# Verif_triangA client of the stack functions

Require VC.Preface. (* Check for the right version of VST *)
Require Import VST.floyd.proofauto.
Require Import VST.floyd.library.
Require Import VC.stack.
Instance CompSpecs : compspecs. make_compspecs prog. Defined.
Definition Vprog : varspecs. mk_varspecs prog. Defined.
Here are some functions (in stack.c) that are clients of the stack ADT. First, push the numbers 1,2,...,n onto a stack, then pop the numbers off the stack and add them up. This computes the nth triangular number, 1+2+...+n = n(n+1)/2.
```void push_increasing (struct stack *st, int n) {
int i;
i=0;
while (i<n) {
i++;
push(st,i);
}
}

int pop_and_add (struct stack *st, int n) {
int i=0;
int t, s=0;
while (i<n) {
t=pop(st);
s += t;
i++;
}
return s;
}

int main (void) {
struct stack *st;
int i,t,s;
st = newstack();
push_increasing(st, 10);
return s;
}
```
Let's verify this program!

## Proofs with integers

The natural numbers have arithmetic axioms that are not very nice. For example, you might expect that a-b+b=a, but that's not true:
¬ ( a b: nat, a-b+b=a)%nat.
Proof.
intros.
intro.
specialize (H 0 1)%nat.
simpl in H. inversion H.
Qed.
This just shows that if the negative numbers did not exist, it would be necessary to construct them! In reasoning about programs, as in many other kinds of mathematics, we should use the integers. In Coq the type is called Z.
a b : Z, a-b+b=a.
Proof. intros. lia. Qed.
The Z type does have an inductive definition . . .
Print Z.
(* Inductive Z : Set := Z0 : Z | Zpos : positive -> Z | Zneg : positive -> Z *)
but we generally prefer to reason abstractly about Z, using the lemmas in the Coq library (that the Coq developers proved from the inductive definition). The induction principle on Z, that's automatically derived from this Inductive definition, is not the one we usually want to use!
Let's consider a recursive function on Z, the function that turns 5 into the list 5::4::3::2::1::nil. In the natural numbers, that's easy to define:

Fixpoint decreasing_nat (n: nat) : list nat :=
match n with S n'n :: decreasing_nat n' | Onil end.
But in the integers Z, we cannot simply pattern-match on successor ...
Fail Fixpoint decreasing_Z (n: Z) : list Z :=
match n with Z.succ n'n :: decreasing_Z n' | 0 ⇒ nil end.
... because Z.succ is a function, not a constructor.
There are two ways we might define a function to produce a decreasing list of Z. First, we might use Z.of_nat and Z.to_nat:

Fixpoint decreasing_Z1_aux (n: nat) : list Z :=
match n with
| S n'Z.of_nat n :: decreasing_Z1_aux n'
| Onil
end.
Definition decreasing_Z1 (n: Z) : list Z :=
decreasing_Z1_aux (Z.to_nat n).
This will work, but in doing proofs the frequent conversion between Z and nat will be awkward. If possible, we'd like to stay in the integers as much as possible. So here's another way:

Check Z_gt_dec. (*   : forall x y : Z, {x > y} + {~ x > y} *)

Function decreasing (n: Z) {measure Z.to_nat n}:=
if Z_gt_dec n 0 then n :: decreasing (n-1) else nil.
Proof.
When you define a Function, you must provide a measure, that is, a function from your argument-type (in this case Z) to the natural numbers, and then you must prove that each recursive call within the function body decreases the measure. In this ecase, there's only one recursive call, so there's just one proof obligation: show that if n>0 then Z.to_nat (n-1) < Z.to_nat n.
lia.
Defined. (* Terminate your Function declarations with Defined instead
of Qed, so that Coq will be able to use your function in computations. *)

#### Exercise: 2 stars, standard (Zinduction)

Coq's standard induction principle for Z is not the one we usually want, so let us define a more natural induction scheme:
Lemma Zinduction: (P: Z Prop),
P 0
( i, 0 < i P (i-1) P i)
n, 0 n P n.
Proof.
intros.
rewrite <- (Z2Nat.id n) in × by lia.
set (j := Z.to_nat n) in ×. clearbody j.
Check inj_S. (* Hint!  this may be useful *)
Print Z.succ. (* Hint!  Z.succ(x) unfolds to x+1 *)
(* FILL IN HERE *) Admitted.

### A theorem about the nth triangular number

#### Exercise: 2 stars, standard (add_list_decreasing)

Theorem: the sum of the list (n)::(n-1):: ... :: 2::1 is n*(n+1)/2.

0 n
(2 × (add_list (decreasing n)))%Z = (n × (n+1))%Z.
Proof.
intros.
pattern n; apply Zinduction.
- reflexivity.
- intros.
WARNING! When using functions defined by Function, don't unfold them! Temporarily remove the (* comment *) brackets from the next line to see what happens!
(*  unfold decreasing. *)
Instead of unfolding decreasing we use the equation that Coq automagically defines for the Function. Try the command Search decreasing. to see all the reasoning principles that Coq defined for the new Function. We will use this one:
Check decreasing_equation.

rewrite decreasing_equation.
during the proof of this lemma, you may find the ring_simplify tactic useful. Read about it in the Coq reference manual. Basically, it takes formulas with multiplication and addition, and simplifies them. But you can do this without ring_simplify, using just ordinary rewriting with lemmas about Z.add and Z.mul.
(* FILL IN HERE *) Admitted.

0 n
add_list (decreasing n) = n × (n+1) / 2.
Proof.
intros.
apply Z.div_unique_exact.
(* FILL IN HERE *) Admitted.

### Definitions copied from Verif_stack.v

We repeat here some material from Verif_stack.v. Normally we would break the .c file into separate modules, and do our Verifiable C proofs in separate modules; but for this example we leave out the modules. Just skip down to "End of the material repeated from Verif_stack.v".
Specification of linked lists in separation logic

Fixpoint listrep (il: list Z) (p: val) : mpred :=
match il with
| i::il'EX y: val,
malloc_token Ews (Tstruct _cons noattr) p ×
data_at Ews (Tstruct _cons noattr) (Vint (Int.repr i),y) p ×
listrep il' y
| nil!! (p = nullval) && emp
end.

Lemma listrep_local_prop: il p, listrep il p |--
!! (is_pointer_or_null p (p=nullval il=nil)).
Proof.
induction il; intro; simpl.
entailer!. intuition.
Intros y.
entailer!.
split; intros. subst.
eapply field_compatible_nullval; eauto.
inversion H3.
Qed.
Hint Resolve listrep_local_prop : saturate_local.

Lemma listrep_valid_pointer:
il p,
listrep il p |-- valid_pointer p.
Proof.
(* FILL IN HERE *) Admitted.
Hint Resolve listrep_valid_pointer : valid_pointer.
Specification of stack data structure

Definition stack (il: list Z) (p: val) :=
EX q: val,
malloc_token Ews (Tstruct _stack noattr) p ×
data_at Ews (Tstruct _stack noattr) q p × listrep il q.

Lemma stack_local_prop: il p, stack il p |-- !! (isptr p).
Proof.
(* FILL IN HERE *) Admitted.
Hint Resolve stack_local_prop : saturate_local.

Lemma stack_valid_pointer:
il p,
stack il p |-- valid_pointer p.
Proof.
(* FILL IN HERE *) Admitted.
Hint Resolve stack_valid_pointer : valid_pointer.

Definition newstack_spec : ident × funspec :=
DECLARE _newstack
WITH gv: globals
PRE [ ]
PROP () PARAMS() GLOBALS(gv) SEP (mem_mgr gv)
POST [ tptr (Tstruct _stack noattr) ]
EX p: val, PROP ( ) RETURN (p) SEP (stack nil p; mem_mgr gv).

Definition push_spec : ident × funspec :=
DECLARE _push
WITH p: val, i: Z, il: list Z, gv: globals
PRE [ tptr (Tstruct _stack noattr), tint ]
PROP (Int.min_signed i Int.max_signed)
PARAMS (p; Vint (Int.repr i)) GLOBALS(gv)
SEP (stack il p; mem_mgr gv)
POST [ tvoid ]
PROP ( ) RETURN() SEP (stack (i::il) p; mem_mgr gv).

Definition pop_spec : ident × funspec :=
DECLARE _pop
WITH p: val, i: Z, il: list Z, gv: globals
PRE [ tptr (Tstruct _stack noattr) ]
PROP ()
PARAMS (p) GLOBALS(gv)
SEP (stack (i::il) p; mem_mgr gv)
POST [ tint ]
PROP ( ) RETURN (Vint (Int.repr i)) SEP (stack il p; mem_mgr gv).
(End of the material repeated from Verif_stack.v)

## Specification of the stack-client functions

Spend a few minutes studying these funspecs, and compare to the implementations in stack.c, until you understand why these might be appropriate specifications.

Definition push_increasing_spec :=
DECLARE _push_increasing
WITH st: val, n: Z, gv: globals
PRE [ tptr (Tstruct _stack noattr), tint ]
PROP (0 n Int.max_signed)
PARAMS (st; Vint (Int.repr n)) GLOBALS(gv)
SEP (stack nil st; mem_mgr gv)
POST [ tvoid ]
PROP() RETURN() SEP (stack (decreasing n) st; mem_mgr gv).

WITH st: val, il: list Z, gv: globals
PRE [ tptr (Tstruct _stack noattr), tint ]
PROP (Zlength il Int.max_signed;
Forall (Z.le 0) il;
PARAMS (st; Vint (Int.repr (Zlength il))) GLOBALS(gv)
SEP (stack il st; mem_mgr gv)
POST [ tint ]
PROP()
SEP (stack nil st; mem_mgr gv).

Definition main_spec :=
DECLARE _main
WITH gv: globals
PRE [ ] main_pre prog tt gv
POST [ tint ]
PROP( ) RETURN (Vint (Int.repr 55)) SEP( TT ).
Putting all the funspecs together

Definition Gprog : funspecs :=
ltac:(with_library prog [
newstack_spec; push_spec; pop_spec;
]).

## Proofs of the stack-client function-bodies

#### Exercise: 3 stars, standard (body_push_increasing)

Lemma body_push_increasing: semax_body Vprog Gprog
f_push_increasing push_increasing_spec.
(* FILL IN HERE *) Admitted.

#### Exercise: 2 stars, standard (add_list_lemmas)

(* FILL IN HERE *) Admitted.

il,
Forall (Z.le 0) il
(* FILL IN HERE *) Admitted.

#### Exercise: 2 stars, standard (add_list_sublist_bounds)

lo hi K il,
0 lo hi
hi Zlength il
Forall (Z.le 0) il
0 add_list (sublist lo hi il) K.
Proof.
Hint: you don't need induction. Useful lemmas are, sublist_same, sublist_split, add_list_nonneg, add_list_app, Forall_sublist, and use the hint tactic to learn when the list_solve tactic will be useful.
(* FILL IN HERE *) Admitted.

#### Exercise: 3 stars, standard (add_another)

Suppose we have a list il of integers, il = [5;4;3;2;1], with Znth 0 il = 5, Znth 4 il = 1, and Zlength il = 5, and we want to add them all up, 5+4+3+2+1=15. Suppose we've already added up the first i of them (let i=2 for example), that is, 5+4=9, and we want to add the next one, that is, the ith one. That is, we want to add 9+3. How do we know that won't overflow the range of C-language signed integer arithmetic?
The proof goes: Every element of the list is nonnegative; the whole list adds up to a number <= Int.max_signed; and any sublist of an all-nonnegative list adds up to less-or-equal to the total of the whole list.

il,
Forall (Z.le 0) il
i : Z,
0 i < Zlength il
Int.min_signed Int.signed (Int.repr (add_list (sublist 0 i il))) +
Int.signed (Int.repr (Znth i il)) Int.max_signed.
Proof.
intros.
(* FILL IN HERE *) admit.
}
assert (0 add_list (sublist 0 i il) Int.max_signed). {
(* FILL IN HERE *) admit.
}
assert (H4: 0 add_list (sublist 0 (i+1) il) Int.max_signed). {
(* FILL IN HERE *) admit.
}
assert (0 Znth i il Int.max_signed). {
replace (Znth i il) with (add_list (sublist i (i+1) il)).
-
(* FILL IN HERE *) admit.
-
(* FILL IN HERE *) admit.
}

Next: Int.signed (Int.repr (add_list (sublist 0 i il))) = add_list (sublist 0 i il). To prove that, we'll use Int.signed_repr:
Check Int.signed_repr.
(*   :   forall z : Z,   Int.min_signed <= z <= Int.max_signed ->
Int.signed (Int.repr z) = z. *)

rewrite Int.signed_repr by rep_lia.
rep_lia is just like lia, but it also knows the numeric values of representation-related constants such as Int.min_signed.
rewrite Int.signed_repr by rep_lia.

rewrite (sublist_split 0 i (i+1)) in H4 by list_solve.
rewrite sublist_len_1 in H4 by list_solve.
simpl in H4.
rep_lia.
(* FILL IN HERE *) Admitted.

#### Exercise: 3 stars, standard (body_pop_and_add)

Proof.
start_function.
forward.
forward.
forward_while (EX i:Z,
PROP(0 i Zlength il)
LOCAL (temp _st st;
temp _i (Vint (Int.repr i));
temp _n (Vint (Int.repr (Zlength il)));
gvars gv)
SEP (stack (sublist i (Zlength il) il) st; mem_mgr gv)).
+ (* Prove that the precondition implies the loop invariant *)
(* FILL IN HERE *) admit.
+ (* "type-check the expression": prove the loop test evaluates *)
entailer!.
+ (* Prove the loop body preserves the loop invariant *)
forward_call (st, Znth i il, sublist (i+1) (Zlength il) il, gv).
This forward_call couldn't quite figure out the "Frame" for the function call. That is, it couldn't match up stack (sublist i (Zlength il) il) st with
stack (Znth i il :: sublist (i + 1) (Zlength il) il) st.
You have to help, by doing some rewrites with sublist_split, sublist_len_1 that prove
sublist i (Zlength il) il = Znth i il :: sublist (i+1) (Zlength il) il.
When you've rewritten the goal into,
stack (Znth i il :: sublist (i + 1) (Zlength il) il) st
|-- stack (Znth i il :: sublist (i + 1) (Zlength il) il) st × fold_right_sepcon Frame
then just do cancel.
(* FILL IN HERE *) admit.
And now we are ready to go forward through the C statement _s = _s + _t;
Fail forward.
oops! we can't go forward through _s = _s + _t; because we forgot to mention temp _s in the loop invariant! Time to start over.
By the way, this statement _s = _s + _t is exactly where forward will ask you to prove a subgoal in which you can use lemma add_another.
Abort.
Into this lemma, paste in the failed proof just above, but adjust the loop invariant: add a LOCAL assertion for _s.
Proof.
Hint: choose the loop invariant for temp _s ??? in such a way that you can make use of Lemma add_another.
(* FILL IN HERE *) Admitted.

#### Exercise: 3 stars, standard (body_main)

Lemma body_main: semax_body Vprog Gprog f_main main_spec.
Proof.
start_function.
We assume that triang.c is linked with an implementation of malloc/free. That assumption is expressed by the create_mem_mgr axiom, which we can sep_apply here. On the other hand, if we want a complete verified system including libraries, then instead of importing floyd.library we would actually link with a malloc/free implementation, but that's beyond the scope of this chapter.
sep_apply (create_mem_mgr gv).
You can see that this has produced the SEP conjunct mem_mgr gv, which is useful to satisfy the precondition of newstack, push, pop, etc. Now you can finish this proof.

(* FILL IN HERE *) Admitted.

(* 2020-09-18 15:39 *)