1197 lines
41 KiB
Coq
1197 lines
41 KiB
Coq
Require Import Coq.ZArith.ZArith.
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Require Import Coq.Bool.Bool.
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Require Import Coq.Strings.String.
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Require Import Coq.Lists.List.
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Require Import Coq.Classes.RelationClasses.
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Require Import Coq.Classes.Morphisms.
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Require Import Coq.micromega.Psatz.
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Require Import Coq.Sorting.Permutation.
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From AUXLib Require Import int_auto Axioms Feq Idents List_lemma VMap.
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Require Import SetsClass.SetsClass. Import SetsNotation.
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From SimpleC.SL Require Import Mem SeparationLogic.
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Require Import Logic.LogicGenerator.demo932.Interface.
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Require Import GmpLib.GmpNumber. Import Internal.
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Local Open Scope Z_scope.
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Local Open Scope sets.
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Local Open Scope string.
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Local Open Scope list.
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Import naive_C_Rules.
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Local Open Scope sac.
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Definition Zmax := Z.max.
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(*----- Function gmp_abs -----*)
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Definition gmp_abs_safety_wit_1 :=
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forall (x_pre: Z) ,
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[| (x_pre < 0) |]
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&& [| (INT_MIN < x_pre) |]
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&& [| (x_pre <= INT_MAX) |]
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&& ((( &( "x" ) )) # Int |-> x_pre)
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|--
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[| (x_pre <> (INT_MIN)) |]
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.
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Definition gmp_abs_return_wit_1_1 :=
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forall (x_pre: Z) ,
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[| (x_pre < 0) |]
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&& [| (INT_MIN < x_pre) |]
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&& [| (x_pre <= INT_MAX) |]
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&& emp
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|--
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[| ((-x_pre) = (Zabs (x_pre))) |]
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&& emp
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.
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Definition gmp_abs_return_wit_1_2 :=
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forall (x_pre: Z) ,
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[| (x_pre >= 0) |]
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&& [| (INT_MIN < x_pre) |]
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&& [| (x_pre <= INT_MAX) |]
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&& emp
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|--
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[| (x_pre = (Zabs (x_pre))) |]
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&& emp
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.
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(*----- Function gmp_max -----*)
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Definition gmp_max_return_wit_1_1 :=
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forall (b_pre: Z) (a_pre: Z) ,
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[| (a_pre <= b_pre) |]
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&& emp
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|--
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[| (b_pre = (Zmax (a_pre) (b_pre))) |]
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&& emp
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.
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Definition gmp_max_return_wit_1_2 :=
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forall (b_pre: Z) (a_pre: Z) ,
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[| (a_pre > b_pre) |]
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&& emp
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|--
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[| (a_pre = (Zmax (a_pre) (b_pre))) |]
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&& emp
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.
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(*----- Function gmp_cmp -----*)
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Definition gmp_cmp_safety_wit_1 :=
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forall (b_pre: Z) (a_pre: Z) ,
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[| (a_pre >= b_pre) |]
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&& [| (a_pre <= b_pre) |]
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&& ((( &( "b" ) )) # Int |-> b_pre)
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** ((( &( "a" ) )) # Int |-> a_pre)
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|--
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[| ((0 - 0 ) <= INT_MAX) |]
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&& [| ((INT_MIN) <= (0 - 0 )) |]
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.
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Definition gmp_cmp_safety_wit_2 :=
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forall (b_pre: Z) (a_pre: Z) ,
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[| (a_pre < b_pre) |]
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&& [| (a_pre <= b_pre) |]
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&& ((( &( "b" ) )) # Int |-> b_pre)
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** ((( &( "a" ) )) # Int |-> a_pre)
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|--
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[| ((0 - 1 ) <= INT_MAX) |]
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&& [| ((INT_MIN) <= (0 - 1 )) |]
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.
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Definition gmp_cmp_safety_wit_3 :=
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forall (b_pre: Z) (a_pre: Z) ,
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[| (a_pre >= b_pre) |]
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&& [| (a_pre > b_pre) |]
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&& ((( &( "b" ) )) # Int |-> b_pre)
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** ((( &( "a" ) )) # Int |-> a_pre)
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|--
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[| ((1 - 0 ) <= INT_MAX) |]
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&& [| ((INT_MIN) <= (1 - 0 )) |]
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.
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Definition gmp_cmp_safety_wit_4 :=
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forall (b_pre: Z) (a_pre: Z) ,
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[| (a_pre < b_pre) |]
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&& [| (a_pre > b_pre) |]
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&& ((( &( "b" ) )) # Int |-> b_pre)
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** ((( &( "a" ) )) # Int |-> a_pre)
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|--
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[| False |]
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.
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Definition gmp_cmp_return_wit_1_1 :=
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forall (b_pre: Z) (a_pre: Z) ,
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[| (a_pre >= b_pre) |]
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&& [| (a_pre > b_pre) |]
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&& emp
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|--
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([| (a_pre < b_pre) |]
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&& [| ((1 - 0 ) = (-1)) |]
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&& emp)
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||
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([| (a_pre = b_pre) |]
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&& [| ((1 - 0 ) = 0) |]
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&& emp)
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||
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([| (a_pre > b_pre) |]
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&& [| ((1 - 0 ) = 1) |]
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&& emp)
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.
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Definition gmp_cmp_return_wit_1_2 :=
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forall (b_pre: Z) (a_pre: Z) ,
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[| (a_pre < b_pre) |]
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&& [| (a_pre <= b_pre) |]
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&& emp
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|--
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([| (a_pre < b_pre) |]
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&& [| ((0 - 1 ) = (-1)) |]
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&& emp)
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||
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([| (a_pre = b_pre) |]
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&& [| ((0 - 1 ) = 0) |]
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&& emp)
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||
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([| (a_pre > b_pre) |]
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&& [| ((0 - 1 ) = 1) |]
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&& emp)
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.
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Definition gmp_cmp_return_wit_1_3 :=
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forall (b_pre: Z) (a_pre: Z) ,
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[| (a_pre >= b_pre) |]
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&& [| (a_pre <= b_pre) |]
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&& emp
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|--
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([| (a_pre < b_pre) |]
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&& [| ((0 - 0 ) = (-1)) |]
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&& emp)
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||
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([| (a_pre = b_pre) |]
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&& [| ((0 - 0 ) = 0) |]
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&& emp)
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||
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([| (a_pre > b_pre) |]
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&& [| ((0 - 0 ) = 1) |]
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&& emp)
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.
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(*----- Function mpn_copyi -----*)
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Definition mpn_copyi_safety_wit_1 :=
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forall (n_pre: Z) (s_pre: Z) (d_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) (l: (@list Z)) ,
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[| ((Zlength (l2)) = cap2) |]
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&& [| (n_pre <= cap1) |]
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&& [| ((Zlength (l)) = n_pre) |]
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&& [| (cap1 <= 100000000) |]
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&& [| (list_store_Z l val ) |]
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&& [| ((Zlength (l2)) = cap2) |]
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&& [| (cap2 >= n_pre) |]
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&& ((( &( "i" ) )) # Int |->_)
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** ((( &( "d" ) )) # Ptr |-> d_pre)
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** (store_uint_array_rec d_pre 0 cap2 l2 )
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** (store_uint_array d_pre 0 nil )
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** ((( &( "n" ) )) # Int |-> n_pre)
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** ((( &( "s" ) )) # Ptr |-> s_pre)
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** (store_uint_array s_pre n_pre l )
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** (store_undef_uint_array_rec s_pre (n_pre + 1 ) cap1 )
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|--
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[| (0 <= INT_MAX) |]
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&& [| ((INT_MIN) <= 0) |]
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.
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Definition mpn_copyi_safety_wit_2 :=
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forall (n_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) (l: (@list Z)) (l': (@list Z)) (d: Z) (s: Z) (l_2: (@list Z)) (n: Z) (i: Z) (a: Z) (l2': (@list Z)) ,
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[| (l' = (cons (a) (l2'))) |]
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&& [| (i < n) |]
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&& [| (n <= cap2) |]
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&& [| (i < n) |]
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&& [| (0 <= i) |]
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&& [| (i <= n) |]
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&& [| ((Zlength (l_2)) = n) |]
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&& [| (list_store_Z l_2 val ) |]
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&& [| (n <= cap1) |]
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&& [| ((Zlength (l2)) = cap2) |]
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&& [| (n_pre <= cap1) |]
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&& [| ((Zlength (l)) = n_pre) |]
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&& [| (cap1 <= 100000000) |]
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&& [| (list_store_Z l val ) |]
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&& [| ((Zlength (l2)) = cap2) |]
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&& [| (cap2 >= n_pre) |]
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&& (store_uint_array d (i + 1 ) (replace_Znth (i) ((Znth i l_2 0)) ((app ((sublist (0) (i) (l_2))) ((cons (a) (nil)))))) )
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** (store_uint_array s n l_2 )
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** ((( &( "i" ) )) # Int |-> i)
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** ((( &( "n" ) )) # Int |-> n)
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** ((( &( "d" ) )) # Ptr |-> d)
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** (store_uint_array_rec d (i + 1 ) cap2 l2' )
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** ((( &( "s" ) )) # Ptr |-> s)
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** (store_undef_uint_array_rec s (n + 1 ) cap1 )
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|--
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[| ((i + 1 ) <= INT_MAX) |]
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&& [| ((INT_MIN) <= (i + 1 )) |]
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.
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Definition mpn_copyi_entail_wit_1 :=
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forall (n_pre: Z) (s_pre: Z) (d_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) (l_2: (@list Z)) ,
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[| ((Zlength (l2)) = cap2) |]
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&& [| (n_pre <= cap1) |]
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&& [| ((Zlength (l_2)) = n_pre) |]
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&& [| (cap1 <= 100000000) |]
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&& [| (list_store_Z l_2 val ) |]
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&& [| ((Zlength (l2)) = cap2) |]
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&& [| (cap2 >= n_pre) |]
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&& (store_uint_array_rec d_pre 0 cap2 l2 )
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** (store_uint_array d_pre 0 nil )
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** (store_uint_array s_pre n_pre l_2 )
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** (store_undef_uint_array_rec s_pre (n_pre + 1 ) cap1 )
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|--
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EX (l': (@list Z)) (l: (@list Z)) ,
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[| (0 <= 0) |]
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&& [| (0 <= n_pre) |]
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&& [| ((Zlength (l)) = n_pre) |]
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&& [| (list_store_Z l val ) |]
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&& [| (n_pre <= cap1) |]
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&& [| ((Zlength (l2)) = cap2) |]
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&& [| (n_pre <= cap1) |]
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&& [| ((Zlength (l_2)) = n_pre) |]
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&& [| (cap1 <= 100000000) |]
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&& [| (list_store_Z l_2 val ) |]
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&& [| ((Zlength (l2)) = cap2) |]
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&& [| (cap2 >= n_pre) |]
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&& (store_uint_array s_pre n_pre l )
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** (store_undef_uint_array_rec s_pre (n_pre + 1 ) cap1 )
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** (store_uint_array d_pre 0 (sublist (0) (0) (l)) )
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** (store_uint_array_rec d_pre 0 cap2 l' )
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.
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Definition mpn_copyi_entail_wit_2 :=
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forall (n_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) (l_2: (@list Z)) (l'_2: (@list Z)) (d: Z) (s: Z) (l_3: (@list Z)) (n: Z) (i: Z) (a: Z) (l2': (@list Z)) ,
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[| (l'_2 = (cons (a) (l2'))) |]
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&& [| (i < n) |]
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&& [| (n <= cap2) |]
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&& [| (i < n) |]
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&& [| (0 <= i) |]
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&& [| (i <= n) |]
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&& [| ((Zlength (l_3)) = n) |]
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&& [| (list_store_Z l_3 val ) |]
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&& [| (n <= cap1) |]
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&& [| ((Zlength (l2)) = cap2) |]
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&& [| (n_pre <= cap1) |]
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&& [| ((Zlength (l_2)) = n_pre) |]
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&& [| (cap1 <= 100000000) |]
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&& [| (list_store_Z l_2 val ) |]
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&& [| ((Zlength (l2)) = cap2) |]
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&& [| (cap2 >= n_pre) |]
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&& (store_uint_array d (i + 1 ) (replace_Znth (i) ((Znth i l_3 0)) ((app ((sublist (0) (i) (l_3))) ((cons (a) (nil)))))) )
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** (store_uint_array s n l_3 )
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** (store_uint_array_rec d (i + 1 ) cap2 l2' )
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** (store_undef_uint_array_rec s (n + 1 ) cap1 )
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|--
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EX (l': (@list Z)) (l: (@list Z)) ,
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[| (0 <= (i + 1 )) |]
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&& [| ((i + 1 ) <= n) |]
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&& [| ((Zlength (l)) = n) |]
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&& [| (list_store_Z l val ) |]
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&& [| (n <= cap1) |]
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&& [| ((Zlength (l2)) = cap2) |]
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&& [| (n_pre <= cap1) |]
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&& [| ((Zlength (l_2)) = n_pre) |]
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&& [| (cap1 <= 100000000) |]
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&& [| (list_store_Z l_2 val ) |]
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&& [| ((Zlength (l2)) = cap2) |]
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&& [| (cap2 >= n_pre) |]
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&& (store_uint_array s n l )
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** (store_undef_uint_array_rec s (n + 1 ) cap1 )
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** (store_uint_array d (i + 1 ) (sublist (0) ((i + 1 )) (l)) )
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** (store_uint_array_rec d (i + 1 ) cap2 l' )
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.
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Definition mpn_copyi_return_wit_1 :=
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forall (n_pre: Z) (s_pre: Z) (d_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) (l_2: (@list Z)) (l': (@list Z)) (d: Z) (s: Z) (l: (@list Z)) (n: Z) (i: Z) ,
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[| (i >= n) |]
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&& [| (0 <= i) |]
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&& [| (i <= n) |]
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&& [| ((Zlength (l)) = n) |]
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&& [| (list_store_Z l val ) |]
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&& [| (n <= cap1) |]
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&& [| ((Zlength (l2)) = cap2) |]
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&& [| (n_pre <= cap1) |]
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&& [| ((Zlength (l_2)) = n_pre) |]
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&& [| (cap1 <= 100000000) |]
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&& [| (list_store_Z l_2 val ) |]
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&& [| ((Zlength (l2)) = cap2) |]
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&& [| (cap2 >= n_pre) |]
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&& (store_uint_array s n l )
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** (store_undef_uint_array_rec s (n + 1 ) cap1 )
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** (store_uint_array d i (sublist (0) (i) (l)) )
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** (store_uint_array_rec d i cap2 l' )
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|--
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(mpd_store_Z s_pre val n_pre cap1 )
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** (mpd_store_Z d_pre val n_pre cap2 )
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.
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Definition mpn_copyi_partial_solve_wit_1 :=
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forall (n_pre: Z) (s_pre: Z) (d_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) ,
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[| ((Zlength (l2)) = cap2) |]
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&& [| (cap2 >= n_pre) |]
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&& (mpd_store_Z s_pre val n_pre cap1 )
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** (store_uint_array d_pre cap2 l2 )
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|--
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[| ((Zlength (l2)) = cap2) |]
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&& [| (cap2 >= n_pre) |]
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&& (mpd_store_Z s_pre val n_pre cap1 )
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** (store_uint_array d_pre cap2 l2 )
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.
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Definition mpn_copyi_partial_solve_wit_2_pure :=
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forall (n_pre: Z) (s_pre: Z) (d_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) (l: (@list Z)) ,
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[| (n_pre <= cap1) |]
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&& [| ((Zlength (l)) = n_pre) |]
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&& [| (cap1 <= 100000000) |]
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&& [| (list_store_Z l val ) |]
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&& [| ((Zlength (l2)) = cap2) |]
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&& [| (cap2 >= n_pre) |]
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&& ((( &( "n" ) )) # Int |-> n_pre)
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** ((( &( "s" ) )) # Ptr |-> s_pre)
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** (store_uint_array s_pre n_pre l )
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** (store_undef_uint_array_rec s_pre (n_pre + 1 ) cap1 )
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** ((( &( "d" ) )) # Ptr |-> d_pre)
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** (store_uint_array d_pre cap2 l2 )
|
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|--
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[| ((Zlength (l2)) = cap2) |]
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.
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Definition mpn_copyi_partial_solve_wit_2_aux :=
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forall (n_pre: Z) (s_pre: Z) (d_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) (l: (@list Z)) ,
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[| (n_pre <= cap1) |]
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&& [| ((Zlength (l)) = n_pre) |]
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&& [| (cap1 <= 100000000) |]
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&& [| (list_store_Z l val ) |]
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&& [| ((Zlength (l2)) = cap2) |]
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&& [| (cap2 >= n_pre) |]
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&& (store_uint_array s_pre n_pre l )
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** (store_undef_uint_array_rec s_pre (n_pre + 1 ) cap1 )
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** (store_uint_array d_pre cap2 l2 )
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|--
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[| ((Zlength (l2)) = cap2) |]
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&& [| (n_pre <= cap1) |]
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&& [| ((Zlength (l)) = n_pre) |]
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&& [| (cap1 <= 100000000) |]
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&& [| (list_store_Z l val ) |]
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&& [| ((Zlength (l2)) = cap2) |]
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&& [| (cap2 >= n_pre) |]
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&& (store_uint_array d_pre cap2 l2 )
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** (store_uint_array s_pre n_pre l )
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** (store_undef_uint_array_rec s_pre (n_pre + 1 ) cap1 )
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.
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Definition mpn_copyi_partial_solve_wit_2 := mpn_copyi_partial_solve_wit_2_pure -> mpn_copyi_partial_solve_wit_2_aux.
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Definition mpn_copyi_partial_solve_wit_3_pure :=
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forall (n_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) (l_2: (@list Z)) (l': (@list Z)) (d: Z) (s: Z) (l: (@list Z)) (n: Z) (i: Z) ,
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[| (i < n) |]
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&& [| (0 <= i) |]
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&& [| (i <= n) |]
|
|
&& [| ((Zlength (l)) = n) |]
|
|
&& [| (list_store_Z l val ) |]
|
|
&& [| (n <= cap1) |]
|
|
&& [| ((Zlength (l2)) = cap2) |]
|
|
&& [| (n_pre <= cap1) |]
|
|
&& [| ((Zlength (l_2)) = n_pre) |]
|
|
&& [| (cap1 <= 100000000) |]
|
|
&& [| (list_store_Z l_2 val ) |]
|
|
&& [| ((Zlength (l2)) = cap2) |]
|
|
&& [| (cap2 >= n_pre) |]
|
|
&& ((( &( "i" ) )) # Int |-> i)
|
|
** ((( &( "n" ) )) # Int |-> n)
|
|
** ((( &( "s" ) )) # Ptr |-> s)
|
|
** (store_uint_array s n l )
|
|
** (store_undef_uint_array_rec s (n + 1 ) cap1 )
|
|
** ((( &( "d" ) )) # Ptr |-> d)
|
|
** (store_uint_array d i (sublist (0) (i) (l)) )
|
|
** (store_uint_array_rec d i cap2 l' )
|
|
|--
|
|
[| (0 <= i) |]
|
|
&& [| (i < n) |]
|
|
&& [| (n <= cap2) |]
|
|
.
|
|
|
|
Definition mpn_copyi_partial_solve_wit_3_aux :=
|
|
forall (n_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) (l: (@list Z)) (l': (@list Z)) (d: Z) (s: Z) (l_2: (@list Z)) (n: Z) (i: Z) ,
|
|
[| (i < n) |]
|
|
&& [| (0 <= i) |]
|
|
&& [| (i <= n) |]
|
|
&& [| ((Zlength (l_2)) = n) |]
|
|
&& [| (list_store_Z l_2 val ) |]
|
|
&& [| (n <= cap1) |]
|
|
&& [| ((Zlength (l2)) = cap2) |]
|
|
&& [| (n_pre <= cap1) |]
|
|
&& [| ((Zlength (l)) = n_pre) |]
|
|
&& [| (cap1 <= 100000000) |]
|
|
&& [| (list_store_Z l val ) |]
|
|
&& [| ((Zlength (l2)) = cap2) |]
|
|
&& [| (cap2 >= n_pre) |]
|
|
&& (store_uint_array s n l_2 )
|
|
** (store_undef_uint_array_rec s (n + 1 ) cap1 )
|
|
** (store_uint_array d i (sublist (0) (i) (l_2)) )
|
|
** (store_uint_array_rec d i cap2 l' )
|
|
|--
|
|
[| (0 <= i) |]
|
|
&& [| (i < n) |]
|
|
&& [| (n <= cap2) |]
|
|
&& [| (i < n) |]
|
|
&& [| (0 <= i) |]
|
|
&& [| (i <= n) |]
|
|
&& [| ((Zlength (l_2)) = n) |]
|
|
&& [| (list_store_Z l_2 val ) |]
|
|
&& [| (n <= cap1) |]
|
|
&& [| ((Zlength (l2)) = cap2) |]
|
|
&& [| (n_pre <= cap1) |]
|
|
&& [| ((Zlength (l)) = n_pre) |]
|
|
&& [| (cap1 <= 100000000) |]
|
|
&& [| (list_store_Z l val ) |]
|
|
&& [| ((Zlength (l2)) = cap2) |]
|
|
&& [| (cap2 >= n_pre) |]
|
|
&& (store_uint_array_rec d i cap2 l' )
|
|
** (store_uint_array d i (sublist (0) (i) (l_2)) )
|
|
** (store_uint_array s n l_2 )
|
|
** (store_undef_uint_array_rec s (n + 1 ) cap1 )
|
|
.
|
|
|
|
Definition mpn_copyi_partial_solve_wit_3 := mpn_copyi_partial_solve_wit_3_pure -> mpn_copyi_partial_solve_wit_3_aux.
|
|
|
|
Definition mpn_copyi_partial_solve_wit_4 :=
|
|
forall (n_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) (l: (@list Z)) (l': (@list Z)) (d: Z) (s: Z) (l_2: (@list Z)) (n: Z) (i: Z) (a: Z) (l2': (@list Z)) ,
|
|
[| (l' = (cons (a) (l2'))) |]
|
|
&& [| (i < n) |]
|
|
&& [| (n <= cap2) |]
|
|
&& [| (i < n) |]
|
|
&& [| (0 <= i) |]
|
|
&& [| (i <= n) |]
|
|
&& [| ((Zlength (l_2)) = n) |]
|
|
&& [| (list_store_Z l_2 val ) |]
|
|
&& [| (n <= cap1) |]
|
|
&& [| ((Zlength (l2)) = cap2) |]
|
|
&& [| (n_pre <= cap1) |]
|
|
&& [| ((Zlength (l)) = n_pre) |]
|
|
&& [| (cap1 <= 100000000) |]
|
|
&& [| (list_store_Z l val ) |]
|
|
&& [| ((Zlength (l2)) = cap2) |]
|
|
&& [| (cap2 >= n_pre) |]
|
|
&& (store_uint_array_rec d (i + 1 ) cap2 l2' )
|
|
** (store_uint_array d (i + 1 ) (app ((sublist (0) (i) (l_2))) ((cons (a) (nil)))) )
|
|
** (store_uint_array s n l_2 )
|
|
** (store_undef_uint_array_rec s (n + 1 ) cap1 )
|
|
|--
|
|
[| (l' = (cons (a) (l2'))) |]
|
|
&& [| (i < n) |]
|
|
&& [| (n <= cap2) |]
|
|
&& [| (i < n) |]
|
|
&& [| (0 <= i) |]
|
|
&& [| (i <= n) |]
|
|
&& [| ((Zlength (l_2)) = n) |]
|
|
&& [| (list_store_Z l_2 val ) |]
|
|
&& [| (n <= cap1) |]
|
|
&& [| ((Zlength (l2)) = cap2) |]
|
|
&& [| (n_pre <= cap1) |]
|
|
&& [| ((Zlength (l)) = n_pre) |]
|
|
&& [| (cap1 <= 100000000) |]
|
|
&& [| (list_store_Z l val ) |]
|
|
&& [| ((Zlength (l2)) = cap2) |]
|
|
&& [| (cap2 >= n_pre) |]
|
|
&& (((s + (i * sizeof(UINT) ) )) # UInt |-> (Znth i l_2 0))
|
|
** (store_uint_array_missing_i_rec s i 0 n l_2 )
|
|
** (store_uint_array_rec d (i + 1 ) cap2 l2' )
|
|
** (store_uint_array d (i + 1 ) (app ((sublist (0) (i) (l_2))) ((cons (a) (nil)))) )
|
|
** (store_undef_uint_array_rec s (n + 1 ) cap1 )
|
|
.
|
|
|
|
Definition mpn_copyi_partial_solve_wit_5 :=
|
|
forall (n_pre: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (val: Z) (l: (@list Z)) (l': (@list Z)) (d: Z) (s: Z) (l_2: (@list Z)) (n: Z) (i: Z) (a: Z) (l2': (@list Z)) ,
|
|
[| (l' = (cons (a) (l2'))) |]
|
|
&& [| (i < n) |]
|
|
&& [| (n <= cap2) |]
|
|
&& [| (i < n) |]
|
|
&& [| (0 <= i) |]
|
|
&& [| (i <= n) |]
|
|
&& [| ((Zlength (l_2)) = n) |]
|
|
&& [| (list_store_Z l_2 val ) |]
|
|
&& [| (n <= cap1) |]
|
|
&& [| ((Zlength (l2)) = cap2) |]
|
|
&& [| (n_pre <= cap1) |]
|
|
&& [| ((Zlength (l)) = n_pre) |]
|
|
&& [| (cap1 <= 100000000) |]
|
|
&& [| (list_store_Z l val ) |]
|
|
&& [| ((Zlength (l2)) = cap2) |]
|
|
&& [| (cap2 >= n_pre) |]
|
|
&& (store_uint_array s n l_2 )
|
|
** (store_uint_array_rec d (i + 1 ) cap2 l2' )
|
|
** (store_uint_array d (i + 1 ) (app ((sublist (0) (i) (l_2))) ((cons (a) (nil)))) )
|
|
** (store_undef_uint_array_rec s (n + 1 ) cap1 )
|
|
|--
|
|
[| (l' = (cons (a) (l2'))) |]
|
|
&& [| (i < n) |]
|
|
&& [| (n <= cap2) |]
|
|
&& [| (i < n) |]
|
|
&& [| (0 <= i) |]
|
|
&& [| (i <= n) |]
|
|
&& [| ((Zlength (l_2)) = n) |]
|
|
&& [| (list_store_Z l_2 val ) |]
|
|
&& [| (n <= cap1) |]
|
|
&& [| ((Zlength (l2)) = cap2) |]
|
|
&& [| (n_pre <= cap1) |]
|
|
&& [| ((Zlength (l)) = n_pre) |]
|
|
&& [| (cap1 <= 100000000) |]
|
|
&& [| (list_store_Z l val ) |]
|
|
&& [| ((Zlength (l2)) = cap2) |]
|
|
&& [| (cap2 >= n_pre) |]
|
|
&& (((d + (i * sizeof(UINT) ) )) # UInt |->_)
|
|
** (store_uint_array_missing_i_rec d i 0 (i + 1 ) (app ((sublist (0) (i) (l_2))) ((cons (a) (nil)))) )
|
|
** (store_uint_array s n l_2 )
|
|
** (store_uint_array_rec d (i + 1 ) cap2 l2' )
|
|
** (store_undef_uint_array_rec s (n + 1 ) cap1 )
|
|
.
|
|
|
|
Definition mpn_copyi_which_implies_wit_1 :=
|
|
forall (cap1: Z) (val: Z) (n: Z) (s: Z) ,
|
|
(mpd_store_Z s val n cap1 )
|
|
|--
|
|
EX (l: (@list Z)) ,
|
|
[| (n <= cap1) |]
|
|
&& [| ((Zlength (l)) = n) |]
|
|
&& [| (cap1 <= 100000000) |]
|
|
&& [| (list_store_Z l val ) |]
|
|
&& (store_uint_array s n l )
|
|
** (store_undef_uint_array_rec s (n + 1 ) cap1 )
|
|
.
|
|
|
|
Definition mpn_copyi_which_implies_wit_2 :=
|
|
forall (cap2: Z) (l2: (@list Z)) (d: Z) ,
|
|
[| ((Zlength (l2)) = cap2) |]
|
|
&& (store_uint_array d cap2 l2 )
|
|
|--
|
|
[| ((Zlength (l2)) = cap2) |]
|
|
&& (store_uint_array_rec d 0 cap2 l2 )
|
|
** (store_uint_array d 0 nil )
|
|
.
|
|
|
|
Definition mpn_copyi_which_implies_wit_3 :=
|
|
forall (cap2: Z) (l': (@list Z)) (l: (@list Z)) (i: Z) (n: Z) (d: Z) ,
|
|
[| (0 <= i) |]
|
|
&& [| (i < n) |]
|
|
&& [| (n <= cap2) |]
|
|
&& (store_uint_array_rec d i cap2 l' )
|
|
** (store_uint_array d i (sublist (0) (i) (l)) )
|
|
|--
|
|
EX (a: Z) (l2': (@list Z)) ,
|
|
[| (l' = (cons (a) (l2'))) |]
|
|
&& [| (i < n) |]
|
|
&& [| (n <= cap2) |]
|
|
&& (store_uint_array_rec d (i + 1 ) cap2 l2' )
|
|
** (store_uint_array d (i + 1 ) (app ((sublist (0) (i) (l))) ((cons (a) (nil)))) )
|
|
.
|
|
|
|
(*----- Function mpn_cmp -----*)
|
|
|
|
Definition mpn_cmp_safety_wit_1 :=
|
|
forall (n_pre: Z) (bp_pre: Z) (ap_pre: Z) (val2: Z) (val1: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (l1: (@list Z)) ,
|
|
[| (list_store_Z l1 val1 ) |]
|
|
&& [| (list_store_Z l2 val2 ) |]
|
|
&& [| (n_pre = (Zlength (l1))) |]
|
|
&& [| (n_pre = (Zlength (l2))) |]
|
|
&& [| (0 < n_pre) |]
|
|
&& [| (n_pre <= cap1) |]
|
|
&& [| (n_pre <= cap2) |]
|
|
&& [| (cap1 <= 100000000) |]
|
|
&& [| (cap2 <= 100000000) |]
|
|
&& (store_uint_array ap_pre n_pre l1 )
|
|
** (store_uint_array bp_pre n_pre l2 )
|
|
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
|
|
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
|
|
** ((( &( "n" ) )) # Int |-> n_pre)
|
|
** ((( &( "bp" ) )) # Ptr |-> bp_pre)
|
|
** ((( &( "ap" ) )) # Ptr |-> ap_pre)
|
|
|--
|
|
[| ((n_pre - 1 ) <= INT_MAX) |]
|
|
&& [| ((INT_MIN) <= (n_pre - 1 )) |]
|
|
.
|
|
|
|
Definition mpn_cmp_safety_wit_2 :=
|
|
forall (n_pre: Z) (bp_pre: Z) (ap_pre: Z) (val2: Z) (val1: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (l1: (@list Z)) (n: Z) ,
|
|
[| ((-1) <= n) |]
|
|
&& [| (n < n_pre) |]
|
|
&& [| (list_store_Z l1 val1 ) |]
|
|
&& [| (list_store_Z l2 val2 ) |]
|
|
&& [| (n_pre = (Zlength (l1))) |]
|
|
&& [| (n_pre = (Zlength (l2))) |]
|
|
&& [| ((sublist ((n + 1 )) (n_pre) (l1)) = (sublist ((n + 1 )) (n_pre) (l2))) |]
|
|
&& [| (list_store_Z l1 val1 ) |]
|
|
&& [| (list_store_Z l2 val2 ) |]
|
|
&& [| (n_pre = (Zlength (l1))) |]
|
|
&& [| (n_pre = (Zlength (l2))) |]
|
|
&& [| (0 < n_pre) |]
|
|
&& [| (n_pre <= cap1) |]
|
|
&& [| (n_pre <= cap2) |]
|
|
&& [| (cap1 <= 100000000) |]
|
|
&& [| (cap2 <= 100000000) |]
|
|
&& ((( &( "n" ) )) # Int |-> n)
|
|
** (store_uint_array ap_pre n_pre l1 )
|
|
** (store_uint_array bp_pre n_pre l2 )
|
|
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
|
|
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
|
|
** ((( &( "bp" ) )) # Ptr |-> bp_pre)
|
|
** ((( &( "ap" ) )) # Ptr |-> ap_pre)
|
|
|--
|
|
[| (0 <= INT_MAX) |]
|
|
&& [| ((INT_MIN) <= 0) |]
|
|
.
|
|
|
|
Definition mpn_cmp_safety_wit_3 :=
|
|
forall (n_pre: Z) (bp_pre: Z) (ap_pre: Z) (val2: Z) (val1: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (l1: (@list Z)) (n: Z) ,
|
|
[| ((Znth n l1 0) <= (Znth n l2 0)) |]
|
|
&& [| ((Znth n l1 0) <> (Znth n l2 0)) |]
|
|
&& [| (n >= 0) |]
|
|
&& [| ((-1) <= n) |]
|
|
&& [| (n < n_pre) |]
|
|
&& [| (list_store_Z l1 val1 ) |]
|
|
&& [| (list_store_Z l2 val2 ) |]
|
|
&& [| (n_pre = (Zlength (l1))) |]
|
|
&& [| (n_pre = (Zlength (l2))) |]
|
|
&& [| ((sublist ((n + 1 )) (n_pre) (l1)) = (sublist ((n + 1 )) (n_pre) (l2))) |]
|
|
&& [| (list_store_Z l1 val1 ) |]
|
|
&& [| (list_store_Z l2 val2 ) |]
|
|
&& [| (n_pre = (Zlength (l1))) |]
|
|
&& [| (n_pre = (Zlength (l2))) |]
|
|
&& [| (0 < n_pre) |]
|
|
&& [| (n_pre <= cap1) |]
|
|
&& [| (n_pre <= cap2) |]
|
|
&& [| (cap1 <= 100000000) |]
|
|
&& [| (cap2 <= 100000000) |]
|
|
&& (store_uint_array bp_pre n_pre l2 )
|
|
** (store_uint_array ap_pre n_pre l1 )
|
|
** ((( &( "n" ) )) # Int |-> n)
|
|
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
|
|
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
|
|
** ((( &( "bp" ) )) # Ptr |-> bp_pre)
|
|
** ((( &( "ap" ) )) # Ptr |-> ap_pre)
|
|
|--
|
|
[| (1 <> (INT_MIN)) |]
|
|
.
|
|
|
|
Definition mpn_cmp_safety_wit_4 :=
|
|
forall (n_pre: Z) (bp_pre: Z) (ap_pre: Z) (val2: Z) (val1: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (l1: (@list Z)) (n: Z) ,
|
|
[| ((Znth n l1 0) <= (Znth n l2 0)) |]
|
|
&& [| ((Znth n l1 0) <> (Znth n l2 0)) |]
|
|
&& [| (n >= 0) |]
|
|
&& [| ((-1) <= n) |]
|
|
&& [| (n < n_pre) |]
|
|
&& [| (list_store_Z l1 val1 ) |]
|
|
&& [| (list_store_Z l2 val2 ) |]
|
|
&& [| (n_pre = (Zlength (l1))) |]
|
|
&& [| (n_pre = (Zlength (l2))) |]
|
|
&& [| ((sublist ((n + 1 )) (n_pre) (l1)) = (sublist ((n + 1 )) (n_pre) (l2))) |]
|
|
&& [| (list_store_Z l1 val1 ) |]
|
|
&& [| (list_store_Z l2 val2 ) |]
|
|
&& [| (n_pre = (Zlength (l1))) |]
|
|
&& [| (n_pre = (Zlength (l2))) |]
|
|
&& [| (0 < n_pre) |]
|
|
&& [| (n_pre <= cap1) |]
|
|
&& [| (n_pre <= cap2) |]
|
|
&& [| (cap1 <= 100000000) |]
|
|
&& [| (cap2 <= 100000000) |]
|
|
&& (store_uint_array bp_pre n_pre l2 )
|
|
** (store_uint_array ap_pre n_pre l1 )
|
|
** ((( &( "n" ) )) # Int |-> n)
|
|
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
|
|
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
|
|
** ((( &( "bp" ) )) # Ptr |-> bp_pre)
|
|
** ((( &( "ap" ) )) # Ptr |-> ap_pre)
|
|
|--
|
|
[| (1 <= INT_MAX) |]
|
|
&& [| ((INT_MIN) <= 1) |]
|
|
.
|
|
|
|
Definition mpn_cmp_safety_wit_5 :=
|
|
forall (n_pre: Z) (bp_pre: Z) (ap_pre: Z) (val2: Z) (val1: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (l1: (@list Z)) (n: Z) ,
|
|
[| ((Znth n l1 0) > (Znth n l2 0)) |]
|
|
&& [| ((Znth n l1 0) <> (Znth n l2 0)) |]
|
|
&& [| (n >= 0) |]
|
|
&& [| ((-1) <= n) |]
|
|
&& [| (n < n_pre) |]
|
|
&& [| (list_store_Z l1 val1 ) |]
|
|
&& [| (list_store_Z l2 val2 ) |]
|
|
&& [| (n_pre = (Zlength (l1))) |]
|
|
&& [| (n_pre = (Zlength (l2))) |]
|
|
&& [| ((sublist ((n + 1 )) (n_pre) (l1)) = (sublist ((n + 1 )) (n_pre) (l2))) |]
|
|
&& [| (list_store_Z l1 val1 ) |]
|
|
&& [| (list_store_Z l2 val2 ) |]
|
|
&& [| (n_pre = (Zlength (l1))) |]
|
|
&& [| (n_pre = (Zlength (l2))) |]
|
|
&& [| (0 < n_pre) |]
|
|
&& [| (n_pre <= cap1) |]
|
|
&& [| (n_pre <= cap2) |]
|
|
&& [| (cap1 <= 100000000) |]
|
|
&& [| (cap2 <= 100000000) |]
|
|
&& (store_uint_array bp_pre n_pre l2 )
|
|
** (store_uint_array ap_pre n_pre l1 )
|
|
** ((( &( "n" ) )) # Int |-> n)
|
|
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
|
|
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
|
|
** ((( &( "bp" ) )) # Ptr |-> bp_pre)
|
|
** ((( &( "ap" ) )) # Ptr |-> ap_pre)
|
|
|--
|
|
[| (1 <= INT_MAX) |]
|
|
&& [| ((INT_MIN) <= 1) |]
|
|
.
|
|
|
|
Definition mpn_cmp_safety_wit_6 :=
|
|
forall (n_pre: Z) (bp_pre: Z) (ap_pre: Z) (val2: Z) (val1: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (l1: (@list Z)) (n: Z) ,
|
|
[| ((Znth n l1 0) = (Znth n l2 0)) |]
|
|
&& [| (n >= 0) |]
|
|
&& [| ((-1) <= n) |]
|
|
&& [| (n < n_pre) |]
|
|
&& [| (list_store_Z l1 val1 ) |]
|
|
&& [| (list_store_Z l2 val2 ) |]
|
|
&& [| (n_pre = (Zlength (l1))) |]
|
|
&& [| (n_pre = (Zlength (l2))) |]
|
|
&& [| ((sublist ((n + 1 )) (n_pre) (l1)) = (sublist ((n + 1 )) (n_pre) (l2))) |]
|
|
&& [| (list_store_Z l1 val1 ) |]
|
|
&& [| (list_store_Z l2 val2 ) |]
|
|
&& [| (n_pre = (Zlength (l1))) |]
|
|
&& [| (n_pre = (Zlength (l2))) |]
|
|
&& [| (0 < n_pre) |]
|
|
&& [| (n_pre <= cap1) |]
|
|
&& [| (n_pre <= cap2) |]
|
|
&& [| (cap1 <= 100000000) |]
|
|
&& [| (cap2 <= 100000000) |]
|
|
&& (store_uint_array bp_pre n_pre l2 )
|
|
** (store_uint_array ap_pre n_pre l1 )
|
|
** ((( &( "n" ) )) # Int |-> n)
|
|
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
|
|
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
|
|
** ((( &( "bp" ) )) # Ptr |-> bp_pre)
|
|
** ((( &( "ap" ) )) # Ptr |-> ap_pre)
|
|
|--
|
|
[| ((n - 1 ) <= INT_MAX) |]
|
|
&& [| ((INT_MIN) <= (n - 1 )) |]
|
|
.
|
|
|
|
Definition mpn_cmp_safety_wit_7 :=
|
|
forall (n_pre: Z) (bp_pre: Z) (ap_pre: Z) (val2: Z) (val1: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (l1: (@list Z)) (n: Z) ,
|
|
[| (n < 0) |]
|
|
&& [| ((-1) <= n) |]
|
|
&& [| (n < n_pre) |]
|
|
&& [| (list_store_Z l1 val1 ) |]
|
|
&& [| (list_store_Z l2 val2 ) |]
|
|
&& [| (n_pre = (Zlength (l1))) |]
|
|
&& [| (n_pre = (Zlength (l2))) |]
|
|
&& [| ((sublist ((n + 1 )) (n_pre) (l1)) = (sublist ((n + 1 )) (n_pre) (l2))) |]
|
|
&& [| (list_store_Z l1 val1 ) |]
|
|
&& [| (list_store_Z l2 val2 ) |]
|
|
&& [| (n_pre = (Zlength (l1))) |]
|
|
&& [| (n_pre = (Zlength (l2))) |]
|
|
&& [| (0 < n_pre) |]
|
|
&& [| (n_pre <= cap1) |]
|
|
&& [| (n_pre <= cap2) |]
|
|
&& [| (cap1 <= 100000000) |]
|
|
&& [| (cap2 <= 100000000) |]
|
|
&& ((( &( "n" ) )) # Int |-> n)
|
|
** (store_uint_array ap_pre n_pre l1 )
|
|
** (store_uint_array bp_pre n_pre l2 )
|
|
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
|
|
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
|
|
** ((( &( "bp" ) )) # Ptr |-> bp_pre)
|
|
** ((( &( "ap" ) )) # Ptr |-> ap_pre)
|
|
|--
|
|
[| (0 <= INT_MAX) |]
|
|
&& [| ((INT_MIN) <= 0) |]
|
|
.
|
|
|
|
Definition mpn_cmp_entail_wit_1 :=
|
|
forall (n_pre: Z) (bp_pre: Z) (ap_pre: Z) (val2: Z) (val1: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (l1: (@list Z)) ,
|
|
[| (list_store_Z l1 val1 ) |]
|
|
&& [| (list_store_Z l2 val2 ) |]
|
|
&& [| (n_pre = (Zlength (l1))) |]
|
|
&& [| (n_pre = (Zlength (l2))) |]
|
|
&& [| (0 < n_pre) |]
|
|
&& [| (n_pre <= cap1) |]
|
|
&& [| (n_pre <= cap2) |]
|
|
&& [| (cap1 <= 100000000) |]
|
|
&& [| (cap2 <= 100000000) |]
|
|
&& (store_uint_array ap_pre n_pre l1 )
|
|
** (store_uint_array bp_pre n_pre l2 )
|
|
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
|
|
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
|
|
|--
|
|
[| ((-1) <= (n_pre - 1 )) |]
|
|
&& [| ((n_pre - 1 ) < n_pre) |]
|
|
&& [| (list_store_Z l1 val1 ) |]
|
|
&& [| (list_store_Z l2 val2 ) |]
|
|
&& [| (n_pre = (Zlength (l1))) |]
|
|
&& [| (n_pre = (Zlength (l2))) |]
|
|
&& [| ((sublist (((n_pre - 1 ) + 1 )) (n_pre) (l1)) = (sublist (((n_pre - 1 ) + 1 )) (n_pre) (l2))) |]
|
|
&& [| (list_store_Z l1 val1 ) |]
|
|
&& [| (list_store_Z l2 val2 ) |]
|
|
&& [| (n_pre = (Zlength (l1))) |]
|
|
&& [| (n_pre = (Zlength (l2))) |]
|
|
&& [| (0 < n_pre) |]
|
|
&& [| (n_pre <= cap1) |]
|
|
&& [| (n_pre <= cap2) |]
|
|
&& [| (cap1 <= 100000000) |]
|
|
&& [| (cap2 <= 100000000) |]
|
|
&& (store_uint_array ap_pre n_pre l1 )
|
|
** (store_uint_array bp_pre n_pre l2 )
|
|
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
|
|
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
|
|
.
|
|
|
|
Definition mpn_cmp_entail_wit_2 :=
|
|
forall (n_pre: Z) (bp_pre: Z) (ap_pre: Z) (val2: Z) (val1: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (l1: (@list Z)) (n: Z) ,
|
|
[| ((Znth n l1 0) = (Znth n l2 0)) |]
|
|
&& [| (n >= 0) |]
|
|
&& [| ((-1) <= n) |]
|
|
&& [| (n < n_pre) |]
|
|
&& [| (list_store_Z l1 val1 ) |]
|
|
&& [| (list_store_Z l2 val2 ) |]
|
|
&& [| (n_pre = (Zlength (l1))) |]
|
|
&& [| (n_pre = (Zlength (l2))) |]
|
|
&& [| ((sublist ((n + 1 )) (n_pre) (l1)) = (sublist ((n + 1 )) (n_pre) (l2))) |]
|
|
&& [| (list_store_Z l1 val1 ) |]
|
|
&& [| (list_store_Z l2 val2 ) |]
|
|
&& [| (n_pre = (Zlength (l1))) |]
|
|
&& [| (n_pre = (Zlength (l2))) |]
|
|
&& [| (0 < n_pre) |]
|
|
&& [| (n_pre <= cap1) |]
|
|
&& [| (n_pre <= cap2) |]
|
|
&& [| (cap1 <= 100000000) |]
|
|
&& [| (cap2 <= 100000000) |]
|
|
&& (store_uint_array bp_pre n_pre l2 )
|
|
** (store_uint_array ap_pre n_pre l1 )
|
|
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
|
|
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
|
|
|--
|
|
[| ((-1) <= (n - 1 )) |]
|
|
&& [| ((n - 1 ) < n_pre) |]
|
|
&& [| (list_store_Z l1 val1 ) |]
|
|
&& [| (list_store_Z l2 val2 ) |]
|
|
&& [| (n_pre = (Zlength (l1))) |]
|
|
&& [| (n_pre = (Zlength (l2))) |]
|
|
&& [| ((sublist (((n - 1 ) + 1 )) (n_pre) (l1)) = (sublist (((n - 1 ) + 1 )) (n_pre) (l2))) |]
|
|
&& [| (list_store_Z l1 val1 ) |]
|
|
&& [| (list_store_Z l2 val2 ) |]
|
|
&& [| (n_pre = (Zlength (l1))) |]
|
|
&& [| (n_pre = (Zlength (l2))) |]
|
|
&& [| (0 < n_pre) |]
|
|
&& [| (n_pre <= cap1) |]
|
|
&& [| (n_pre <= cap2) |]
|
|
&& [| (cap1 <= 100000000) |]
|
|
&& [| (cap2 <= 100000000) |]
|
|
&& (store_uint_array ap_pre n_pre l1 )
|
|
** (store_uint_array bp_pre n_pre l2 )
|
|
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
|
|
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
|
|
.
|
|
|
|
Definition mpn_cmp_return_wit_1_1 :=
|
|
forall (n_pre: Z) (bp_pre: Z) (ap_pre: Z) (val2: Z) (val1: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (l1: (@list Z)) (n: Z) ,
|
|
[| ((Znth n l1 0) <= (Znth n l2 0)) |]
|
|
&& [| ((Znth n l1 0) <> (Znth n l2 0)) |]
|
|
&& [| (n >= 0) |]
|
|
&& [| ((-1) <= n) |]
|
|
&& [| (n < n_pre) |]
|
|
&& [| (list_store_Z l1 val1 ) |]
|
|
&& [| (list_store_Z l2 val2 ) |]
|
|
&& [| (n_pre = (Zlength (l1))) |]
|
|
&& [| (n_pre = (Zlength (l2))) |]
|
|
&& [| ((sublist ((n + 1 )) (n_pre) (l1)) = (sublist ((n + 1 )) (n_pre) (l2))) |]
|
|
&& [| (list_store_Z l1 val1 ) |]
|
|
&& [| (list_store_Z l2 val2 ) |]
|
|
&& [| (n_pre = (Zlength (l1))) |]
|
|
&& [| (n_pre = (Zlength (l2))) |]
|
|
&& [| (0 < n_pre) |]
|
|
&& [| (n_pre <= cap1) |]
|
|
&& [| (n_pre <= cap2) |]
|
|
&& [| (cap1 <= 100000000) |]
|
|
&& [| (cap2 <= 100000000) |]
|
|
&& (store_uint_array bp_pre n_pre l2 )
|
|
** (store_uint_array ap_pre n_pre l1 )
|
|
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
|
|
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
|
|
|--
|
|
([| (val1 < val2) |]
|
|
&& [| ((-1) = (-1)) |]
|
|
&& (mpd_store_Z ap_pre val1 n_pre cap1 )
|
|
** (mpd_store_Z bp_pre val2 n_pre cap2 ))
|
|
||
|
|
([| (val1 = val2) |]
|
|
&& [| ((-1) = 0) |]
|
|
&& (mpd_store_Z ap_pre val1 n_pre cap1 )
|
|
** (mpd_store_Z bp_pre val2 n_pre cap2 ))
|
|
||
|
|
([| (val1 > val2) |]
|
|
&& [| ((-1) = 1) |]
|
|
&& (mpd_store_Z ap_pre val1 n_pre cap1 )
|
|
** (mpd_store_Z bp_pre val2 n_pre cap2 ))
|
|
.
|
|
|
|
Definition mpn_cmp_return_wit_1_2 :=
|
|
forall (n_pre: Z) (bp_pre: Z) (ap_pre: Z) (val2: Z) (val1: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (l1: (@list Z)) (n: Z) ,
|
|
[| ((Znth n l1 0) > (Znth n l2 0)) |]
|
|
&& [| ((Znth n l1 0) <> (Znth n l2 0)) |]
|
|
&& [| (n >= 0) |]
|
|
&& [| ((-1) <= n) |]
|
|
&& [| (n < n_pre) |]
|
|
&& [| (list_store_Z l1 val1 ) |]
|
|
&& [| (list_store_Z l2 val2 ) |]
|
|
&& [| (n_pre = (Zlength (l1))) |]
|
|
&& [| (n_pre = (Zlength (l2))) |]
|
|
&& [| ((sublist ((n + 1 )) (n_pre) (l1)) = (sublist ((n + 1 )) (n_pre) (l2))) |]
|
|
&& [| (list_store_Z l1 val1 ) |]
|
|
&& [| (list_store_Z l2 val2 ) |]
|
|
&& [| (n_pre = (Zlength (l1))) |]
|
|
&& [| (n_pre = (Zlength (l2))) |]
|
|
&& [| (0 < n_pre) |]
|
|
&& [| (n_pre <= cap1) |]
|
|
&& [| (n_pre <= cap2) |]
|
|
&& [| (cap1 <= 100000000) |]
|
|
&& [| (cap2 <= 100000000) |]
|
|
&& (store_uint_array bp_pre n_pre l2 )
|
|
** (store_uint_array ap_pre n_pre l1 )
|
|
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
|
|
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
|
|
|--
|
|
([| (val1 < val2) |]
|
|
&& [| (1 = (-1)) |]
|
|
&& (mpd_store_Z ap_pre val1 n_pre cap1 )
|
|
** (mpd_store_Z bp_pre val2 n_pre cap2 ))
|
|
||
|
|
([| (val1 = val2) |]
|
|
&& [| (1 = 0) |]
|
|
&& (mpd_store_Z ap_pre val1 n_pre cap1 )
|
|
** (mpd_store_Z bp_pre val2 n_pre cap2 ))
|
|
||
|
|
([| (val1 > val2) |]
|
|
&& [| (1 = 1) |]
|
|
&& (mpd_store_Z ap_pre val1 n_pre cap1 )
|
|
** (mpd_store_Z bp_pre val2 n_pre cap2 ))
|
|
.
|
|
|
|
Definition mpn_cmp_return_wit_2 :=
|
|
forall (n_pre: Z) (bp_pre: Z) (ap_pre: Z) (val2: Z) (val1: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (l1: (@list Z)) (n: Z) ,
|
|
[| (n < 0) |]
|
|
&& [| ((-1) <= n) |]
|
|
&& [| (n < n_pre) |]
|
|
&& [| (list_store_Z l1 val1 ) |]
|
|
&& [| (list_store_Z l2 val2 ) |]
|
|
&& [| (n_pre = (Zlength (l1))) |]
|
|
&& [| (n_pre = (Zlength (l2))) |]
|
|
&& [| ((sublist ((n + 1 )) (n_pre) (l1)) = (sublist ((n + 1 )) (n_pre) (l2))) |]
|
|
&& [| (list_store_Z l1 val1 ) |]
|
|
&& [| (list_store_Z l2 val2 ) |]
|
|
&& [| (n_pre = (Zlength (l1))) |]
|
|
&& [| (n_pre = (Zlength (l2))) |]
|
|
&& [| (0 < n_pre) |]
|
|
&& [| (n_pre <= cap1) |]
|
|
&& [| (n_pre <= cap2) |]
|
|
&& [| (cap1 <= 100000000) |]
|
|
&& [| (cap2 <= 100000000) |]
|
|
&& (store_uint_array ap_pre n_pre l1 )
|
|
** (store_uint_array bp_pre n_pre l2 )
|
|
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
|
|
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
|
|
|--
|
|
([| (val1 < val2) |]
|
|
&& [| (0 = (-1)) |]
|
|
&& (mpd_store_Z ap_pre val1 n_pre cap1 )
|
|
** (mpd_store_Z bp_pre val2 n_pre cap2 ))
|
|
||
|
|
([| (val1 = val2) |]
|
|
&& [| (0 = 0) |]
|
|
&& (mpd_store_Z ap_pre val1 n_pre cap1 )
|
|
** (mpd_store_Z bp_pre val2 n_pre cap2 ))
|
|
||
|
|
([| (val1 > val2) |]
|
|
&& [| (0 = 1) |]
|
|
&& (mpd_store_Z ap_pre val1 n_pre cap1 )
|
|
** (mpd_store_Z bp_pre val2 n_pre cap2 ))
|
|
.
|
|
|
|
Definition mpn_cmp_partial_solve_wit_1 :=
|
|
forall (n_pre: Z) (bp_pre: Z) (ap_pre: Z) (val2: Z) (val1: Z) (cap2: Z) (cap1: Z) ,
|
|
[| (0 < n_pre) |]
|
|
&& [| (n_pre <= cap1) |]
|
|
&& [| (n_pre <= cap2) |]
|
|
&& [| (cap1 <= 100000000) |]
|
|
&& [| (cap2 <= 100000000) |]
|
|
&& (mpd_store_Z ap_pre val1 n_pre cap1 )
|
|
** (mpd_store_Z bp_pre val2 n_pre cap2 )
|
|
|--
|
|
[| (0 < n_pre) |]
|
|
&& [| (n_pre <= cap1) |]
|
|
&& [| (n_pre <= cap2) |]
|
|
&& [| (cap1 <= 100000000) |]
|
|
&& [| (cap2 <= 100000000) |]
|
|
&& (mpd_store_Z ap_pre val1 n_pre cap1 )
|
|
** (mpd_store_Z bp_pre val2 n_pre cap2 )
|
|
.
|
|
|
|
Definition mpn_cmp_partial_solve_wit_2 :=
|
|
forall (n_pre: Z) (bp_pre: Z) (ap_pre: Z) (val2: Z) (val1: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (l1: (@list Z)) (n: Z) ,
|
|
[| (n >= 0) |]
|
|
&& [| ((-1) <= n) |]
|
|
&& [| (n < n_pre) |]
|
|
&& [| (list_store_Z l1 val1 ) |]
|
|
&& [| (list_store_Z l2 val2 ) |]
|
|
&& [| (n_pre = (Zlength (l1))) |]
|
|
&& [| (n_pre = (Zlength (l2))) |]
|
|
&& [| ((sublist ((n + 1 )) (n_pre) (l1)) = (sublist ((n + 1 )) (n_pre) (l2))) |]
|
|
&& [| (list_store_Z l1 val1 ) |]
|
|
&& [| (list_store_Z l2 val2 ) |]
|
|
&& [| (n_pre = (Zlength (l1))) |]
|
|
&& [| (n_pre = (Zlength (l2))) |]
|
|
&& [| (0 < n_pre) |]
|
|
&& [| (n_pre <= cap1) |]
|
|
&& [| (n_pre <= cap2) |]
|
|
&& [| (cap1 <= 100000000) |]
|
|
&& [| (cap2 <= 100000000) |]
|
|
&& (store_uint_array ap_pre n_pre l1 )
|
|
** (store_uint_array bp_pre n_pre l2 )
|
|
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
|
|
** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
|
|
|--
|
|
[| (n >= 0) |]
|
|
&& [| ((-1) <= n) |]
|
|
&& [| (n < n_pre) |]
|
|
&& [| (list_store_Z l1 val1 ) |]
|
|
&& [| (list_store_Z l2 val2 ) |]
|
|
&& [| (n_pre = (Zlength (l1))) |]
|
|
&& [| (n_pre = (Zlength (l2))) |]
|
|
&& [| ((sublist ((n + 1 )) (n_pre) (l1)) = (sublist ((n + 1 )) (n_pre) (l2))) |]
|
|
&& [| (list_store_Z l1 val1 ) |]
|
|
&& [| (list_store_Z l2 val2 ) |]
|
|
&& [| (n_pre = (Zlength (l1))) |]
|
|
&& [| (n_pre = (Zlength (l2))) |]
|
|
&& [| (0 < n_pre) |]
|
|
&& [| (n_pre <= cap1) |]
|
|
&& [| (n_pre <= cap2) |]
|
|
&& [| (cap1 <= 100000000) |]
|
|
&& [| (cap2 <= 100000000) |]
|
|
&& (((ap_pre + (n * sizeof(UINT) ) )) # UInt |-> (Znth n l1 0))
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** (store_uint_array_missing_i_rec ap_pre n 0 n_pre l1 )
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** (store_uint_array bp_pre n_pre l2 )
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** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
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** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
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.
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Definition mpn_cmp_partial_solve_wit_3 :=
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forall (n_pre: Z) (bp_pre: Z) (ap_pre: Z) (val2: Z) (val1: Z) (cap2: Z) (cap1: Z) (l2: (@list Z)) (l1: (@list Z)) (n: Z) ,
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[| (n >= 0) |]
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&& [| ((-1) <= n) |]
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&& [| (n < n_pre) |]
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&& [| (list_store_Z l1 val1 ) |]
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&& [| (list_store_Z l2 val2 ) |]
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&& [| (n_pre = (Zlength (l1))) |]
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&& [| (n_pre = (Zlength (l2))) |]
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&& [| ((sublist ((n + 1 )) (n_pre) (l1)) = (sublist ((n + 1 )) (n_pre) (l2))) |]
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&& [| (list_store_Z l1 val1 ) |]
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&& [| (list_store_Z l2 val2 ) |]
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&& [| (n_pre = (Zlength (l1))) |]
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&& [| (n_pre = (Zlength (l2))) |]
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&& [| (0 < n_pre) |]
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&& [| (n_pre <= cap1) |]
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&& [| (n_pre <= cap2) |]
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&& [| (cap1 <= 100000000) |]
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&& [| (cap2 <= 100000000) |]
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&& (store_uint_array ap_pre n_pre l1 )
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** (store_uint_array bp_pre n_pre l2 )
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** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
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** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
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|--
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[| (n >= 0) |]
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&& [| ((-1) <= n) |]
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&& [| (n < n_pre) |]
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&& [| (list_store_Z l1 val1 ) |]
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&& [| (list_store_Z l2 val2 ) |]
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&& [| (n_pre = (Zlength (l1))) |]
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&& [| (n_pre = (Zlength (l2))) |]
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&& [| ((sublist ((n + 1 )) (n_pre) (l1)) = (sublist ((n + 1 )) (n_pre) (l2))) |]
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&& [| (list_store_Z l1 val1 ) |]
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&& [| (list_store_Z l2 val2 ) |]
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&& [| (n_pre = (Zlength (l1))) |]
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&& [| (n_pre = (Zlength (l2))) |]
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&& [| (0 < n_pre) |]
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&& [| (n_pre <= cap1) |]
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&& [| (n_pre <= cap2) |]
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&& [| (cap1 <= 100000000) |]
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&& [| (cap2 <= 100000000) |]
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&& (((bp_pre + (n * sizeof(UINT) ) )) # UInt |-> (Znth n l2 0))
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** (store_uint_array_missing_i_rec bp_pre n 0 n_pre l2 )
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|
** (store_uint_array ap_pre n_pre l1 )
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|
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
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** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
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|
.
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|
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Definition mpn_cmp_which_implies_wit_1 :=
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forall (n_pre: Z) (bp_pre: Z) (ap_pre: Z) (val2: Z) (val1: Z) (cap2: Z) (cap1: Z) ,
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(mpd_store_Z ap_pre val1 n_pre cap1 )
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** (mpd_store_Z bp_pre val2 n_pre cap2 )
|
|
|--
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EX (l2: (@list Z)) (l1: (@list Z)) ,
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[| (list_store_Z l1 val1 ) |]
|
|
&& [| (list_store_Z l2 val2 ) |]
|
|
&& [| (n_pre = (Zlength (l1))) |]
|
|
&& [| (n_pre = (Zlength (l2))) |]
|
|
&& (store_uint_array ap_pre n_pre l1 )
|
|
** (store_uint_array bp_pre n_pre l2 )
|
|
** (store_undef_uint_array_rec ap_pre (n_pre + 1 ) cap1 )
|
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** (store_undef_uint_array_rec bp_pre (n_pre + 1 ) cap2 )
|
|
.
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|
|
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Module Type VC_Correct.
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Axiom proof_of_gmp_abs_safety_wit_1 : gmp_abs_safety_wit_1.
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Axiom proof_of_gmp_abs_return_wit_1_1 : gmp_abs_return_wit_1_1.
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Axiom proof_of_gmp_abs_return_wit_1_2 : gmp_abs_return_wit_1_2.
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Axiom proof_of_gmp_max_return_wit_1_1 : gmp_max_return_wit_1_1.
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Axiom proof_of_gmp_max_return_wit_1_2 : gmp_max_return_wit_1_2.
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Axiom proof_of_gmp_cmp_safety_wit_1 : gmp_cmp_safety_wit_1.
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Axiom proof_of_gmp_cmp_safety_wit_2 : gmp_cmp_safety_wit_2.
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Axiom proof_of_gmp_cmp_safety_wit_3 : gmp_cmp_safety_wit_3.
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Axiom proof_of_gmp_cmp_safety_wit_4 : gmp_cmp_safety_wit_4.
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Axiom proof_of_gmp_cmp_return_wit_1_1 : gmp_cmp_return_wit_1_1.
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Axiom proof_of_gmp_cmp_return_wit_1_2 : gmp_cmp_return_wit_1_2.
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Axiom proof_of_gmp_cmp_return_wit_1_3 : gmp_cmp_return_wit_1_3.
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Axiom proof_of_mpn_copyi_safety_wit_1 : mpn_copyi_safety_wit_1.
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Axiom proof_of_mpn_copyi_safety_wit_2 : mpn_copyi_safety_wit_2.
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Axiom proof_of_mpn_copyi_entail_wit_1 : mpn_copyi_entail_wit_1.
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Axiom proof_of_mpn_copyi_entail_wit_2 : mpn_copyi_entail_wit_2.
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Axiom proof_of_mpn_copyi_return_wit_1 : mpn_copyi_return_wit_1.
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Axiom proof_of_mpn_copyi_partial_solve_wit_1 : mpn_copyi_partial_solve_wit_1.
|
|
Axiom proof_of_mpn_copyi_partial_solve_wit_2_pure : mpn_copyi_partial_solve_wit_2_pure.
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|
Axiom proof_of_mpn_copyi_partial_solve_wit_2 : mpn_copyi_partial_solve_wit_2.
|
|
Axiom proof_of_mpn_copyi_partial_solve_wit_3_pure : mpn_copyi_partial_solve_wit_3_pure.
|
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Axiom proof_of_mpn_copyi_partial_solve_wit_3 : mpn_copyi_partial_solve_wit_3.
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|
Axiom proof_of_mpn_copyi_partial_solve_wit_4 : mpn_copyi_partial_solve_wit_4.
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|
Axiom proof_of_mpn_copyi_partial_solve_wit_5 : mpn_copyi_partial_solve_wit_5.
|
|
Axiom proof_of_mpn_copyi_which_implies_wit_1 : mpn_copyi_which_implies_wit_1.
|
|
Axiom proof_of_mpn_copyi_which_implies_wit_2 : mpn_copyi_which_implies_wit_2.
|
|
Axiom proof_of_mpn_copyi_which_implies_wit_3 : mpn_copyi_which_implies_wit_3.
|
|
Axiom proof_of_mpn_cmp_safety_wit_1 : mpn_cmp_safety_wit_1.
|
|
Axiom proof_of_mpn_cmp_safety_wit_2 : mpn_cmp_safety_wit_2.
|
|
Axiom proof_of_mpn_cmp_safety_wit_3 : mpn_cmp_safety_wit_3.
|
|
Axiom proof_of_mpn_cmp_safety_wit_4 : mpn_cmp_safety_wit_4.
|
|
Axiom proof_of_mpn_cmp_safety_wit_5 : mpn_cmp_safety_wit_5.
|
|
Axiom proof_of_mpn_cmp_safety_wit_6 : mpn_cmp_safety_wit_6.
|
|
Axiom proof_of_mpn_cmp_safety_wit_7 : mpn_cmp_safety_wit_7.
|
|
Axiom proof_of_mpn_cmp_entail_wit_1 : mpn_cmp_entail_wit_1.
|
|
Axiom proof_of_mpn_cmp_entail_wit_2 : mpn_cmp_entail_wit_2.
|
|
Axiom proof_of_mpn_cmp_return_wit_1_1 : mpn_cmp_return_wit_1_1.
|
|
Axiom proof_of_mpn_cmp_return_wit_1_2 : mpn_cmp_return_wit_1_2.
|
|
Axiom proof_of_mpn_cmp_return_wit_2 : mpn_cmp_return_wit_2.
|
|
Axiom proof_of_mpn_cmp_partial_solve_wit_1 : mpn_cmp_partial_solve_wit_1.
|
|
Axiom proof_of_mpn_cmp_partial_solve_wit_2 : mpn_cmp_partial_solve_wit_2.
|
|
Axiom proof_of_mpn_cmp_partial_solve_wit_3 : mpn_cmp_partial_solve_wit_3.
|
|
Axiom proof_of_mpn_cmp_which_implies_wit_1 : mpn_cmp_which_implies_wit_1.
|
|
|
|
End VC_Correct.
|