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finish carry proof

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ZhuangYumin
2025-06-21 07:50:20 +00:00
parent f462570ccd
commit 0656f30a16
4 changed files with 171 additions and 39 deletions
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6
projects/lib/GmpAux.v
View File

@ -21,6 +21,12 @@ Local Open Scope sac.
Module Aux. 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_of_nat_succ: forall (n: nat), Lemma Z_of_nat_succ: forall (n: nat),
Z.of_nat (S n) = Z.of_nat n + 1. Z.of_nat (S n) = Z.of_nat n + 1.
Proof. lia. Qed. Proof. lia. Qed.

116
projects/lib/gmp_goal.v
View File

@ -1787,6 +1787,8 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
&& [| (i < n_pre) |] && [| (i < n_pre) |]
&& [| (n_pre <= cap2) |] && [| (n_pre <= cap2) |]
&& [| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) >= b) |] && [| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) >= b) |]
&& [| (0 <= b) |]
&& [| (b <= UINT_MAX) |]
&& [| (i < n_pre) |] && [| (i < n_pre) |]
&& [| (0 <= i) |] && [| (0 <= i) |]
&& [| (i <= n_pre) |] && [| (i <= n_pre) |]
@ -1810,10 +1812,10 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
&& (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)))))) ) && (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) ** ((( &( "i" ) )) # Int |-> i)
** (store_uint_array_rec rp_pre (i + 1 ) cap2 l''' ) ** (store_uint_array_rec rp_pre (i + 1 ) cap2 l''' )
** ((( &( "b" ) )) # UInt |-> 0)
** (store_uint_array ap_pre n_pre l_2 ) ** (store_uint_array ap_pre n_pre l_2 )
** ((( &( "r" ) )) # UInt |-> (unsigned_last_nbits (((Znth i l_2 0) + b )) (32))) ** ((( &( "r" ) )) # UInt |-> (unsigned_last_nbits (((Znth i l_2 0) + b )) (32)))
** (store_undef_uint_array_rec ap_pre n_pre cap1 ) ** (store_undef_uint_array_rec ap_pre n_pre cap1 )
** ((( &( "b" ) )) # UInt |-> 0)
** ((( &( "n" ) )) # Int |-> n_pre) ** ((( &( "n" ) )) # Int |-> n_pre)
** ((( &( "ap" ) )) # Ptr |-> ap_pre) ** ((( &( "ap" ) )) # Ptr |-> ap_pre)
** ((( &( "rp" ) )) # Ptr |-> rp_pre) ** ((( &( "rp" ) )) # Ptr |-> rp_pre)
@ -1829,6 +1831,8 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
&& [| (i < n_pre) |] && [| (i < n_pre) |]
&& [| (n_pre <= cap2) |] && [| (n_pre <= cap2) |]
&& [| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) < b) |] && [| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) < b) |]
&& [| (0 <= b) |]
&& [| (b <= UINT_MAX) |]
&& [| (i < n_pre) |] && [| (i < n_pre) |]
&& [| (0 <= i) |] && [| (0 <= i) |]
&& [| (i <= n_pre) |] && [| (i <= n_pre) |]
@ -1852,10 +1856,10 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
&& (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)))))) ) && (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) ** ((( &( "i" ) )) # Int |-> i)
** (store_uint_array_rec rp_pre (i + 1 ) cap2 l''' ) ** (store_uint_array_rec rp_pre (i + 1 ) cap2 l''' )
** ((( &( "b" ) )) # UInt |-> 1)
** (store_uint_array ap_pre n_pre l_2 ) ** (store_uint_array ap_pre n_pre l_2 )
** ((( &( "r" ) )) # UInt |-> (unsigned_last_nbits (((Znth i l_2 0) + b )) (32))) ** ((( &( "r" ) )) # UInt |-> (unsigned_last_nbits (((Znth i l_2 0) + b )) (32)))
** (store_undef_uint_array_rec ap_pre n_pre cap1 ) ** (store_undef_uint_array_rec ap_pre n_pre cap1 )
** ((( &( "b" ) )) # UInt |-> 1)
** ((( &( "n" ) )) # Int |-> n_pre) ** ((( &( "n" ) )) # Int |-> n_pre)
** ((( &( "ap" ) )) # Ptr |-> ap_pre) ** ((( &( "ap" ) )) # Ptr |-> ap_pre)
** ((( &( "rp" ) )) # Ptr |-> rp_pre) ** ((( &( "rp" ) )) # Ptr |-> rp_pre)
@ -1908,13 +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'' ) ** (store_uint_array_rec rp_pre 0 cap2 l'' )
. .
Definition mpn_add_1_entail_wit_2_1 := 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_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 |-> 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 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_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)) , 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'''))) |] [| (l''_2 = (cons (a) (l'''))) |]
&& [| (0 <= i) |] && [| (0 <= i) |]
&& [| (i < n_pre) |] && [| (i < n_pre) |]
&& [| (n_pre <= cap2) |] && [| (n_pre <= cap2) |]
&& [| ((unsigned_last_nbits (((Znth i l_3 0) + b )) (32)) < b) |] && [| ((unsigned_last_nbits (((Znth i l_3 0) + b )) (32)) < b) |]
&& [| (0 <= b) |]
&& [| (b <= UINT_MAX) |]
&& [| (i < n_pre) |] && [| (i < n_pre) |]
&& [| (0 <= i) |] && [| (0 <= i) |]
&& [| (i <= n_pre) |] && [| (i <= n_pre) |]
@ -1966,13 +2039,15 @@ 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 + 1 ) cap2 l'' ) ** (store_uint_array_rec rp_pre (i + 1 ) cap2 l'' )
. .
Definition mpn_add_1_entail_wit_2_2 := 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)) , 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'''))) |] [| (l''_2 = (cons (a) (l'''))) |]
&& [| (0 <= i) |] && [| (0 <= i) |]
&& [| (i < n_pre) |] && [| (i < n_pre) |]
&& [| (n_pre <= cap2) |] && [| (n_pre <= cap2) |]
&& [| ((unsigned_last_nbits (((Znth i l_3 0) + b )) (32)) >= b) |] && [| ((unsigned_last_nbits (((Znth i l_3 0) + b )) (32)) >= b) |]
&& [| (0 <= b) |]
&& [| (b <= UINT_MAX) |]
&& [| (i < n_pre) |] && [| (i < n_pre) |]
&& [| (0 <= i) |] && [| (0 <= i) |]
&& [| (i <= n_pre) |] && [| (i <= n_pre) |]
@ -2193,6 +2268,8 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
Definition mpn_add_1_partial_solve_wit_4_pure := 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) , 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) |] [| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) >= b) |]
&& [| (0 <= b) |]
&& [| (b <= UINT_MAX) |]
&& [| (i < n_pre) |] && [| (i < n_pre) |]
&& [| (0 <= i) |] && [| (0 <= i) |]
&& [| (i <= n_pre) |] && [| (i <= n_pre) |]
@ -2213,13 +2290,13 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
&& [| (cap2 <= 100000000) |] && [| (cap2 <= 100000000) |]
&& [| (n_pre > 0) |] && [| (n_pre > 0) |]
&& [| (n_pre <= cap1) |] && [| (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))) ** ((( &( "r" ) )) # UInt |-> (unsigned_last_nbits (((Znth i l_2 0) + b )) (32)))
** ((( &( "i" ) )) # Int |-> i) ** ((( &( "i" ) )) # Int |-> i)
** (store_undef_uint_array_rec ap_pre n_pre cap1 ) ** (store_undef_uint_array_rec ap_pre n_pre cap1 )
** (store_uint_array rp_pre i l' ) ** (store_uint_array rp_pre i l' )
** (store_uint_array_rec rp_pre i cap2 l'' ) ** (store_uint_array_rec rp_pre i cap2 l'' )
** ((( &( "b" ) )) # UInt |-> 0)
** ((( &( "n" ) )) # Int |-> n_pre) ** ((( &( "n" ) )) # Int |-> n_pre)
** ((( &( "ap" ) )) # Ptr |-> ap_pre) ** ((( &( "ap" ) )) # Ptr |-> ap_pre)
** ((( &( "rp" ) )) # Ptr |-> rp_pre) ** ((( &( "rp" ) )) # Ptr |-> rp_pre)
@ -2232,6 +2309,8 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
Definition mpn_add_1_partial_solve_wit_4_aux := 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) , 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) |] [| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) >= b) |]
&& [| (0 <= b) |]
&& [| (b <= UINT_MAX) |]
&& [| (i < n_pre) |] && [| (i < n_pre) |]
&& [| (0 <= i) |] && [| (0 <= i) |]
&& [| (i <= n_pre) |] && [| (i <= n_pre) |]
@ -2261,6 +2340,8 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
&& [| (i < n_pre) |] && [| (i < n_pre) |]
&& [| (n_pre <= cap2) |] && [| (n_pre <= cap2) |]
&& [| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) >= b) |] && [| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) >= b) |]
&& [| (0 <= b) |]
&& [| (b <= UINT_MAX) |]
&& [| (i < n_pre) |] && [| (i < n_pre) |]
&& [| (0 <= i) |] && [| (0 <= i) |]
&& [| (i <= n_pre) |] && [| (i <= n_pre) |]
@ -2292,6 +2373,8 @@ Definition mpn_add_1_partial_solve_wit_4 := mpn_add_1_partial_solve_wit_4_pure -
Definition mpn_add_1_partial_solve_wit_5_pure := 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) , 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) |] [| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) < b) |]
&& [| (0 <= b) |]
&& [| (b <= UINT_MAX) |]
&& [| (i < n_pre) |] && [| (i < n_pre) |]
&& [| (0 <= i) |] && [| (0 <= i) |]
&& [| (i <= n_pre) |] && [| (i <= n_pre) |]
@ -2312,13 +2395,13 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
&& [| (cap2 <= 100000000) |] && [| (cap2 <= 100000000) |]
&& [| (n_pre > 0) |] && [| (n_pre > 0) |]
&& [| (n_pre <= cap1) |] && [| (n_pre <= cap1) |]
&& (store_uint_array ap_pre n_pre l_2 ) && ((( &( "b" ) )) # UInt |-> 1)
** (store_uint_array ap_pre n_pre l_2 )
** ((( &( "r" ) )) # UInt |-> (unsigned_last_nbits (((Znth i l_2 0) + b )) (32))) ** ((( &( "r" ) )) # UInt |-> (unsigned_last_nbits (((Znth i l_2 0) + b )) (32)))
** ((( &( "i" ) )) # Int |-> i) ** ((( &( "i" ) )) # Int |-> i)
** (store_undef_uint_array_rec ap_pre n_pre cap1 ) ** (store_undef_uint_array_rec ap_pre n_pre cap1 )
** (store_uint_array rp_pre i l' ) ** (store_uint_array rp_pre i l' )
** (store_uint_array_rec rp_pre i cap2 l'' ) ** (store_uint_array_rec rp_pre i cap2 l'' )
** ((( &( "b" ) )) # UInt |-> 1)
** ((( &( "n" ) )) # Int |-> n_pre) ** ((( &( "n" ) )) # Int |-> n_pre)
** ((( &( "ap" ) )) # Ptr |-> ap_pre) ** ((( &( "ap" ) )) # Ptr |-> ap_pre)
** ((( &( "rp" ) )) # Ptr |-> rp_pre) ** ((( &( "rp" ) )) # Ptr |-> rp_pre)
@ -2331,6 +2414,8 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
Definition mpn_add_1_partial_solve_wit_5_aux := 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) , 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) |] [| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) < b) |]
&& [| (0 <= b) |]
&& [| (b <= UINT_MAX) |]
&& [| (i < n_pre) |] && [| (i < n_pre) |]
&& [| (0 <= i) |] && [| (0 <= i) |]
&& [| (i <= n_pre) |] && [| (i <= n_pre) |]
@ -2360,6 +2445,8 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
&& [| (i < n_pre) |] && [| (i < n_pre) |]
&& [| (n_pre <= cap2) |] && [| (n_pre <= cap2) |]
&& [| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) < b) |] && [| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) < b) |]
&& [| (0 <= b) |]
&& [| (b <= UINT_MAX) |]
&& [| (i < n_pre) |] && [| (i < n_pre) |]
&& [| (0 <= i) |] && [| (0 <= i) |]
&& [| (i <= n_pre) |] && [| (i <= n_pre) |]
@ -2395,6 +2482,8 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
&& [| (i < n_pre) |] && [| (i < n_pre) |]
&& [| (n_pre <= cap2) |] && [| (n_pre <= cap2) |]
&& [| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) < b) |] && [| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) < b) |]
&& [| (0 <= b) |]
&& [| (b <= UINT_MAX) |]
&& [| (i < n_pre) |] && [| (i < n_pre) |]
&& [| (0 <= i) |] && [| (0 <= i) |]
&& [| (i <= n_pre) |] && [| (i <= n_pre) |]
@ -2425,6 +2514,8 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
&& [| (i < n_pre) |] && [| (i < n_pre) |]
&& [| (n_pre <= cap2) |] && [| (n_pre <= cap2) |]
&& [| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) < b) |] && [| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) < b) |]
&& [| (0 <= b) |]
&& [| (b <= UINT_MAX) |]
&& [| (i < n_pre) |] && [| (i < n_pre) |]
&& [| (0 <= i) |] && [| (0 <= i) |]
&& [| (i <= n_pre) |] && [| (i <= n_pre) |]
@ -2459,6 +2550,8 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
&& [| (i < n_pre) |] && [| (i < n_pre) |]
&& [| (n_pre <= cap2) |] && [| (n_pre <= cap2) |]
&& [| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) >= b) |] && [| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) >= b) |]
&& [| (0 <= b) |]
&& [| (b <= UINT_MAX) |]
&& [| (i < n_pre) |] && [| (i < n_pre) |]
&& [| (0 <= i) |] && [| (0 <= i) |]
&& [| (i <= n_pre) |] && [| (i <= n_pre) |]
@ -2489,6 +2582,8 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
&& [| (i < n_pre) |] && [| (i < n_pre) |]
&& [| (n_pre <= cap2) |] && [| (n_pre <= cap2) |]
&& [| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) >= b) |] && [| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) >= b) |]
&& [| (0 <= b) |]
&& [| (b <= UINT_MAX) |]
&& [| (i < n_pre) |] && [| (i < n_pre) |]
&& [| (0 <= i) |] && [| (0 <= i) |]
&& [| (i <= n_pre) |] && [| (i <= n_pre) |]
@ -2630,8 +2725,9 @@ 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_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_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_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 : mpn_add_1_entail_wit_2.
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_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_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_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_pure : mpn_add_1_partial_solve_wit_2_pure.

87
projects/lib/gmp_proof_manual.v
View File

@ -422,14 +422,20 @@ Proof.
+ simpl. tauto. + simpl. tauto.
Qed. Qed.
Lemma proof_of_mpn_add_1_entail_wit_2_1 : mpn_add_1_entail_wit_2_1. Lemma proof_of_mpn_add_1_entail_wit_2 : mpn_add_1_entail_wit_2.
Proof.
pre_process.
entailer!.
Admitted.
Lemma proof_of_mpn_add_1_entail_wit_3_1 : mpn_add_1_entail_wit_3_1.
Proof. Proof.
pre_process. pre_process.
rewrite replace_Znth_app_r. rewrite replace_Znth_app_r.
- Exists l'''. - Exists l'''.
rewrite H12. rewrite H14.
assert (i - i = 0) by lia. assert (i - i = 0) by lia.
rewrite H24. rewrite H26.
set (new_b := (unsigned_last_nbits (Znth i l_3 0 + b) 32)). set (new_b := (unsigned_last_nbits (Znth i l_3 0 + b) 32)).
rewrite replace_Znth_nothing; try lia. rewrite replace_Znth_nothing; try lia.
assert (replace_Znth 0 new_b (a :: nil) = new_b :: nil). { assert (replace_Znth 0 new_b (a :: nil) = new_b :: nil). {
@ -438,85 +444,108 @@ Proof.
unfold replace_nth. unfold replace_nth.
reflexivity. reflexivity.
} }
rewrite H25. rewrite H27.
Exists (l'_2 ++ new_b :: nil). Exists (l'_2 ++ new_b :: nil).
Exists (val2_2 + new_b * (UINT_MOD^ i)). Exists (val2_2 + new_b * (UINT_MOD^ i)).
Exists (val1_2 + (Znth i l_3 0) * (UINT_MOD^ i)). Exists (val1_2 + (Znth i l_3 0) * (UINT_MOD^ i)).
Exists l_3. Exists l_3.
entailer!. entailer!.
+ rewrite Zlength_app. + rewrite Zlength_app.
rewrite H12. rewrite H14.
unfold Zlength. unfold Zlength.
unfold Zlength_aux. unfold Zlength_aux.
lia. 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. + 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 H26. rewrite H28.
rewrite <- H11. rewrite <- H13.
assert (Znth i l_3 0 + b = new_b + UINT_MOD). assert (Znth i l_3 0 + b = new_b + UINT_MOD).
{ {
subst new_b. subst new_b.
unfold unsigned_last_nbits. unfold unsigned_last_nbits.
unfold unsigned_last_nbits in H3. unfold unsigned_last_nbits in H3.
assert (2^32 = 4294967296). { nia. } assert (2^32 = 4294967296). { nia. }
rewrite H27 in *. rewrite H29 in *.
admit. 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.
} }
admit. 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). + pose proof (__list_store_Z_concat_r l'_2 val2_2 new_b).
apply H26 in H10. apply H28 in H12.
rewrite H12 in H10. rewrite H14 in H12.
assert (new_b * 4294967296 ^ i + val2_2 = (val2_2 + new_b * 4294967296 ^ i)) by lia. assert (new_b * 4294967296 ^ i + val2_2 = (val2_2 + new_b * 4294967296 ^ i)) by lia.
rewrite H27 in H10. rewrite H29 in H12.
tauto. tauto.
subst new_b. subst new_b.
unfold unsigned_last_nbits. unfold unsigned_last_nbits.
assert (2 ^ 32 = 4294967296). { nia. } assert (2 ^ 32 = 4294967296). { nia. }
rewrite H27. rewrite H29.
apply Z.mod_pos_bound. apply Z.mod_pos_bound.
lia. lia.
+ assert (l_2=l_3). + assert (l_2=l_3).
{ {
pose proof (list_store_Z_compact_reverse_injection l_2 l_3 val val). pose proof (list_store_Z_compact_reverse_injection l_2 l_3 val val).
apply H26 in H7; try tauto. apply H28 in H9; try tauto.
} }
assert (i < Zlength l_3). { assert (i < Zlength l_3). {
subst l_3. subst l_3.
rewrite H15. rewrite H17.
tauto. tauto.
} }
assert((sublist 0 (i + 1) l_3) = (sublist 0 i l_3) ++ (Znth i l_3 0) :: nil). { 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_split 0 (i+1) i l_3).
pose proof (sublist_single i l_3 0). pose proof (sublist_single i l_3 0).
rewrite <-H29. rewrite <-H31.
apply H28. apply H30.
lia. lia.
subst l_3. subst l_3.
rewrite Zlength_correct in H27. rewrite Zlength_correct in H29.
lia. lia.
rewrite Zlength_correct in H27. rewrite Zlength_correct in H29.
lia. lia.
} }
rewrite H28.
pose proof (__list_store_Z_concat_r (sublist 0 i l_3) val1_2 (Znth i l_3 0)).
apply H29 in H9.
rewrite Zlength_sublist0 in H9.
assert (val1_2 + Znth i l_3 0 * 4294967296 ^ i = Znth i l_3 0 * 4294967296 ^ i + val1_2) by lia.
rewrite H30. 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. tauto.
subst l_3. subst l_3.
rewrite H15. rewrite H17.
lia. lia.
apply list_within_bound_Znth. apply list_within_bound_Znth.
lia. lia.
unfold list_store_Z_compact in H7. unfold list_store_Z_compact in H9.
tauto. tauto.
- pose proof (Zlength_sublist0 i l'_2). - pose proof (Zlength_sublist0 i l'_2).
lia. lia.
Admitted. Qed.
Lemma proof_of_mpn_add_1_entail_wit_2_2 : mpn_add_1_entail_wit_2_2. Lemma proof_of_mpn_add_1_entail_wit_3_2 : mpn_add_1_entail_wit_3_2.
Proof. Proof.
pre_process. pre_process.
Admitted. Admitted.

1
projects/mini-gmp.c
View File

@ -293,6 +293,7 @@ mpn_add_1 (unsigned int *rp, unsigned int *ap, int n, unsigned int b)
Given l l' l'' val1 val2 Given l l' l'' val1 val2
*/ */
unsigned int r = ap[i] + b; unsigned int r = ap[i] + b;
/*@ 0 <= b && b <= UINT_MAX by local */
b = (r < b); b = (r < b);
/*@ /*@
0 <= i && i < n@pre && n@pre <= cap2 && 0 <= i && i < n@pre && n@pre <= cap2 &&