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Require Import Coq.Bool.Bool.
Require Import Coq.Strings.String.
Require Import Coq.Strings.Ascii.
Require Import Coq.Lists.List.
Require Import Coq.Classes.RelationClasses.
Require Import Coq.Classes.Morphisms.
Require Import Coq.micromega.Psatz.
Require Import Coq.Sorting.Permutation.
From AUXLib Require Import int_auto Axioms Feq Idents ListLib VMap.
Require Import SetsClass.SetsClass. Import SetsNotation.
From SimpleC.SL Require Import Mem SeparationLogic.
Require Import Logic.LogicGenerator.demo932.Interface.
Local Open Scope Z_scope.
Local Open Scope sets.
Local Open Scope string_scope.
Local Open Scope list.
Import naive_C_Rules.
Require Import PVbench.Data_structures.stack.rocq.spec_lib.
Local Open Scope sac.
(*----- Function push -----*)
Definition push_entail_wit_1 :=
(
forall (n_pre: Z) (stack_pre: Z) (before: sll) (PreH1 : (n_pre < stack_capacity)) ,
(store_stack stack_pre before n_pre )
** (IntArray.undef_seg stack_pre n_pre (n_pre + 1 ) )
|--
EX (concrete: (@list Z)) ,
“ (0 <= n_pre) ”
&& “ (n_pre < stack_capacity) ”
&& “ (StackConcreteView before concrete n_pre ) ”
&& (IntArray.full stack_pre n_pre concrete )
** (IntArray.undef_seg stack_pre n_pre (n_pre + 1 ) )
) \/
(
forall (n_pre: Z) (stack_pre: Z) (before: sll) (PreH1 : (n_pre < stack_capacity)) ,
(store_stack stack_pre before n_pre )
|--
EX (concrete: (@list Z)) ,
“ (0 <= n_pre) ”
&& “ (n_pre < stack_capacity) ”
&& “ (StackConcreteView before concrete n_pre ) ”
&& (IntArray.full stack_pre n_pre concrete )
).
Definition push_entail_wit_2 :=
(
forall (x_pre: Z) (n_pre: Z) (stack_pre: Z) (before: sll) (concrete: (@list Z)) (PreH1 : (0 <= n_pre)) (PreH2 : (n_pre < stack_capacity)) (PreH3 : (StackConcreteView before concrete n_pre )) ,
(IntArray.full stack_pre (n_pre + 1 ) (app (concrete) ((cons (x_pre) ((@nil Z))))) )
|--
“ (0 <= n_pre) ”
&& “ (n_pre < stack_capacity) ”
&& (store_stack stack_pre (sll_cons (x_pre) (before)) (n_pre + 1 ) )
) \/
(
forall (x_pre: Z) (n_pre: Z) (stack_pre: Z) (before: sll) (concrete: (@list Z)) (PreH1 : (0 <= n_pre)) (PreH2 : (n_pre < stack_capacity)) (PreH3 : (StackConcreteView before concrete n_pre )) ,
(IntArray.full stack_pre (n_pre + 1 ) (app (concrete) ((cons (x_pre) ((@nil Z))))) )
|--
(store_stack stack_pre (sll_cons (x_pre) (before)) (n_pre + 1 ) )
).
Definition push_entail_wit_2_split_goal_spatial :=
forall (x_pre: Z) (n_pre: Z) (stack_pre: Z) (before: sll) (concrete: (@list Z)) (PreH1 : (0 <= n_pre)) (PreH2 : (n_pre < stack_capacity)) (PreH3 : (StackConcreteView before concrete n_pre )) ,
(IntArray.full stack_pre (n_pre + 1 ) (app (concrete) ((cons (x_pre) ((@nil Z))))) )
|--
(store_stack stack_pre (sll_cons (x_pre) (before)) (n_pre + 1 ) )
.
Definition push_return_wit_1 :=
forall (x_pre: Z) (n_pre: Z) (stack_pre: Z) (before: sll) (PreH1 : (0 <= n_pre)) (PreH2 : (n_pre < stack_capacity)) ,
(store_stack stack_pre (sll_cons (x_pre) (before)) (n_pre + 1 ) )
|--
(store_stack stack_pre (sll_cons (x_pre) (before)) (n_pre + 1 ) )
.
Definition push_partial_solve_wit_1 :=
forall (n_pre: Z) (stack_pre: Z) (before: sll) (concrete: (@list Z)) (PreH1 : (0 <= n_pre)) (PreH2 : (n_pre < stack_capacity)) (PreH3 : (StackConcreteView before concrete n_pre )) ,
(IntArray.full stack_pre n_pre concrete )
** (IntArray.undef_seg stack_pre n_pre (n_pre + 1 ) )
|--
“ (0 <= n_pre) ”
&& “ (n_pre < stack_capacity) ”
&& “ (StackConcreteView before concrete n_pre ) ”
&& (((stack_pre + (n_pre * sizeof(INT)))) # Int |->_)
** (IntArray.full stack_pre n_pre concrete )
.
(*----- Function pop -----*)
Definition pop_safety_wit_1 :=
forall (n_pre: Z) (stack_pre: Z) (rest: sll) (top: Z) (concrete: (@list Z)) (PreH1 : (1 <= n_pre)) (PreH2 : (n_pre <= stack_capacity)) (PreH3 : (StackConcreteView (sll_cons (top) (rest)) concrete n_pre )) ,
((( &( "ret" ) )) # Int |->_)
** ((( &( "stack" ) )) # Ptr |-> stack_pre)
** ((( &( "n" ) )) # Int |-> n_pre)
** (IntArray.full stack_pre n_pre concrete )
|--
“ ((n_pre - 1 ) <= INT_MAX) ”
&& “ ((INT_MIN) <= (n_pre - 1 )) ”
.
Definition pop_safety_wit_2 :=
forall (n_pre: Z) (stack_pre: Z) (rest: sll) (top: Z) (concrete: (@list Z)) (PreH1 : (1 <= n_pre)) (PreH2 : (n_pre <= stack_capacity)) (PreH3 : (StackConcreteView (sll_cons (top) (rest)) concrete n_pre )) ,
((( &( "ret" ) )) # Int |->_)
** ((( &( "stack" ) )) # Ptr |-> stack_pre)
** ((( &( "n" ) )) # Int |-> n_pre)
** (IntArray.full stack_pre n_pre concrete )
|--
“ (1 <= INT_MAX) ”
&& “ ((INT_MIN) <= 1) ”
.
Definition pop_entail_wit_1 :=
(
forall (n_pre: Z) (stack_pre: Z) (rest: sll) (top: Z) (PreH1 : (1 <= n_pre)) ,
(store_stack stack_pre (sll_cons (top) (rest)) n_pre )
|--
EX (concrete: (@list Z)) ,
“ (1 <= n_pre) ”
&& “ (n_pre <= stack_capacity) ”
&& “ (StackConcreteView (sll_cons (top) (rest)) concrete n_pre ) ”
&& (IntArray.full stack_pre n_pre concrete )
) \/
(
forall (n_pre: Z) (stack_pre: Z) (rest: sll) (top: Z) (PreH1 : (1 <= n_pre)) ,
(store_stack stack_pre (sll_cons (top) (rest)) n_pre )
|--
EX (concrete: (@list Z)) ,
“ (1 <= n_pre) ”
&& “ (n_pre <= stack_capacity) ”
&& “ (StackConcreteView (sll_cons (top) (rest)) concrete n_pre ) ”
&& (IntArray.full stack_pre n_pre concrete )
).
Definition pop_entail_wit_2 :=
(
forall (n_pre: Z) (stack_pre: Z) (rest: sll) (top: Z) (concrete: (@list Z)) (PreH1 : (1 <= n_pre)) (PreH2 : (n_pre <= stack_capacity)) (PreH3 : (StackConcreteView (sll_cons (top) (rest)) concrete n_pre )) ,
(IntArray.full stack_pre n_pre concrete )
** ((( &( "ret" ) )) # Int |-> (Znth (n_pre - 1 ) concrete 0))
|--
“ (1 <= n_pre) ”
&& “ (n_pre <= stack_capacity) ”
&& ((( &( "ret" ) )) # Int |-> top)
** (store_stack stack_pre rest (n_pre - 1 ) )
** (IntArray.undef_seg stack_pre (n_pre - 1 ) n_pre )
) \/
(
forall (n_pre: Z) (stack_pre: Z) (rest: sll) (top: Z) (concrete: (@list Z)) (PreH1 : (1 <= n_pre)) (PreH2 : (n_pre <= stack_capacity)) (PreH3 : (StackConcreteView (sll_cons (top) (rest)) concrete n_pre )) ,
(IntArray.full stack_pre n_pre concrete )
|--
“ ((Znth (n_pre - 1 ) concrete 0) = top) ”
&& (store_stack stack_pre rest (n_pre - 1 ) )
** (IntArray.undef_seg stack_pre (n_pre - 1 ) n_pre )
).
Definition pop_entail_wit_2_split_goal_1 :=
forall (n_pre: Z) (stack_pre: Z) (rest: sll) (top: Z) (concrete: (@list Z)) (PreH1 : (1 <= n_pre)) (PreH2 : (n_pre <= stack_capacity)) (PreH3 : (StackConcreteView (sll_cons (top) (rest)) concrete n_pre )) ,
(IntArray.full stack_pre n_pre concrete )
|--
“ ((Znth (n_pre - 1 ) concrete 0) = top) ”
.
Definition pop_entail_wit_2_split_goal_spatial :=
forall (n_pre: Z) (stack_pre: Z) (rest: sll) (top: Z) (concrete: (@list Z)) (PreH1 : (1 <= n_pre)) (PreH2 : (n_pre <= stack_capacity)) (PreH3 : (StackConcreteView (sll_cons (top) (rest)) concrete n_pre )) ,
(IntArray.full stack_pre n_pre concrete )
|--
(store_stack stack_pre rest (n_pre - 1 ) )
** (IntArray.undef_seg stack_pre (n_pre - 1 ) n_pre )
.
Definition pop_return_wit_1 :=
forall (n_pre: Z) (stack_pre: Z) (rest: sll) (top: Z) (PreH1 : (1 <= n_pre)) (PreH2 : (n_pre <= stack_capacity)) ,
(store_stack stack_pre rest (n_pre - 1 ) )
** (IntArray.undef_seg stack_pre (n_pre - 1 ) n_pre )
|--
“ (top = top) ”
&& (store_stack stack_pre rest (n_pre - 1 ) )
** (IntArray.undef_seg stack_pre (n_pre - 1 ) n_pre )
.
Definition pop_partial_solve_wit_1 :=
forall (n_pre: Z) (stack_pre: Z) (rest: sll) (top: Z) (concrete: (@list Z)) (PreH1 : (1 <= n_pre)) (PreH2 : (n_pre <= stack_capacity)) (PreH3 : (StackConcreteView (sll_cons (top) (rest)) concrete n_pre )) ,
(IntArray.full stack_pre n_pre concrete )
|--
“ (1 <= n_pre) ”
&& “ (n_pre <= stack_capacity) ”
&& “ (StackConcreteView (sll_cons (top) (rest)) concrete n_pre ) ”
&& (((stack_pre + ((n_pre - 1 ) * sizeof(INT)))) # Int |-> (Znth (n_pre - 1 ) concrete 0))
** (IntArray.missing_i stack_pre (n_pre - 1 ) 0 n_pre concrete )
.
(*----- Function build -----*)
Definition build_safety_wit_1 :=
forall (n_pre: Z) (stack_pre: Z) (input: (@list Z)) (PreH1 : (0 <= n_pre)) (PreH2 : (n_pre <= stack_capacity)) (PreH3 : ((Zlength (input)) = n_pre)) ,
((( &( "i" ) )) # Int |->_)
** ((( &( "stack" ) )) # Ptr |-> stack_pre)
** ((( &( "n" ) )) # Int |-> n_pre)
** (IntArray.full stack_pre n_pre input )
|--
“ (1 <= INT_MAX) ”
&& “ ((INT_MIN) <= 1) ”
.
Definition build_safety_wit_2 :=
forall (n_pre: Z) (stack_pre: Z) (input: (@list Z)) (i: Z) (PreH1 : (i < n_pre)) (PreH2 : (n_pre = 0)) (PreH3 : (i = 1)) (PreH4 : ((Zlength (input)) = n_pre)) ,
((( &( "stack" ) )) # Ptr |-> stack_pre)
** ((( &( "n" ) )) # Int |-> n_pre)
** ((( &( "i" ) )) # Int |-> i)
** (IntArray.full stack_pre n_pre input )
|--
“ False ”
.
Definition build_safety_wit_3 :=
forall (n_pre: Z) (stack_pre: Z) (input: (@list Z)) (next: sll) (i: Z) (x: Z) (PreH1 : (1 <= n_pre)) (PreH2 : (n_pre <= stack_capacity)) (PreH3 : (1 <= i)) (PreH4 : (i < n_pre)) (PreH5 : (x = (Znth i input 0))) (PreH6 : ((Zlength (input)) = n_pre)) (PreH7 : (BuildStackPrefix next input (i + 1 ) )) ,
((( &( "stack" ) )) # Ptr |-> stack_pre)
** ((( &( "n" ) )) # Int |-> n_pre)
** ((( &( "i" ) )) # Int |-> i)
** (store_stack stack_pre next (i + 1 ) )
** (IntArray.seg stack_pre (i + 1 ) n_pre (sublist ((i + 1 )) (n_pre) (input)) )
|--
“ ((i + 1 ) <= INT_MAX) ”
&& “ ((INT_MIN) <= (i + 1 )) ”
.
Definition build_entail_wit_1 :=
forall (n_pre: Z) (stack_pre: Z) (input: (@list Z)) (PreH1 : (0 <= n_pre)) (PreH2 : (n_pre <= stack_capacity)) (PreH3 : ((Zlength (input)) = n_pre)) ,
(IntArray.full stack_pre n_pre input )
|--
(“ (n_pre = 0) ”
&& “ (1 = 1) ”
&& “ ((Zlength (input)) = n_pre) ”
&& (IntArray.full stack_pre n_pre input ))
||
(EX (prefix: sll) ,
“ (1 <= n_pre) ”
&& “ (n_pre <= stack_capacity) ”
&& “ (1 <= 1) ”
&& “ (1 <= n_pre) ”
&& “ ((Zlength (input)) = n_pre) ”
&& “ (BuildStackPrefix prefix input 1 ) ”
&& (store_stack stack_pre prefix 1 )
** (IntArray.seg stack_pre 1 n_pre (sublist (1) (n_pre) (input)) ))
.
Definition build_entail_wit_2 :=
(
forall (n_pre: Z) (stack_pre: Z) (input: (@list Z)) (prefix_2: sll) (i: Z) (PreH1 : (i < n_pre)) (PreH2 : (1 <= n_pre)) (PreH3 : (n_pre <= stack_capacity)) (PreH4 : (1 <= i)) (PreH5 : (i <= n_pre)) (PreH6 : ((Zlength (input)) = n_pre)) (PreH7 : (BuildStackPrefix prefix_2 input i )) ,
(IntArray.seg stack_pre i n_pre (sublist (i) (n_pre) (input)) )
** (store_stack stack_pre prefix_2 i )
|--
EX (prefix: sll) ,
“ (1 <= n_pre) ”
&& “ (n_pre <= stack_capacity) ”
&& “ (1 <= i) ”
&& “ (i < n_pre) ”
&& “ ((Znth (i - i ) (sublist (i) (n_pre) (input)) 0) = (Znth i input 0)) ”
&& “ ((Zlength (input)) = n_pre) ”
&& “ (BuildStackPrefix prefix input i ) ”
&& (store_stack stack_pre prefix i )
** (IntArray.undef_seg stack_pre i (i + 1 ) )
** (IntArray.seg stack_pre (i + 1 ) n_pre (sublist ((i + 1 )) (n_pre) (input)) )
) \/
(
forall (n_pre: Z) (stack_pre: Z) (input: (@list Z)) (prefix_2: sll) (i: Z) (PreH1 : (i < n_pre)) (PreH2 : (1 <= n_pre)) (PreH3 : (n_pre <= stack_capacity)) (PreH4 : (1 <= i)) (PreH5 : (i <= n_pre)) (PreH6 : ((Zlength (input)) = n_pre)) (PreH7 : (BuildStackPrefix prefix_2 input i )) ,
(IntArray.seg stack_pre i n_pre (sublist (i) (n_pre) (input)) )
** (store_stack stack_pre prefix_2 i )
|--
EX (prefix: sll) ,
“ (1 <= n_pre) ”
&& “ (n_pre <= stack_capacity) ”
&& “ (1 <= i) ”
&& “ (i < n_pre) ”
&& “ ((Znth (i - i ) (sublist (i) (n_pre) (input)) 0) = (Znth i input 0)) ”
&& “ ((Zlength (input)) = n_pre) ”
&& “ (BuildStackPrefix prefix input i ) ”
&& (store_stack stack_pre prefix i )
** (IntArray.undef_seg stack_pre i (i + 1 ) )
** (IntArray.seg stack_pre (i + 1 ) n_pre (sublist ((i + 1 )) (n_pre) (input)) )
).
Definition build_entail_wit_3 :=
(
forall (n_pre: Z) (stack_pre: Z) (input: (@list Z)) (prefix: sll) (i: Z) (x: Z) (PreH1 : (1 <= n_pre)) (PreH2 : (n_pre <= stack_capacity)) (PreH3 : (1 <= i)) (PreH4 : (i < n_pre)) (PreH5 : (x = (Znth i input 0))) (PreH6 : ((Zlength (input)) = n_pre)) (PreH7 : (BuildStackPrefix prefix input i )) ,
(store_stack stack_pre (sll_cons (x) (prefix)) (i + 1 ) )
** (IntArray.seg stack_pre (i + 1 ) n_pre (sublist ((i + 1 )) (n_pre) (input)) )
|--
EX (next: sll) ,
“ (1 <= n_pre) ”
&& “ (n_pre <= stack_capacity) ”
&& “ (1 <= i) ”
&& “ (i < n_pre) ”
&& “ (x = (Znth i input 0)) ”
&& “ ((Zlength (input)) = n_pre) ”
&& “ (BuildStackPrefix next input (i + 1 ) ) ”
&& (store_stack stack_pre next (i + 1 ) )
** (IntArray.seg stack_pre (i + 1 ) n_pre (sublist ((i + 1 )) (n_pre) (input)) )
) \/
(
forall (n_pre: Z) (input: (@list Z)) (prefix: sll) (i: Z) (x: Z) (PreH1 : (1 <= n_pre)) (PreH2 : (n_pre <= stack_capacity)) (PreH3 : (1 <= i)) (PreH4 : (i < n_pre)) (PreH5 : (x = (Znth i input 0))) (PreH6 : ((Zlength (input)) = n_pre)) (PreH7 : (BuildStackPrefix prefix input i )) ,
TT && emp
|--
“ (BuildStackPrefix (sll_cons (x) (prefix)) input (i + 1 ) ) ”
&& emp
).
Definition build_entail_wit_3_split_goal_1 :=
forall (n_pre: Z) (input: (@list Z)) (prefix: sll) (i: Z) (x: Z) (PreH1 : (1 <= n_pre)) (PreH2 : (n_pre <= stack_capacity)) (PreH3 : (1 <= i)) (PreH4 : (i < n_pre)) (PreH5 : (x = (Znth i input 0))) (PreH6 : ((Zlength (input)) = n_pre)) (PreH7 : (BuildStackPrefix prefix input i )) ,
(BuildStackPrefix (sll_cons (x) (prefix)) input (i + 1 ) )
.
Definition build_entail_wit_4 :=
forall (n_pre: Z) (stack_pre: Z) (input: (@list Z)) (next: sll) (i: Z) (x: Z) (PreH1 : (1 <= n_pre)) (PreH2 : (n_pre <= stack_capacity)) (PreH3 : (1 <= i)) (PreH4 : (i < n_pre)) (PreH5 : (x = (Znth i input 0))) (PreH6 : ((Zlength (input)) = n_pre)) (PreH7 : (BuildStackPrefix next input (i + 1 ) )) ,
(store_stack stack_pre next (i + 1 ) )
** (IntArray.seg stack_pre (i + 1 ) n_pre (sublist ((i + 1 )) (n_pre) (input)) )
|--
EX (prefix: sll) ,
“ (1 <= n_pre) ”
&& “ (n_pre <= stack_capacity) ”
&& “ (1 <= (i + 1 )) ”
&& “ ((i + 1 ) <= n_pre) ”
&& “ ((Zlength (input)) = n_pre) ”
&& “ (BuildStackPrefix prefix input (i + 1 ) ) ”
&& (store_stack stack_pre prefix (i + 1 ) )
** (IntArray.seg stack_pre (i + 1 ) n_pre (sublist ((i + 1 )) (n_pre) (input)) )
.
Definition build_entail_wit_5_1 :=
(
forall (n_pre: Z) (stack_pre: Z) (input: (@list Z)) (i: Z) (PreH1 : (i >= n_pre)) (PreH2 : (n_pre = 0)) (PreH3 : (i = 1)) (PreH4 : ((Zlength (input)) = n_pre)) ,
(IntArray.full stack_pre n_pre input )
|--
“ (0 <= n_pre) ”
&& “ (n_pre <= stack_capacity) ”
&& “ ((Zlength (input)) = n_pre) ”
&& (store_stack stack_pre (sll_from_array (input)) n_pre )
) \/
(
forall (n_pre: Z) (stack_pre: Z) (input: (@list Z)) (i: Z) (PreH1 : (i >= n_pre)) (PreH2 : (n_pre = 0)) (PreH3 : (i = 1)) (PreH4 : ((Zlength (input)) = n_pre)) ,
(IntArray.full stack_pre n_pre input )
|--
“ (0 <= stack_capacity) ”
&& (store_stack stack_pre (sll_from_array (input)) n_pre )
).
Definition build_entail_wit_5_1_split_goal_1 :=
forall (n_pre: Z) (stack_pre: Z) (input: (@list Z)) (i: Z) (PreH1 : (i >= n_pre)) (PreH2 : (n_pre = 0)) (PreH3 : (i = 1)) (PreH4 : ((Zlength (input)) = n_pre)) ,
(IntArray.full stack_pre n_pre input )
|--
“ (0 <= stack_capacity) ”
.
Definition build_entail_wit_5_1_split_goal_spatial :=
forall (n_pre: Z) (stack_pre: Z) (input: (@list Z)) (i: Z) (PreH1 : (i >= n_pre)) (PreH2 : (n_pre = 0)) (PreH3 : (i = 1)) (PreH4 : ((Zlength (input)) = n_pre)) ,
(IntArray.full stack_pre n_pre input )
|--
(store_stack stack_pre (sll_from_array (input)) n_pre )
.
Definition build_entail_wit_5_2 :=
(
forall (n_pre: Z) (stack_pre: Z) (input: (@list Z)) (prefix: sll) (i: Z) (PreH1 : (i >= n_pre)) (PreH2 : (1 <= n_pre)) (PreH3 : (n_pre <= stack_capacity)) (PreH4 : (1 <= i)) (PreH5 : (i <= n_pre)) (PreH6 : ((Zlength (input)) = n_pre)) (PreH7 : (BuildStackPrefix prefix input i )) ,
(store_stack stack_pre prefix i )
** (IntArray.seg stack_pre i n_pre (sublist (i) (n_pre) (input)) )
|--
“ (0 <= n_pre) ”
&& “ (n_pre <= stack_capacity) ”
&& “ ((Zlength (input)) = n_pre) ”
&& (store_stack stack_pre (sll_from_array (input)) n_pre )
) \/
(
forall (n_pre: Z) (stack_pre: Z) (input: (@list Z)) (prefix: sll) (i: Z) (PreH1 : (i >= n_pre)) (PreH2 : (1 <= n_pre)) (PreH3 : (n_pre <= stack_capacity)) (PreH4 : (1 <= i)) (PreH5 : (i <= n_pre)) (PreH6 : ((Zlength (input)) = n_pre)) (PreH7 : (BuildStackPrefix prefix input i )) ,
(store_stack stack_pre prefix i )
** (IntArray.seg stack_pre i n_pre (sublist (i) (n_pre) (input)) )
|--
(store_stack stack_pre (sll_from_array (input)) n_pre )
).
Definition build_entail_wit_5_2_split_goal_spatial :=
forall (n_pre: Z) (stack_pre: Z) (input: (@list Z)) (prefix: sll) (i: Z) (PreH1 : (i >= n_pre)) (PreH2 : (1 <= n_pre)) (PreH3 : (n_pre <= stack_capacity)) (PreH4 : (1 <= i)) (PreH5 : (i <= n_pre)) (PreH6 : ((Zlength (input)) = n_pre)) (PreH7 : (BuildStackPrefix prefix input i )) ,
(store_stack stack_pre prefix i )
** (IntArray.seg stack_pre i n_pre (sublist (i) (n_pre) (input)) )
|--
(store_stack stack_pre (sll_from_array (input)) n_pre )
.
Definition build_return_wit_1 :=
forall (n_pre: Z) (stack_pre: Z) (input: (@list Z)) (PreH1 : (0 <= n_pre)) (PreH2 : (n_pre <= stack_capacity)) (PreH3 : ((Zlength (input)) = n_pre)) ,
(store_stack stack_pre (sll_from_array (input)) n_pre )
|--
(store_stack stack_pre (sll_from_array (input)) n_pre )
.
Definition build_partial_solve_wit_1 :=
forall (n_pre: Z) (stack_pre: Z) (input: (@list Z)) (prefix: sll) (i: Z) (PreH1 : (i < n_pre)) (PreH2 : (1 <= n_pre)) (PreH3 : (n_pre <= stack_capacity)) (PreH4 : (1 <= i)) (PreH5 : (i <= n_pre)) (PreH6 : ((Zlength (input)) = n_pre)) (PreH7 : (BuildStackPrefix prefix input i )) ,
(store_stack stack_pre prefix i )
** (IntArray.seg stack_pre i n_pre (sublist (i) (n_pre) (input)) )
|--
“ (i < n_pre) ”
&& “ (1 <= n_pre) ”
&& “ (n_pre <= stack_capacity) ”
&& “ (1 <= i) ”
&& “ (i <= n_pre) ”
&& “ ((Zlength (input)) = n_pre) ”
&& “ (BuildStackPrefix prefix input i ) ”
&& (((stack_pre + (i * sizeof(INT)))) # Int |-> (Znth (i - i ) (sublist (i) (n_pre) (input)) 0))
** (IntArray.missing_i stack_pre i i n_pre (sublist (i) (n_pre) (input)) )
** (store_stack stack_pre prefix i )
.
Definition build_partial_solve_wit_2_pure :=
forall (n_pre: Z) (stack_pre: Z) (input: (@list Z)) (prefix: sll) (i: Z) (x: Z) (PreH1 : (1 <= n_pre)) (PreH2 : (n_pre <= stack_capacity)) (PreH3 : (1 <= i)) (PreH4 : (i < n_pre)) (PreH5 : (x = (Znth i input 0))) (PreH6 : ((Zlength (input)) = n_pre)) (PreH7 : (BuildStackPrefix prefix input i )) ,
((( &( "stack" ) )) # Ptr |-> stack_pre)
** ((( &( "n" ) )) # Int |-> n_pre)
** ((( &( "i" ) )) # Int |-> i)
** ((( &( "x" ) )) # Int |-> x)
** (store_stack stack_pre prefix i )
** (IntArray.undef_seg stack_pre i (i + 1 ) )
** (IntArray.seg stack_pre (i + 1 ) n_pre (sublist ((i + 1 )) (n_pre) (input)) )
|--
“ (i < stack_capacity) ”
.
Definition build_partial_solve_wit_2_aux :=
forall (n_pre: Z) (stack_pre: Z) (input: (@list Z)) (prefix: sll) (i: Z) (x: Z) (PreH1 : (1 <= n_pre)) (PreH2 : (n_pre <= stack_capacity)) (PreH3 : (1 <= i)) (PreH4 : (i < n_pre)) (PreH5 : (x = (Znth i input 0))) (PreH6 : ((Zlength (input)) = n_pre)) (PreH7 : (BuildStackPrefix prefix input i )) ,
(store_stack stack_pre prefix i )
** (IntArray.undef_seg stack_pre i (i + 1 ) )
** (IntArray.seg stack_pre (i + 1 ) n_pre (sublist ((i + 1 )) (n_pre) (input)) )
|--
“ (i < stack_capacity) ”
&& “ (1 <= n_pre) ”
&& “ (n_pre <= stack_capacity) ”
&& “ (1 <= i) ”
&& “ (i < n_pre) ”
&& “ (x = (Znth i input 0)) ”
&& “ ((Zlength (input)) = n_pre) ”
&& “ (BuildStackPrefix prefix input i ) ”
&& (store_stack stack_pre prefix i )
** (IntArray.undef_seg stack_pre i (i + 1 ) )
** (IntArray.seg stack_pre (i + 1 ) n_pre (sublist ((i + 1 )) (n_pre) (input)) )
.
Definition build_partial_solve_wit_2 := build_partial_solve_wit_2_pure -> build_partial_solve_wit_2_aux.
Module Type VC_Correct.
Axiom proof_of_push_entail_wit_1 : push_entail_wit_1.
Axiom proof_of_push_entail_wit_2 : push_entail_wit_2.
Axiom proof_of_push_return_wit_1 : push_return_wit_1.
Axiom proof_of_push_partial_solve_wit_1 : push_partial_solve_wit_1.
Axiom proof_of_pop_safety_wit_1 : pop_safety_wit_1.
Axiom proof_of_pop_safety_wit_2 : pop_safety_wit_2.
Axiom proof_of_pop_entail_wit_1 : pop_entail_wit_1.
Axiom proof_of_pop_entail_wit_2 : pop_entail_wit_2.
Axiom proof_of_pop_return_wit_1 : pop_return_wit_1.
Axiom proof_of_pop_partial_solve_wit_1 : pop_partial_solve_wit_1.
Axiom proof_of_build_safety_wit_1 : build_safety_wit_1.
Axiom proof_of_build_safety_wit_2 : build_safety_wit_2.
Axiom proof_of_build_safety_wit_3 : build_safety_wit_3.
Axiom proof_of_build_entail_wit_1 : build_entail_wit_1.
Axiom proof_of_build_entail_wit_2 : build_entail_wit_2.
Axiom proof_of_build_entail_wit_3 : build_entail_wit_3.
Axiom proof_of_build_entail_wit_4 : build_entail_wit_4.
Axiom proof_of_build_entail_wit_5_1 : build_entail_wit_5_1.
Axiom proof_of_build_entail_wit_5_2 : build_entail_wit_5_2.
Axiom proof_of_build_return_wit_1 : build_return_wit_1.
Axiom proof_of_build_partial_solve_wit_1 : build_partial_solve_wit_1.
Axiom proof_of_build_partial_solve_wit_2_pure : build_partial_solve_wit_2_pure.
Axiom proof_of_build_partial_solve_wit_2 : build_partial_solve_wit_2.
End VC_Correct.
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