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feat(cmp): Proved correctness of mpn_cmp.

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This commit is contained in:
xiaoh105
2025-06-11 16:54:36 +08:00
parent 4c0b0e98fa
commit 36204b8877
7 changed files with 824 additions and 18 deletions
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11
projects/lib/GmpAux.v
View File

@ -101,6 +101,17 @@ Proof.
reflexivity.
Qed.
Lemma list_app_single_l: forall (l: list Z) (a: Z),
([a] ++ l)%list = a :: l.
Proof.
intros.
induction l.
+ simpl; reflexivity.
+ rewrite list_app_cons.
simpl.
reflexivity.
Qed.
Lemma store_array_rec_false: forall x storeA lo hi (l: list Z),
lo > hi ->
store_array_rec storeA x lo hi l |-- [| False |].

100
projects/lib/GmpNumber.v
View File

@ -49,6 +49,19 @@ 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 |].
Lemma list_store_Z_injection: forall l1 l2 n1 n2,
list_store_Z l1 n1 ->
list_store_Z l2 n2 ->
l1 = l2 -> n1 = n2.
Proof.
intros.
unfold list_store_Z in *.
destruct H, H0.
rewrite <-H, <-H0.
rewrite <-H1.
reflexivity.
Qed.
Lemma __list_within_bound_concat_r: forall (l1: list Z) (a: Z),
list_within_bound l1 ->
0 <= a < UINT_MOD ->
@ -350,6 +363,93 @@ Proof.
reflexivity.
Qed.
Lemma list_store_Z_cmp: forall l1 l2 n1 n2 i,
0 <= i < Zlength l1 ->
Zlength l1 = Zlength l2 ->
list_store_Z l1 n1 ->
list_store_Z l2 n2 ->
sublist (i + 1) (Zlength l1) l1 = sublist (i + 1) (Zlength l2) l2 ->
Znth i l1 0 < Znth i l2 0 ->
n1 < n2.
Proof.
intros.
assert (Zlength l1 = Z.of_nat (Datatypes.length l1)). { apply Zlength_correct. }
pose proof (sublist_split 0 (Zlength l1) i l1 ltac:(lia) ltac:(lia)).
assert (Zlength l2 = Z.of_nat (Datatypes.length l2)). { apply Zlength_correct. }
pose proof (sublist_split 0 (Zlength l2) i l2 ltac:(lia) ltac:(lia)).
clear H5 H7.
rewrite (sublist_self l1 (Zlength l1) ltac:(lia)) in H6.
rewrite (sublist_self l2 (Zlength l2) ltac:(lia)) in H8.
rewrite H6 in H1.
rewrite H8 in H2.
apply list_store_Z_split in H1, H2.
remember (Zlength l1) as n eqn:Hn.
assert (Zlength l2 = n). { lia. }
rewrite H5 in *.
rewrite Zlength_sublist0 in H1, H2; try lia.
destruct H1, H2.
remember (n1 mod UINT_MOD ^ i) as n1_lo eqn:Hn1_lo.
remember (n1 / UINT_MOD ^ i) as n1_hi eqn:Hn1_hi.
remember (n2 mod UINT_MOD ^ i) as n2_lo eqn:Hn2_lo.
remember (n2 / UINT_MOD ^ i) as n2_hi eqn:Hn2_hi.
remember (sublist 0 i l1) as l1_lo eqn:Hl1_lo.
remember (sublist i n l1) as l1_hi eqn:Hl1_hi.
remember (sublist 0 i l2) as l2_lo eqn:Hl2_lo.
remember (sublist i n l2) as l2_hi eqn:Hl2_hi.
assert (n1_lo - n2_lo < UINT_MOD ^ i). {
pose proof (list_store_Z_bound l1_lo n1_lo H1).
pose proof (list_store_Z_bound l2_lo n2_lo H2).
rewrite Hl1_lo, Zlength_sublist0 in H10; lia.
}
assert (n2_hi - n1_hi >= 1). {
assert (Zlength l1_hi >= 1 /\ Zlength l2_hi >= 1). {
pose proof (Zlength_sublist i n l1 ltac:(lia)).
pose proof (Zlength_sublist i n l2 ltac:(lia)).
rewrite <-Hl1_hi in H11.
rewrite <-Hl2_hi in H12.
lia.
}
destruct H11.
destruct l1_hi, l2_hi; try rewrite Zlength_nil in *; try lia.
unfold list_store_Z in H7, H9.
destruct H7, H9.
simpl in H7, H9.
assert (forall a (l0 l: list Z),
a :: l0 = sublist i n l ->
Zlength l = n ->
l0 = sublist (i + 1) n l /\ a = Znth i l 0
). {
intros.
assert (n = Z.of_nat(Datatypes.length l)). { rewrite <-Zlength_correct. lia. }
pose proof (sublist_split i n (i + 1) l ltac:(lia) ltac:(lia)).
rewrite (sublist_single i l 0) in H18; try lia.
rewrite <-H15 in H18.
rewrite Aux.list_app_single_l in H18.
injection H18; intros.
rewrite H19, H20.
split; reflexivity.
}
pose proof (H15 z0 l2_hi l2 Hl2_hi ltac:(lia)).
specialize (H15 z l1_hi l1 Hl1_hi ltac:(lia)).
destruct H15, H16.
rewrite H15, H17 in H7.
rewrite H16, H18 in H9.
rewrite <-H3 in H9.
lia.
}
pose proof (Zmod_eq_full n1 (UINT_MOD ^ i) ltac:(lia)).
pose proof (Zmod_eq_full n2 (UINT_MOD ^ i) ltac:(lia)).
rewrite <-Hn1_lo, <-Hn1_hi in H12.
rewrite <-Hn2_lo, <-Hn2_hi in H13.
assert (n2_hi * UINT_MOD ^ i - n1_hi * UINT_MOD ^ i >= UINT_MOD ^ i). {
rewrite <-Z.mul_sub_distr_r.
pose proof (Zmult_ge_compat_r (n2_hi - n1_hi) 1 (UINT_MOD ^ i) ltac:(lia) ltac:(lia)).
rewrite Z.mul_1_l in H14.
lia.
}
lia.
Qed.
End Internal.
Record bigint_ent: Type := {

574
projects/lib/gmp_goal.v
View File

@ -9,8 +9,8 @@ Require Import Coq.Sorting.Permutation.
From AUXLib Require Import int_auto Axioms Feq Idents List_lemma VMap.
Require Import SetsClass.SetsClass. Import SetsNotation.
From SimpleC.SL Require Import Mem SeparationLogic.
Require Import GmpLib.GmpNumber. Import Internal.
Require Import Logic.LogicGenerator.demo932.Interface.
Require Import GmpLib.GmpNumber. Import Internal.
Local Open Scope Z_scope.
Local Open Scope sets.
Local Open Scope string.
@ -591,6 +591,562 @@ forall (cap2: Z) (l': (@list Z)) (l: (@list Z)) (i: Z) (n: Z) (d: Z) ,
** (store_uint_array d (i + 1 ) (app ((sublist (0) (i) (l))) ((cons (a) (nil)))) )
.
(*----- Function mpn_cmp -----*)
Definition mpn_cmp_safety_wit_1 :=
forall (n_pre: Z) (bp_pre: Z) (ap_pre: Z) (val2: Z) (val1: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (l1: (@list Z)) ,
[| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| (0 < n_pre) |]
&& [| (n_pre <= cap1) |]
&& [| (n_pre <= cap2) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& (store_uint_array ap_pre n_pre l1 )
** (store_uint_array bp_pre n_pre l2 )
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
** ((( &( "n" ) )) # Int |-> n_pre)
** ((( &( "bp" ) )) # Ptr |-> bp_pre)
** ((( &( "ap" ) )) # Ptr |-> ap_pre)
|--
[| ((n_pre - 1 ) <= INT_MAX) |]
&& [| ((INT_MIN) <= (n_pre - 1 )) |]
.
Definition mpn_cmp_safety_wit_2 :=
forall (n_pre: Z) (bp_pre: Z) (ap_pre: Z) (val2: Z) (val1: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (l1: (@list Z)) (n: Z) ,
[| ((-1) <= n) |]
&& [| (n < n_pre) |]
&& [| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| ((sublist ((n + 1 )) (n_pre) (l1)) = (sublist ((n + 1 )) (n_pre) (l2))) |]
&& [| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| (0 < n_pre) |]
&& [| (n_pre <= cap1) |]
&& [| (n_pre <= cap2) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& ((( &( "n" ) )) # Int |-> n)
** (store_uint_array ap_pre n_pre l1 )
** (store_uint_array bp_pre n_pre l2 )
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
** ((( &( "bp" ) )) # Ptr |-> bp_pre)
** ((( &( "ap" ) )) # Ptr |-> ap_pre)
|--
[| (0 <= INT_MAX) |]
&& [| ((INT_MIN) <= 0) |]
.
Definition mpn_cmp_safety_wit_3 :=
forall (n_pre: Z) (bp_pre: Z) (ap_pre: Z) (val2: Z) (val1: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (l1: (@list Z)) (n: Z) ,
[| ((Znth n l1 0) <= (Znth n l2 0)) |]
&& [| ((Znth n l1 0) <> (Znth n l2 0)) |]
&& [| (n >= 0) |]
&& [| ((-1) <= n) |]
&& [| (n < n_pre) |]
&& [| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| ((sublist ((n + 1 )) (n_pre) (l1)) = (sublist ((n + 1 )) (n_pre) (l2))) |]
&& [| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| (0 < n_pre) |]
&& [| (n_pre <= cap1) |]
&& [| (n_pre <= cap2) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& (store_uint_array bp_pre n_pre l2 )
** (store_uint_array ap_pre n_pre l1 )
** ((( &( "n" ) )) # Int |-> n)
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
** ((( &( "bp" ) )) # Ptr |-> bp_pre)
** ((( &( "ap" ) )) # Ptr |-> ap_pre)
|--
[| (1 <> (INT_MIN)) |]
.
Definition mpn_cmp_safety_wit_4 :=
forall (n_pre: Z) (bp_pre: Z) (ap_pre: Z) (val2: Z) (val1: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (l1: (@list Z)) (n: Z) ,
[| ((Znth n l1 0) <= (Znth n l2 0)) |]
&& [| ((Znth n l1 0) <> (Znth n l2 0)) |]
&& [| (n >= 0) |]
&& [| ((-1) <= n) |]
&& [| (n < n_pre) |]
&& [| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| ((sublist ((n + 1 )) (n_pre) (l1)) = (sublist ((n + 1 )) (n_pre) (l2))) |]
&& [| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| (0 < n_pre) |]
&& [| (n_pre <= cap1) |]
&& [| (n_pre <= cap2) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& (store_uint_array bp_pre n_pre l2 )
** (store_uint_array ap_pre n_pre l1 )
** ((( &( "n" ) )) # Int |-> n)
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
** ((( &( "bp" ) )) # Ptr |-> bp_pre)
** ((( &( "ap" ) )) # Ptr |-> ap_pre)
|--
[| (1 <= INT_MAX) |]
&& [| ((INT_MIN) <= 1) |]
.
Definition mpn_cmp_safety_wit_5 :=
forall (n_pre: Z) (bp_pre: Z) (ap_pre: Z) (val2: Z) (val1: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (l1: (@list Z)) (n: Z) ,
[| ((Znth n l1 0) > (Znth n l2 0)) |]
&& [| ((Znth n l1 0) <> (Znth n l2 0)) |]
&& [| (n >= 0) |]
&& [| ((-1) <= n) |]
&& [| (n < n_pre) |]
&& [| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| ((sublist ((n + 1 )) (n_pre) (l1)) = (sublist ((n + 1 )) (n_pre) (l2))) |]
&& [| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| (0 < n_pre) |]
&& [| (n_pre <= cap1) |]
&& [| (n_pre <= cap2) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& (store_uint_array bp_pre n_pre l2 )
** (store_uint_array ap_pre n_pre l1 )
** ((( &( "n" ) )) # Int |-> n)
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
** ((( &( "bp" ) )) # Ptr |-> bp_pre)
** ((( &( "ap" ) )) # Ptr |-> ap_pre)
|--
[| (1 <= INT_MAX) |]
&& [| ((INT_MIN) <= 1) |]
.
Definition mpn_cmp_safety_wit_6 :=
forall (n_pre: Z) (bp_pre: Z) (ap_pre: Z) (val2: Z) (val1: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (l1: (@list Z)) (n: Z) ,
[| ((Znth n l1 0) = (Znth n l2 0)) |]
&& [| (n >= 0) |]
&& [| ((-1) <= n) |]
&& [| (n < n_pre) |]
&& [| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| ((sublist ((n + 1 )) (n_pre) (l1)) = (sublist ((n + 1 )) (n_pre) (l2))) |]
&& [| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| (0 < n_pre) |]
&& [| (n_pre <= cap1) |]
&& [| (n_pre <= cap2) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& (store_uint_array bp_pre n_pre l2 )
** (store_uint_array ap_pre n_pre l1 )
** ((( &( "n" ) )) # Int |-> n)
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
** ((( &( "bp" ) )) # Ptr |-> bp_pre)
** ((( &( "ap" ) )) # Ptr |-> ap_pre)
|--
[| ((n - 1 ) <= INT_MAX) |]
&& [| ((INT_MIN) <= (n - 1 )) |]
.
Definition mpn_cmp_safety_wit_7 :=
forall (n_pre: Z) (bp_pre: Z) (ap_pre: Z) (val2: Z) (val1: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (l1: (@list Z)) (n: Z) ,
[| (n < 0) |]
&& [| ((-1) <= n) |]
&& [| (n < n_pre) |]
&& [| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| ((sublist ((n + 1 )) (n_pre) (l1)) = (sublist ((n + 1 )) (n_pre) (l2))) |]
&& [| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| (0 < n_pre) |]
&& [| (n_pre <= cap1) |]
&& [| (n_pre <= cap2) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& ((( &( "n" ) )) # Int |-> n)
** (store_uint_array ap_pre n_pre l1 )
** (store_uint_array bp_pre n_pre l2 )
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
** ((( &( "bp" ) )) # Ptr |-> bp_pre)
** ((( &( "ap" ) )) # Ptr |-> ap_pre)
|--
[| (0 <= INT_MAX) |]
&& [| ((INT_MIN) <= 0) |]
.
Definition mpn_cmp_entail_wit_1 :=
forall (n_pre: Z) (bp_pre: Z) (ap_pre: Z) (val2: Z) (val1: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (l1: (@list Z)) ,
[| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| (0 < n_pre) |]
&& [| (n_pre <= cap1) |]
&& [| (n_pre <= cap2) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& (store_uint_array ap_pre n_pre l1 )
** (store_uint_array bp_pre n_pre l2 )
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
|--
[| ((-1) <= (n_pre - 1 )) |]
&& [| ((n_pre - 1 ) < n_pre) |]
&& [| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| ((sublist (((n_pre - 1 ) + 1 )) (n_pre) (l1)) = (sublist (((n_pre - 1 ) + 1 )) (n_pre) (l2))) |]
&& [| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| (0 < n_pre) |]
&& [| (n_pre <= cap1) |]
&& [| (n_pre <= cap2) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& (store_uint_array ap_pre n_pre l1 )
** (store_uint_array bp_pre n_pre l2 )
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
.
Definition mpn_cmp_entail_wit_2 :=
forall (n_pre: Z) (bp_pre: Z) (ap_pre: Z) (val2: Z) (val1: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (l1: (@list Z)) (n: Z) ,
[| ((Znth n l1 0) = (Znth n l2 0)) |]
&& [| (n >= 0) |]
&& [| ((-1) <= n) |]
&& [| (n < n_pre) |]
&& [| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| ((sublist ((n + 1 )) (n_pre) (l1)) = (sublist ((n + 1 )) (n_pre) (l2))) |]
&& [| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| (0 < n_pre) |]
&& [| (n_pre <= cap1) |]
&& [| (n_pre <= cap2) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& (store_uint_array bp_pre n_pre l2 )
** (store_uint_array ap_pre n_pre l1 )
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
|--
[| ((-1) <= (n - 1 )) |]
&& [| ((n - 1 ) < n_pre) |]
&& [| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| ((sublist (((n - 1 ) + 1 )) (n_pre) (l1)) = (sublist (((n - 1 ) + 1 )) (n_pre) (l2))) |]
&& [| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| (0 < n_pre) |]
&& [| (n_pre <= cap1) |]
&& [| (n_pre <= cap2) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& (store_uint_array ap_pre n_pre l1 )
** (store_uint_array bp_pre n_pre l2 )
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
.
Definition mpn_cmp_return_wit_1_1 :=
forall (n_pre: Z) (bp_pre: Z) (ap_pre: Z) (val2: Z) (val1: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (l1: (@list Z)) (n: Z) ,
[| ((Znth n l1 0) <= (Znth n l2 0)) |]
&& [| ((Znth n l1 0) <> (Znth n l2 0)) |]
&& [| (n >= 0) |]
&& [| ((-1) <= n) |]
&& [| (n < n_pre) |]
&& [| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| ((sublist ((n + 1 )) (n_pre) (l1)) = (sublist ((n + 1 )) (n_pre) (l2))) |]
&& [| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| (0 < n_pre) |]
&& [| (n_pre <= cap1) |]
&& [| (n_pre <= cap2) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& (store_uint_array bp_pre n_pre l2 )
** (store_uint_array ap_pre n_pre l1 )
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
|--
([| (val1 < val2) |]
&& [| ((-1) = (-1)) |]
&& (mpd_store_Z ap_pre val1 n_pre cap1 )
** (mpd_store_Z bp_pre val2 n_pre cap2 ))
||
([| (val1 = val2) |]
&& [| ((-1) = 0) |]
&& (mpd_store_Z ap_pre val1 n_pre cap1 )
** (mpd_store_Z bp_pre val2 n_pre cap2 ))
||
([| (val1 > val2) |]
&& [| ((-1) = 1) |]
&& (mpd_store_Z ap_pre val1 n_pre cap1 )
** (mpd_store_Z bp_pre val2 n_pre cap2 ))
.
Definition mpn_cmp_return_wit_1_2 :=
forall (n_pre: Z) (bp_pre: Z) (ap_pre: Z) (val2: Z) (val1: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (l1: (@list Z)) (n: Z) ,
[| ((Znth n l1 0) > (Znth n l2 0)) |]
&& [| ((Znth n l1 0) <> (Znth n l2 0)) |]
&& [| (n >= 0) |]
&& [| ((-1) <= n) |]
&& [| (n < n_pre) |]
&& [| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| ((sublist ((n + 1 )) (n_pre) (l1)) = (sublist ((n + 1 )) (n_pre) (l2))) |]
&& [| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| (0 < n_pre) |]
&& [| (n_pre <= cap1) |]
&& [| (n_pre <= cap2) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& (store_uint_array bp_pre n_pre l2 )
** (store_uint_array ap_pre n_pre l1 )
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
|--
([| (val1 < val2) |]
&& [| (1 = (-1)) |]
&& (mpd_store_Z ap_pre val1 n_pre cap1 )
** (mpd_store_Z bp_pre val2 n_pre cap2 ))
||
([| (val1 = val2) |]
&& [| (1 = 0) |]
&& (mpd_store_Z ap_pre val1 n_pre cap1 )
** (mpd_store_Z bp_pre val2 n_pre cap2 ))
||
([| (val1 > val2) |]
&& [| (1 = 1) |]
&& (mpd_store_Z ap_pre val1 n_pre cap1 )
** (mpd_store_Z bp_pre val2 n_pre cap2 ))
.
Definition mpn_cmp_return_wit_2 :=
forall (n_pre: Z) (bp_pre: Z) (ap_pre: Z) (val2: Z) (val1: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (l1: (@list Z)) (n: Z) ,
[| (n < 0) |]
&& [| ((-1) <= n) |]
&& [| (n < n_pre) |]
&& [| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| ((sublist ((n + 1 )) (n_pre) (l1)) = (sublist ((n + 1 )) (n_pre) (l2))) |]
&& [| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| (0 < n_pre) |]
&& [| (n_pre <= cap1) |]
&& [| (n_pre <= cap2) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& (store_uint_array ap_pre n_pre l1 )
** (store_uint_array bp_pre n_pre l2 )
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
|--
([| (val1 < val2) |]
&& [| (0 = (-1)) |]
&& (mpd_store_Z ap_pre val1 n_pre cap1 )
** (mpd_store_Z bp_pre val2 n_pre cap2 ))
||
([| (val1 = val2) |]
&& [| (0 = 0) |]
&& (mpd_store_Z ap_pre val1 n_pre cap1 )
** (mpd_store_Z bp_pre val2 n_pre cap2 ))
||
([| (val1 > val2) |]
&& [| (0 = 1) |]
&& (mpd_store_Z ap_pre val1 n_pre cap1 )
** (mpd_store_Z bp_pre val2 n_pre cap2 ))
.
Definition mpn_cmp_partial_solve_wit_1 :=
forall (n_pre: Z) (bp_pre: Z) (ap_pre: Z) (val2: Z) (val1: Z) (cap2: Z) (cap1: Z) ,
[| (0 < n_pre) |]
&& [| (n_pre <= cap1) |]
&& [| (n_pre <= cap2) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& (mpd_store_Z ap_pre val1 n_pre cap1 )
** (mpd_store_Z bp_pre val2 n_pre cap2 )
|--
[| (0 < n_pre) |]
&& [| (n_pre <= cap1) |]
&& [| (n_pre <= cap2) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& (mpd_store_Z ap_pre val1 n_pre cap1 )
** (mpd_store_Z bp_pre val2 n_pre cap2 )
.
Definition mpn_cmp_partial_solve_wit_2 :=
forall (n_pre: Z) (bp_pre: Z) (ap_pre: Z) (val2: Z) (val1: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (l1: (@list Z)) (n: Z) ,
[| (n >= 0) |]
&& [| ((-1) <= n) |]
&& [| (n < n_pre) |]
&& [| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| ((sublist ((n + 1 )) (n_pre) (l1)) = (sublist ((n + 1 )) (n_pre) (l2))) |]
&& [| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| (0 < n_pre) |]
&& [| (n_pre <= cap1) |]
&& [| (n_pre <= cap2) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& (store_uint_array ap_pre n_pre l1 )
** (store_uint_array bp_pre n_pre l2 )
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
|--
[| (n >= 0) |]
&& [| ((-1) <= n) |]
&& [| (n < n_pre) |]
&& [| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| ((sublist ((n + 1 )) (n_pre) (l1)) = (sublist ((n + 1 )) (n_pre) (l2))) |]
&& [| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| (0 < n_pre) |]
&& [| (n_pre <= cap1) |]
&& [| (n_pre <= cap2) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& (((ap_pre + (n * sizeof(UINT) ) )) # UInt |-> (Znth n l1 0))
** (store_uint_array_missing_i_rec ap_pre n 0 n_pre l1 )
** (store_uint_array bp_pre n_pre l2 )
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
.
Definition mpn_cmp_partial_solve_wit_3 :=
forall (n_pre: Z) (bp_pre: Z) (ap_pre: Z) (val2: Z) (val1: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (l1: (@list Z)) (n: Z) ,
[| (n >= 0) |]
&& [| ((-1) <= n) |]
&& [| (n < n_pre) |]
&& [| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| ((sublist ((n + 1 )) (n_pre) (l1)) = (sublist ((n + 1 )) (n_pre) (l2))) |]
&& [| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| (0 < n_pre) |]
&& [| (n_pre <= cap1) |]
&& [| (n_pre <= cap2) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& (store_uint_array ap_pre n_pre l1 )
** (store_uint_array bp_pre n_pre l2 )
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
|--
[| (n >= 0) |]
&& [| ((-1) <= n) |]
&& [| (n < n_pre) |]
&& [| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| ((sublist ((n + 1 )) (n_pre) (l1)) = (sublist ((n + 1 )) (n_pre) (l2))) |]
&& [| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& [| (0 < n_pre) |]
&& [| (n_pre <= cap1) |]
&& [| (n_pre <= cap2) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& (((bp_pre + (n * sizeof(UINT) ) )) # UInt |-> (Znth n l2 0))
** (store_uint_array_missing_i_rec bp_pre n 0 n_pre l2 )
** (store_uint_array ap_pre n_pre l1 )
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
.
Definition mpn_cmp_which_implies_wit_1 :=
forall (n_pre: Z) (bp_pre: Z) (ap_pre: Z) (val2: Z) (val1: Z) (cap2: Z) (cap1: Z) ,
(mpd_store_Z ap_pre val1 n_pre cap1 )
** (mpd_store_Z bp_pre val2 n_pre cap2 )
|--
EX (l2: (@list Z)) (l1: (@list Z)) ,
[| (list_store_Z l1 val1 ) |]
&& [| (list_store_Z l2 val2 ) |]
&& [| (n_pre = (Zlength (l1))) |]
&& [| (n_pre = (Zlength (l2))) |]
&& (store_uint_array ap_pre n_pre l1 )
** (store_uint_array bp_pre n_pre l2 )
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
.
Module Type VC_Correct.
Axiom proof_of_gmp_abs_safety_wit_1 : gmp_abs_safety_wit_1.
@ -620,5 +1176,21 @@ Axiom proof_of_mpn_copyi_partial_solve_wit_5 : mpn_copyi_partial_solve_wit_5.
Axiom proof_of_mpn_copyi_which_implies_wit_1 : mpn_copyi_which_implies_wit_1.
Axiom proof_of_mpn_copyi_which_implies_wit_2 : mpn_copyi_which_implies_wit_2.
Axiom proof_of_mpn_copyi_which_implies_wit_3 : mpn_copyi_which_implies_wit_3.
Axiom proof_of_mpn_cmp_safety_wit_1 : mpn_cmp_safety_wit_1.
Axiom proof_of_mpn_cmp_safety_wit_2 : mpn_cmp_safety_wit_2.
Axiom proof_of_mpn_cmp_safety_wit_3 : mpn_cmp_safety_wit_3.
Axiom proof_of_mpn_cmp_safety_wit_4 : mpn_cmp_safety_wit_4.
Axiom proof_of_mpn_cmp_safety_wit_5 : mpn_cmp_safety_wit_5.
Axiom proof_of_mpn_cmp_safety_wit_6 : mpn_cmp_safety_wit_6.
Axiom proof_of_mpn_cmp_safety_wit_7 : mpn_cmp_safety_wit_7.
Axiom proof_of_mpn_cmp_entail_wit_1 : mpn_cmp_entail_wit_1.
Axiom proof_of_mpn_cmp_entail_wit_2 : mpn_cmp_entail_wit_2.
Axiom proof_of_mpn_cmp_return_wit_1_1 : mpn_cmp_return_wit_1_1.
Axiom proof_of_mpn_cmp_return_wit_1_2 : mpn_cmp_return_wit_1_2.
Axiom proof_of_mpn_cmp_return_wit_2 : mpn_cmp_return_wit_2.
Axiom proof_of_mpn_cmp_partial_solve_wit_1 : mpn_cmp_partial_solve_wit_1.
Axiom proof_of_mpn_cmp_partial_solve_wit_2 : mpn_cmp_partial_solve_wit_2.
Axiom proof_of_mpn_cmp_partial_solve_wit_3 : mpn_cmp_partial_solve_wit_3.
Axiom proof_of_mpn_cmp_which_implies_wit_1 : mpn_cmp_which_implies_wit_1.
End VC_Correct.

4
projects/lib/gmp_goal_check.v
View File

@ -1,6 +1,6 @@
From Require Import gmp_goal gmp_proof_auto gmp_proof_manual.
From Require Import gmp_goal gmp_proof_auto gmp_proof_manual_tmp.
Module VC_Correctness : VC_Correct.
Include gmp_proof_auto.
Include gmp_proof_manual.
Include gmp_proof_manual_tmp.
End VC_Correctness.

30
projects/lib/gmp_proof_auto.v
View File

@ -63,3 +63,33 @@ Proof. Admitted.
Lemma proof_of_mpn_copyi_partial_solve_wit_5 : mpn_copyi_partial_solve_wit_5.
Proof. Admitted.
Lemma proof_of_mpn_cmp_safety_wit_1 : mpn_cmp_safety_wit_1.
Proof. Admitted.
Lemma proof_of_mpn_cmp_safety_wit_2 : mpn_cmp_safety_wit_2.
Proof. Admitted.
Lemma proof_of_mpn_cmp_safety_wit_3 : mpn_cmp_safety_wit_3.
Proof. Admitted.
Lemma proof_of_mpn_cmp_safety_wit_4 : mpn_cmp_safety_wit_4.
Proof. Admitted.
Lemma proof_of_mpn_cmp_safety_wit_5 : mpn_cmp_safety_wit_5.
Proof. Admitted.
Lemma proof_of_mpn_cmp_safety_wit_6 : mpn_cmp_safety_wit_6.
Proof. Admitted.
Lemma proof_of_mpn_cmp_safety_wit_7 : mpn_cmp_safety_wit_7.
Proof. Admitted.
Lemma proof_of_mpn_cmp_partial_solve_wit_1 : mpn_cmp_partial_solve_wit_1.
Proof. Admitted.
Lemma proof_of_mpn_cmp_partial_solve_wit_2 : mpn_cmp_partial_solve_wit_2.
Proof. Admitted.
Lemma proof_of_mpn_cmp_partial_solve_wit_3 : mpn_cmp_partial_solve_wit_3.
Proof. Admitted.

91
projects/lib/gmp_proof_manual.v
View File

@ -149,4 +149,95 @@ Proof.
rewrite H5 in H4; clear H5.
rewrite <-H4.
sep_apply store_uint_array_rec_tail_merge; [ reflexivity | lia ].
Qed.
Lemma proof_of_mpn_cmp_safety_wit_1 : mpn_cmp_safety_wit_1.
Proof.
pre_process.
Qed.
Lemma proof_of_mpn_cmp_entail_wit_1 : mpn_cmp_entail_wit_1.
Proof.
pre_process.
entailer!.
assert (n_pre - 1 + 1 = n_pre). { lia. }
rewrite H8; clear H8.
pose proof (Zlength_sublist n_pre n_pre l1 ltac:(lia)).
pose proof (Zlength_nil_inv (sublist n_pre n_pre l1) ltac:(lia)).
rewrite H9.
pose proof (Zlength_sublist n_pre n_pre l2 ltac:(lia)).
pose proof (Zlength_nil_inv (sublist n_pre n_pre l2) ltac:(lia)).
rewrite H11.
reflexivity.
Qed.
Lemma proof_of_mpn_cmp_entail_wit_2 : mpn_cmp_entail_wit_2.
Proof.
pre_process.
entailer!; try lia.
assert (n - 1 + 1 = n). { lia. }
rewrite H17; clear H17.
assert (n_pre <= Z.of_nat (Datatypes.length l1)). {
rewrite <-Zlength_correct.
lia.
}
assert (n_pre <= Z.of_nat (Datatypes.length l2)). {
rewrite <-Zlength_correct.
lia.
}
rewrite (sublist_split n n_pre (n + 1) l1 ltac:(lia) ltac:(lia)).
rewrite (sublist_split n n_pre (n + 1) l2 ltac:(lia) ltac:(lia)).
rewrite (sublist_single n l1 0 ltac:(lia)).
rewrite (sublist_single n l2 0 ltac:(lia)).
rewrite H.
rewrite H7.
reflexivity.
Qed.
Lemma proof_of_mpn_cmp_return_wit_1_1 : mpn_cmp_return_wit_1_1.
Proof.
pre_process.
entailer!.
repeat rewrite <-derivable1_orp_intros1.
entailer!.
+ unfold mpd_store_Z.
Exists l1 l2.
unfold mpd_store_list.
entailer!.
rewrite <-H6, <-H7.
entailer!.
+ assert (Znth n l1 0 < Znth n l2 0). { lia. }
clear H H0.
apply (list_store_Z_cmp l1 l2 val1 val2 n ltac:(lia) ltac:(lia) H4 H5).
- rewrite <-H6, <-H7.
tauto.
- lia.
Qed.
Lemma proof_of_mpn_cmp_return_wit_1_2 : mpn_cmp_return_wit_1_2.
Proof.
pre_process.
rewrite <-derivable1_orp_intros2.
entailer!.
+ unfold mpd_store_Z, mpd_store_list.
Exists l1 l2.
rewrite <-H6, <-H7.
entailer!.
+ pose proof (list_store_Z_cmp l2 l1 val2 val1 n ltac:(lia) ltac:(lia) H5 H4).
rewrite <-H6, <-H7 in H18.
rewrite H8 in H18.
specialize (H18 ltac:(reflexivity) ltac:(lia)).
lia.
Qed.
Lemma proof_of_mpn_cmp_which_implies_wit_1 : mpn_cmp_which_implies_wit_1.
Proof.
pre_process.
unfold mpd_store_Z.
Intros l1 l2.
Exists l2 l1.
unfold mpd_store_list.
entailer!.
rewrite <-H4, <-H6.
entailer!.
Qed.

32
projects/mini-gmp.c
View File

@ -104,44 +104,46 @@ mpn_cmp (unsigned int *ap, unsigned int *bp, int n)
Require
mpd_store_Z(ap, val1, n, cap1) *
mpd_store_Z(bp, val2, n, cap2) &&
n <= cap1 && n <= cap2
0 < n && n <= cap1 && n <= cap2 &&
cap1 <= 100000000 && cap2 <= 100000000
Ensure
val1 > val2 && __return == 1 ||
(val1 > val2 && __return == 1 ||
val1 == val2 && __return == 0 ||
val1 < val2 && __return == -1
val1 < val2 && __return == -1) &&
mpd_store_Z(ap@pre, val1, n@pre, cap1) *
mpd_store_Z(bp@pre, val2, n@pre, cap2)
*/
{
/*@
mpd_store_Z(ap, val1, n, cap1) * mpd_store_Z(bp, val2, n, cap2)
mpd_store_Z(ap@pre, val1, n@pre, cap1) * mpd_store_Z(bp@pre, val2, n@pre, cap2)
which implies
exists l1 l2,
mpd_store_list(ap, l1, cap1) * mpd_store_list(bp, l2, cap2) &&
store_uint_array(ap@pre, n@pre, l1) * store_uint_array(bp@pre, n@pre, l2) *
store_undef_uint_array_rec(ap@pre, n@pre + 1, cap1) *
store_undef_uint_array_rec(bp@pre, n@pre + 1, cap2) &&
list_store_Z(l1, val1) && list_store_Z(l2, val2) &&
n == Zlength(l1) && n == Zlength(l2)
n@pre == Zlength(l1) && n@pre == Zlength(l2)
*/
/*@
Given l1 l2
*/
--n;
/*@Inv
mpd_store_list(ap, l1, cap1) * mpd_store_list(bp, l2, cap2) &&
-1 <= n && n < n@pre &&
store_uint_array(ap@pre, n@pre, l1) * store_uint_array(bp@pre, n@pre, l2) *
store_undef_uint_array_rec(ap@pre, n@pre + 1, cap1) *
store_undef_uint_array_rec(bp@pre, n@pre + 1, cap2) &&
list_store_Z(l1, val1) && list_store_Z(l2, val2) &&
n@pre == Zlength(l1) && n@pre == Zlength(l2) &&
sublist(n, n@pre, l1) == sublist(n, n@pre, l2)
sublist(n + 1, n@pre, l1) == sublist(n + 1, n@pre, l2)
*/
while (n >= 0)
{
/*@
mpd_store_list(ap, l1, cap1) * mpd_store_list(bp, l2, cap2)
which implies
store_uint_array(ap, n, l1) * store_uint_array(bp, n, l2) *
store_undef_uint_array(ap, n + 1, cap1) * store_uint_array(bp, n + 1, cap2) &&
*/
if (ap[n] != bp[n])
return ap[n] > bp[n] ? 1 : -1;
--n;
}
// Note: The parser cannot parse "--n" in loop so we paraphrased it.
// Note: The parser cannot parse "--n" in loop condition so we paraphrased it.
/*
while (--n >= 0)
{