytes_unsigned_le'.
cbn.
f_equal.
{
rewrite _get_byte_lor _get_set_later_bytes_irrelevant.
2: lia.
rewrite Z.lor_0_r _get_set_byte_roundtrip _get_byte_0_small_id Z2N.inj_mod.
2: lia.
2: lia.
by rewrite N2Z.id N.mod_small.
}
move: (IHcnt (S idx) tl small_tlH cnt_eq).
rewrite /_Z_to_bytes_unsigned_le' /_Z_from_bytes_unsigned_le'.
move => IH.
rewrite -{3}IH.
apply: map_ext_Forall.
rewrite Forall_seq.
move => ? ?.
cbn.
f_equal.
rewrite _get_byte_lor _get_set_byte_no_overlap.
2: lia.
by rewrite Z.lor_0_l.
Qed.
Lemma _Z_to_from_bytes_unsigned_le_roundtrip (bytes : list N) (cnt : nat) :
Forall (fun b => b < 256)%N bytes ->
cnt = length bytes ->
(_Z_to_bytes_unsigned_le cnt $ _Z_from_bytes_unsigned_le bytes) = bytes.
Proof.
move => ? ?.
by rewrite/_Z_to_bytes_unsigned_le/_Z_from_bytes_unsigned_le
_Z_to_from_bytes_unsigned_le'_roundtrip.
Qed.
Lemma _Z_from_to_bytes_le_roundtrip:
forall (cnt: nat) (sgn: signed) (v: Z),
match sgn with
| Signed => -2^((8*cnt)-1) <= v /\ v <= 2^((8*cnt)-1) - 1
| Unsigned => 0 <= v /\ v < 2^(8*cnt)
end%Z ->
_Z_from_bytes_le sgn (_Z_to_bytes_le cnt sgn v) = v.
Proof.
destruct sgn; intros v [Hlower Hupper];
[ | by apply _Z_from_to_bytes_unsigned_le_roundtrip].
rewrite /_Z_from_bytes_le /_Z_to_bytes_le _Z_to_bytes_unsigned_le_length.
assert (v < 0 \/ 0 <= v)%Z as [Hv | Hv] by lia.
- rewrite _Z_from_to_bytes_unsigned_le_roundtrip.
+ apply to_signed_unsigned_bits_roundtrip; intuition;
replace (Z.of_N (8 * N.of_nat cnt) - 1)%Z
with (8 * cnt - 1)%Z; lia.
+ rewrite /trim; apply Z.mod_pos;
apply Z.pow_pos_nonneg; lia.
+ rewrite /trim.
replace (Z.of_N (8 * N.of_nat cnt))
with (8 * cnt)%Z by lia.
by pose proof (Z_mod_lt v (2^(8*cnt))%Z
ltac:(apply Z.lt_gt; apply Z.pow_pos_nonneg; lia))
as [? ?].
- rewrite /trim Zmod_small; intuition; try lia.
+ rewrite _Z_from_to_bytes_unsigned_le_roundtrip; try lia.
* apply to_signed_bits_id; intuition.
eapply Z.le_lt_trans; eauto.
replace (Z.of_N (8 * N.of_nat cnt) - 1)%Z
with (8 * cnt - 1)%Z; lia.
* eapply Z.le_lt_trans; eauto.
match goal with
| |- (_ < ?r)%Z => replace r with (r - 0)%Z by lia
end.
apply Z.sub_lt_le_mono; try apply Z.pow_lt_mono_r; lia.
+ eapply Z.le_lt_trans; eauto.
match goal with
| |- (_ < ?r)%Z => replace r with (r - 0)%Z by lia
end.
apply Z.sub_lt_le_mono; try apply Z.pow_lt_mono_r; lia.
Qed.
Lemma _Z_from_unsigned_to_signed_bytes_le:
forall (cnt: nat) (v: Z),
(-2^((8*cnt)-1) <= v)%Z ->
(v <= 2^((8*cnt)-1) - 1)%Z ->
_Z_from_bytes_le Unsigned (_Z_to_bytes_le cnt Signed v) =
to_unsigned_bits (8*N.of_nat cnt) v.
Proof.
move=> cnt v Hlower Hupper.
rewrite /trim /_Z_from_bytes_le /_Z_to_bytes_le.
rewrite _Z_from_to_bytes_unsigned_le_roundtrip /trim //.
- apply Z_mod_lt; rewrite Z.gt_lt_iff;
apply Z.pow_pos_nonneg; lia.
- replace (Z.of_N (8 * N.of_nat cnt)) with (8 * cnt)%Z by lia;
apply Z_mod_lt; rewrite Z.gt_lt_iff;
apply Z.pow_pos_nonneg; lia.
Qed.
Lemma _Z_from_signed_to_unsigned_bytes_le:
forall (cnt: nat) (v: Z),
(0 <= v)%Z ->
(v < 2^(8*cnt))%Z ->
_Z_from_bytes_le Signed (_Z_to_bytes_le cnt Unsigned v) =
to_signed_bits (8*N.of_nat cnt) v.
Proof.
move=> cnt v Hlower Hupper.
rewrite /_Z_from_bytes_le /_Z_to_bytes_le
_Z_to_bytes_unsigned_le_length
_Z_from_to_bytes_unsigned_le_roundtrip //.
Qed.
Lemma _Z_from_to_bytes_def_roundtrip:
forall (cnt: nat) (endianness: endian) (sgn: signed) (v: Z),
match sgn with
| Signed => -2^((8*cnt)-1) <= v /\ v <= 2^((8*cnt)-1) - 1
| Unsigned => 0 <= v /\ v < 2^(8*cnt)
end%Z ->
_Z_from_bytes_def endianness sgn (_Z_to_bytes_def cnt endianness sgn v) = v.
Proof.
rewrite /_Z_from_bytes_def /_Z_to_bytes_def;
intros cnt [ | ] sgn v H;
try rewrite rev_involutive;
by apply _Z_from_to_bytes_le_roundtrip.
Qed.
Lemma _Z_to_from_bytes_le_round_trip sgn cnt bytes :
Forall (λ b : N, (b < 256)%N) bytes ->
cnt = length bytes ->
_Z_to_bytes_le cnt sgn (_Z_from_bytes_le sgn bytes) = bytes.
Proof.
move => small cntH.
move: sgn => [].
2: {
cbn.
rewrite/_Z_to_bytes_le.
by apply: _Z_to_from_bytes_unsigned_le_roundtrip.
}
rewrite/_Z_to_bytes_le.
cbn.
rewrite !cntH.
rewrite trim_to_signed_bits_agree.
rewrite to_unsigned_bits_id.
{ by apply: _Z_to_from_bytes_unsigned_le_roundtrip. }
apply: _Z_from_bytes_unsigned_le_bounds.
rewrite -(Z.pow_mul_r 2 8).
2: done.
2: lia.
apply: Z.eq_le_incl.
f_equal.
rewrite -nat_N_Z.
case (N.of_nat (length bytes)).
{ lia. }
move => ?.
lia.
Qed.
Lemma _Z_to_from_bytes_def_roundtrip bytes sgn endianness cnt :
Forall (λ b : N, (b < 256)%N) bytes →
cnt = length bytes →
_Z_to_bytes_def cnt endianness sgn (_Z_from_bytes_def endianness sgn bytes) = bytes.
Proof.
move => small cntH.
rewrite/_Z_from_bytes_def/_Z_to_bytes_def.
move: endianness => [].
{ exact: _Z_to_from_bytes_le_round_trip. }
rewrite _Z_to_from_bytes_le_round_trip.
{ exact: rev_involutive. }
{ exact: Forall_rev. }
by rewrite length_rev.
Qed.
Lemma _Z_from_unsigned_to_signed_bytes_def:
forall (cnt: nat) (endianness: endian) (v: Z),
(-2^((8*cnt)-1) <= v)%Z ->
(v <= 2^((8*cnt)-1) - 1)%Z ->
_Z_from_bytes_def endianness Unsigned (_Z_to_bytes_def cnt endianness Signed v) =
to_unsigned_bits (8*N.of_nat cnt) v.
Proof.
rewrite /_Z_from_bytes_def /_Z_to_bytes_def;
intros cnt [ | ] v Hlower Hupper;
try rewrite rev_involutive;
by apply _Z_from_unsigned_to_signed_bytes_le.
Qed.
Lemma _Z_from_signed_to_unsigned_bytes_def:
forall (cnt: nat) (endianness: endian) (v: Z),
(0 <= v)%Z ->
(v < 2^(8*cnt))%Z ->
_Z_from_bytes_def endianness Signed (_Z_to_bytes_def cnt endianness Unsigned v) =
to_signed_bits (8*N.of_nat cnt) v.
Proof.
rewrite /_Z_from_bytes_def /_Z_to_bytes_def;
intros cnt [ | ] v Hlower Hupper;
try rewrite rev_involutive;
by apply _Z_from_signed_to_unsigned_bytes_le.
Qed.
Lemma _Z_from_bytes_Unsigned_bound_internal (endianness : endian) :
forall lst,
(0 <= _Z_from_bytes_def endianness Unsigned (rev lst) < 256 ^ (length $ rev lst))%Z.
Proof.
rewrite/_Z_from_bytes_def.
rewrite/_Z_from_bytes_le.
move => lst.
apply _Z_from_bytes_unsigned_le_bounds.
case_match; by rewrite !length_rev.
Qed.
End FromToFacts_internal.
Section ToBytes_external.
Definition _Z_to_bytes_aux : seal (_Z_to_bytes_def). Proof. by eexists. Qed.
Definition _Z_to_bytes := _Z_to_bytes_aux.(unseal).
Definition _Z_to_bytes_eq: _Z_to_bytes = _ := _Z_to_bytes_aux.(seal_eq).
End ToBytes_external.
Section ToBytesFacts_external.
Lemma _Z_to_bytes_length:
forall (cnt: nat) endianness sgn v,
length (_Z_to_bytes cnt endianness sgn v) = cnt.
Proof. move=> *; rewrite _Z_to_bytes_eq; apply _Z_to_bytes_def_length. Qed.
Lemma _Z_to_bytes_lengthN:
forall (cnt: nat) endianness sgn v,
lengthN (_Z_to_bytes cnt endianness sgn v) = N.of_nat cnt.
Proof.
intros.
by rewrite/lengthN _Z_to_bytes_length.
Qed.
Lemma _Z_to_bytes_0_bytes:
forall sgn endianness v,
_Z_to_bytes 0 endianness sgn v = [].
Proof. move=> *; rewrite _Z_to_bytes_eq; apply _Z_to_bytes_def_0_bytes. Qed.
Lemma _Z_to_bytes_0_value:
forall (cnt: nat) endianness sgn,
_Z_to_bytes cnt endianness sgn 0%Z = repeat 0%N cnt.
Proof. move=> *; rewrite _Z_to_bytes_eq; apply _Z_to_bytes_def_0_value. Qed.
Lemma _Z_to_bytes_unsigned_le_range cnt v :
Forall (λ v : N, (v < 256)%N) (_Z_to_bytes_unsigned_le cnt v).
Proof.
rewrite/_Z_to_bytes_unsigned_le/_Z_to_bytes_unsigned_le'.
rewrite Forall_map.
apply: Forall_true.
move => x.
move: (_get_byte_bound v x).
cbn.
lia.
Qed.
Lemma _Z_to_bytes_le_range z cnt sign :
Forall (λ v : N, v < 256)%N (_Z_to_bytes_le cnt sign z).
Proof.
rewrite/_Z_to_bytes_le.
move: sign => []; exact: _Z_to_bytes_unsigned_le_range.
Qed.
Lemma _Z_to_bytes_range z cnt endianness sign:
Forall (λ v : N, v < 256)%N (_Z_to_bytes cnt endianness sign z).
Proof.
rewrite _Z_to_bytes_eq/_Z_to_bytes_def.
move: endianness => [].
{ exact: _Z_to_bytes_le_range. }
apply: Forall_rev.
exact: _Z_to_bytes_le_range.
Qed.
End ToBytesFacts_external.
Section FromBytes_external.
Definition _Z_from_bytes_aux: seal _Z_from_bytes_def. Proof. by eexists. Qed.
Definition _Z_from_bytes := _Z_from_bytes_aux.(unseal).
Definition _Z_from_bytes_eq: @_Z_from_bytes = _ := _Z_from_bytes_aux.(seal_eq).
End FromBytes_external.
Section FromBytesFacts_external.
Lemma _Z_from_bytes_nil:
forall endianness sgn,
_Z_from_bytes endianness sgn [] = 0%Z.
Proof. move=> *; rewrite _Z_from_bytes_eq; apply _Z_from_bytes_def_nil. Qed.
Lemma _Z_from_bytes_0s:
forall endianness sgn (cnt: nat),
_Z_from_bytes endianness sgn (repeat 0%N cnt) = 0%Z.
Proof. move=> *; rewrite _Z_from_bytes_eq; apply _Z_from_bytes_def_0s. Qed.
Lemma _Z_from_bytes_Unsigned_bound (endianness : endian) :
forall lst,
(0 <= _Z_from_bytes endianness Unsigned lst < 256 ^ (length lst))%Z.
Proof.
move => lst.
rewrite -[lst]rev_involutive.
rewrite _Z_from_bytes_eq.
exact: _Z_from_bytes_Unsigned_bound_internal.
Qed.
Lemma _Z_from_bytes_Unsigned_boundN (endianness : endian) :
forall lst,
(0 <= _Z_from_bytes endianness Unsigned lst < 256 ^ (lengthN lst))%Z.
Proof.
move=> *; rewrite/lengthN length_lengthN Nat2N.id nat_N_Z.
exact: _Z_from_bytes_Unsigned_bound.
Qed.
Lemma _Z_from_bytes_singleton endianness sgn a :
_Z_from_bytes endianness sgn [a] =
match sgn with
| Signed => to_signed_bits 8 (_get_byte a 0)
| Unsigned => _get_byte a 0
end.
Proof.
rewrite _Z_from_bytes_eq.
exact: _Z_from_bytes_def_singleton.
Qed.
End FromBytesFacts_external.
Section FromToFacts_external.
Lemma _Z_from_to_bytes_roundtrip:
forall (cnt: nat) (endianness: endian) (sgn: signed) (v: Z),
match sgn with
| Signed => -2^((8*cnt)-1) <= v /\ v <= 2^((8*cnt)-1) - 1
| Unsigned => 0 <= v /\ v < 2^(8*cnt)
end%Z ->
_Z_from_bytes endianness sgn (_Z_to_bytes cnt endianness sgn v) = v.
Proof.
move=> *; rewrite _Z_from_bytes_eq _Z_to_bytes_eq;
by apply _Z_from_to_bytes_def_roundtrip.
Qed.
Lemma _Z_to_from_bytes_roundtrip bytes sgn endianness cnt :
Forall (λ b : N, (b < 256)%N) bytes →
cnt = length bytes →
_Z_to_bytes cnt endianness sgn (_Z_from_bytes endianness sgn bytes) = bytes.
Proof.
move=> *; rewrite _Z_from_bytes_eq _Z_to_bytes_eq;
by apply _Z_to_from_bytes_def_roundtrip.
Qed.
Lemma _Z_from_unsigned_to_signed_bytes:
forall (cnt: nat) (endianness: endian) (v: Z),
(-2^((8*cnt)-1) <= v)%Z ->
(v <= 2^((8*cnt)-1) - 1)%Z ->
_Z_from_bytes endianness Unsigned (_Z_to_bytes cnt endianness Signed v) =
to_unsigned_bits (8*N.of_nat cnt) v.
Proof.
move=> *; rewrite _Z_from_bytes_eq _Z_to_bytes_eq;
by apply _Z_from_unsigned_to_signed_bytes_def.
Qed.
Lemma _Z_from_signed_to_unsigned_bytes:
forall (cnt: nat) (endianness: endian) (v: Z),
(0 <= v)%Z ->
(v < 2^(8*cnt))%Z ->
_Z_from_bytes endianness Signed (_Z_to_bytes cnt endianness Unsigned v) =
to_signed_bits (8*N.of_nat cnt) v.
Proof.
move=> *; rewrite _Z_from_bytes_eq _Z_to_bytes_eq;
by apply _Z_from_signed_to_unsigned_bytes_def.
Qed.
End FromToFacts_external.
End FromToBytes.