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ready to finalize proof_of_mpn_add_1_entail_wit_2_1

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ZhuangYumin
2025-06-21 05:45:27 +00:00
parent 49848bd048
commit f462570ccd
4 changed files with 435 additions and 52 deletions
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projects/lib/gmp_goal.v
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@ -1781,8 +1781,12 @@ 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)) (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) |]
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) |]
&& [| (i < n_pre) |]
&& [| (0 <= i) |]
&& [| (i <= n_pre) |]
@ -1803,12 +1807,12 @@ 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 )) (32))) (l'')) )
&& (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_uint_array_rec rp_pre (i + 1 ) cap2 l''' )
** (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' )
** ((( &( "b" ) )) # UInt |-> 0)
** ((( &( "n" ) )) # Int |-> n_pre)
** ((( &( "ap" ) )) # Ptr |-> ap_pre)
@ -1819,8 +1823,12 @@ 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)) (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) |]
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) |]
&& [| (i < n_pre) |]
&& [| (0 <= i) |]
&& [| (i <= n_pre) |]
@ -1841,12 +1849,12 @@ 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 )) (32))) (l'')) )
&& (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_uint_array_rec rp_pre (i + 1 ) cap2 l''' )
** (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' )
** ((( &( "b" ) )) # UInt |-> 1)
** ((( &( "n" ) )) # Int |-> n_pre)
** ((( &( "ap" ) )) # Ptr |-> ap_pre)
@ -1901,8 +1909,12 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
.
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)) (b: 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 )) (32)) < b) |]
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) |]
&& [| (i < n_pre) |]
&& [| (0 <= i) |]
&& [| (i <= n_pre) |]
@ -1923,10 +1935,10 @@ 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 )) (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 )
|--
EX (l'': (@list Z)) (l': (@list Z)) (val2: Z) (val1: Z) (l: (@list Z)) ,
[| (0 <= (i + 1 )) |]
@ -1955,8 +1967,12 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
.
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)) (b: 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 )) (32)) >= b) |]
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) |]
&& [| (i < n_pre) |]
&& [| (0 <= i) |]
&& [| (i <= n_pre) |]
@ -1977,10 +1993,10 @@ 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 )) (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 )
|--
EX (l'': (@list Z)) (l': (@list Z)) (val2: Z) (val1: Z) (l: (@list Z)) ,
[| (0 <= (i + 1 )) |]
@ -2174,9 +2190,9 @@ 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'' )
.
Definition mpn_add_1_partial_solve_wit_4 :=
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) |]
[| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) >= b) |]
&& [| (i < n_pre) |]
&& [| (0 <= i) |]
&& [| (i <= n_pre) |]
@ -2198,39 +2214,22 @@ forall (b_pre: Z) (n_pre: Z) (ap_pre: Z) (rp_pre: Z) (cap2: Z) (cap1: Z) (l2: (@
&& [| (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 |-> 0)
** ((( &( "n" ) )) # Int |-> n_pre)
** ((( &( "ap" ) )) # Ptr |-> ap_pre)
** ((( &( "rp" ) )) # Ptr |-> rp_pre)
|--
[| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) < b) |]
[| (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 * (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 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 :=
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) |]
&& [| (i < n_pre) |]
@ -2258,7 +2257,174 @@ 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 )) (32)) >= b) |]
[| (0 <= i) |]
&& [| (i < n_pre) |]
&& [| (n_pre <= cap2) |]
&& [| ((unsigned_last_nbits (((Znth i l_2 0) + b )) (32)) >= b) |]
&& [| (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_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) |]
&& [| (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 |-> 1)
** ((( &( "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) |]
&& [| (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) |]
&& [| (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) |]
&& [| (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) |]
&& [| (i < n_pre) |]
&& [| (0 <= i) |]
&& [| (i <= n_pre) |]
@ -2280,10 +2446,74 @@ 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) |]
&& [| (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) |]
&& [| (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 :=
@ -2309,6 +2539,23 @@ 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.
@ -2390,9 +2637,14 @@ 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.

115
projects/lib/gmp_proof_manual.v
View File

@ -407,13 +407,119 @@ 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.
Proof.
pre_process.
rewrite replace_Znth_app_r.
- Exists l'''.
rewrite H12.
assert (i - i = 0) by lia.
rewrite H24.
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 H25.
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 H12.
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 H26.
rewrite <- H11.
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 H27 in *.
admit.
}
admit.
+ pose proof (__list_store_Z_concat_r l'_2 val2_2 new_b).
apply H26 in H10.
rewrite H12 in H10.
assert (new_b * 4294967296 ^ i + val2_2 = (val2_2 + new_b * 4294967296 ^ i)) by lia.
rewrite H27 in H10.
tauto.
subst new_b.
unfold unsigned_last_nbits.
assert (2 ^ 32 = 4294967296). { nia. }
rewrite H27.
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 H26 in H7; try tauto.
}
assert (i < Zlength l_3). {
subst l_3.
rewrite H15.
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 <-H29.
apply H28.
lia.
subst l_3.
rewrite Zlength_correct in H27.
lia.
rewrite Zlength_correct in H27.
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.
tauto.
subst l_3.
rewrite H15.
lia.
apply list_within_bound_Znth.
lia.
unfold list_store_Z_compact in H7.
tauto.
- pose proof (Zlength_sublist0 i l'_2).
lia.
Admitted.
Lemma proof_of_mpn_add_1_entail_wit_2_2 : mpn_add_1_entail_wit_2_2.
Proof. Admitted.
Proof.
pre_process.
Admitted.
Lemma proof_of_mpn_add_1_return_wit_1 : mpn_add_1_return_wit_1.
Proof.
@ -456,3 +562,6 @@ Proof. Admitted.
Lemma proof_of_mpn_add_1_which_implies_wit_2 : mpn_add_1_which_implies_wit_2.
Proof. Admitted.
Lemma proof_of_mpn_add_1_which_implies_wit_3 : mpn_add_1_which_implies_wit_3.
Proof. Admitted.

10
projects/mini-gmp.c
View File

@ -294,6 +294,16 @@ mpn_add_1 (unsigned int *rp, unsigned int *ap, int n, unsigned int b)
*/
unsigned int r = ap[i] + b;
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;
}