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Merge pull request #1 from xiaoh105/mpn_add_1

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Mpn add 1
This commit is contained in:
happyZYM
2025-06-21 19:49:18 +08:00
committed by GitHub
parent c206e4165f 5c69a81a44
commit 77e1460ebc
7 changed files with 911 additions and 120 deletions
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17
projects/lib/GmpAux.v
View File

@ -21,6 +21,18 @@ Local Open Scope sac.
Module Aux.
Lemma Z_mod_add_carry: forall (a b m: Z),
m > 0 -> 0 <= a < m -> 0 <= b < m ->
(a + b) mod m < b ->
a + b = (a + b) mod m + m.
Proof. Admitted.
Lemma Z_mod_add_uncarry: forall (a b m: Z),
m > 0 -> 0 <= a < m -> 0 <= b < m ->
(a + b) mod m >= b ->
a + b = (a + b) mod m.
Proof. Admitted.
Lemma Z_of_nat_succ: forall (n: nat),
Z.of_nat (S n) = Z.of_nat n + 1.
Proof. lia. Qed.
@ -314,6 +326,11 @@ Proof.
split; tauto.
Qed.
Lemma store_uint_array_rec_def2undef: forall x a b l,
store_uint_array_rec x a b l |--
store_undef_uint_array_rec x a b.
Proof. Admitted.
Lemma store_undef_uint_array_rec_divide: forall x l mid r,
0 <= l <= r ->
l <= mid <= r ->

6
projects/lib/GmpNumber.v
View File

@ -89,6 +89,12 @@ Proof.
reflexivity.
Qed.
Lemma list_store_Z_compact_reverse_injection: forall l1 l2 n1 n2,
list_store_Z_compact l1 n1 ->
list_store_Z_compact l2 n2 ->
n1 = n2 -> l1 = l2.
Proof. Admitted.
Lemma __list_within_bound_concat_r: forall (l1: list Z) (a: Z),
list_within_bound l1 ->
0 <= a < UINT_MOD ->

574
projects/lib/gmp_goal.v
View File

@ -1781,8 +1781,14 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
.
Definition mpn_add_1_safety_wit_2 :=
forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) (l: (@list Z)) (l'': (@list Z)) (l': (@list Z)) (val2: Z) (val1: Z) (l_2: (@list Z)) (i: Z) ,
[| ((unsigned_last_nbits (((Znth i l_2 0) + b_pre )) (32)) >= b_pre) |]
forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) (l: (@list Z)) (b: Z) (l'': (@list Z)) (l': (@list Z)) (val2: Z) (val1: Z) (l_2: (@list Z)) (i: Z) (a: Z) (l''': (@list Z)) ,
[| (l'' = (cons (a) (l'''))) |]
&& [| (0 <= i) |]
&& [| (i < n_pre) |]
&& [| (n_pre <= cap2) |]
&& [| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) >= b) |]
&& [| (0 <= b) |]
&& [| (b <= UINT_MAX) |]
&& [| (i < n_pre) |]
&& [| (0 <= i) |]
&& [| (i <= n_pre) |]
@ -1790,7 +1796,8 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
&& [| (n_pre <= cap1) |]
&& [| (list_store_Z (sublist (0) (i) (l_2)) val1 ) |]
&& [| (list_store_Z l' val2 ) |]
&& [| ((val2 + (b_pre * (Z.lxor UINT_MOD i) ) ) = (val1 + b_pre )) |]
&& [| ((val2 + (b * (Z.pow (UINT_MOD) (i)) ) ) = (val1 + b_pre )) |]
&& [| ((Zlength (l')) = i) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (n_pre <= cap1) |]
&& [| ((Zlength (l)) = n_pre) |]
@ -1802,13 +1809,13 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
&& [| (cap2 <= 100000000) |]
&& [| (n_pre > 0) |]
&& [| (n_pre <= cap1) |]
&& (store_uint_array_rec rp_pre i cap2 (replace_Znth ((i - i )) ((unsigned_last_nbits (((Znth i l_2 0) + b_pre )) (32))) (l'')) )
** (store_uint_array ap_pre n_pre l_2 )
** ((( &( "r" ) )) # UInt |-> (unsigned_last_nbits (((Znth i l_2 0) + b_pre )) (32)))
&& (store_uint_array rp_pre (i + 1 ) (replace_Znth (i) ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32))) ((app (l') ((cons (a) (nil)))))) )
** ((( &( "i" ) )) # Int |-> i)
** (store_undef_uint_array_rec ap_pre n_pre cap1 )
** (store_uint_array rp_pre i l' )
** (store_uint_array_rec rp_pre (i + 1 ) cap2 l''' )
** ((( &( "b" ) )) # UInt |-> 0)
** (store_uint_array ap_pre n_pre l_2 )
** ((( &( "r" ) )) # UInt |-> (unsigned_last_nbits (((Znth i l_2 0) + b )) (32)))
** (store_undef_uint_array_rec ap_pre n_pre cap1 )
** ((( &( "n" ) )) # Int |-> n_pre)
** ((( &( "ap" ) )) # Ptr |-> ap_pre)
** ((( &( "rp" ) )) # Ptr |-> rp_pre)
@ -1818,8 +1825,14 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
.
Definition mpn_add_1_safety_wit_3 :=
forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) (l: (@list Z)) (l'': (@list Z)) (l': (@list Z)) (val2: Z) (val1: Z) (l_2: (@list Z)) (i: Z) ,
[| ((unsigned_last_nbits (((Znth i l_2 0) + b_pre )) (32)) < b_pre) |]
forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) (l: (@list Z)) (b: Z) (l'': (@list Z)) (l': (@list Z)) (val2: Z) (val1: Z) (l_2: (@list Z)) (i: Z) (a: Z) (l''': (@list Z)) ,
[| (l'' = (cons (a) (l'''))) |]
&& [| (0 <= i) |]
&& [| (i < n_pre) |]
&& [| (n_pre <= cap2) |]
&& [| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) < b) |]
&& [| (0 <= b) |]
&& [| (b <= UINT_MAX) |]
&& [| (i < n_pre) |]
&& [| (0 <= i) |]
&& [| (i <= n_pre) |]
@ -1827,7 +1840,8 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
&& [| (n_pre <= cap1) |]
&& [| (list_store_Z (sublist (0) (i) (l_2)) val1 ) |]
&& [| (list_store_Z l' val2 ) |]
&& [| ((val2 + (b_pre * (Z.lxor UINT_MOD i) ) ) = (val1 + b_pre )) |]
&& [| ((val2 + (b * (Z.pow (UINT_MOD) (i)) ) ) = (val1 + b_pre )) |]
&& [| ((Zlength (l')) = i) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (n_pre <= cap1) |]
&& [| ((Zlength (l)) = n_pre) |]
@ -1839,13 +1853,13 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
&& [| (cap2 <= 100000000) |]
&& [| (n_pre > 0) |]
&& [| (n_pre <= cap1) |]
&& (store_uint_array_rec rp_pre i cap2 (replace_Znth ((i - i )) ((unsigned_last_nbits (((Znth i l_2 0) + b_pre )) (32))) (l'')) )
** (store_uint_array ap_pre n_pre l_2 )
** ((( &( "r" ) )) # UInt |-> (unsigned_last_nbits (((Znth i l_2 0) + b_pre )) (32)))
&& (store_uint_array rp_pre (i + 1 ) (replace_Znth (i) ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32))) ((app (l') ((cons (a) (nil)))))) )
** ((( &( "i" ) )) # Int |-> i)
** (store_undef_uint_array_rec ap_pre n_pre cap1 )
** (store_uint_array rp_pre i l' )
** (store_uint_array_rec rp_pre (i + 1 ) cap2 l''' )
** ((( &( "b" ) )) # UInt |-> 1)
** (store_uint_array ap_pre n_pre l_2 )
** ((( &( "r" ) )) # UInt |-> (unsigned_last_nbits (((Znth i l_2 0) + b )) (32)))
** (store_undef_uint_array_rec ap_pre n_pre cap1 )
** ((( &( "n" ) )) # Int |-> n_pre)
** ((( &( "ap" ) )) # Ptr |-> ap_pre)
** ((( &( "rp" ) )) # Ptr |-> rp_pre)
@ -1879,7 +1893,8 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
&& [| (n_pre <= cap1) |]
&& [| (list_store_Z (sublist (0) (0) (l)) val1 ) |]
&& [| (list_store_Z l' val2 ) |]
&& [| ((val2 + (b_pre * (Z.lxor UINT_MOD 0) ) ) = (val1 + b_pre )) |]
&& [| ((val2 + (b_pre * (Z.pow (UINT_MOD) (0)) ) ) = (val1 + b_pre )) |]
&& [| ((Zlength (l')) = 0) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (n_pre <= cap1) |]
&& [| ((Zlength (l_2)) = n_pre) |]
@ -1897,63 +1912,82 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
** (store_uint_array_rec rp_pre 0 cap2 l'' )
.
Definition mpn_add_1_entail_wit_2_1 :=
forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) (l_2: (@list Z)) (l''_2: (@list Z)) (l'_2: (@list Z)) (val2_2: Z) (val1_2: Z) (l_3: (@list Z)) (i: Z) ,
[| ((unsigned_last_nbits (((Znth i l_3 0) + b_pre )) (32)) < b_pre) |]
Definition mpn_add_1_entail_wit_2 :=
forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) (l: (@list Z)) (b: Z) (l'': (@list Z)) (l': (@list Z)) (val2: Z) (val1: Z) (l_2: (@list Z)) (i: Z) ,
[| (i < n_pre) |]
&& [| (0 <= i) |]
&& [| (i <= n_pre) |]
&& [| (list_store_Z_compact l_2 val ) |]
&& [| (n_pre <= cap1) |]
&& [| (list_store_Z (sublist (0) (i) (l_2)) val1 ) |]
&& [| (list_store_Z l' val2 ) |]
&& [| ((val2 + (b * (Z.pow (UINT_MOD) (i)) ) ) = (val1 + b_pre )) |]
&& [| ((Zlength (l')) = i) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (n_pre <= cap1) |]
&& [| ((Zlength (l)) = n_pre) |]
&& [| (cap1 <= 100000000) |]
&& [| (list_store_Z_compact l val ) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (cap2 >= n_pre) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& [| (n_pre > 0) |]
&& [| (n_pre <= cap1) |]
&& (store_uint_array ap_pre n_pre l_2 )
** ((( &( "r" ) )) # UInt |-> (unsigned_last_nbits (((Znth i l_2 0) + b )) (32)))
** ((( &( "i" ) )) # Int |-> i)
** (store_undef_uint_array_rec ap_pre n_pre cap1 )
** (store_uint_array rp_pre i l' )
** (store_uint_array_rec rp_pre i cap2 l'' )
** ((( &( "b" ) )) # UInt |-> b)
** ((( &( "n" ) )) # Int |-> n_pre)
** ((( &( "ap" ) )) # Ptr |-> ap_pre)
** ((( &( "rp" ) )) # Ptr |-> rp_pre)
|--
[| (0 <= b) |]
&& [| (b <= UINT_MAX) |]
&& [| (i < n_pre) |]
&& [| (0 <= i) |]
&& [| (i <= n_pre) |]
&& [| (list_store_Z_compact l_3 val ) |]
&& [| (n_pre <= cap1) |]
&& [| (list_store_Z (sublist (0) (i) (l_3)) val1_2 ) |]
&& [| (list_store_Z l'_2 val2_2 ) |]
&& [| ((val2_2 + (b_pre * (Z.lxor UINT_MOD i) ) ) = (val1_2 + b_pre )) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (n_pre <= cap1) |]
&& [| ((Zlength (l_2)) = n_pre) |]
&& [| (cap1 <= 100000000) |]
&& [| (list_store_Z_compact l_2 val ) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (cap2 >= n_pre) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& [| (n_pre > 0) |]
&& [| (n_pre <= cap1) |]
&& (store_uint_array_rec rp_pre i cap2 (replace_Znth ((i - i )) ((unsigned_last_nbits (((Znth i l_3 0) + b_pre )) (32))) (l''_2)) )
** (store_uint_array ap_pre n_pre l_3 )
** (store_undef_uint_array_rec ap_pre n_pre cap1 )
** (store_uint_array rp_pre i l'_2 )
** ((( &( "b" ) )) # UInt |-> 1)
|--
EX (l'': (@list Z)) (l': (@list Z)) (val2: Z) (val1: Z) (l: (@list Z)) ,
[| (0 <= (i + 1 )) |]
&& [| ((i + 1 ) <= n_pre) |]
&& [| (list_store_Z_compact l val ) |]
&& [| (n_pre <= cap1) |]
&& [| (list_store_Z (sublist (0) ((i + 1 )) (l)) val1 ) |]
&& [| (list_store_Z (sublist (0) (i) (l_2)) val1 ) |]
&& [| (list_store_Z l' val2 ) |]
&& [| ((val2 + (b_pre * (Z.lxor UINT_MOD (i + 1 )) ) ) = (val1 + b_pre )) |]
&& [| ((val2 + (b * (Z.pow (UINT_MOD) (i)) ) ) = (val1 + b_pre )) |]
&& [| ((Zlength (l')) = i) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (n_pre <= cap1) |]
&& [| ((Zlength (l_2)) = n_pre) |]
&& [| ((Zlength (l)) = n_pre) |]
&& [| (cap1 <= 100000000) |]
&& [| (list_store_Z_compact l_2 val ) |]
&& [| (list_store_Z_compact l val ) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (cap2 >= n_pre) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& [| (n_pre > 0) |]
&& [| (n_pre <= cap1) |]
&& (store_uint_array ap_pre n_pre l )
&& ((( &( "b" ) )) # UInt |-> b)
** (store_uint_array ap_pre n_pre l_2 )
** ((( &( "r" ) )) # UInt |-> (unsigned_last_nbits (((Znth i l_2 0) + b )) (32)))
** ((( &( "i" ) )) # Int |-> i)
** (store_undef_uint_array_rec ap_pre n_pre cap1 )
** (store_uint_array rp_pre (i + 1 ) l' )
** (store_uint_array_rec rp_pre (i + 1 ) cap2 l'' )
** ((( &( "b" ) )) # UInt |-> b_pre)
** (store_uint_array rp_pre i l' )
** (store_uint_array_rec rp_pre i cap2 l'' )
** ((( &( "n" ) )) # Int |-> n_pre)
** ((( &( "ap" ) )) # Ptr |-> ap_pre)
** ((( &( "rp" ) )) # Ptr |-> rp_pre)
.
Definition mpn_add_1_entail_wit_2_2 :=
forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) (l_2: (@list Z)) (l''_2: (@list Z)) (l'_2: (@list Z)) (val2_2: Z) (val1_2: Z) (l_3: (@list Z)) (i: Z) ,
[| ((unsigned_last_nbits (((Znth i l_3 0) + b_pre )) (32)) >= b_pre) |]
Definition mpn_add_1_entail_wit_3_1 :=
forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) (l_2: (@list Z)) (b: Z) (l''_2: (@list Z)) (l'_2: (@list Z)) (val2_2: Z) (val1_2: Z) (l_3: (@list Z)) (i: Z) (a: Z) (l''': (@list Z)) ,
[| (l''_2 = (cons (a) (l'''))) |]
&& [| (0 <= i) |]
&& [| (i < n_pre) |]
&& [| (n_pre <= cap2) |]
&& [| ((unsigned_last_nbits (((Znth i l_3 0) + b )) (32)) < b) |]
&& [| (0 <= b) |]
&& [| (b <= UINT_MAX) |]
&& [| (i < n_pre) |]
&& [| (0 <= i) |]
&& [| (i <= n_pre) |]
@ -1961,7 +1995,8 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
&& [| (n_pre <= cap1) |]
&& [| (list_store_Z (sublist (0) (i) (l_3)) val1_2 ) |]
&& [| (list_store_Z l'_2 val2_2 ) |]
&& [| ((val2_2 + (b_pre * (Z.lxor UINT_MOD i) ) ) = (val1_2 + b_pre )) |]
&& [| ((val2_2 + (b * (Z.pow (UINT_MOD) (i)) ) ) = (val1_2 + b_pre )) |]
&& [| ((Zlength (l'_2)) = i) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (n_pre <= cap1) |]
&& [| ((Zlength (l_2)) = n_pre) |]
@ -1973,11 +2008,10 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
&& [| (cap2 <= 100000000) |]
&& [| (n_pre > 0) |]
&& [| (n_pre <= cap1) |]
&& (store_uint_array_rec rp_pre i cap2 (replace_Znth ((i - i )) ((unsigned_last_nbits (((Znth i l_3 0) + b_pre )) (32))) (l''_2)) )
&& (store_uint_array rp_pre (i + 1 ) (replace_Znth (i) ((unsigned_last_nbits (((Znth i l_3 0) + b )) (32))) ((app (l'_2) ((cons (a) (nil)))))) )
** (store_uint_array_rec rp_pre (i + 1 ) cap2 l''' )
** (store_uint_array ap_pre n_pre l_3 )
** (store_undef_uint_array_rec ap_pre n_pre cap1 )
** (store_uint_array rp_pre i l'_2 )
** ((( &( "b" ) )) # UInt |-> 0)
|--
EX (l'': (@list Z)) (l': (@list Z)) (val2: Z) (val1: Z) (l: (@list Z)) ,
[| (0 <= (i + 1 )) |]
@ -1986,7 +2020,68 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
&& [| (n_pre <= cap1) |]
&& [| (list_store_Z (sublist (0) ((i + 1 )) (l)) val1 ) |]
&& [| (list_store_Z l' val2 ) |]
&& [| ((val2 + (b_pre * (Z.lxor UINT_MOD (i + 1 )) ) ) = (val1 + b_pre )) |]
&& [| ((val2 + (1 * (Z.pow (UINT_MOD) ((i + 1 ))) ) ) = (val1 + b_pre )) |]
&& [| ((Zlength (l')) = (i + 1 )) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (n_pre <= cap1) |]
&& [| ((Zlength (l_2)) = n_pre) |]
&& [| (cap1 <= 100000000) |]
&& [| (list_store_Z_compact l_2 val ) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (cap2 >= n_pre) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& [| (n_pre > 0) |]
&& [| (n_pre <= cap1) |]
&& (store_uint_array ap_pre n_pre l )
** (store_undef_uint_array_rec ap_pre n_pre cap1 )
** (store_uint_array rp_pre (i + 1 ) l' )
** (store_uint_array_rec rp_pre (i + 1 ) cap2 l'' )
.
Definition mpn_add_1_entail_wit_3_2 :=
forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) (l_2: (@list Z)) (b: Z) (l''_2: (@list Z)) (l'_2: (@list Z)) (val2_2: Z) (val1_2: Z) (l_3: (@list Z)) (i: Z) (a: Z) (l''': (@list Z)) ,
[| (l''_2 = (cons (a) (l'''))) |]
&& [| (0 <= i) |]
&& [| (i < n_pre) |]
&& [| (n_pre <= cap2) |]
&& [| ((unsigned_last_nbits (((Znth i l_3 0) + b )) (32)) >= b) |]
&& [| (0 <= b) |]
&& [| (b <= UINT_MAX) |]
&& [| (i < n_pre) |]
&& [| (0 <= i) |]
&& [| (i <= n_pre) |]
&& [| (list_store_Z_compact l_3 val ) |]
&& [| (n_pre <= cap1) |]
&& [| (list_store_Z (sublist (0) (i) (l_3)) val1_2 ) |]
&& [| (list_store_Z l'_2 val2_2 ) |]
&& [| ((val2_2 + (b * (Z.pow (UINT_MOD) (i)) ) ) = (val1_2 + b_pre )) |]
&& [| ((Zlength (l'_2)) = i) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (n_pre <= cap1) |]
&& [| ((Zlength (l_2)) = n_pre) |]
&& [| (cap1 <= 100000000) |]
&& [| (list_store_Z_compact l_2 val ) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (cap2 >= n_pre) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& [| (n_pre > 0) |]
&& [| (n_pre <= cap1) |]
&& (store_uint_array rp_pre (i + 1 ) (replace_Znth (i) ((unsigned_last_nbits (((Znth i l_3 0) + b )) (32))) ((app (l'_2) ((cons (a) (nil)))))) )
** (store_uint_array_rec rp_pre (i + 1 ) cap2 l''' )
** (store_uint_array ap_pre n_pre l_3 )
** (store_undef_uint_array_rec ap_pre n_pre cap1 )
|--
EX (l'': (@list Z)) (l': (@list Z)) (val2: Z) (val1: Z) (l: (@list Z)) ,
[| (0 <= (i + 1 )) |]
&& [| ((i + 1 ) <= n_pre) |]
&& [| (list_store_Z_compact l val ) |]
&& [| (n_pre <= cap1) |]
&& [| (list_store_Z (sublist (0) ((i + 1 )) (l)) val1 ) |]
&& [| (list_store_Z l' val2 ) |]
&& [| ((val2 + (0 * (Z.pow (UINT_MOD) ((i + 1 ))) ) ) = (val1 + b_pre )) |]
&& [| ((Zlength (l')) = (i + 1 )) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (n_pre <= cap1) |]
&& [| ((Zlength (l_2)) = n_pre) |]
@ -2002,11 +2097,10 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
** (store_undef_uint_array_rec ap_pre n_pre cap1 )
** (store_uint_array rp_pre (i + 1 ) l' )
** (store_uint_array_rec rp_pre (i + 1 ) cap2 l'' )
** ((( &( "b" ) )) # UInt |-> b_pre)
.
Definition mpn_add_1_return_wit_1 :=
forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) (l_2: (@list Z)) (l'': (@list Z)) (l': (@list Z)) (val2: Z) (val1: Z) (l: (@list Z)) (i: Z) ,
forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) (l_2: (@list Z)) (b: Z) (l'': (@list Z)) (l': (@list Z)) (val2: Z) (val1: Z) (l: (@list Z)) (i: Z) ,
[| (i >= n_pre) |]
&& [| (0 <= i) |]
&& [| (i <= n_pre) |]
@ -2014,7 +2108,8 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
&& [| (n_pre <= cap1) |]
&& [| (list_store_Z (sublist (0) (i) (l)) val1 ) |]
&& [| (list_store_Z l' val2 ) |]
&& [| ((val2 + (b_pre * (Z.lxor UINT_MOD i) ) ) = (val1 + b_pre )) |]
&& [| ((val2 + (b * (Z.pow (UINT_MOD) (i)) ) ) = (val1 + b_pre )) |]
&& [| ((Zlength (l')) = i) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (n_pre <= cap1) |]
&& [| ((Zlength (l_2)) = n_pre) |]
@ -2032,9 +2127,9 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
** (store_uint_array_rec rp_pre i cap2 l'' )
|--
EX (val': Z) ,
[| ((val' + (b_pre * (Z.lxor UINT_MOD n_pre) ) ) = (val + b_pre )) |]
[| ((val' + (b * (Z.pow (UINT_MOD) (n_pre)) ) ) = (val + b_pre )) |]
&& (mpd_store_Z_compact ap_pre val n_pre cap1 )
** (mpd_store_Z_compact rp_pre val' n_pre cap2 )
** (mpd_store_Z rp_pre val' n_pre cap2 )
.
Definition mpn_add_1_partial_solve_wit_1 :=
@ -2117,7 +2212,7 @@ forall (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (v
Definition mpn_add_1_partial_solve_wit_2 := mpn_add_1_partial_solve_wit_2_pure -> mpn_add_1_partial_solve_wit_2_aux.
Definition mpn_add_1_partial_solve_wit_3 :=
forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) (l: (@list Z)) (l'': (@list Z)) (l': (@list Z)) (val2: Z) (val1: Z) (l_2: (@list Z)) (i: Z) ,
forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) (l: (@list Z)) (b: Z) (l'': (@list Z)) (l': (@list Z)) (val2: Z) (val1: Z) (l_2: (@list Z)) (i: Z) ,
[| (i < n_pre) |]
&& [| (0 <= i) |]
&& [| (i <= n_pre) |]
@ -2125,7 +2220,8 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
&& [| (n_pre <= cap1) |]
&& [| (list_store_Z (sublist (0) (i) (l_2)) val1 ) |]
&& [| (list_store_Z l' val2 ) |]
&& [| ((val2 + (b_pre * (Z.lxor UINT_MOD i) ) ) = (val1 + b_pre )) |]
&& [| ((val2 + (b * (Z.pow (UINT_MOD) (i)) ) ) = (val1 + b_pre )) |]
&& [| ((Zlength (l')) = i) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (n_pre <= cap1) |]
&& [| ((Zlength (l)) = n_pre) |]
@ -2149,7 +2245,8 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
&& [| (n_pre <= cap1) |]
&& [| (list_store_Z (sublist (0) (i) (l_2)) val1 ) |]
&& [| (list_store_Z l' val2 ) |]
&& [| ((val2 + (b_pre * (Z.lxor UINT_MOD i) ) ) = (val1 + b_pre )) |]
&& [| ((val2 + (b * (Z.pow (UINT_MOD) (i)) ) ) = (val1 + b_pre )) |]
&& [| ((Zlength (l')) = i) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (n_pre <= cap1) |]
&& [| ((Zlength (l)) = n_pre) |]
@ -2168,9 +2265,11 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
** (store_uint_array_rec rp_pre i cap2 l'' )
.
Definition mpn_add_1_partial_solve_wit_4 :=
forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) (l: (@list Z)) (l'': (@list Z)) (l': (@list Z)) (val2: Z) (val1: Z) (l_2: (@list Z)) (i: Z) ,
[| ((unsigned_last_nbits (((Znth i l_2 0) + b_pre )) (32)) < b_pre) |]
Definition mpn_add_1_partial_solve_wit_4_pure :=
forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) (l: (@list Z)) (b: Z) (l'': (@list Z)) (l': (@list Z)) (val2: Z) (val1: Z) (l_2: (@list Z)) (i: Z) ,
[| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) >= b) |]
&& [| (0 <= b) |]
&& [| (b <= UINT_MAX) |]
&& [| (i < n_pre) |]
&& [| (0 <= i) |]
&& [| (i <= n_pre) |]
@ -2178,7 +2277,8 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
&& [| (n_pre <= cap1) |]
&& [| (list_store_Z (sublist (0) (i) (l_2)) val1 ) |]
&& [| (list_store_Z l' val2 ) |]
&& [| ((val2 + (b_pre * (Z.lxor UINT_MOD i) ) ) = (val1 + b_pre )) |]
&& [| ((val2 + (b * (Z.pow (UINT_MOD) (i)) ) ) = (val1 + b_pre )) |]
&& [| ((Zlength (l')) = i) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (n_pre <= cap1) |]
&& [| ((Zlength (l)) = n_pre) |]
@ -2190,41 +2290,27 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
&& [| (cap2 <= 100000000) |]
&& [| (n_pre > 0) |]
&& [| (n_pre <= cap1) |]
&& (store_uint_array ap_pre n_pre l_2 )
&& ((( &( "b" ) )) # UInt |-> 0)
** (store_uint_array ap_pre n_pre l_2 )
** ((( &( "r" ) )) # UInt |-> (unsigned_last_nbits (((Znth i l_2 0) + b )) (32)))
** ((( &( "i" ) )) # Int |-> i)
** (store_undef_uint_array_rec ap_pre n_pre cap1 )
** (store_uint_array rp_pre i l' )
** (store_uint_array_rec rp_pre i cap2 l'' )
** ((( &( "n" ) )) # Int |-> n_pre)
** ((( &( "ap" ) )) # Ptr |-> ap_pre)
** ((( &( "rp" ) )) # Ptr |-> rp_pre)
|--
[| ((unsigned_last_nbits (((Znth i l_2 0) + b_pre )) (32)) < b_pre) |]
[| (0 <= i) |]
&& [| (i < n_pre) |]
&& [| (0 <= i) |]
&& [| (i <= n_pre) |]
&& [| (list_store_Z_compact l_2 val ) |]
&& [| (n_pre <= cap1) |]
&& [| (list_store_Z (sublist (0) (i) (l_2)) val1 ) |]
&& [| (list_store_Z l' val2 ) |]
&& [| ((val2 + (b_pre * (Z.lxor UINT_MOD i) ) ) = (val1 + b_pre )) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (n_pre <= cap1) |]
&& [| ((Zlength (l)) = n_pre) |]
&& [| (cap1 <= 100000000) |]
&& [| (list_store_Z_compact l val ) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (cap2 >= n_pre) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& [| (n_pre > 0) |]
&& [| (n_pre <= cap1) |]
&& (((rp_pre + (i * sizeof(UINT) ) )) # UInt |->_)
** (store_uint_array_missing_i_rec rp_pre i i cap2 l'' )
** (store_uint_array ap_pre n_pre l_2 )
** (store_undef_uint_array_rec ap_pre n_pre cap1 )
** (store_uint_array rp_pre i l' )
&& [| (n_pre <= cap2) |]
.
Definition mpn_add_1_partial_solve_wit_5 :=
forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) (l: (@list Z)) (l'': (@list Z)) (l': (@list Z)) (val2: Z) (val1: Z) (l_2: (@list Z)) (i: Z) ,
[| ((unsigned_last_nbits (((Znth i l_2 0) + b_pre )) (32)) >= b_pre) |]
Definition mpn_add_1_partial_solve_wit_4_aux :=
forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) (l: (@list Z)) (b: Z) (l'': (@list Z)) (l': (@list Z)) (val2: Z) (val1: Z) (l_2: (@list Z)) (i: Z) ,
[| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) >= b) |]
&& [| (0 <= b) |]
&& [| (b <= UINT_MAX) |]
&& [| (i < n_pre) |]
&& [| (0 <= i) |]
&& [| (i <= n_pre) |]
@ -2232,7 +2318,8 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
&& [| (n_pre <= cap1) |]
&& [| (list_store_Z (sublist (0) (i) (l_2)) val1 ) |]
&& [| (list_store_Z l' val2 ) |]
&& [| ((val2 + (b_pre * (Z.lxor UINT_MOD i) ) ) = (val1 + b_pre )) |]
&& [| ((val2 + (b * (Z.pow (UINT_MOD) (i)) ) ) = (val1 + b_pre )) |]
&& [| ((Zlength (l')) = i) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (n_pre <= cap1) |]
&& [| ((Zlength (l)) = n_pre) |]
@ -2249,7 +2336,12 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
** (store_uint_array rp_pre i l' )
** (store_uint_array_rec rp_pre i cap2 l'' )
|--
[| ((unsigned_last_nbits (((Znth i l_2 0) + b_pre )) (32)) >= b_pre) |]
[| (0 <= i) |]
&& [| (i < n_pre) |]
&& [| (n_pre <= cap2) |]
&& [| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) >= b) |]
&& [| (0 <= b) |]
&& [| (b <= UINT_MAX) |]
&& [| (i < n_pre) |]
&& [| (0 <= i) |]
&& [| (i <= n_pre) |]
@ -2257,7 +2349,182 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
&& [| (n_pre <= cap1) |]
&& [| (list_store_Z (sublist (0) (i) (l_2)) val1 ) |]
&& [| (list_store_Z l' val2 ) |]
&& [| ((val2 + (b_pre * (Z.lxor UINT_MOD i) ) ) = (val1 + b_pre )) |]
&& [| ((val2 + (b * (Z.pow (UINT_MOD) (i)) ) ) = (val1 + b_pre )) |]
&& [| ((Zlength (l')) = i) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (n_pre <= cap1) |]
&& [| ((Zlength (l)) = n_pre) |]
&& [| (cap1 <= 100000000) |]
&& [| (list_store_Z_compact l val ) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (cap2 >= n_pre) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& [| (n_pre > 0) |]
&& [| (n_pre <= cap1) |]
&& (store_uint_array rp_pre i l' )
** (store_uint_array_rec rp_pre i cap2 l'' )
** (store_uint_array ap_pre n_pre l_2 )
** (store_undef_uint_array_rec ap_pre n_pre cap1 )
.
Definition mpn_add_1_partial_solve_wit_4 := mpn_add_1_partial_solve_wit_4_pure -> mpn_add_1_partial_solve_wit_4_aux.
Definition mpn_add_1_partial_solve_wit_5_pure :=
forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) (l: (@list Z)) (b: Z) (l'': (@list Z)) (l': (@list Z)) (val2: Z) (val1: Z) (l_2: (@list Z)) (i: Z) ,
[| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) < b) |]
&& [| (0 <= b) |]
&& [| (b <= UINT_MAX) |]
&& [| (i < n_pre) |]
&& [| (0 <= i) |]
&& [| (i <= n_pre) |]
&& [| (list_store_Z_compact l_2 val ) |]
&& [| (n_pre <= cap1) |]
&& [| (list_store_Z (sublist (0) (i) (l_2)) val1 ) |]
&& [| (list_store_Z l' val2 ) |]
&& [| ((val2 + (b * (Z.pow (UINT_MOD) (i)) ) ) = (val1 + b_pre )) |]
&& [| ((Zlength (l')) = i) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (n_pre <= cap1) |]
&& [| ((Zlength (l)) = n_pre) |]
&& [| (cap1 <= 100000000) |]
&& [| (list_store_Z_compact l val ) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (cap2 >= n_pre) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& [| (n_pre > 0) |]
&& [| (n_pre <= cap1) |]
&& ((( &( "b" ) )) # UInt |-> 1)
** (store_uint_array ap_pre n_pre l_2 )
** ((( &( "r" ) )) # UInt |-> (unsigned_last_nbits (((Znth i l_2 0) + b )) (32)))
** ((( &( "i" ) )) # Int |-> i)
** (store_undef_uint_array_rec ap_pre n_pre cap1 )
** (store_uint_array rp_pre i l' )
** (store_uint_array_rec rp_pre i cap2 l'' )
** ((( &( "n" ) )) # Int |-> n_pre)
** ((( &( "ap" ) )) # Ptr |-> ap_pre)
** ((( &( "rp" ) )) # Ptr |-> rp_pre)
|--
[| (0 <= i) |]
&& [| (i < n_pre) |]
&& [| (n_pre <= cap2) |]
.
Definition mpn_add_1_partial_solve_wit_5_aux :=
forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) (l: (@list Z)) (b: Z) (l'': (@list Z)) (l': (@list Z)) (val2: Z) (val1: Z) (l_2: (@list Z)) (i: Z) ,
[| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) < b) |]
&& [| (0 <= b) |]
&& [| (b <= UINT_MAX) |]
&& [| (i < n_pre) |]
&& [| (0 <= i) |]
&& [| (i <= n_pre) |]
&& [| (list_store_Z_compact l_2 val ) |]
&& [| (n_pre <= cap1) |]
&& [| (list_store_Z (sublist (0) (i) (l_2)) val1 ) |]
&& [| (list_store_Z l' val2 ) |]
&& [| ((val2 + (b * (Z.pow (UINT_MOD) (i)) ) ) = (val1 + b_pre )) |]
&& [| ((Zlength (l')) = i) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (n_pre <= cap1) |]
&& [| ((Zlength (l)) = n_pre) |]
&& [| (cap1 <= 100000000) |]
&& [| (list_store_Z_compact l val ) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (cap2 >= n_pre) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& [| (n_pre > 0) |]
&& [| (n_pre <= cap1) |]
&& (store_uint_array ap_pre n_pre l_2 )
** (store_undef_uint_array_rec ap_pre n_pre cap1 )
** (store_uint_array rp_pre i l' )
** (store_uint_array_rec rp_pre i cap2 l'' )
|--
[| (0 <= i) |]
&& [| (i < n_pre) |]
&& [| (n_pre <= cap2) |]
&& [| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) < b) |]
&& [| (0 <= b) |]
&& [| (b <= UINT_MAX) |]
&& [| (i < n_pre) |]
&& [| (0 <= i) |]
&& [| (i <= n_pre) |]
&& [| (list_store_Z_compact l_2 val ) |]
&& [| (n_pre <= cap1) |]
&& [| (list_store_Z (sublist (0) (i) (l_2)) val1 ) |]
&& [| (list_store_Z l' val2 ) |]
&& [| ((val2 + (b * (Z.pow (UINT_MOD) (i)) ) ) = (val1 + b_pre )) |]
&& [| ((Zlength (l')) = i) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (n_pre <= cap1) |]
&& [| ((Zlength (l)) = n_pre) |]
&& [| (cap1 <= 100000000) |]
&& [| (list_store_Z_compact l val ) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (cap2 >= n_pre) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& [| (n_pre > 0) |]
&& [| (n_pre <= cap1) |]
&& (store_uint_array rp_pre i l' )
** (store_uint_array_rec rp_pre i cap2 l'' )
** (store_uint_array ap_pre n_pre l_2 )
** (store_undef_uint_array_rec ap_pre n_pre cap1 )
.
Definition mpn_add_1_partial_solve_wit_5 := mpn_add_1_partial_solve_wit_5_pure -> mpn_add_1_partial_solve_wit_5_aux.
Definition mpn_add_1_partial_solve_wit_6 :=
forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) (l: (@list Z)) (b: Z) (l'': (@list Z)) (l': (@list Z)) (val2: Z) (val1: Z) (l_2: (@list Z)) (i: Z) (a: Z) (l''': (@list Z)) ,
[| (l'' = (cons (a) (l'''))) |]
&& [| (0 <= i) |]
&& [| (i < n_pre) |]
&& [| (n_pre <= cap2) |]
&& [| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) < b) |]
&& [| (0 <= b) |]
&& [| (b <= UINT_MAX) |]
&& [| (i < n_pre) |]
&& [| (0 <= i) |]
&& [| (i <= n_pre) |]
&& [| (list_store_Z_compact l_2 val ) |]
&& [| (n_pre <= cap1) |]
&& [| (list_store_Z (sublist (0) (i) (l_2)) val1 ) |]
&& [| (list_store_Z l' val2 ) |]
&& [| ((val2 + (b * (Z.pow (UINT_MOD) (i)) ) ) = (val1 + b_pre )) |]
&& [| ((Zlength (l')) = i) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (n_pre <= cap1) |]
&& [| ((Zlength (l)) = n_pre) |]
&& [| (cap1 <= 100000000) |]
&& [| (list_store_Z_compact l val ) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (cap2 >= n_pre) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& [| (n_pre > 0) |]
&& [| (n_pre <= cap1) |]
&& (store_uint_array_rec rp_pre (i + 1 ) cap2 l''' )
** (store_uint_array rp_pre (i + 1 ) (app (l') ((cons (a) (nil)))) )
** (store_uint_array ap_pre n_pre l_2 )
** (store_undef_uint_array_rec ap_pre n_pre cap1 )
|--
[| (l'' = (cons (a) (l'''))) |]
&& [| (0 <= i) |]
&& [| (i < n_pre) |]
&& [| (n_pre <= cap2) |]
&& [| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) < b) |]
&& [| (0 <= b) |]
&& [| (b <= UINT_MAX) |]
&& [| (i < n_pre) |]
&& [| (0 <= i) |]
&& [| (i <= n_pre) |]
&& [| (list_store_Z_compact l_2 val ) |]
&& [| (n_pre <= cap1) |]
&& [| (list_store_Z (sublist (0) (i) (l_2)) val1 ) |]
&& [| (list_store_Z l' val2 ) |]
&& [| ((val2 + (b * (Z.pow (UINT_MOD) (i)) ) ) = (val1 + b_pre )) |]
&& [| ((Zlength (l')) = i) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (n_pre <= cap1) |]
&& [| ((Zlength (l)) = n_pre) |]
@ -2270,10 +2537,78 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
&& [| (n_pre > 0) |]
&& [| (n_pre <= cap1) |]
&& (((rp_pre + (i * sizeof(UINT) ) )) # UInt |->_)
** (store_uint_array_missing_i_rec rp_pre i i cap2 l'' )
** (store_uint_array_missing_i_rec rp_pre i 0 (i + 1 ) (app (l') ((cons (a) (nil)))) )
** (store_uint_array_rec rp_pre (i + 1 ) cap2 l''' )
** (store_uint_array ap_pre n_pre l_2 )
** (store_undef_uint_array_rec ap_pre n_pre cap1 )
.
Definition mpn_add_1_partial_solve_wit_7 :=
forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) (l: (@list Z)) (b: Z) (l'': (@list Z)) (l': (@list Z)) (val2: Z) (val1: Z) (l_2: (@list Z)) (i: Z) (a: Z) (l''': (@list Z)) ,
[| (l'' = (cons (a) (l'''))) |]
&& [| (0 <= i) |]
&& [| (i < n_pre) |]
&& [| (n_pre <= cap2) |]
&& [| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) >= b) |]
&& [| (0 <= b) |]
&& [| (b <= UINT_MAX) |]
&& [| (i < n_pre) |]
&& [| (0 <= i) |]
&& [| (i <= n_pre) |]
&& [| (list_store_Z_compact l_2 val ) |]
&& [| (n_pre <= cap1) |]
&& [| (list_store_Z (sublist (0) (i) (l_2)) val1 ) |]
&& [| (list_store_Z l' val2 ) |]
&& [| ((val2 + (b * (Z.pow (UINT_MOD) (i)) ) ) = (val1 + b_pre )) |]
&& [| ((Zlength (l')) = i) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (n_pre <= cap1) |]
&& [| ((Zlength (l)) = n_pre) |]
&& [| (cap1 <= 100000000) |]
&& [| (list_store_Z_compact l val ) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (cap2 >= n_pre) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& [| (n_pre > 0) |]
&& [| (n_pre <= cap1) |]
&& (store_uint_array_rec rp_pre (i + 1 ) cap2 l''' )
** (store_uint_array rp_pre (i + 1 ) (app (l') ((cons (a) (nil)))) )
** (store_uint_array ap_pre n_pre l_2 )
** (store_undef_uint_array_rec ap_pre n_pre cap1 )
|--
[| (l'' = (cons (a) (l'''))) |]
&& [| (0 <= i) |]
&& [| (i < n_pre) |]
&& [| (n_pre <= cap2) |]
&& [| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) >= b) |]
&& [| (0 <= b) |]
&& [| (b <= UINT_MAX) |]
&& [| (i < n_pre) |]
&& [| (0 <= i) |]
&& [| (i <= n_pre) |]
&& [| (list_store_Z_compact l_2 val ) |]
&& [| (n_pre <= cap1) |]
&& [| (list_store_Z (sublist (0) (i) (l_2)) val1 ) |]
&& [| (list_store_Z l' val2 ) |]
&& [| ((val2 + (b * (Z.pow (UINT_MOD) (i)) ) ) = (val1 + b_pre )) |]
&& [| ((Zlength (l')) = i) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (n_pre <= cap1) |]
&& [| ((Zlength (l)) = n_pre) |]
&& [| (cap1 <= 100000000) |]
&& [| (list_store_Z_compact l val ) |]
&& [| ((Zlength (l2)) = cap2) |]
&& [| (cap2 >= n_pre) |]
&& [| (cap1 <= 100000000) |]
&& [| (cap2 <= 100000000) |]
&& [| (n_pre > 0) |]
&& [| (n_pre <= cap1) |]
&& (((rp_pre + (i * sizeof(UINT) ) )) # UInt |->_)
** (store_uint_array_missing_i_rec rp_pre i 0 (i + 1 ) (app (l') ((cons (a) (nil)))) )
** (store_uint_array_rec rp_pre (i + 1 ) cap2 l''' )
** (store_uint_array ap_pre n_pre l_2 )
** (store_undef_uint_array_rec ap_pre n_pre cap1 )
** (store_uint_array rp_pre i l' )
.
Definition mpn_add_1_which_implies_wit_1 :=
@ -2299,6 +2634,23 @@ forall (rp_pre: Z) (cap2: Z) (l2: (@list Z)) ,
** (store_uint_array rp_pre 0 nil )
.
Definition mpn_add_1_which_implies_wit_3 :=
forall (n_pre: Z) (rp_pre: Z) (cap2: Z) (l'': (@list Z)) (l': (@list Z)) (i: Z) ,
[| (0 <= i) |]
&& [| (i < n_pre) |]
&& [| (n_pre <= cap2) |]
&& (store_uint_array rp_pre i l' )
** (store_uint_array_rec rp_pre i cap2 l'' )
|--
EX (a: Z) (l''': (@list Z)) ,
[| (l'' = (cons (a) (l'''))) |]
&& [| (0 <= i) |]
&& [| (i < n_pre) |]
&& [| (n_pre <= cap2) |]
&& (store_uint_array_rec rp_pre (i + 1 ) cap2 l''' )
** (store_uint_array rp_pre (i + 1 ) (app (l') ((cons (a) (nil)))) )
.
Module Type VC_Correct.
Axiom proof_of_gmp_abs_safety_wit_1 : gmp_abs_safety_wit_1.
@ -2373,16 +2725,22 @@ Axiom proof_of_mpn_add_1_safety_wit_1 : mpn_add_1_safety_wit_1.
Axiom proof_of_mpn_add_1_safety_wit_2 : mpn_add_1_safety_wit_2.
Axiom proof_of_mpn_add_1_safety_wit_3 : mpn_add_1_safety_wit_3.
Axiom proof_of_mpn_add_1_entail_wit_1 : mpn_add_1_entail_wit_1.
Axiom proof_of_mpn_add_1_entail_wit_2_1 : mpn_add_1_entail_wit_2_1.
Axiom proof_of_mpn_add_1_entail_wit_2_2 : mpn_add_1_entail_wit_2_2.
Axiom proof_of_mpn_add_1_entail_wit_2 : mpn_add_1_entail_wit_2.
Axiom proof_of_mpn_add_1_entail_wit_3_1 : mpn_add_1_entail_wit_3_1.
Axiom proof_of_mpn_add_1_entail_wit_3_2 : mpn_add_1_entail_wit_3_2.
Axiom proof_of_mpn_add_1_return_wit_1 : mpn_add_1_return_wit_1.
Axiom proof_of_mpn_add_1_partial_solve_wit_1 : mpn_add_1_partial_solve_wit_1.
Axiom proof_of_mpn_add_1_partial_solve_wit_2_pure : mpn_add_1_partial_solve_wit_2_pure.
Axiom proof_of_mpn_add_1_partial_solve_wit_2 : mpn_add_1_partial_solve_wit_2.
Axiom proof_of_mpn_add_1_partial_solve_wit_3 : mpn_add_1_partial_solve_wit_3.
Axiom proof_of_mpn_add_1_partial_solve_wit_4_pure : mpn_add_1_partial_solve_wit_4_pure.
Axiom proof_of_mpn_add_1_partial_solve_wit_4 : mpn_add_1_partial_solve_wit_4.
Axiom proof_of_mpn_add_1_partial_solve_wit_5_pure : mpn_add_1_partial_solve_wit_5_pure.
Axiom proof_of_mpn_add_1_partial_solve_wit_5 : mpn_add_1_partial_solve_wit_5.
Axiom proof_of_mpn_add_1_partial_solve_wit_6 : mpn_add_1_partial_solve_wit_6.
Axiom proof_of_mpn_add_1_partial_solve_wit_7 : mpn_add_1_partial_solve_wit_7.
Axiom proof_of_mpn_add_1_which_implies_wit_1 : mpn_add_1_which_implies_wit_1.
Axiom proof_of_mpn_add_1_which_implies_wit_2 : mpn_add_1_which_implies_wit_2.
Axiom proof_of_mpn_add_1_which_implies_wit_3 : mpn_add_1_which_implies_wit_3.
End VC_Correct.

12
projects/lib/gmp_proof_auto.v
View File

@ -164,9 +164,21 @@ Proof. Admitted.
Lemma proof_of_mpn_add_1_partial_solve_wit_3 : mpn_add_1_partial_solve_wit_3.
Proof. Admitted.
Lemma proof_of_mpn_add_1_partial_solve_wit_4_pure : mpn_add_1_partial_solve_wit_4_pure.
Proof. Admitted.
Lemma proof_of_mpn_add_1_partial_solve_wit_4 : mpn_add_1_partial_solve_wit_4.
Proof. Admitted.
Lemma proof_of_mpn_add_1_partial_solve_wit_5_pure : mpn_add_1_partial_solve_wit_5_pure.
Proof. Admitted.
Lemma proof_of_mpn_add_1_partial_solve_wit_5 : mpn_add_1_partial_solve_wit_5.
Proof. Admitted.
Lemma proof_of_mpn_add_1_partial_solve_wit_6 : mpn_add_1_partial_solve_wit_6.
Proof. Admitted.
Lemma proof_of_mpn_add_1_partial_solve_wit_7 : mpn_add_1_partial_solve_wit_7.
Proof. Admitted.

403
projects/lib/gmp_proof_manual.v
View File

@ -11,7 +11,7 @@ Require Import SetsClass.SetsClass. Import SetsNotation.
From SimpleC.SL Require Import Mem SeparationLogic.
From GmpLib Require Import gmp_goal.
Require Import GmpLib.GmpNumber. Import Internal.
Require Import GmpLib.GmpAux.
Require Import GmpLib.GmpAux. Import Aux.
Require Import Logic.LogicGenerator.demo932.Interface.
Local Open Scope Z_scope.
Local Open Scope sets.
@ -407,19 +407,404 @@ Proof.
Qed.
Lemma proof_of_mpn_add_1_entail_wit_1 : mpn_add_1_entail_wit_1.
Proof. Admitted.
Proof.
pre_process.
Exists l2 nil 0 0 l_2.
entailer!.
- unfold list_store_Z.
split.
+ simpl. tauto.
+ simpl. tauto.
- rewrite (sublist_nil l_2 0 0); try lia.
unfold list_store_Z.
split.
+ simpl. tauto.
+ simpl. tauto.
Qed.
Lemma proof_of_mpn_add_1_entail_wit_2_1 : mpn_add_1_entail_wit_2_1.
Proof. Admitted.
Lemma proof_of_mpn_add_1_entail_wit_2 : mpn_add_1_entail_wit_2.
Proof.
pre_process.
prop_apply (store_uint_range &("b") b).
entailer!.
Qed.
Lemma proof_of_mpn_add_1_entail_wit_2_2 : mpn_add_1_entail_wit_2_2.
Proof. Admitted.
Lemma proof_of_mpn_add_1_entail_wit_3_1 : mpn_add_1_entail_wit_3_1.
Proof.
pre_process.
rewrite replace_Znth_app_r.
- Exists l'''.
rewrite H14.
assert (i - i = 0) by lia.
rewrite H26.
set (new_b := (unsigned_last_nbits (Znth i l_3 0 + b) 32)).
rewrite replace_Znth_nothing; try lia.
assert (replace_Znth 0 new_b (a :: nil) = new_b :: nil). {
unfold replace_Znth.
unfold Z.to_nat.
unfold replace_nth.
reflexivity.
}
rewrite H27.
Exists (l'_2 ++ new_b :: nil).
Exists (val2_2 + new_b * (UINT_MOD^ i)).
Exists (val1_2 + (Znth i l_3 0) * (UINT_MOD^ i)).
Exists l_3.
entailer!.
+ rewrite Zlength_app.
rewrite H14.
unfold Zlength.
unfold Zlength_aux.
lia.
+ assert (val1_2 + Znth i l_3 0 * 4294967296 ^ i + b_pre = (val1_2 + b_pre) + Znth i l_3 0 * 4294967296 ^ i) by lia.
rewrite H28.
rewrite <- H13.
assert (Znth i l_3 0 + b = new_b + UINT_MOD).
{
subst new_b.
unfold unsigned_last_nbits.
unfold unsigned_last_nbits in H3.
assert (2^32 = 4294967296). { nia. }
rewrite H29 in *.
assert (0 <= Znth i l_3 0 < 4294967296). {
assert (l_2=l_3).
{
pose proof (list_store_Z_compact_reverse_injection l_2 l_3 val val).
apply H30 in H9; try tauto.
}
assert (i < Zlength l_3). {
subst l_3.
rewrite H17.
tauto.
}
unfold list_store_Z_compact in H9.
apply list_within_bound_Znth.
lia.
tauto.
}
apply Z_mod_add_carry; try lia; try tauto.
}
assert (b * 4294967296 ^ i + Znth i l_3 0 * 4294967296 ^ i = new_b * 4294967296 ^ i + 1 * 4294967296 ^ (i + 1)).
{
subst new_b.
Search [ Zmult Zplus "distr" ].
rewrite <- Z.mul_add_distr_r.
rewrite (Zpow_add_1 4294967296 i); try lia.
}
lia.
+ pose proof (__list_store_Z_concat_r l'_2 val2_2 new_b).
apply H28 in H12.
rewrite H14 in H12.
assert (new_b * 4294967296 ^ i + val2_2 = (val2_2 + new_b * 4294967296 ^ i)) by lia.
rewrite H29 in H12.
tauto.
subst new_b.
unfold unsigned_last_nbits.
assert (2 ^ 32 = 4294967296). { nia. }
rewrite H29.
apply Z.mod_pos_bound.
lia.
+ assert (l_2=l_3).
{
pose proof (list_store_Z_compact_reverse_injection l_2 l_3 val val).
apply H28 in H9; try tauto.
}
assert (i < Zlength l_3). {
subst l_3.
rewrite H17.
tauto.
}
assert((sublist 0 (i + 1) l_3) = (sublist 0 i l_3) ++ (Znth i l_3 0) :: nil). {
pose proof (sublist_split 0 (i+1) i l_3).
pose proof (sublist_single i l_3 0).
rewrite <-H31.
apply H30.
lia.
subst l_3.
rewrite Zlength_correct in H29.
lia.
rewrite Zlength_correct in H29.
lia.
}
rewrite H30.
pose proof (__list_store_Z_concat_r (sublist 0 i l_3) val1_2 (Znth i l_3 0)).
apply H31 in H11.
rewrite Zlength_sublist0 in H11.
assert (val1_2 + Znth i l_3 0 * 4294967296 ^ i = Znth i l_3 0 * 4294967296 ^ i + val1_2) by lia.
rewrite H32.
tauto.
subst l_3.
rewrite H17.
lia.
apply list_within_bound_Znth.
lia.
unfold list_store_Z_compact in H9.
tauto.
- pose proof (Zlength_sublist0 i l'_2).
lia.
Qed.
Lemma proof_of_mpn_add_1_entail_wit_3_2 : mpn_add_1_entail_wit_3_2.
Proof.
pre_process.
rewrite replace_Znth_app_r.
- Exists l'''.
rewrite H14.
assert (i - i = 0) by lia.
rewrite H26.
set (new_b := (unsigned_last_nbits (Znth i l_3 0 + b) 32)).
rewrite replace_Znth_nothing; try lia.
assert (replace_Znth 0 new_b (a :: nil) = new_b :: nil). {
unfold replace_Znth.
unfold Z.to_nat.
unfold replace_nth.
reflexivity.
}
rewrite H27.
Exists (l'_2 ++ new_b :: nil).
Exists (val2_2 + new_b * (UINT_MOD^ i)).
Exists (val1_2 + (Znth i l_3 0) * (UINT_MOD^ i)).
Exists l_3.
entailer!.
+ rewrite Zlength_app.
rewrite H14.
unfold Zlength.
unfold Zlength_aux.
lia.
+ assert (val1_2 + Znth i l_3 0 * 4294967296 ^ i + b_pre = (val1_2 + b_pre) + Znth i l_3 0 * 4294967296 ^ i) by lia.
rewrite H28.
rewrite <- H13.
assert (Znth i l_3 0 + b = new_b).
{
subst new_b.
unfold unsigned_last_nbits.
unfold unsigned_last_nbits in H3.
assert (2^32 = 4294967296). { nia. }
rewrite H29 in *.
assert (0 <= Znth i l_3 0 < 4294967296). {
assert (l_2=l_3).
{
pose proof (list_store_Z_compact_reverse_injection l_2 l_3 val val).
apply H30 in H9; try tauto.
}
assert (i < Zlength l_3). {
subst l_3.
rewrite H17.
tauto.
}
unfold list_store_Z_compact in H9.
apply list_within_bound_Znth.
lia.
tauto.
}
apply Z_mod_add_uncarry; try lia; try tauto.
}
assert (b * 4294967296 ^ i + Znth i l_3 0 * 4294967296 ^ i = new_b * 4294967296 ^ i + 0 * 4294967296 ^ (i + 1)).
{
subst new_b.
Search [ Zmult Zplus "distr" ].
rewrite <- Z.mul_add_distr_r.
rewrite (Zpow_add_1 4294967296 i); try lia.
}
lia.
+ pose proof (__list_store_Z_concat_r l'_2 val2_2 new_b).
apply H28 in H12.
rewrite H14 in H12.
assert (new_b * 4294967296 ^ i + val2_2 = (val2_2 + new_b * 4294967296 ^ i)) by lia.
rewrite H29 in H12.
tauto.
subst new_b.
unfold unsigned_last_nbits.
assert (2 ^ 32 = 4294967296). { nia. }
rewrite H29.
apply Z.mod_pos_bound.
lia.
+ assert (l_2=l_3).
{
pose proof (list_store_Z_compact_reverse_injection l_2 l_3 val val).
apply H28 in H9; try tauto.
}
assert (i < Zlength l_3). {
subst l_3.
rewrite H17.
tauto.
}
assert((sublist 0 (i + 1) l_3) = (sublist 0 i l_3) ++ (Znth i l_3 0) :: nil). {
pose proof (sublist_split 0 (i+1) i l_3).
pose proof (sublist_single i l_3 0).
rewrite <-H31.
apply H30.
lia.
subst l_3.
rewrite Zlength_correct in H29.
lia.
rewrite Zlength_correct in H29.
lia.
}
rewrite H30.
pose proof (__list_store_Z_concat_r (sublist 0 i l_3) val1_2 (Znth i l_3 0)).
apply H31 in H11.
rewrite Zlength_sublist0 in H11.
assert (val1_2 + Znth i l_3 0 * 4294967296 ^ i = Znth i l_3 0 * 4294967296 ^ i + val1_2) by lia.
rewrite H32.
tauto.
subst l_3.
rewrite H17.
lia.
apply list_within_bound_Znth.
lia.
unfold list_store_Z_compact in H9.
tauto.
- pose proof (Zlength_sublist0 i l'_2).
lia.
Qed.
Lemma proof_of_mpn_add_1_return_wit_1 : mpn_add_1_return_wit_1.
Proof. Admitted.
Proof.
pre_process.
unfold mpd_store_Z_compact.
unfold mpd_store_list.
Exists val2.
pose proof (list_store_Z_compact_reverse_injection l l_2 val val).
apply H19 in H2; try tauto.
rewrite <-H2 in H10.
assert (i = n_pre) by lia.
rewrite H20 in H4.
rewrite <- H10 in H4.
rewrite (sublist_self l (Zlength l)) in H4; try tauto.
rewrite <-H2 in H12.
assert (list_store_Z l val). { apply list_store_Z_compact_to_normal. tauto. }
pose proof (list_store_Z_injection l l val1 val).
apply H22 in H4; try tauto.
rewrite H4 in H6.
entailer!.
Exists l.
entailer!.
entailer!; try rewrite H20; try tauto.
- rewrite H10.
entailer!.
unfold mpd_store_Z.
unfold mpd_store_list.
Exists l'.
rewrite H7.
subst i.
entailer!.
rewrite H20.
entailer!.
apply store_uint_array_rec_def2undef.
- rewrite <- H20. tauto.
Qed.
Lemma proof_of_mpn_add_1_which_implies_wit_1 : mpn_add_1_which_implies_wit_1.
Proof. Admitted.
Proof.
pre_process.
unfold mpd_store_Z_compact.
Intros l.
Exists l.
unfold mpd_store_list.
entailer!.
subst n_pre.
entailer!.
Qed.
Lemma proof_of_mpn_add_1_which_implies_wit_2 : mpn_add_1_which_implies_wit_2.
Proof. Admitted.
Proof.
pre_process.
pose proof (store_uint_array_divide rp_pre cap2 l2 0).
pose proof (Zlength_nonneg l2).
specialize (H0 ltac:(lia) ltac:(lia)).
destruct H0 as [H0 _].
simpl in H0.
entailer!.
rewrite (sublist_nil l2 0 0) in H0; [ | lia].
sep_apply H0.
entailer!.
unfold store_uint_array, store_uint_array_rec.
unfold store_array.
rewrite (sublist_self l2 cap2); [ | lia ].
assert (rp_pre + 0 = rp_pre). { lia. }
rewrite H2; clear H2.
assert (cap2 - 0 = cap2). { lia. }
rewrite H2; clear H2.
reflexivity.
Qed.
Lemma proof_of_mpn_add_1_which_implies_wit_3 : mpn_add_1_which_implies_wit_3.
Proof.
pre_process.
destruct l''. {
unfold store_uint_array_rec.
simpl.
entailer!.
}
pose proof (store_uint_array_rec_cons rp_pre i cap2 z l'' ltac:(lia)).
sep_apply H2.
Exists z l''.
entailer!.
assert (i = 0 \/ i > 0). { lia. }
destruct H3.
+ subst.
simpl.
entailer!.
simpl in H2.
assert (rp_pre + 0 = rp_pre). { lia. }
rewrite H3.
rewrite H3 in H2.
clear H3.
pose proof (store_uint_array_empty rp_pre l').
sep_apply H3.
rewrite logic_equiv_andp_comm.
rewrite logic_equiv_coq_prop_andp_sepcon.
Intros.
subst l'.
rewrite app_nil_l.
unfold store_uint_array.
unfold store_array.
unfold store_array_rec.
simpl.
assert (rp_pre + 0 = rp_pre). { lia. }
rewrite H4; clear H4.
entailer!.
+ pose proof (Aux.uint_array_rec_to_uint_array rp_pre 0 i (sublist 0 i l') ltac:(lia)).
destruct H4 as [_ H4].
assert (rp_pre + sizeof(UINT) * 0 = rp_pre). { lia. }
rewrite H5 in H4; clear H5.
assert (i - 0 = i). { lia. }
rewrite H5 in H4; clear H5.
pose proof (Aux.uint_array_rec_to_uint_array rp_pre 0 (i + 1) (sublist 0 i l' ++ z :: nil) ltac:(lia)).
destruct H5 as [H5 _].
assert (i + 1 - 0 = i + 1). { lia. }
rewrite H6 in H5; clear H6.
assert (rp_pre + sizeof(UINT) * 0 = rp_pre). { lia. }
rewrite H6 in H5; clear H6.
pose proof (uint_array_rec_to_uint_array rp_pre 0 i l').
specialize (H6 H).
assert ((rp_pre + sizeof ( UINT ) * 0) = rp_pre) by lia.
rewrite H7 in H6; clear H7.
assert ((i-0) = i) by lia.
rewrite H7 in H6; clear H7.
destruct H6 as [_ H6].
sep_apply H6.
(* pose proof (uint_array_rec_to_uint_array rp_pre 0 (i+1) (l' ++ z :: nil)).
assert (H_i_plus_1 : 0 <= i + 1) by lia.
specialize (H7 H_i_plus_1); clear H_i_plus_1.
destruct H7 as [H7 _].
assert (i + 1 - 0 = i + 1) by lia.
rewrite H8 in H7; clear H8.
assert ((rp_pre + sizeof ( UINT ) * 0) = rp_pre) by lia.
rewrite H8 in H7; clear H8.
rewrite <-H7.
clear H6.
clear H7. *)
pose proof (store_uint_array_divide_rec rp_pre (i+1) (l' ++ z :: nil) i).
assert (H_tmp: 0 <= i <= i+1) by lia.
specialize (H7 H_tmp); clear H_tmp.
rewrite <- store_uint_array_single.
sep_apply store_uint_array_rec_divide_rev.
entailer!.
lia.
Qed.

18
projects/mini-gmp.c
View File

@ -238,8 +238,8 @@ mpn_add_1 (unsigned int *rp, unsigned int *ap, int n, unsigned int b)
Ensure
exists val',
mpd_store_Z_compact(ap@pre, val, n@pre, cap1) *
mpd_store_Z_compact(rp@pre, val', n@pre, cap2) &&
(val' + __return * (UINT_MOD ^ n@pre) == val + b@pre)
mpd_store_Z(rp@pre, val', n@pre, cap2) &&
(val' + __return * Z::pow(UINT_MOD, n@pre) == val + b@pre)
*/
{
/*@
@ -285,14 +285,26 @@ mpn_add_1 (unsigned int *rp, unsigned int *ap, int n, unsigned int b)
list_store_Z(l', val2) &&
store_uint_array(rp@pre, i, l') *
store_uint_array_rec(rp@pre, i, cap2, l'') &&
(val2 + b@pre * (UINT_MOD ^ i) == val1 + b@pre)
(val2 + b * Z::pow(UINT_MOD, i) == val1 + b@pre) &&
Zlength(l') == i
*/
while (i<n) {
/*@
Given l l' l'' val1 val2
*/
unsigned int r = ap[i] + b;
/*@ 0 <= b && b <= UINT_MAX by local */
b = (r < b);
/*@
0 <= i && i < n@pre && n@pre <= cap2 &&
store_uint_array(rp@pre, i, l') *
store_uint_array_rec(rp@pre, i, cap2, l'')
which implies
exists a l''',
l'' == cons(a, l''') && 0<= i && i<n@pre && n@pre <=cap2 &&
store_uint_array_rec(rp@pre, i+1, cap2, l''') *
store_uint_array(rp@pre, i+1, app(l', cons(a, nil)))
*/
rp[i] = r;
++i;
}

1
projects/mini-gmp.h
View File

@ -64,6 +64,7 @@ void mpz_set (mpz_t, const mpz_t);
/*@
Extern Coq (Zabs : Z -> Z)
(Z::max : Z -> Z -> Z)
(Z::pow : Z -> Z -> Z)
(mpd_store_Z : Z -> Z -> Z -> Z -> Assertion)
(mpd_store_Z_compact: Z -> Z -> Z -> Z -> Assertion)
(mpd_store_list : Z -> list Z -> Z -> Assertion)