Implementation Defined Annotations

This appendix lists all the uses of aspect or pragma Annotate for GNATprove. Aspect or pragma Annotate can also be used to control other AdaCore tools. The uses of such annotations are explained in the User’s Guide of each tool.

Annotations in GNATprove are useful in two cases:

  1. for justifying check messages using Direct Justification with Pragma Annotate, typically using a pragma rather than an aspect, as the justification is generally associated to a statement or declaration.

  2. for influencing the generation of proof obligations, typically using an aspect rather than a pragma, as the annotation is generally associated to an entity in that case. Some of these uses can be seen in SPARK Libraries for example. Some of these annotations introduce additional assumptions which are not verified by the GNATprove tool, and thus should be used with care.

When the annotation is associated to an entity, both the pragma and aspect form can be used and are equivalent, for example on a subprogram:

function Func (X : T) return T
  with Annotate => (GNATprove, <annotation name>);

or

function Func (X : T) return T;
pragma Annotate (GNATprove, <annotation name>, Func);

In the following, we use the aspect form whenever possible.

Annotation for Justifying Check Messages

You can use annotations of the form

pragma Annotate (GNATprove, False_Positive,
                 "message to be justified", "reason");

to justify an unproved check message that cannot be proved by other means. See the section Direct Justification with Pragma Annotate for more details about this use of pragma Annotate.

Annotation for Skipping Parts of the Analysis for an Entity

Subprograms, packages, tasks, entries and protected subprograms can be annotated to skip flow analysis, and to skip generating proof obligations for their implementation, and the implementations of all such entities defined inside.

procedure P
  with Annotate => (GNATprove, Skip_Proof);

procedure P
  with Annotate => (GNATprove, Skip_Flow_And_Proof);

If an entity is annotated with Skip_Proof, no messages related to possible run-time errors and functional contracts are issued for this entity and any contained entities. This is similar to specifying --mode=flow on the command line (see Effect of Mode on Output), except that the effect is limited to this entity (and enclosed entities).

If an entity is annotated with Skip_Flow_And_Proof, in addition, no messages related to global contracts, initialization and dependency relations are issues for this entity and any contained entities. This is similar to specifying --mode=check-all on the command line, expect that the effect is limited to this entity (and enclosed entities).

Note that the Skip_Proof annotation cannot be used if an enclosing subprogram already has the Skip_Flow_And_Proof annotation.

Annotation for Overflow Checking on Modular Types

The standard semantics of arithmetic on modular types is that operations wrap around, hence GNATprove issues no overflow checks on such operations. You can instruct it to issue such checks (hence detecting possible wrap-around) using annotations of the form:

type T is mod 2**32
  with Annotate => (GNATprove, No_Wrap_Around);

or on a derived type:

type T is new U
  with Annotate => (GNATprove, No_Wrap_Around);

This annotation is inherited by derived types. It must be specified on a type declaration (and cannot be specified on a subtype declaration). All four binary arithmetic operations + - * ** are checked for possible overflows. Division cannot lead to overflow. Unary negation is checked for possible non-nullity of its argument, which leads to overflow. The predecessor attribute 'Pred and successor attribute 'Succ are also checked for possible overflows.

Annotation for Simplifying Iteration for Proof

The translation presented in :ref:Aspect and Pragma Iterable` may produce complicated proofs, especially when verifying complex properties over sets. The |GNATprove| annotation ``Iterable_For_Proof can be used to change the way for ... of quantification is translated. More precisely, it allows to provide GNATprove with a Contains function, that will be used for quantification. For example, on our sets, we could write:

function Mem (S : Set; E : Element_Type) return Boolean;
pragma Annotate (GNATprove, Iterable_For_Proof, "Contains", Mem);

With this annotation, the postcondition of Intersection is translated in a simpler way, using logic quantification directly over elements:

(for all E : Element_Type =>
     (if Mem (Intersection'Result, E) then Mem (S1, E) and Mem (S2, E)))
and (for all E : Element_Type =>
     (if Mem (S1, E) then
            (if Mem (S2, E) then Mem (Intersection'Result, E))))

Note that care should be taken to provide an appropriate function contains, which returns true if and only if the element E is present in S. This assumption will not be verified by GNATprove.

The annotation Iterable_For_Proof can also be used in another case. Operations over complex data structures are sometimes specified using operations over a simpler model type. In this case, it may be more appropriate to translate for ... of quantification as quantification over the model’s cursors. As an example, let us consider a package of linked lists that is specified using a sequence that allows accessing the element stored at each position:

package Lists with SPARK_Mode is

 type Sequence is private with
   Ghost,
   Iterable => (...,
                Element     => Get);
 function Length (M : Sequence) return Natural with Ghost;
 function Get (M : Sequence; P : Positive) return Element_Type with
   Ghost,
   Pre => P <= Length (M);

 type Cursor is private;
 type List is private with
   Iterable => (...,
                Element     => Element);

 function Position (L : List; C : Cursor) return Positive with Ghost;
 function Model (L : List) return Sequence with
   Ghost,
   Post => (for all I in 1 .. Length (Model'Result) =>
                (for some C in L => Position (L, C) = I));

 function Element (L : List; C : Cursor) return Element_Type with
   Pre  => Has_Element (L, C),
   Post => Element'Result = Get (Model (L), Position (L, C));

 function Has_Element (L : List; C : Cursor) return Boolean with
   Post => Has_Element'Result = (Position (L, C) in 1 .. Length (Model (L)));

 procedure Append (L : in out List; E : Element_Type) with
   Post => length (Model (L)) = Length (Model (L))'Old + 1
   and Get (Model (L), Length (Model (L))) = E
   and (for all I in 1 .. Length (Model (L))'Old =>
          Get (Model (L), I) = Get (Model (L'Old), I));

 function Init (N : Natural; E : Element_Type) return List with
   Post => length (Model (Init'Result)) = N
     and (for all F of Init'Result => F = E);

Elements of lists can only be accessed through cursors. To specify easily the effects of position-based operations such as Append, we introduce a ghost type Sequence, that is used to represent logically the content of the linked list in specifications. The sequence associated to a list can be constructed using the Model function. Following the usual translation scheme for quantified expressions, the last line of the postcondition of Init is translated for proof as:

(for all C : Cursor =>
    (if Has_Element (Init'Result, C) then Element (Init'Result, C) = E));

Using the definition of Element and Has_Element, it can then be refined further into:

(for all C : Cursor =>
    (if Position (Init'Result, C) in 1 .. Length (Model (Init'Result))
     then Get (Model (Init'Result), Position (Init'Result, C)) = E));

To be able to link this property with other properties specified directly on models, like the postcondition of Append, it needs to be lifted to iterate over positions instead of cursors. This can be done using the postcondition of Model that states that there is a valid cursor in L for each position of its model. This lifting requires a lot of quantifier reasoning from the prover, thus making proofs more difficult.

The GNATprove Iterable_For_Proof annotation can be used to provide GNATprove with a Model function, that will be to translate quantification on complex containers toward quantification on their model. For example, on our lists, we could write:

function Model (L : List) return Sequence;
pragma Annotate (GNATprove, Iterable_For_Proof, "Model", Entity => Model);

With this annotation, the postcondition of Init is translated directly as a quantification on the elements of the result’s model:

(for all I : Positive =>
   (if I in 1 .. Length (Model (Init'Result)) then
      Get (Model (Init'Result), I) = E));

Like with the previous annotation, care should be taken to define the model function such that it always return a model containing exactly the same elements as L.

Note

It is not possible to specify more than one Iterable_For_Proof annotation per container type with an Iterable aspect.

Annotation for Inlining Functions for Proof

Contracts for functions are generally translated by GNATprove as axioms on otherwise undefined functions. As an example, consider the following function:

function Increment (X : Integer) return Integer with
  Post => Increment'Result >= X;

It will be translated by GNATprove as follows:

function Increment (X : Integer) return Integer;

axiom : (for all X : Integer. Increment (X) >= X);

For internal reasons due to ordering issues, expression functions are also defined using axioms. For example:

function Is_Positive (X : Integer) return Boolean is (X > 0);

will be translated exactly as if its definition was given through a postcondition, namely:

function Is_Positive (X : Integer) return Boolean;

axiom : (for all X : Integer. Is_Positive (X) = (X > 0));

This encoding may sometimes cause difficulties to the underlying solvers, especially for quantifier instantiation heuristics. This can cause strange behaviors, where an assertion is proven when some calls to expression functions are manually inlined but not without this inlining.

If such a case occurs, it is sometimes possible to instruct the tool to inline the definition of expression functions using pragma Annotate Inline_For_Proof. When such a pragma is provided for an expression function, a direct definition will be used for the function instead of an axiom:

function Is_Positive (X : Integer) return Boolean is (X > 0);
pragma Annotate (GNATprove, Inline_For_Proof, Is_Positive);

The same pragma will also allow to inline a regular function, if its postcondition is simply an equality between its result and an expression:

function Is_Positive (X : Integer) return Boolean with
  Post => Is_Positive'Result = (X > 0);
pragma Annotate (GNATprove, Inline_For_Proof, Is_Positive);

In this case, GNATprove will introduce a check when verifying the body of Is_Positive to make sure that the inline annotation is correct, namely, that Is_Positive (X) and X > 0 always yield the same result. This check may not be redundant with the verification of the postcondition of Is_Positive if the = symbol on booleans has been overridden.

Note that, since the translation through axioms is necessary for ordering issues, this annotation can sometimes lead to a crash in GNATprove. It is the case for example when the definition of the function uses quantification over a container using the Iterable aspect.

Annotation for Referring to a Value at the End of a Local Borrow

Local borrowers are objects of an anonymous access-to-variable type. At their declaration, the ownership of (a part of) an existing data-structure is temporarily transferred to the new object. The borrowed data-structure will regain ownership afterward. During the lifetime of the borrower, the borrowed object can be accessed indirectly through the borrower. It is forbidden to modify or even read the borrowed object during the borrow. As an example, let us consider a recursive type of doubly-linked lists:

type List;
type List_Acc is access List;
type List is record
   Val  : Integer;
   Next : List_Acc;
end record;

Using this type, let us construct a list X which stores the numbers from 1 to 5:

X := new List'(1, null);
X.Next := new List'(2, null);
X.Next.Next := new List'(3, null);
X.Next.Next.Next := new List'(4, null);
X.Next.Next.Next.Next := new List'(5, null);

We can borrow the structure designated by X in a local borrower Y:

declare
   Y : access List := X;
begin
   V := Y.Val; --  OK, ownership has been transferred to Y temporarily
   V := X.Val; --  Illegal, X does not have ownership during the scope of Y
end;

While in the scope of Y, the ownership of the list designated by X is transferred to Y, so that it is not allowed to access it from X anymore. After the end of the declare block, ownership is restored to X, which can again be accessed or modified directly.

During the lifetime of the borrower, the borrowed object can be modified indirectly through the borrower. Therefore, when the borrower goes out of scope and ownership is transferred back to the borrowed object, GNATprove needs to reconstruct the new value of the borrowed object from the value of the borrower at the end of the borrow. In general, it can be done entirely automatically. However, it can happen that the exact relation between the values of the borrowed object and the borrower at the end of the borrow is lost by the tool. In particular, it is the case when the borrower is created inside a (traversal) function, as proof is modular on a per subprogram basis. It also happens when the borrower is modified inside a loop, as analysing loops involves cutpoints. In this case, GNATprove relies on the user to adequately describe the link between the values of the borrowed object and the borrower at the end of the borrow inside annotations - postconditions of traversal functions or loop invariants.

To this effect, it is possible to refer to the value of a local borrower or a borrowed expression at the end of the borrow using a ghost identity function annotated with At_End_Borrow. Calls to these functions are interpreted by the tool as markers of references to values at the end of the borrow:

function At_End_Borrow (E : access constant List_Acc) return access constant List_Acc is
  (E)
with Ghost,
  Annotate => (GNATprove, At_End_Borrow);

Note that the name of the function could be something other than At_End_Borrow. GNATprove will check that a function associated with the At_End_Borrow annotation is a ghost expression function which takes a single parameter of an access-to-constant type and returns it. At_End_Borrow functions can only be called inside regular assertions or contracts, within a parameter of a call to a lemma subprogram, or within the initialization of a (ghost) constant of anonymous access-to-constant type.

When GNATprove encounters a call to such a function, it checks that the actual parameter of the call is either a local borrower or an expression which is borrowed in the current scope. It does not interpret it as the current value of the expression, but rather as what is usually called a prophecy variable in the literature, namely, an imprecise value representing the value that the expression will have at the end of the borrow. As GNATprove does not do any look-ahead, nothing will be known about the actual value of a local borrower at the end of the borrow. However, the tool will still be aware of the relation between this final value and the final value of the expression it borrows. As an example, consider a local borrower Y of the list X as defined above. The At_End_Borrow function can be used to give properties that are known to hold during the scope of Y. For example, since Y and X designate the same value, GNATprove can verify that no matter what happens during the scope of Y, at the end of the borrow, the Val component of X will be the Val component of Y:

pragma Assert (At_End_Borrow (X).Val = At_End_Borrow (Y).Val);

However, even though at the beginning of the declare block, the first value of X is 1, it is not correct to assert that it will necessarily be so at the end of the borrow:

pragma Assert (Y.Val = 1);                 --  proved
pragma Assert (At_End_Borrow (X).Val = 1); --  incorrect

Indeed, Y could be modified later so that X.Val is not 1 anymore:

declare
   Y : access List := X;
begin
   Y.Val := 2;
end;
pragma Assert (X.Val = 2);

Note that the assertion above is invalid even if Y.Val is not modified in the following statements. It needs to be provable only from the information available at the assertion point, not knowing what will actually happen later in the scope of the borrow. The analysis performed by GNATprove only considers those statements that occur before the assertion to be proved; GNATprove does not consider statements that occur later in the control flow. In other words, there is no lookahead when an assertion is to be proved.

Note

Since At_End_Borrow functions are identity functions, the current values of the borrower and borrowed expression are used when executing assertions containing prophecy variables. This is sound. Indeed, GNATprove will show that the assertion holds for all possible modifications of the borrower. As not modifying the borrower is a valid senario, this is enough to ensure that the assertion necessarily evaluates to True at runtime.

Let us now consider a case where X is not borrowed completely. In the declaration of Y, we can decide to borrow only the last three elements of the list:

declare
   Y : access List := X.Next.Next;
begin
   pragma Assert (At_End_Borrow (X.Next.Next).Val = At_End_Borrow (Y).Val);
   pragma Assert (At_End_Borrow (X.Next.Next) /= null);
   -- Proved, this follows from the relationship between X and Y

   pragma Assert (At_End_Borrow (X.Next.Next.Val) = 3);
   -- Incorrect, X could be modified through Y

   X.Val := 42;
end;

Here, like in the previous example, we can state that, at the end of the borrow, X.Next.Next.Val is Y.Val, and then X.Next.Next cannot be set to null. We cannot assume anything about the part of X designated by Y, so we won’t be able to prove that X.Next.Next.Val will remain 3. Note that we cannot get the value at the end of the borrow of an expression which is not borrowed in the current scope. Here, even if X.Next.Next is borrowed, X and X.Next are not. As a result, calls to At_End_Borrow on them will be rejected by the tool.

Inside the scope of Y, it is possible to modify the variable Y itself, as opposed to modifying the structure it designates, so that it gives access to a subcomponent of the borrowed structure. It is called a reborrow. During a reborrow, the part of the structure designated by the borrower is reduced, so the prophecy variable giving the value of the borrower at the end of the borrow is reduced as well. The part of the borrowed expression which is no longer accessible through the borrower cannot be modified anymore for the rest of the borrow. It is said to be frozen and its final value is known. For example, let’s use Y to borrow X entirely and then modify it to only designate X.Next.Next:

declare
   Y : access List := X;
begin
   Y := Y.Next.Next; -- reborrow

   pragma Assert (At_End_Borrow (X).Next.Next /= null);
   pragma Assert (At_End_Borrow (X).Val = 1);
   pragma Assert (At_End_Borrow (X).Next.Val = 2);
   pragma Assert (At_End_Borrow (X).Next.Next.Val = 3);      --  incorrect
   pragma Assert (At_End_Borrow (X).Next.Next.Next /= null); --  incorrect
end;

After the reborrow, the part of X still accessible from the borrower is reduced, but since X was borrowed entirely to begin with, the ownership policy of SPARK still forbids direct access to any components of X while in the scope of Y. As a result, we have more information about the final value of X than in the previous case. We still know that X will hold at least three elements, that is X.Next.Next /= null. Additionally, the first and second components of X are no longer accessible from Y, and since they cannot be accessed directly through X, we know that they will keep their current values. This is why we can now assert that, at the end of the borrow, X.Val is 1 and X.Next.Val is 2. However, we still cannot know anything about the part of X still accessible from Y as these properties could be modified later in the borrow:

Y.Val := 42;
Y.Next := null;

During sequences of re-borrows, it is additionally possible to use constants of anonymous access-to-constant type in order to save prophecy variables for intermediate values of an access variable, as in the following example:

declare
   Y : access List := X;
begin
   Y := Y.Next; --  first reborrow
   declare
      Z : constant access constant List := At_End_Borrow (Y) with Ghost;
      --  At_End_Borrow is Ghost, so Z too.
   begin
      Y := Y.Next; -- second reborrow
      pragma Assert (At_End_Borrow (X).Next.Val = Z.Val);
      --  Z match prophecy of first reborrow of Y
      pragma Assert (Z.Next.Val = At_End_Borrow (Y).Val);
   end
end

As Z serves to save a prophecy variable, it is subject to the same usage restrictions as the corresponding At_End_Borrow (Y) call, in place of the usual rules for local observers.

As said earlier, in general, GNATprove can handle local borrows without any additional user written annotations. Therefore, At_End_Borrow functions are mostly useful at places where information is lost by the tool: in postconditions of borrowing traversal functions (which return a local borrower of their first parameter) and in loop invariants if the loop involves a reborrow (in this case the value of the borrower at the end of the borrow is modified inside the loop and needs to be described in the invariant). Let us consider the following example:

 1with SPARK.Big_Integers;  use SPARK.Big_Integers;
 2with SPARK.Big_Intervals; use SPARK.Big_Intervals;
 3with Lists;               use Lists;
 4
 5procedure List_Borrows with SPARK_Mode is
 6
 7   function Length (L : access constant List) return Big_Natural is
 8     (if L = null then 0 else Length (L.Next) + 1)
 9   with Subprogram_Variant => (Structural => L);
10
11   function Get (L : access constant List; P : Big_Positive) return Integer is
12     (if P = Length (L) then L.Val else Get (L.Next, P))
13   with Subprogram_Variant => (Structural => L),
14     Pre => P <= Length (L);
15
16   function Eq (L, R : access constant List) return Boolean is
17     (Length (L) = Length (R)
18      and then (for all P in Interval'(1, Length (L)) => Get (L, P) = Get (R, P)))
19   with Annotate => (GNATprove, Inline_For_Proof);
20
21   function Tail (L : access List) return access List with
22     Contract_Cases =>
23       (L = null =>
24          Tail'Result = null and At_End_Borrow (L) = null,
25        others   => At_End_Borrow (L).Val = L.Val
26          and Eq (L.Next, Tail'Result)
27          and Eq (At_End_Borrow (L).Next, At_End_Borrow (Tail'Result)));
28   --  No matter what is done later with the result of Tail, at the end of the
29   --  borrow L.Val will be the current value of L.Val and the remainder of the
30   --  list will be the (updated) value of Tail'Result.
31
32   function Tail (L : access List) return access List is
33   begin
34      if L = null then
35         return null;
36      else
37         return L.Next;
38      end if;
39   end Tail;
40
41   procedure Set_All_To_Zero (L : access List) with
42     Post => Length (L) = Length (L)'Old
43     and then (for all P in Interval'(1, Length (L)) => Get (L, P) = 0);
44
45   procedure Set_All_To_Zero (L : access List) is
46      X : access List := L;
47      C : Big_Natural := 0 with Ghost;
48      --  Number of traversed elements
49
50   begin
51      while X /= null loop
52         pragma Loop_Invariant (C = Length (X)'Loop_Entry - Length (X));
53         --  C is the number of traversed elements
54         pragma Loop_Invariant
55           (Length (At_End_Borrow (L)) = C + Length (At_End_Borrow (X)));
56         --  At the end of the borrow, L will have C more elements than X
57         pragma Loop_Invariant
58           (for all P in Interval'(1, Length (At_End_Borrow (L))) =>
59                (if P <= Length (At_End_Borrow (X))
60                 then Get (At_End_Borrow (L), P) = Get (At_End_Borrow (X), P)
61                 else Get (At_End_Borrow (L), P) = 0));
62         --  At the end of the borrow, the tail of L will be X and the rest
63         --  will contain zeros.
64
65         X.Val := 0;
66         X := X.Next; --  Reborrow
67         C := C + 1;
68      end loop;
69   end Set_All_To_Zero;
70
71   X : List_Acc :=
72     new List'(1, new List'(2, new List'(3, new List'(4, new List'(5, null)))));
73begin
74   declare
75      Y : access List := Tail (X.Next);
76   begin
77      Y.Val := 42;
78   end;
79
80   pragma Assert (X.Val = 1);
81   pragma Assert (X.Next.Val = 2);
82   pragma Assert (X.Next.Next.Val = 42);
83   pragma Assert (X.Next.Next.Next.Val = 4);
84end List_Borrows;

The function Tail is a borrowing traversal function. It returns a local borrower of its parameter L. As GNATprove works modularly on a per subprogram basis, it is necessary to specify in its postcondition how the value of the borrowed parameter L can be reconstructed from the value of the borrower Tail'Result at the end of the borrow. Otherwise, GNATprove would not be able to recompute the value of the borrowed parameter after the returned borrower goes out of scope in the caller.

The Tail function returns the Next component of a list if there is one, and null otherwise. As pointer equality is not allowed in SPARK, we define our own equality function Eq which compares the elements of the list one by one. Note that the Get function indexes the list from the end (the first element of the list is accessed by Get (L, Length (L))). This is done to avoid arithmetic in the recursive definition of Get as it slows the proofs down. In the postcondition of Tail, we consider the two cases, and, in each case, specify both the value returned by the function and how the parameter L is related to the returned borrower:

  • If L is null then Tail returns null and L will stay null for the duration of the borrow.

  • Otherwise, Tail returns L.Next, the first element of L will stay as it was at the time of call, and the rest of L stays equal to the object returned by Tail.

Thanks to this postcondition, GNATprove can verify a program which borrows a part of L using the Tail function and modifies L through this borrower, as can be seen in the body of List_Borrows.

Postconditions of borrowing traversal functions systematically need to provide two properties: one specifying the result, and another specifying how the parameter is related to the borrower. This is generally redundant, as by nature the parameter/borrower relation always holds at the point of return of the function. For example, on the post of Tail, Eq (L.Next, Tail'Result) repeats Eq (At_End_Borrow (L).Next, At_End_Borrow (Tail'Result)).

The tool limits that redundancy by letting the user write only the parameter/borrower relation. Properties of the result are automatically derived by duplicating the postcondition, with calls to At_End_Borrow replaced by their arguments. This covers most (if not all) properties of the result, and additional properties of the result can be explicitly written if needed. This means we get equivalent behavior for function Tail by removing the second conjunct of the postcondition.

At_End_Borrow functions are also useful to write loop invariants in loops involving reborrows. This is exemplified in the Set_All_To_Zero procedure which traverses a list and sets all its elements to 0. The variable X borrows the whole input list L at the beginning of the function. Inside the loop, X is used to modify the structure designated by L. At the end of the procedure, ownership is transferred back to L automatically. To prove the postcondition of Set_All_To_Zero, GNATprove needs to know precisely how to reconstruct the value of L at this point. As X is reborrowed in the loop, the relation between its value and the value of L at the end of the borrow (the end of the scope of X) changes at each iteration. At the beginning of the loop, X is an alias of L, so the value designated by L is equal to the value designated by X at the end of the borrow. At each iteration, an element is dropped from X, so the value designated by the current value of X at the end of the borrow shrinks. At the same time, we get more information about the value designated by L at the end of the borrow as more and more elements are frozen and therefore definitely set to their current value, that is, 0.

Because proof uses cutpoints to reason about loops, it is necessary to supply all this information in a loop invariant. This is what is done in the body of Set_All_To_Zero. To help readability, a ghost variable C is introduced to count the number of iterations in the loop. The first invariant is a regular invariant, it maintains the value of C at each iteration. The two following ones are used to describe how L can be reconstructed from X at the end of the borrow: L will be made of C zeros followed by the final value of X. Note that, in the invariant, no assumption is made about the changes that can be made to X during the rest of the borrow, there is no look ahead. Both Tail and Set_All_To_Zero can be entirely verified by GNATprove:

 1list_borrows.adb:7:13: info: implicit aspect Always_Terminates on "Length" has been proved, subprogram will terminate
 2list_borrows.adb:8:24: info: predicate check proved
 3list_borrows.adb:8:31: info: subprogram variant proved
 4list_borrows.adb:8:31: info: predicate check proved
 5list_borrows.adb:8:40: info: pointer dereference check proved
 6list_borrows.adb:8:47: info: predicate check proved
 7list_borrows.adb:8:49: info: predicate check proved
 8list_borrows.adb:11:13: info: implicit aspect Always_Terminates on "Get" has been proved, subprogram will terminate
 9list_borrows.adb:12:10: info: predicate check proved
10list_borrows.adb:12:14: info: predicate check proved
11list_borrows.adb:12:31: info: pointer dereference check proved
12list_borrows.adb:12:41: info: subprogram variant proved
13list_borrows.adb:12:41: info: precondition proved
14list_borrows.adb:12:47: info: pointer dereference check proved
15list_borrows.adb:14:13: info: predicate check proved
16list_borrows.adb:14:18: info: predicate check proved
17list_borrows.adb:16:13: info: implicit aspect Always_Terminates on "Eq" has been proved, subprogram will terminate
18list_borrows.adb:17:07: info: predicate check proved
19list_borrows.adb:17:20: info: predicate check proved
20list_borrows.adb:18:58: info: precondition proved
21list_borrows.adb:18:66: info: predicate check proved
22list_borrows.adb:18:71: info: precondition proved
23list_borrows.adb:18:79: info: predicate check proved
24list_borrows.adb:21:13: info: implicit aspect Always_Terminates on "Tail" has been proved, subprogram will terminate
25list_borrows.adb:23:18: info: contract case proved
26list_borrows.adb:25:18: info: contract case proved
27list_borrows.adb:25:38: info: pointer dereference check proved
28list_borrows.adb:25:46: info: pointer dereference check proved
29list_borrows.adb:26:20: info: pointer dereference check proved
30list_borrows.adb:27:36: info: pointer dereference check proved
31list_borrows.adb:37:18: info: pointer dereference check proved
32list_borrows.adb:42:14: info: predicate check proved
33list_borrows.adb:42:14: info: postcondition proved
34list_borrows.adb:42:37: info: predicate check proved
35list_borrows.adb:43:57: info: precondition proved
36list_borrows.adb:43:65: info: predicate check proved
37list_borrows.adb:47:26: info: predicate check proved
38list_borrows.adb:52:33: info: predicate check proved
39list_borrows.adb:52:33: info: loop invariant preservation proved
40list_borrows.adb:52:33: info: loop invariant initialization proved
41list_borrows.adb:52:47: info: predicate check proved
42list_borrows.adb:52:61: info: predicate check proved
43list_borrows.adb:55:13: info: predicate check proved
44list_borrows.adb:55:13: info: loop invariant initialization proved
45list_borrows.adb:55:13: info: loop invariant preservation proved
46list_borrows.adb:55:42: info: predicate check proved
47list_borrows.adb:55:46: info: predicate check proved
48list_borrows.adb:58:13: info: loop invariant initialization proved
49list_borrows.adb:58:13: info: loop invariant preservation proved
50list_borrows.adb:59:21: info: predicate check proved
51list_borrows.adb:59:26: info: predicate check proved
52list_borrows.adb:60:23: info: precondition proved
53list_borrows.adb:60:47: info: predicate check proved
54list_borrows.adb:60:52: info: precondition proved
55list_borrows.adb:60:76: info: predicate check proved
56list_borrows.adb:61:23: info: precondition proved
57list_borrows.adb:61:47: info: predicate check proved
58list_borrows.adb:65:11: info: pointer dereference check proved
59list_borrows.adb:66:16: info: pointer dereference check proved
60list_borrows.adb:67:15: info: predicate check proved
61list_borrows.adb:67:17: info: predicate check proved
62list_borrows.adb:67:19: info: predicate check proved
63
64list_borrows.adb:71:04: medium: resource or memory leak might occur at end of scope
65   71>|   X : List_Acc :=
66   72 |     new List'(1, new List'(2, new List'(3, new List'(4, new List'(5, null)))));
67list_borrows.adb:75:33: info: pointer dereference check proved
68  in reconstructed value at the end of the borrow at list_borrows.adb:75
69list_borrows.adb:77:08: info: pointer dereference check proved
70list_borrows.adb:80:19: info: assertion proved
71list_borrows.adb:80:20: info: pointer dereference check proved
72list_borrows.adb:81:19: info: assertion proved
73list_borrows.adb:81:20: info: pointer dereference check proved
74list_borrows.adb:81:25: info: pointer dereference check proved
75list_borrows.adb:82:19: info: assertion proved
76list_borrows.adb:82:20: info: pointer dereference check proved
77list_borrows.adb:82:25: info: pointer dereference check proved
78list_borrows.adb:82:30: info: pointer dereference check proved
79list_borrows.adb:83:19: info: assertion proved
80list_borrows.adb:83:20: info: pointer dereference check proved
81list_borrows.adb:83:25: info: pointer dereference check proved
82list_borrows.adb:83:30: info: pointer dereference check proved
83list_borrows.adb:83:35: info: pointer dereference check proved

Annotation for Accessing the Logical Equality for a Type

In Ada, the equality is not the logical equality in general. In particular, arrays are considered to be equal if they contain the same elements, even with different bounds, +0.0 and -0.0 are considered equal…

It is possible to use a pragma Annotate (GNATprove, Logical_Equal) to ask GNATprove to interpret a function with an equality profile as the logical equality for the type. If the function body is visible in SPARK, a check will be emitted to ensure that it indeed checks for logical equality as understood by the proof engine (which depends on the underlying model used by the tool, see below). It comes in handy for example to express that a (part of a) data-structure is left unchanged by a procedure, as is done in the example below:

subtype Length is Natural range 0 .. 100;
type Word (L : Length := 0) is record
   Value : String (1 .. L);
end record;
function Real_Eq (W1, W2 : String) return Boolean with
  Ghost,
  Import,
  Annotate => (GNATprove, Logical_Equal);
type Dictionary is array (Positive range <>) of Word;

procedure Set (D : in out Dictionary; I : Positive; W : String) with
  Pre  => I in D'Range and W'Length <= 100,
  Post => D (I).Value = W
  and then (for all J in D'Range =>
              (if I /= J then Real_Eq (D (J).Value, D'Old (J).Value)))
is
begin
   D (I) := (L => W'Length, Value => W);
end Set;

The actual interpretation of logical equality is highly dependent on the underlying model used for formal verification. However, the following properties are always valid, no matter the actual type on which a logical equality function applies:

  • Logical equality functions are equivalence relations, that is, they are reflexive, symmetric, and transitive. This implies that such functions can always be used to express that something is preserved as done above.

  • There is no way to tell the difference between between two values which are logically equal. Said otherwise, all SPARK compatible functions will return the same result on logically equal inputs. Note that Ada functions which do not follow SPARK assumptions may not, for example, if they compare the address of pointers which are not modelled by GNATprove. Such functions should never be used inside SPARK code as they can introduce soundness issues.

  • Logical equality is handled efficiently, in a builtin way, by the underlying solvers. This is different from the regular Ada equality which is basically handled as a function call, using potentially complex defining axioms (in particular Ada equality over arrays involves quantifiers).

Note

In Ada, copying an expression might not necessarily return a logically equal value. This happens for example when a value is assigned into a variable or returned by a function. Indeed, such copies might involve for example sliding (for arrays) or a partial copy (for view conversions of tagged types).

It might happen that the underlying model of values of an Ada type contain components which are not present in the Ada value. This makes it impossible to implement a comparison function in Ada which would compute logical equality on such types. This is the case in particular for arrays and discriminated records with variant parts. More precisely, logical equality can be implemented using the regular Ada equality for discrete types and fixed point types. For floating point types, both the value and the sign need to be compared (to tell the difference between +0.0 and -0.0). For pointers, as the address is not modeled in SPARK, it is enough to check for nullity and compare the designated value. Logical equality on untagged record with no variant parts can be achieved by comparing the components. For other composite types, it cannot be implemented and has to remain non-executable as in the example above. Additionally, a user can safely annotate a comparison function on private types whose full view is not in SPARK using the Logical_Equal annotation if it behaves as expected, namely, it is an equivalence relation, and there is no way, using the API provided in the public part of the enclosing package, to tell the difference between two values on which this function returns True.

Note

For private types whose full view is not in SPARK, GNATprove will peek inside the full view to try and determine whether or not to interpret the primitive equality on such types as the logical equality.

Note that, for types on which implementing the logical equality in Ada is impossible, GNATprove might not be able to prove that two Ada values are logically equal even if there is no way to tell the difference in SPARK. For example, it might not be able to prove that two arrays are logically equal even if they have the same bounds and the same value. It is because it is not necessarily true in the underlying model, where values outside of the array bounds are represented. Therefore, using an assumption to state that two objects which are equal-in-Ada are logically equal might introduce an unsoundness, in particular in the presence of slices. It is demonstrated in the example below where GNATprove can prove that two strings are not logically equal even though they have the same bounds and the same elements. However, logical equality can be used safely as long as everything is proved correct (no assumption is used).

procedure Test is
   S1 : constant String := "foo1";
   S2 : constant String := "foo2";

begin
   pragma Assert (S1 (1 .. 3) = S2 (1 .. 3));
   pragma Assert (not Real_Eq (S1 (1 .. 3), S2 (1 .. 3)));
end Test;

Annotation for Enforcing Ownership Checking on a Private Type

Private types whose full view is not in SPARK can be annotated with a pragma Annotate (GNATprove, Ownership, ...). Such a type is handled by GNATprove in accordance to the Memory Ownership Policy of SPARK. For example, the type T declared in the procedure Simple_Ownership below has an ownership annotation. As a result, GNATprove will reject the second call to Read in the body of Simple_Ownership, because the value designated by X has been moved by the assignment to Y.

 1procedure Simple_Ownership with SPARK_Mode is
 2   package Nested is
 3      type T is private with
 4        Default_Initial_Condition,
 5        Annotate => (GNATprove, Ownership);
 6
 7      function Read (X : T) return Boolean;
 8   private
 9      pragma SPARK_Mode (Off);
10      type T is null record;
11      function Read (X : T) return Boolean is (True);
12   end Nested;
13   use Nested;
14
15   X : T;
16   Y : T;
17
18begin
19   pragma Assert (Read (X)); -- OK
20
21   Y := X;
22
23   pragma Assert (Read (X)); -- Error, X has been moved
24end Simple_Ownership;

In addition, it is possible to state that a type needs reclamation with a pragma Annotate (GNATprove, Onwership, "Needs_Reclamation", ...). In this case, GNATprove emits checks to ensure that the resource is not leaked. For these checks to be handled precisely, the user should annotate a function taking a value of this type as a parameter and returning a boolean with a pragma Annotate (GNATprove, Onwership, "Needs_Reclamation", ...) or pragma Annotate (GNATprove, Onwership, "Is_Reclaimed", ...). This function will be used to check that the resource cannot be leaked. A function annotated with "Needs_Reclamation" shall return True when its input holds some resource while a function annotated with "Is_Reclaimed" shall return True when its input has already been reclaimed. Only one such function shall be provided for a given type. Two examples of use are given below.

 1package Hidden_Pointers with
 2  SPARK_Mode,
 3  Always_Terminates
 4is
 5   type Pool_Specific_Access is private with
 6     Default_Initial_Condition => Is_Null (Pool_Specific_Access),
 7     Annotate => (GNATprove, Ownership, "Needs_Reclamation");
 8
 9   function Is_Null (A : Pool_Specific_Access) return Boolean with
10     Global => null,
11     Annotate => (GNATprove, Ownership, "Is_Reclaimed");
12   function Deref (A : Pool_Specific_Access) return Integer with
13     Global => null,
14     Pre => not Is_Null (A);
15
16   function Allocate (X : Integer) return Pool_Specific_Access with
17     Global => null,
18     Post => not Is_Null (Allocate'Result) and then Deref (Allocate'Result) = X;
19   procedure Deallocate (A : in out Pool_Specific_Access) with
20     Global => null,
21     Post => Is_Null (A);
22
23private
24   pragma SPARK_Mode (Off);
25   type Pool_Specific_Access is access Integer;
26end Hidden_Pointers;

The package Hidden_Pointers defines a type Pool_Specific_Access which is really a pool specific access type. The ownership annotation on Pool_Specific_Access instructs GNATprove that objects of this type should abide by the Memory Ownership Policy of SPARK. It also states that the type needs reclamation when a value is finalized. Because of the annotation on the Is_Null function, GNATprove will attempt to prove that Is_Null returns True when an object of type Pool_Specific_Access is finalized unless it was moved.

The mechanism can also be used for resources which are not linked to memory management. For example, the package Text_IO defines a limited type File_Descriptor and uses ownership annotations to force GNATprove to verify that all file descriptors are closed before being finalized.

 1package Text_IO with
 2  SPARK_Mode,
 3  Always_Terminates
 4is
 5   type File_Descriptor is limited private with
 6     Default_Initial_Condition => not Is_Open (File_Descriptor),
 7     Annotate => (GNATprove, Ownership, "Needs_Reclamation");
 8
 9   function Is_Open (F : File_Descriptor) return Boolean with
10     Global => null,
11     Annotate => (GNATprove, Ownership, "Needs_Reclamation");
12   function Read_Line (F : File_Descriptor) return String with
13     Global => null,
14     Pre => Is_Open (F);
15
16   function Open (N : String) return File_Descriptor with
17     Global => null,
18     Post => Is_Open (Open'Result);
19   procedure Close (F : in out File_Descriptor) with
20     Global => null,
21     Post => not Is_Open (F);
22
23private
24   pragma SPARK_Mode (Off);
25   type Text;
26   type File_Descriptor is access all Text;
27end Text_IO;

Annotation for Instantiating Lemma Procedures Automatically

As featured in Manual Proof Using User Lemmas, it is possible to write lemmas in SPARK as ghost procedures. However, actual calls to the procedure need to be manually inserted in the program whenever an instance of the lemma is necessary. It is possible to use a pragma Annotate to instruct GNATprove that an axiom should be introduced for a lemma procedure so manual instantiations are no longer necessary. This annotation is called Automatic_Instantiation. As an example, the Equivalent function below is an equivalence relation. This is expressed using three lemma procedures which should be instantiated automatically:

function Equivalent (X, Y : Integer) return Boolean with
  Global => null;

procedure Lemma_Reflexive (X : Integer) with
  Ghost,
  Global => null,
  Annotate => (GNATprove, Automatic_Instantiation),
  Post => Equivalent (X, X);

procedure Lemma_Symmetric (X, Y : Integer) with
  Ghost,
  Global => null,
  Annotate => (GNATprove, Automatic_Instantiation),
  Pre  => Equivalent (X, Y),
  Post => Equivalent (Y, X);

procedure Lemma_Transitive (X, Y, Z : Integer) with
  Ghost,
  Global => null,
  Annotate => (GNATprove, Automatic_Instantiation),
  Pre  => Equivalent (X, Y) and Equivalent (Y, Z),
  Post => Equivalent (X, Z);

Such lemmas should be declared directly after a function declaration, here the Equivalent function. The axiom will only be available when the associated function is used in the proof context.

Annotation for Hiding Information

By default, when verifying a part of a program, GNATprove makes all information about used entities available in the context. For example, it assumes values of global constants, postconditions of called subprograms, bodies of expression functions… While in general this behavior is desirable, it might result in untractable proof contexts on large programs. It is possible to use an annotation to manually add or remove information from the proof context. For the time being, only bodies of expression functions can be handled in this way.

Information hiding is decided at the level of an entity for which checks are generated, namely, a subprogram or entry, or the elaboration of a package. It cannot be refined in a smaller scope. To state that the body of an expression function should be hidden when verifying an entity E, a pragma with the Hide_Info annotation should be used either at the begining of the body of E or just after its specification or body. In the following example, the body of the expression function F is hidden when verifying its callers, making it impossible to prove their postconditions:

function F (X, Y : Boolean) return Boolean is (X and Y);

function Call_F (X, Y : Boolean) return Boolean is
  (F (X, Y))
with Post => Call_F'Result = (X and Y); --  Unprovable, F is hidden
pragma Annotate (GNATprove, Hide_Info, "Expression_Function_Body", F);

function Call_F_2 (X, Y : Boolean) return Boolean with
  Post => Call_F_2'Result = (X and Y); --  Unprovable, F is hidden

function Call_F_2 (X, Y : Boolean) return Boolean is
begin
   return F (X, Y);
end Call_F_2;
pragma Annotate (GNATprove, Hide_Info, "Expression_Function_Body", F);

function Call_F_3 (X, Y : Boolean) return Boolean with
  Post => Call_F_3'Result = (X and Y); --  Unprovable, F is hidden

function Call_F_3 (X, Y : Boolean) return Boolean is
   pragma Annotate (GNATprove, Hide_Info, "Expression_Function_Body", F);
begin
   return F (X, Y);
end Call_F_3;

It is also possible to hide information by default and then use an annotation to disclose it when needed. A Hide_Info annotation located on the entity which is hidden is considered to provide a default. For example, the body of the expression function G is hidden by default. The postcondition of its caller Call_G cannot be proved.

function G (X, Y : Boolean) return Boolean is (X and Y) with
  Annotate => (GNATprove, Hide_Info, "Expression_Function_Body");
--  G is hidden by default

function Call_G (X, Y : Boolean) return Boolean is
  (G (X, Y))
with Post => Call_G'Result = (X and Y); --  Unprovable, G is hidden

When information is hidden by default, it is possible to disclose it while verifying an entity using the Unhide_Info annotation. This allows proving the Call_G_2 function below:

function Call_G_2 (X, Y : Boolean) return Boolean is
  (G (X, Y))
with Post => Call_G_2'Result = (X and Y); --  Provable, G is no longer hidden
pragma Annotate (GNATprove, Unhide_Info, "Expression_Function_Body", G);

Remark that, when information is hidden by default, it is even hidden during the verification of the entity whose information we are hiding. For example, when verifying a recursive expression function whose body is hidden by default, the body of recursive calls is not available. If necessary, it can be disclosed using an Unhide_Info annotation:

--  Rec_F is hidden for its recursive calls

function Rec_F (X, Y : Boolean) return Boolean is
  (if not X then False elsif X = Y then True else Rec_F (Y, X))
    with
      Subprogram_Variant => (Decreases => X),
      Post => (if X then Rec_F'Result = Y), --  Unprovable, Rec_F is hidden
      Annotate => (GNATprove, Hide_Info, "Expression_Function_Body");

--  The second annotation overrides the default for recursive calls

function Rec_F_2 (X, Y : Boolean) return Boolean is
  (if not X then False elsif X = Y then True else Rec_F_2 (Y, X))
    with
      Subprogram_Variant => (Decreases => X),
      Post => (if X then Rec_F_2'Result = Y), --  Provable, Rec_F_2 is visible
      Annotate => (GNATprove, Hide_Info, "Expression_Function_Body");
pragma Annotate (GNATprove, Unhide_Info, "Expression_Function_Body", Rec_F_2);

Annotation for Handling Specially Higher Order Functions

Functions for higher order programming can be expressed using parameters of an anonymous access-to-function type. As an example, here is a function Map returning a new array whose elements are the result of the application a function F to the elements of its input array parameter:

function Map
  (A : Nat_Array;
   F : not null access function (N : Natural) return Natural)
   return Nat_Array
with
  Post => Map'Result'First = A'First and Map'Result'Last = A'Last
    and (for all I in A'Range => Map'Result (I) = F (A (I)));

In a functional programming style, these functions are often called with an access to a local function as a parameter. Unfortunately, as GNATprove generally handles access-to-subprograms using refinement semantics, it is not possible to use a function accessing global data as an actual for an anonymous access-to-function parameter (see Subprogram Pointers). To workaround this limitation, it is possible to annotate the function Map with a pragma or aspect Annotate => (GNATprove, Higher_Order_Specialization). When such a function is called on a reference to the Access attribute of a function, it will benefit from a partial exemption from the checks usually performed on the creation of such an access. Namely, the function will be allowed to access data, and it might even have a precondition if it can be proved to always hold at the point of call. We say that such a call is specialized for a particular value of its access-to-function parameter. As an example, consider the function Divide_All defined below. As the function Map is annotated with Higher_Order_Specialization, it can be called on the function Divide, even though it accesses some global data (the parameter N) and has a precondition.

function Map
  (A : Nat_Array;
   F : not null access function (N : Natural) return Natural)
   return Nat_Array
with
  Annotate => (GNATprove, Higher_Order_Specialization),
  Post => Map'Result'First = A'First and Map'Result'Last = A'Last
    and (for all I in A'Range => Map'Result (I) = F (A (I)));

function Divide_All (A : Nat_Array; N : Natural) return Nat_Array with
  Pre  => N /= 0,
  Post => (for all I in A'Range => Divide_All'Result (I) = A (I) / N);

function Divide_All (A : Nat_Array; N : Natural) return Nat_Array is
   function Divide (E : Natural) return Natural is
     (E / N)
   with Pre => N /= 0;
begin
   return Map (A, Divide'Access);
end Divide_All;

Note

It might not be possible to annotate a function with Higher_Order_Specialization if the access value of its parameter is used in the contract of the function, as opposed to only its dereference. This happens in particular if the parameter is used in a call to a function which does not have the Higher_Order_Specialization annotation.

Because the analysis done by GNATprove stays modular, the precondition of the referenced function has to be provable on all possible parameters (but the specialized access-to-function parameters) and not only on the ones on which it is actually called. For example, even if we know that the precondition of Add holds for all the elements of the input array A, there will still be a failed check on the call to Map in Add_All defined below. Indeed, GNATprove does not peek into the body of Map and therefore has no way to know that its parameter F is only called on elements of A.

function Add_All (A : Nat_Array; N : Natural) return Nat_Array with
  Pre  => (for all E of A => E <= Natural'Last - N),
  Post => (for all I in A'Range => Add_All'Result (I) = A (I) + N);

function Add_All (A : Nat_Array; N : Natural) return Nat_Array is
   function Add (E : Natural) return Natural is
     (E + N)
   with Pre => E <= Natural'Last - N;
begin
   return Map (A, Add'Access);
end Add_All;

To avoid this kind of issue, it is possible to write a function with no preconditions and a fallback value as done below. Remark that this does not prevent the tool from proving the postcondition of Add_All.

function Add_All (A : Nat_Array; N : Natural) return Nat_Array is
   function Add (E : Natural) return Natural is
     (if E <= Natural'Last - N then E + N else 0);
begin
   return Map (A, Add'Access);
end Add_All;

Note

Only the regular contract of functions is available on specialized calls. Bodies of expression functions and refined postconditions will be ignored.

The Higher_Order_Specialization annotation can also be supplied on a lemma procedure (see Manual Proof Using User Lemmas). If this procedure has an Automatic_Instantiation annotation (see Annotation for Instantiating Lemma Procedures Automatically) and its associated function also has an Higher_Order_Specialization annotation, the lemma will generally be instantiated automatically on specialized calls to the function. As an example, the function Count defined below returns the number of elements with a property in an array. The function Count is specified in a recursive way in its postcondition. The two lemmas Lemma_Count_All and Lemma_Count_None give the value of Count when all the elements of A or no elements of A fulfill the property. They will be automatically instantiated on all specializations of Count.

function Count
  (A : Nat_Array;
   F : not null access function (N : Natural) return Boolean)
   return Natural
with
  Annotate => (GNATprove, Higher_Order_Specialization),
  Subprogram_Variant => (Decreases => A'Length),
  Post =>
    (if A'Length = 0 then Count'Result = 0
     else Count'Result =
         (if F (A (A'Last)) then 1 else 0) +
           Count (A (A'First .. A'Last - 1), F))
    and Count'Result <= A'Length;

procedure Lemma_Count_All
  (A : Nat_Array;
   F : not null access function (N : Natural) return Boolean)
with
  Ghost,
  Annotate => (GNATprove, Automatic_Instantiation),
  Annotate => (GNATprove, Higher_Order_Specialization),
  Pre  => (for all E of A => F (E)),
  Post => Count (A, F) = A'Length;

procedure Lemma_Count_None
  (A : Nat_Array;
   F : not null access function (N : Natural) return Boolean)
with
  Ghost,
  Annotate => (GNATprove, Automatic_Instantiation),
  Annotate => (GNATprove, Higher_Order_Specialization),
  Pre  => (for all E of A => not F (E)),
  Post => Count (A, F) = 0;

Note

It can happen that some lemmas cannot be specialized with their associated function. It is the case in particular if the lemma contains several calls to the function with different access-to-function parameters. In this case, a warning will be emitted on the lemma declaration.

Annotation for Handlers

Some programs define handlers, subprograms which might be called asynchronously to perform specific treatments, for example when an interrupt occurs. These handlers are often registered using access-to-subprogram types. In general, access-to-subprograms are not allowed to access or modify global data in SPARK. However, this is too constraining for handlers which tend to work by side-effects. To alleviate this limitation, it is possible to annotate an access-to-subprogram type with a pragma or aspect Annotate => (GNATprove, Handler). This instructs GNATprove that these access-to-subprograms will be called asynchronously. It is possible to create a value of such a type using a reference to the Access attribute on a subprogram accessing or modifying global data, but only when this global data is synchronized (see SPARK RM 9.1). However, GNATprove will make sure that the subprograms designated by objects of these types are never called from SPARK code, as it could result in a missing data dependency.

For example, consider the following procedure which resets to zero a global variable Counter:

Counter : Natural := 0 with Atomic;

procedure Reset is
begin
   Counter := 0;
end Reset;

It is not possible to store an access to Reset in a regular access-to-procedure type, as it has a global effect. However, it can be stored in a type annotated with an aspect Annotate => (GNATprove, Handler) like below:

type No_Param_Proc is access procedure with
  Annotate => (GNATprove, Handler);

P : No_Param_Proc := Reset'Access;

However, it is not possible to call P from SPARK code as the effect to the Counter variable would be missed.

Note

As handlers are called asynchronously, GNATprove does not allow providing pre and postconditions for the access-to-subprogram type. As a result, a check will be emitted to ensure that the precondition of each specific subprogram designated by these handlers is provable in the empty context.

Annotation for Container Aggregates

The Container_Aggregates annotation allows specifying aggregates on types annotated with the Aspect Aggregate either directly, using predefined aggregate patterns, or through a model. The second option is the easiest. It is enough to provide a model function returning a type which supports compatible aggregates. When an aggregate is encountered, GNATprove deduces the value of its model using the Container_Aggregates annotation on the model type. In particular, functional containers from the SPARK Libraries are annotated with the Aggregate aspect. Other container libraries can take advantage of this support to specify aggregates on their own library by providing a model function returning a functional container. As an example, sequences can be used as a model to provide aggregates for a linked list of integers:

 1with SPARK.Containers.Functional.Infinite_Sequences;
 2
 3package Through_Model with SPARK_Mode is
 4   package My_Seqs is new SPARK.Containers.Functional.Infinite_Sequences (Integer);
 5   use My_Seqs;
 6
 7   type List is private with
 8     Aggregate => (Empty => Empty_List, Add_Unnamed => Add),
 9     Annotate  => (GNATprove, Container_Aggregates, "From_Model");
10
11   Empty_List : constant List;
12
13   function Model (L : List) return Sequence with
14     Subprogram_Variant => (Structural => L),
15     Annotate => (GNATprove, Container_Aggregates, "Model");
16
17   procedure Add (L : in out List; E : Integer) with
18     Always_Terminates,
19     Post => Model (L) = Add (Model (L)'Old, E);
20
21private
22
23   type List_Cell;
24   type List is access List_Cell;
25   type List_Cell is record
26      Data : Integer;
27      Next : List;
28   end record;
29
30   function Model (L : List) return Sequence is
31     (if L = null then Empty_Sequence
32      else Add (Model (L.Next), L.Data));
33
34   Empty_List : constant List := null;
35end Through_Model;

The Container_Aggregates annotation on type List indicates that its aggregates are specified using a model function. The annotation on the Model function identifies it as the function that should be used to model aggregates of type List. It is then possible to use aggregates of type List in SPARK as in the Main procedure below. GNATprove will use the handling of aggregates on infinite sequences to determine the value of the model of the aggregate and use it to prove the program.

1with Through_Model; use Through_Model;
2
3procedure Main with SPARK_Mode is
4   L : constant List := [1, 2, 3, 4, 5, 6];
5begin
6   pragma Assert (My_Seqs.Get (Model (L), 4) = 4);
7end Main;

It is also possible to specify directly how aggregates should be handled, without going through a model. To do this, GNATprove provides three different predefined patterns: one for vectors and sequences, one for sets, and one for maps. The chosen pattern is specified inside the Container_Aggregates annotation on the type. Depending on this pattern, different functions can be provided to describe it. As an example, aggregates on our linked list of integers can also be described directly by supplying the three functions required for predefined sequence aggregates - First, Last, and Get:

 1with SPARK.Big_Integers;  use SPARK.Big_Integers;
 2with SPARK.Big_Intervals; use SPARK.Big_Intervals;
 3
 4package Predefined with SPARK_Mode is
 5   pragma Unevaluated_Use_Of_Old (Allow);
 6   type List is private with
 7     Aggregate => (Empty => Empty_List, Add_Unnamed => Add),
 8     Annotate  => (GNATprove, Container_Aggregates, "Predefined_Sequences");
 9
10   Empty_List : constant List;
11
12   function First return Big_Positive is (1) with
13     Annotate => (GNATprove, Container_Aggregates, "First");
14
15   function Length (L : List) return Big_Natural with
16     Subprogram_Variant => (Structural => L),
17     Annotate => (GNATprove, Container_Aggregates, "Last");
18
19   function Get (L : List; P : Big_Positive) return Integer with
20     Pre => P <= Length (L),
21     Subprogram_Variant => (Structural => L),
22     Annotate => (GNATprove, Container_Aggregates, "Get");
23
24   function Eq (L1, L2 : List) return Boolean is
25     (Length (L1) = Length (L2)
26      and then
27        (for all I in Interval'(1, Length (L1)) =>
28              Get (L1, I) = Get (L2, I)));
29
30   function Copy (L : List) return List with
31     Subprogram_Variant => (Structural => L),
32     Post => Eq (L, Copy'Result);
33
34   procedure Add (L : in out List; E : Integer) with
35     Always_Terminates,
36     Post => Length (L) = Length (L)'Old + 1
37     and then Get (L, Length (L)) = E
38     and then (for all I in Interval'(1, Length (L) - 1) =>
39                 Get (L, I) = Get (Copy (L)'Old, I));
40
41private
42   type List_Cell;
43   type List is access List_Cell;
44   type List_Cell is record
45      Data : Integer;
46      Next : List;
47   end record;
48
49   function Length (L : List) return Big_Natural is
50     (if L = null then 0 else 1 + Length (L.Next));
51
52   function Get (L : List; P : Big_Positive) return Integer is
53     (if P = Length (L) then L.Data else Get (L.Next, P));
54
55   Empty_List : constant List := null;
56end Predefined;

For all types annotated with Container_Aggregates, GNATprove performs consistency checks to make sure that the description induced by the annotation is compatible with the subprograms supplied by its Aggregate aspect. For example, on the List type defined above, GNATprove generates checks to ensure that Length returns First - 1 on Empty_List. It also checks that calling Add increases the result of Length by one and that its parameter E is associated to this last position. These checks succeed on our example thanks to the postcondition of Add as made explicit by the info messages on line 8:

predefined.ads:8:06: info: Container_Aggregates annotation proved
  when establishing invariant on Length at predefined.ads:15
  after a call to Empty_List at predefined.ads:10
predefined.ads:8:06: info: Container_Aggregates annotation proved
  when reestablishing invariant on Get at predefined.ads:19
  after a call to Add at predefined.ads:34
predefined.ads:8:06: info: Container_Aggregates annotation proved
  when reestablishing invariant on Length at predefined.ads:15
  after a call to Add at predefined.ads:34

The Container_Aggregates annotation might also induce restrictions on aggregate usage. For example, if we had chosen a signed integer type for the return type of Length in Predefined, GNATprove would have introduced checks on all aggregates of type List to make sure that the number of elements doesn’t exceed the number of indexes.

When a model is used to specify aggregates, both the restrictions on aggregate usage and the consistency checks of the model are inherited. For example, GNATprove attempts to prove the consistency of Empty_List and Add from Through_Model with the functions supplied for the model. These checks are proved thanks to the precise postcondition on Add. In the same way, if we had chosen to use a functional vector indexed by a signed integer type instead of an infinite sequence as a model in Through_Model, we would have been limited by the size of the integer type for our aggregates.

All types with a Container_Aggregates annotation can be supplied an additional Capacity function. In general, this function does not have any parameters. It provides a constant bound on the number of elements allowed in the container. If a Capacity function is supplied on a type using models for its aggregates, it overrides the Capacity function of the model if any. If present, the Capacity function is used both as an additional restriction on aggregate usage and as an additional assumption for the consistency checks: GNATprove assumes that the container holds strictly less than Capacity elements before a call to Add and it makes sure that the number of elements in actual aggregates never exceeds the capacity.

To accomodate containers which can have a different capacity depending on the object, Ada allows providing an Empty function which takes an integer value for the capacity as a parameter in the Aggregate aspect. If a Capacity function is specified for such a type, it shall take the container as a parameter since the capacity is specific to each container. In this case, it is the upper bound of the parameter type of the Empty function which is used as a restriction on aggregate usage. Examples of containers with a capacity function (both with and without parameters) can be found in formal container packages in the SPARK Libraries.

There are three kinds of predefined aggregates patterns: Predefined_Sequences like in the Predefined package, Predefined_Sets, and Predefined_Maps. The selected pattern should be provided as a string to the Container_Aggregates annotation specified on the container type. Additional Container_Aggregates annotations are necessary on each specific function supported by the pattern. They also require a string encoding its intended use. This usage is examplified in the Predefined package.

The Predefined_Sequences pattern can be applied to containers with positional aggregates. It supports three kinds of function annotations:

  • The function First returns the index or position that should be used for the first element in a container.

  • The function Last returns the index or position of the last element in a specific container.

  • The function Get returns the element associated to a valid index or position in the container.

All three functions are mandatory. The return types of First and Last should be subtypes of the same signed integer type, or possibly of Big_Integer.

The consistency checks ensure that:

  • Empty returns a container C such that Last (C) = First - 1,

  • Add can be called on a container C such that Last (C) < Index_Type'Last, and

  • after a call to Add, the result of Last (C) has been incremented by one, the result of calling Get on all previous indexes is unchanged, and calling Get on the last index returns the added element.

The contracts of Empty and Append below ensure conformance to these consistency checks:

type T is private with
  Aggregate => (Empty       => Empty,
                Add_Unnamed => Append),
  Annotate => (GNATprove, Container_Aggregates, "Predefined_Sequences");

function Empty return T with
  Post => Last (Empty'Result) = First - 1;
procedure Append (X : in out T; E : Element_Type) with
  Always_Terminates,
  Pre  => Last (X) < Index_Type'Last,
  Post => Last (X) = Last (X)'Old + 1 and Get (X, Last (X)) = E
  and (for all I in First .. Last (X) - 1 => Get (X, I) = Get (X'Old, I));

function Last (X : T) return Ext_Index with
  Annotate => (GNATprove, Container_Aggregates, "Last");

function First return Index_Type with
  Annotate => (GNATprove, Container_Aggregates, "First");

function Get (X : T; I : Index_Type) return Element_Type with
  Annotate => (GNATprove, Container_Aggregates, "Get"),
  Pre => I <= Last (X);

If Last returns a signed integer type, there is a restriction on predefined sequence aggregates usage: GNATprove will make sure that the number of elements in an aggregate never exceeds the maximum value of the return type of Last.

When an aggregate C is encountered, GNATprove automatically infers that:

  • Last (C) - First + 1 is the number of elements in the aggregate and

  • Get (First + N - 1) returns a copy of the N’th element.

The Predefined_Sets pattern can be applied to containers with positional aggregates. It supports three kinds of function annotations:

  • The Contains function returns True if and only if its element parameter is in the container.

  • Equivalent_Elements is an equivalence relation such that Contains always returns the same value on two equivalent elements.

  • The Length function returns the number of elements in the set.

Contains and Equivalent_Elements are mandatory, Length is optional. If it is supplied, the Length function should return a signed integer type or a subtype of Big_Integer.

The consistency checks ensure that:

  • Contains returns False for all elements on the result of Empty and Length, if specified, returns 0,

  • Add can be called on a container C and an element E such that Contains (C, E) returns False,

  • after a call to Add on a container C and an element E, Contains returns True on E and on all elements which were in C before the call, plus potential additional elements equivalent to E,

  • after a call to Add, the result of the Length function if any is incremented by one.

The contracts of Empty and Insert below ensure conformance to these consistency checks:

type T is private with
  Aggregate => (Empty       => Empty,
                Add_Unnamed => Insert),
  Annotate => (GNATprove, Container_Aggregates, "Predefined_Sets");

function Empty return T with
  Post => Length (Empty'Result) = 0
  and then (for all E in Element_Type => not Contains (Empty'Result, E));
procedure Insert (X : in out T; E : Element_Type) with
  Always_Terminates,
  Pre  => not Contains (X, E),
  Post => Length (X) = Length (X'Old) + 1 and Contains (X, E)
  and (for all F in Element_Type =>
         (if Contains (X'Old, F) then Contains (X, F)))
  and (for all F in Element_Type =>
         (if Contains (X, F) then Contains (X'Old, F) or Eq_Elem (F, E)));

function Eq_Elem (X, Y : Element_Type) return Boolean with
  Annotate => (GNATprove, Container_Aggregates, "Equivalent_Elements");

function Contains (X : T; E : Element_Type) return Boolean with
  Annotate => (GNATprove, Container_Aggregates, "Contains");

function Length (X : T) return Natural with
  Annotate => (GNATprove, Container_Aggregates, "Length");

Aggregates of types annotated with Predefined_Sets cannot contain duplicates, that is, two elements on which Equivalent_Elements returns True. This restriction on aggregate usage is enforced by GNATprove. It allows the tool to properly assess the cardinality of the resulting set. If a Length function is supplied and returns a signed integer type, and no Capacity function applies to the type, GNATprove also makes sure that the number of elements in the aggregate does not exceed the upper bound of the return type of Length. This last check is replaced by a check on the capacity if there is one.

When an aggregate C is encountered, GNATprove automatically infers that:

  • Contains (C, E) returns True for every element E of the aggregate,

  • Contains (C, E) returns False for every element E which is not equivalent to any element in the aggregate, and

  • Length (C), if supplied, is the number of elements in the aggregate.

The Predefined_Maps pattern can be applied to containers with named aggregates. It supports five kinds of function annotations:

  • The Default_Item function returns the element associated by default to keys of total maps.

  • The Has_Key function returns True if and only if its key parameter has an association in a partial map.

  • The Get function returns the element associated to a key in the container.

  • Equivalent_Keys is an equivalence relation such that Has_Key and Get return the same value on two equivalent keys.

  • The Length function returns the number of associations in a partial map.

Get and Equivalent_Keys are mandatory, exactly one of Default_Item and Has_Key should be supplied (depending on whether the map is total - it associates a value to all keys - or partial) , and Length is optional and can only be supplied for partial maps. If it is supplied, the Length function should return a signed integer type or a subtype of Big_Integer.

The consistency checks ensure that:

  • if Default_Item is supplied, Get returns Default_Item for all keys on the result of Empty,

  • if Has_Key is supplied, it returns False for all keys on the result of Empty, and Length, if specified, returns 0,

  • Add can be called on a container C and a key K such that Has_Key (C, K) returns False (for patial maps) or Get (C, K) = Default_Item (for total maps),

  • after a call to Add on a container C, a key K, and an element E, Has_Key, if any, returns True on K and on all keys which were in C before the call, plus potential additional keys equivalent to K,

  • after a call to Add on a container C, a key K, and an element E, Get returns a copy of E on K and its association in C before the call on keys which are not equivalent to K, and

  • after a call to Add, the result of the Length function if any is incremented by one.

The contracts of Empty and Insert below ensure conformance to these consistency checks for total maps:

type T is private with
  Aggregate => (Empty     => Empty,
                Add_Named => Insert),
  Annotate => (GNATprove, Container_Aggregates, "Predefined_Maps");

function Empty return T with
  Post => (for all K in Key_Type => Get (Empty'Result, K) = Default_Value);
procedure Insert (X : in out T; K : Key_Type; E : Element_Type) with
  Always_Terminates,
  Pre  => Get (X, K) = Default_Value,
  Post => Get (X, K) = E
  and (for all F in Key_Type =>
         (if not Eq_Key (F, K) then Get (X, F) = Get (X'Old, F)));

function Eq_Key (X, Y : Key_Type) return Boolean with
  Annotate => (GNATprove, Container_Aggregates, "Equivalent_Keys");

function Default_Value return Element_Type with
  Annotate => (GNATprove, Container_Aggregates, "Default_Item");

function Get (X : T; K : Key_Type) return Element_Type with
  Annotate => (GNATprove, Container_Aggregates, "Get";

And here is a version for partial maps:

type T is private with
  Aggregate => (Empty     => Empty,
                Add_Named => Insert),
  Annotate => (GNATprove, Container_Aggregates, "Predefined_Maps");

function Empty return T with
  Post => Length (Empty'Result) = 0
  and (for all K in Key_Type => not Has_Key (Empty'Result, K));
procedure Insert (X : in out T; K : Key_Type; E : Element_Type) with
  Always_Terminates,
  Pre  => not Has_Key (X, K),
  Post => Length (X) - 1 = Length (X)'Old
  and Has_Key (X, K)
  and (for all F in Key_Type =>
         (if Has_Key (X'Old, F) then Has_Key (X, F)))
  and (for all L in Key_Type =>
         (if Has_Key (X, L) then Has_Key (X'Old, L) or Eq_Key (L, K)))
  and Get (X, K) = E
  and (for all F in Key_Type =>
         (if Has_Key (X'Old, F) then Get (X, F) = Get (X'Old, F)));

function Eq_Key (X, Y : Key_Type) return Boolean with
  Annotate => (GNATprove, Container_Aggregates, "Equivalent_Keys");

function Has_Key (X : T; K : Key_Type) return Boolean with
  Annotate => (GNATprove, Container_Aggregates, "Has_Key");

function Get (X : T; K : Key_Type) return Element_Type with
  Pre => Has_Key (X, K),
  Annotate => (GNATprove, Container_Aggregates, "Get");

function Length (X : T) return Natural with
  Annotate => (GNATprove, Container_Aggregates, "Length");

Aggregates of types annotated with Predefined_Maps cannot contain duplicates, that is, two keys on which Equivalent_Keys returns True. This restriction on aggregate usage is enforced by GNATprove. It allows the tool to determine the associated element uniquely and to assess the cardinality of the resulting map. If a Length function is supplied and returns a signed integer type, and no Capacity function applies to the type, GNATprove also makes sure that the number of elements in the aggregate does not exceed the upper bound of the return type of Length. This last check is replaced by a check on the capacity if there is one.

When an aggregate C is encountered, GNATprove automatically infers that:

  • if Has_Key is supplied, Has_Key (C, K) returns True for every association K => _ in the aggregate,

  • Get (C, K) returns a copy of E for every association K => E in the aggregate,

  • for every key K which is not equivalent to any key in the aggregate either Has_Key (C, K) returns False (for patial maps) or Get (C, K) returns Default_Item (for total maps), and

  • Length (C), if supplied, is the number of elements in the aggregate.