ZYM/MiniGmp-Verification
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feat: rewrite certain definitions of internal structures to make proof easier.

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xiaoh105
2025-05-30 00:41:29 +08:00
parent 47ca3e72bd
commit f6bb7e9a66
2 changed files with 269 additions and 160 deletions
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projects/lib/gmp_aux.v Normal file → Executable file
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@ -35,6 +35,30 @@ Proof.
rewrite (Z.pow_add_r a b 1); lia.
Qed.
Lemma Zmul_mod_cancel: forall (n a b: Z),
n >= 0 -> a > 0 -> b >= 0 ->
(n * a) mod (a ^ (b + 1)) = a * (n mod (a ^ b)).
Proof.
intros.
pose proof (Z_div_mod_eq_full n (a ^ b)).
pose proof (Z.mod_bound_pos n (a ^ b) ltac:(lia) ltac:(nia)).
remember (n / a ^ b) as q eqn:Hq.
remember (n mod a ^ b) as rem eqn:Hrem.
rewrite H2.
rewrite Z.mul_add_distr_r.
rewrite (Z.mul_comm (a ^ b) q); rewrite <-Z.mul_assoc.
rewrite <-Zpow_add_1; try lia.
assert (0 <= rem * a < a ^ (b + 1)). {
rewrite Zpow_add_1; try lia.
nia.
}
rewrite <-(Zmod_unique_full (q * a ^ (b + 1) + rem * a) (a ^ (b + 1)) q (rem * a)).
+ lia.
+ unfold Remainder.
lia.
+ lia.
Qed.
Lemma Zdiv_mod_pow: forall (n a b: Z),
a > 0 -> b >= 0 -> n >= 0 ->
(n / a) mod (a ^ b) = (n mod (a ^ (b + 1))) / a.
@ -62,6 +86,22 @@ Proof.
exists q.
lia.
Qed.
Lemma list_app_cons: forall (l1 l2: list Z) (a: Z),
app l1 (a :: l2) = app (app l1 (a :: nil)) l2.
Proof.
intros.
revert a l2.
induction l1.
+ intros.
rewrite app_nil_l.
reflexivity.
+ intros.
simpl in *.
specialize (IHl1 a0 l2).
rewrite IHl1.
reflexivity.
Qed.
End Aux.
@ -72,194 +112,263 @@ Definition mpd_store_list (ptr: addr) (data: list Z) (cap: Z): Assertion :=
store_uint_array ptr (Zlength data) data &&
store_undef_uint_array_rec ptr ((Zlength data) + 1) cap.
Fixpoint list_store_Z (data: list Z) (n: Z): Assertion :=
Fixpoint list_to_Z (data: list Z): Z :=
match data with
| nil => [| n = 0 |]
| a :: l0 => [| (n mod UINT_MOD) = a |] && list_store_Z l0 (n / UINT_MOD)
| nil => 0
| a :: l0 => (list_to_Z l0) * UINT_MOD + a
end.
Fixpoint list_within_bound (data: list Z): Prop :=
match data with
| nil => True
| a :: l0 => 0 <= a < UINT_MOD /\ (list_within_bound l0)
end.
Definition list_store_Z (data: list Z) (n: Z): Prop :=
list_to_Z data = n /\ list_within_bound data.
Definition mpd_store_Z (ptr: addr) (n: Z) (size: Z) (cap: Z): Assertion :=
EX data,
mpd_store_list ptr data cap && list_store_Z data n && [| size = Zlength data |].
mpd_store_list ptr data cap && [| list_store_Z data n|] && [| size = Zlength data |].
Fixpoint list_store_Z_pref (data: list Z) (n: Z) (len: nat): Assertion :=
match len with
| O => [| n = 0 |]
| S len' =>
EX a l0,
[| data = a :: l0 |] && [| (n mod UINT_MOD) = a |] && list_store_Z_pref l0 (n / UINT_MOD) len'
end.
Definition mpd_store_Z_pref (ptr: addr) (n: Z) (size: Z) (cap: Z) (len: nat): Assertion :=
EX data,
mpd_store_list ptr data cap && list_store_Z_pref data n len && [| size = Zlength data |].
Lemma list_store_Z_split: forall (data: list Z) (n: Z) (len: nat),
n >= 0 ->
Z.of_nat len < Zlength data ->
list_store_Z data n |--
list_store_Z_pref data (n mod (UINT_MOD ^ Z.of_nat len)) len.
Lemma __list_within_bound_concat_r: forall (l1: list Z) (a: Z),
list_within_bound l1 ->
0 <= a < UINT_MOD ->
list_within_bound (l1 ++ [a]).
Proof.
intros.
revert n H data H0.
induction len.
+ intros.
induction l1.
+ rewrite app_nil_l.
simpl.
entailer!.
apply Z.mod_1_r.
+ intros.
assert (Zlength data >= 1); [ lia | ].
destruct data; [ unfold Zlength, Zlength_aux in H1; lia | ].
simpl.
Exists z data.
entailer!.
sep_apply IHlen; try tauto.
- pose proof (Aux.Zdiv_mod_pow n UINT_MOD (Z.of_nat len) ltac:(lia) ltac:(lia) ltac:(lia)).
rewrite H5.
pose proof (Aux.Z_of_nat_succ len).
rewrite <-H6.
reflexivity.
- pose proof (Aux.Z_of_nat_succ len).
pose proof (Zlength_cons z data).
lia.
- pose proof (Z.div_pos n UINT_MOD ltac:(lia) ltac:(lia)).
lia.
- rewrite <-H2.
pose proof (Znumtheory.Zmod_div_mod UINT_MOD (Z.pow_pos UINT_MOD (Pos.of_succ_nat len)) n ltac:(lia) ltac:(lia)).
rewrite H3; try tauto.
unfold Z.divide.
destruct len.
* exists 1.
simpl.
lia.
* exists (Z.pow_pos UINT_MOD (Pos.of_succ_nat len)).
assert (Pos.of_succ_nat (S len) = Pos.add (Pos.of_succ_nat len) xH). { lia. }
rewrite H4.
apply Zpower_pos_is_exp.
lia.
+ simpl in *; repeat split; try tauto.
Qed.
Lemma list_store_Z_pref_full: forall (data: list Z) (n: Z),
list_store_Z_pref data n (Z.to_nat (Zlength data)) --||-- list_store_Z data n.
Lemma list_within_bound_concat: forall (l1 l2: list Z),
list_within_bound l1 ->
list_within_bound l2 ->
list_within_bound (l1 ++ l2).
Proof.
intros.
revert n.
induction data.
revert l1 H.
induction l2.
+ intros.
simpl.
split; entailer!.
rewrite app_nil_r.
tauto.
+ intros.
pose proof (Zlength_cons a data).
rewrite H.
pose proof (Z2Nat.inj_succ (Zlength data) (Zlength_nonneg data)).
rewrite H0.
simpl.
split.
- Intros a0 l0.
injection H1; intros.
subst.
specialize (IHdata (n / UINT_MOD)).
destruct IHdata.
sep_apply H2.
entailer!.
- Exists a data.
entailer!.
specialize (IHdata (n / UINT_MOD)).
destruct IHdata.
sep_apply H3.
entailer!.
simpl in H0.
destruct H0.
rewrite Aux.list_app_cons.
pose proof (__list_within_bound_concat_r l1 a H H0).
specialize (IHl2 H1 (app l1 [a]) H2).
tauto.
Qed.
Lemma list_store_Z_pref_extend_data: forall (data: list Z) (a: Z) (n: Z) (len: nat),
list_store_Z_pref data n len |--
list_store_Z_pref (data ++ (a :: nil)) n len.
Lemma __list_within_bound_split_r: forall (l1: list Z) (a: Z),
list_within_bound (l1 ++ [a]) ->
list_within_bound l1 /\ 0 <= a < UINT_MOD.
Proof.
intros.
revert data n.
induction len.
induction l1.
+ rewrite app_nil_l in H.
simpl in *.
tauto.
+ simpl in *.
destruct H.
specialize (IHl1 H0).
tauto.
Qed.
Lemma list_within_bound_split: forall (l1 l2: list Z),
list_within_bound (l1 ++ l2) ->
list_within_bound l1 /\ list_within_bound l2.
Proof.
intros.
revert l1 H.
induction l2.
+ intros.
simpl.
entailer!.
rewrite app_nil_r in H.
tauto.
+ intros.
simpl.
Intros a0 l0.
Exists a0 (app l0 (cons a nil)).
entailer!.
rewrite Aux.list_app_cons in H.
specialize (IHl2 (app l1 [a]) H).
destruct IHl2.
apply __list_within_bound_split_r in H0.
tauto.
Qed.
Lemma __list_store_Z_concat_r: forall (l1: list Z) (n1 a: Z),
list_store_Z l1 n1 ->
0 <= a < UINT_MOD ->
list_store_Z (l1 ++ [a]) (a * (UINT_MOD ^ (Zlength l1)) + n1).
Proof.
induction l1; intros.
+ rewrite app_nil_l.
unfold Zlength, Zlength_aux.
rewrite Z.pow_0_r.
unfold list_store_Z in H; destruct H.
simpl in H.
subst.
reflexivity.
Qed.
Search list.
Lemma list_store_Z_pref_extend: forall (data: list Z) (a: Z) (n: Z) (len: nat),
n >= 0 ->
Zlength data = Z.of_nat len ->
list_store_Z_pref data (n mod (UINT_MOD ^ Z.of_nat len)) len &&
[| a = (n / (UINT_MOD ^ Z.of_nat len)) mod UINT_MOD |] |--
list_store_Z_pref (data ++ (cons a nil)) (n mod (UINT_MOD ^ (Z.of_nat len + 1))) (S len).
Proof.
intros.
entailer!.
simpl.
revert a data H0 H1.
induction len.
+ intros.
Exists a nil.
simpl.
entailer!.
- intros.
unfold Z.pow_pos, Pos.iter; simpl.
apply Z.mod_div; lia.
- unfold Z.pow_pos, Pos.iter; simpl.
simpl in H1; rewrite Z.div_1_r in H1.
rewrite Z.mod_mod; lia.
- pose proof (Zlength_nil_inv data H0).
rewrite H2.
reflexivity.
+ intros.
simpl.
Intros a0 l0.
Exists a0 (app l0 [a]).
assert (Zlength l0 = Z.of_nat len). {
rewrite H2 in H0.
rewrite Zlength_cons in H0.
lia.
unfold list_store_Z; simpl.
lia.
+ unfold list_store_Z in H; destruct H.
simpl in H.
simpl in H1.
assert (list_store_Z l1 ((n1 - a) / UINT_MOD)). {
unfold list_store_Z; split; try simpl; try tauto.
apply Z.div_unique_exact; lia.
}
specialize (IHlen ((n / UINT_MOD ^ (Z.of_nat len)) mod UINT_MOD) l0 H4 ltac:(lia)).
sep_apply IHlen.
Admitted. (* Unfinished. *)
Lemma mpd_store_Z_split: forall (ptr: addr) (n: Z) (size: Z) (cap: Z) (len: nat),
n >= 0 ->
Z.of_nat len < size ->
mpd_store_Z ptr n size cap |--
mpd_store_Z_pref ptr (n mod (UINT_MOD ^ Z.of_nat len)) size cap len.
Proof.
intros.
unfold mpd_store_Z, mpd_store_Z_pref.
Intros data.
Exists data.
unfold mpd_store_list.
Intros.
entailer!.
sep_apply (list_store_Z_split data n len).
+ entailer!.
+ lia.
+ lia.
specialize (IHl1 ((n1 - a) / UINT_MOD) a0 H2 ltac:(lia)).
unfold list_store_Z; split.
- simpl.
unfold list_store_Z in IHl1; destruct IHl1.
rewrite Zlength_cons; unfold Z.succ.
rewrite H3.
assert ((n1 - a) / UINT_MOD * UINT_MOD = n1 - a). {
rewrite <-(Z.div_unique_exact (n1 - a) UINT_MOD (list_to_Z l1)); lia.
}
rewrite Z.mul_add_distr_r.
rewrite H5.
rewrite Aux.Zpow_add_1; try lia.
pose proof (Zlength_nonneg l1).
lia.
- apply list_within_bound_concat; try simpl; try lia; try tauto.
Qed.
Lemma mpd_store_Z_pref_full: forall (ptr: addr) (n: Z) (size: Z) (cap: Z),
mpd_store_Z ptr n size cap --||-- mpd_store_Z_pref ptr n size cap (Z.to_nat size).
Lemma list_store_Z_concat: forall (l1 l2: list Z) (n1 n2: Z),
list_store_Z l1 n1 ->
list_store_Z l2 n2 ->
list_store_Z (l1 ++ l2) (n1 + n2 * (UINT_MOD ^ (Zlength l1))).
Proof.
unfold list_store_Z.
intros.
split; [ | apply list_within_bound_concat; tauto].
revert n1 l1 n2 H H0.
induction l2.
+ intros.
simpl in *.
subst.
rewrite app_nil_r.
nia.
+ intros.
destruct H0.
destruct H.
simpl in H0.
rewrite Aux.list_app_cons.
specialize (IHl2 (n1 + a * UINT_MOD ^ (Zlength l1)) (app l1 [a]) ((n2 - a) / UINT_MOD)).
rewrite IHl2.
- rewrite Zlength_app; rewrite Zlength_cons; simpl.
assert ((n2 - a) / UINT_MOD * UINT_MOD = n2 - a). {
rewrite <-(Z.div_unique_exact (n2 - a) UINT_MOD (list_to_Z l2)); try lia.
}
rewrite Aux.Zpow_add_1; try lia.
pose proof (Zlength_nonneg l1); lia.
- pose proof (__list_store_Z_concat_r l1 n1 a).
assert (list_store_Z l1 n1). { unfold list_store_Z; tauto. }
simpl in H1.
specialize (H3 H4 ltac:(lia)).
unfold list_store_Z in H3.
destruct H3.
split; [ lia | tauto].
- simpl in H1.
split; [ | tauto].
apply Z.div_unique_exact; lia.
Qed.
Lemma list_store_Z_bound: forall (l1: list Z) (n: Z),
list_store_Z l1 n -> 0 <= n < UINT_MOD ^ (Zlength l1).
Proof.
induction l1; intros.
+ rewrite Zlength_nil; rewrite Z.pow_0_r.
unfold list_store_Z in H; destruct H; simpl in *.
lia.
+ rewrite Zlength_cons; unfold Z.succ.
unfold list_store_Z in *; destruct H; simpl in *.
assert (list_to_Z l1 = (n - a) / UINT_MOD /\ list_within_bound l1). {
rewrite (Z.div_unique_exact (n - a) UINT_MOD (list_to_Z l1)); try lia; try tauto.
}
specialize (IHl1 ((n - a) / UINT_MOD) H1).
rewrite Aux.Zpow_add_1; try lia.
pose proof (Zlength_nonneg l1); lia.
Qed.
Lemma list_store_Z_split: forall (l1 l2: list Z) (n: Z),
list_store_Z (l1 ++ l2) n ->
list_store_Z l1 (n mod UINT_MOD ^ (Zlength l1)) /\
list_store_Z l2 (n / UINT_MOD ^ (Zlength l1)).
Proof.
intros.
unfold mpd_store_Z, mpd_store_Z_pref.
pose proof list_store_Z_pref_full.
split; Intros data; Exists data; entailer!; specialize (H data n); destruct H.
+ sep_apply H1.
subst.
entailer!.
+ subst.
sep_apply H.
entailer!.
revert n H.
induction l1; split.
+ intros.
rewrite app_nil_l in H.
rewrite Zlength_nil; rewrite Z.pow_0_r; rewrite Z.mod_1_r.
unfold list_store_Z; simpl; lia.
+ rewrite app_nil_l in H.
unfold list_store_Z; simpl.
rewrite Z.div_1_r.
tauto.
+ rewrite Zlength_cons; unfold Z.succ.
unfold list_store_Z in H; simpl in H; destruct H.
unfold list_store_Z in IHl1.
assert (list_to_Z (l1 ++ l2) = (n - a) / UINT_MOD /\ list_within_bound (l1 ++ l2)). {
rewrite (Z.div_unique_exact (n - a) UINT_MOD (list_to_Z (app l1 l2))); try lia; try tauto.
}
specialize (IHl1 ((n - a) / UINT_MOD) H1).
unfold list_store_Z; simpl; split.
- destruct IHl1; destruct H2, H3.
rewrite H2.
destruct H1.
rewrite <-H1, <-H.
remember (list_to_Z (l1 ++ l2)) as n' eqn:Hn'.
rewrite Zplus_mod.
rewrite Aux.Zmul_mod_cancel; try lia.
* assert (UINT_MOD ^ (Zlength l1 + 1) >= UINT_MOD). {
pose proof (Zlength_nonneg l1).
rewrite Aux.Zpow_add_1; try tauto; try lia.
}
rewrite (Z.mod_small a (UINT_MOD ^ (Zlength l1 + 1)) ltac:(lia)).
assert (list_store_Z (l1 ++ l2) n'). { unfold list_store_Z; split; [ lia | tauto]. }
pose proof (list_store_Z_bound (l1 ++ l2) n' H8).
pose proof (Zlength_nonneg l1).
pose proof (Z.mod_bound_pos n' (UINT_MOD ^ (Zlength l1)) ltac:(lia) ltac:(lia)).
assert (UINT_MOD * (n' mod UINT_MOD ^ (Zlength l1)) + a < UINT_MOD ^ (Zlength l1 + 1)). {
rewrite Aux.Zpow_add_1; lia.
}
rewrite (Z.mod_small (UINT_MOD * (n' mod UINT_MOD ^ (Zlength l1)) + a) (UINT_MOD ^ (Zlength l1 + 1)) ltac:(lia)).
lia.
* assert (list_store_Z (l1 ++ l2) n'). { unfold list_store_Z; split; [ lia | tauto]. }
pose proof (list_store_Z_bound (l1 ++ l2) n' H7).
lia.
* pose proof (Zlength_nonneg l1); lia.
- split; [ lia | tauto].
+ unfold list_store_Z in *.
simpl in *.
pose proof (list_within_bound_split l1 l2 ltac:(tauto)).
split; [ | tauto].
rewrite Zlength_cons; unfold Z.succ.
destruct H as [H [H1 H2]].
assert (list_to_Z (l1 ++ l2) = (n - a) / UINT_MOD /\ list_within_bound (l1 ++ l2)). {
rewrite (Z.div_unique_exact (n - a) UINT_MOD (list_to_Z (l1 ++ l2))); try lia; try tauto.
}
specialize (IHl1 ((n - a) / UINT_MOD) H3).
destruct IHl1 as [[H4 H5] [H6 H7]].
rewrite H6.
rewrite Aux.Zpow_add_1; try lia.
- rewrite Z.mul_comm.
rewrite <-Zdiv_Zdiv; try lia.
destruct H3.
assert ((n - a) / UINT_MOD = n / UINT_MOD). {
apply (Zdiv_unique_full n UINT_MOD ((n - a) / UINT_MOD) a).
+ unfold Remainder; lia.
+ lia.
}
rewrite H9.
reflexivity.
- pose proof (Zlength_nonneg l1); lia.
Qed.
End Internal.
@ -281,8 +390,8 @@ Definition store_bigint_ent (x: addr) (n: bigint_ent): Assertion :=
Internal.mpd_store_list p (data n) (cap n).
Definition bigint_ent_store_Z (n: bigint_ent) (x: Z): Assertion :=
[| sign n |] && Internal.list_store_Z (data n) (-x) ||
[| ~(sign n) |] && Internal.list_store_Z (data n) x.
[| sign n |] && [| Internal.list_store_Z (data n) (-x) |] ||
[| ~(sign n) |] && [| Internal.list_store_Z (data n) x |].
Definition store_Z (x: addr) (n: Z): Assertion :=
EX ent,