Require Import Coq.ZArith.ZArith.
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.