5.5. Specification Features¶

SPARK contains many features for specifying the intended behavior of programs. Some of these features come from Ada 2012 (Attribute Old and Expression Functions for example). Other features are specific to SPARK (Attribute Loop_Entry and Ghost Code for example). In this section, we describe these features and their impact on execution and formal verification.

5.5.1. Aspect `Constant_After_Elaboration`¶

[SPARK]

Aspect Constant_After_Elaboration can be specified on a library level variable that has an initialization expression. When specified, the corresponding variable can only be changed during the elaboration of its enclosing package. SPARK ensures that users of the package do not change the variable. This feature can be particularly useful in tasking code since variables that are Constant_After_Elaboration are guaranteed to prevent unsynchronized modifications (see Tasks and Data Races).

package CAE is
   Var : Integer := 0 with
     Constant_After_Elaboration;

   --  The following is illegal because users of CAE could call Illegal
   --  and that would cause an update of Var after CAE has been
   --  elaborated.
   procedure Illegal with
      Global => (Output => Var);
end CAE;

package body CAE is
   procedure Illegal is
   begin
      Var := 10;
   end Illegal;

   --  The following subprogram is legal because it is declared inside
   --  the body of CAE and therefore it cannot be directly called
   --  from a user of CAE.
   procedure Legal is
   begin
      Var := Var + 2;
   end Legal;

begin
   --  The following statements are legal since they take place during
   --  the elaboration of CAE.
   Var := Var + 1;
   Legal;
end CAE;

5.5.2. Aspect `No_Caching`¶

[SPARK]

Aspect No_Caching can be specified for a volatile variable to indicate that this variable can be analyzed as non-volatile by GNATprove. This is typically used to hold the value of local variables guarding the access to some critical section of the code. To defend against fault injection attacks, a common practice is to duplicate the test guarding the critical section, and the variable is marked as volatile to prevent the compiler from optimizing out the duplicate tests. For example:

Cond : Boolean with Volatile, No_Caching := Some_Computation;

if not Cond then
    return;
end if;

if not Cond then
    return;
end if;

if Cond then
    -- here do some critical work
end if;

Without No_Caching, the volatile variable is assumed to be used for Interfaces to the Physical World, GNATprove analyses it specially and one cannot declare it inside a subprogram.

5.5.3. Aspect `Relaxed_Initialization` and Attribute `Initialized`¶

[SPARK]

Modes on parameters and data dependency contracts in SPARK have a stricter meaning than in Ada (see Data Initialization Policy). In general, this allows GNATprove to ensure correct initialization of data in a quick and scalable way through flow analysis, without the need for user-supplied annotations. However, in some cases, the initialization policy may be considered too constraining. In particular, it does not permit initializing composite objects by part through different subprograms, or leaving data uninitialized on return if an error occurred.

5.5.3.1. Aspect `Relaxed_Initialization`¶

To handle these cases, it is possible to relax the standard data initialization policy of SPARK using the Relaxed_Initialization aspect. This aspect can be used:

on objects, to state that the object should not be subjected to the initialization policy of SPARK,
on types, so that it applies to every object or component of the type, or
on subprograms, to annotate the parameters or result.

Here are some examples:

type My_Rec is record
   F, G : Positive;
end record;

G : My_Rec with Relaxed_Initialization;
procedure Init_G_If_No_Errors (Error : out Boolean) with
   Global => (Output => G);
--  G is only initialized if the Error flag is False

In the snippet above, the aspect Relaxed_Initialization is used to annotate the object G so that SPARK will allow returning from Init_G_If_No_Errors with an uninitialized value in G in case of errors in the initialization routine.

On a subprogram, the Relaxed_Initialization aspect expects some parameters to specify to which objects it applies. For example, the parameter X of the procedures below is concerned by the aspect:

procedure Init_Only_F (X : out My_Rec) with
  Relaxed_Initialization => X;
--  Initialize the F component of X,
--  X.G should not be read after the call.

procedure Init_Only_G (X : in out My_Rec) with
  Relaxed_Initialization => X;
--  Initialize the G component of X,
--  X.F can be read after the call if it was already initialized.

The procedures Init_Only_F and Init_Only_G above differ only by the mode of parameter X. Just like for Init_G_If_No_Errors, the mode out in Init_Only_F does not mean that X should be entirely initialized by the call. Its purpose is mostly for data dependencies (see Data Dependencies). It states that the value on entry of the procedure call should not leak into the parts of the output value which are read after the call. To ensure that, GNATprove considers that out parameters may not be copied when entering a procedure call, and so, even for parameters which are in fact passed by reference.

To exempt the value returned by a function from the data initialization policy of SPARK, the result attribute can be specified as a parameter of the Relaxed_Initialization aspect, as in Read_G below. It is also possible to give several objects to the aspect using an aggregate notation:

procedure Copy (Source : My_Rec; Target : out My_Rec) with
  Relaxed_Initialization => (Source, Target);
--  Can copy a partially initialized record

function Read_G return My_Rec with
  Relaxed_Initialization => Read_G'Result;
--  The result of Read_G might not be initialized

Note

The Relaxed_Initialization aspect has no effect on subprogram parameters or function results of a scalar type with relaxed initialization. Indeed, the Ada semantics mandates a copy of scalars on entry and return of subprograms, which is considered to be an error if the object was not initialized.

Finally, if we want to exempt all objects of a type from the data initialization policy of SPARK, it is possible to specify the Relaxed_Initialization aspect on a type. This also allows to exempt a single component of a record, like in the following example:

type Content_Type is array (Positive range 1 .. 100) of Integer with
  Relaxed_Initialization;
type Stack is record
   Top     : Natural := 0;
   Content : Content_Type;
end record
  with Predicate => Top in 0 .. 100;
--  Elements located after Top in Content do not need to be initialized

A stack is made of two components: an array Content storing the actual content of the stack, and the index Top of the topmost element currently allocated on the stack. If the stack is initialized, the Top component necessarily holds a meaningful value. However, because of the API of the stack, it is not possible to read a value stored above the Top index in Content without writing it first. For this reason, it is not necessary to initialize all elements of the stack at creation. To express that, we use in the type Stack, which itself is subjected to the standard initialization policy, an array with the Relaxed_Initialization aspect for the Content field.

Note

The Relaxed_Initialization aspect is not allowed on subtypes, so a derived type is necessary to add the aspect to an existing type.

5.5.3.2. Attribute `Initialized`¶

As explained above, the standard data initialization policy does not apply to objects annotated with the Relaxed_Initialization aspect. As a result, it becomes necessary to annotate which parts of accessed objects are initialized on entry and exit of subprograms in contracts. This can be done using the Initialized attribute. This attribute can be applied to (parts of) objects annotated with the Relaxed_Initialization aspect. If the object is completely initialized, this attribute evaluates to True.

Note

It is not true that the Initialized aspect necessarily evaluates to False on uninitialized data. This is to comply with execution, where some values may happen to be valid even if they have not been initialized. However, it is not possible to prove that the Initialized aspect evaluates to True if the object has not been entirely initialized.

As an example, let’s add some contracts to the subprograms presented in the previous example to replace the comments. The case of Init_G_If_No_Errors is straightforward:

procedure Init_G_If_No_Errors (Error : out Boolean) with
  Post => (if not Error then G'Initialized);

It states that if no errors have occurred (Error is False on exit), G has been initialized by the call.

The postcondition of Read_G is a bit more complicated. We want to state that the function returns the value stored in G. However, we cannot use equality, as it would evaluate the components of both operands and fail if G is not entirely initialized. What we really want to say is that each component of the result of Read_G will be initialized if and only if the corresponding component in G is initialized, and then that the values of the components necessarily match in this case. To express that, we introduce safe accessors for the record components, which check whether the field is initialized before returning it. If the component is not initialized, they return 0 which is an invalid value since both components of My_Rec are of type Positive. This allows to encode both the initialization status and the value of the field in one go:

function Get_F (X : My_Rec) return Integer is
   (if X.F'Initialized then X.F else 0)
with Ghost,
  Relaxed_Initialization => X;

function Get_G (X : My_Rec) return Integer is
   (if X.G'Initialized then X.G else 0)
with Ghost,
  Relaxed_Initialization => X;

Using these accessors, we can define an equality which can safely be called on uninitialized data, and use it in the postcondition of Read_G:

function Safe_Eq (X, Y : My_Rec) return Boolean is
  (Get_F (X) = Get_F (Y) and Get_G (X) = Get_G (Y))
with Ghost,
  Relaxed_Initialization => (X, Y);

function Read_G return My_Rec with
  Relaxed_Initialization => Read_G'Result,
  Post => Safe_Eq (Read_G'Result, G);

The same safe equality function can be used for the postcondition of Copy:

procedure Copy (Source : My_Rec; Target : out My_Rec) with
  Relaxed_Initialization => (Source, Target),
  Post => Safe_Eq (Source, Target);

Remain the procedures Init_Only_F and Init_Only_G. We reflect the asymmetry of their parameter modes in their postconditions:

procedure Init_Only_F (X : out My_Rec) with
  Relaxed_Initialization => X,
  Post => X.F'Initialized;

procedure Init_Only_G (X : in out My_Rec) with
  Relaxed_Initialization => X,
  Post => X.G'Initialized and Get_F (X) = Get_F (X)'Old;

The procedure Init_Only_G preserves the value of X.F whereas Init_Only_F does not preserve X.G. Note that a postcondition similar to the one of Init_Only_G would be proved on Init_Only_F, but it will be of no use as out parameters are considered to be havocked at the beginning of procedure calls, so Get_G (X)'Old wouldn’t actually refer to the value of G before the call.

Finally, let’s consider the type Stack defined above. We have annotated the array type used for its content with the Relaxed_Initialization aspect, so that we do not need to initialize all of its components at declaration. However, we still need to know that elements up to Top have been initialized to ensure that poping an element returns an initialized value. This can be stated by extending the subtype predicate of Stack in the following way:

type Stack is record
   Top     : Natural := 0;
   Content : Content_Type;
end record
  with Predicate => Top in 0 .. 100
  and then (for all I in 1 .. Top => Content (I)'Initialized);

Note

When the Relaxed_Initialization aspect is used, correct initialization is verified by proof (--mode=all or --mode=silver), and not flow analysis (--mode=flow or --mode=bronze).

It is possible to annotate an object with the Relaxed_Initialization aspect to use proof to verify its initialization. For example, it allows to workaround limitations in flow analysis with respect to initialization of arrays. However, if this initialization goes through a loop, using the Initialized attribute in a loop invariant might be required for proof to verify the program.

5.5.4. Attribute `Loop_Entry`¶

[SPARK]

It is sometimes convenient to refer to the value of variables at loop entry. In many cases, the variable has not been modified between the subprogram entry and the start of the loop, so this value is the same as the value at subprogram entry. But Attribute Old cannot be used in that case. Instead, we can use attribute Loop_Entry. For example, we can express that after J iterations of the loop, the value of parameter array X at index J is equal to its value at loop entry plus one:

procedure Increment_Array (X : in out Integer_Array) is
begin
   for J in X'Range loop
      X(J) := X(J) + 1;
      pragma Assert (X(J) = X'Loop_Entry(J) + 1);
   end loop
end Increment_Array;

At run time, a copy of the variable X is made when entering the loop. This copy is then read when evaluating the expression X'Loop_Entry. No copy is made if the loop is never entered. Because it requires copying the value of X, the type of X cannot be limited.

Attribute Loop_Entry can only be used in top-level Assertion Pragmas inside a loop. It is mostly useful for expressing complex Loop Invariants which relate the value of a variable at a given iteration of the loop and its value at loop entry. For example, we can express that after J iterations of the loop, the value of parameter array X at all indexes already seen is equal to its value at loop entry plus one, and that its value at all indexes not yet seen is unchanged, using Quantified Expressions:

procedure Increment_Array (X : in out Integer_Array) is
begin
   for J in X'Range loop
      X(J) := X(J) + 1;
      pragma Loop_Invariant (for all K in X'First .. J => X(K) = X'Loop_Entry(K) + 1);
      pragma Loop_Invariant (for all K in J + 1 .. X'Last => X(K) = X'Loop_Entry(K));
   end loop;
end Increment_Array;

Attribute Loop_Entry may be indexed by the name of the loop to which it applies, which is useful to refer to the value of a variable on entry to an outter loop. When used without loop name, the attribute applies to the closest enclosing loop. For examples, X'Loop_Entry is the same as X'Loop_Entry(Inner) in the loop below, which is not the same as X'Loop_Entry(Outter) (although all three assertions are true):

procedure Increment_Matrix (X : in out Integer_Matrix) is
begin
   Outter: for J in X'Range(1) loop
      Inner: for K in X'Range(2) loop
         X(J,K) := X(J,K) + 1;
         pragma Assert (X(J,K) = X'Loop_Entry(J,K) + 1);
         pragma Assert (X(J,K) = X'Loop_Entry(Inner)(J,K) + 1);
         pragma Assert (X(J,K) = X'Loop_Entry(Outter)(J,K) + 1);
      end loop Inner;
   end loop Outter;
end Increment_Matrix;

By default, similar restrictions exist for the use of attribute Loop_Entry and the use of attribute Old In a Potentially Unevaluated Expression. The same solutions apply here, in particular the use of GNAT pragma Unevaluated_Use_Of_Old.

5.5.5. Attribute `Old`¶

[Ada 2012]

5.5.5.1. In a Postcondition¶

Inside Postconditions, attribute Old refers to the values that expressions had at subprogram entry. For example, the postcondition of procedure Increment might specify that the value of parameter X upon returning from the procedure has been incremented:

procedure Increment (X : in out Integer) with
  Post => X = X'Old + 1;

At run time, a copy of the variable X is made when entering the subprogram. This copy is then read when evaluating the expression X'Old in the postcondition. Because it requires copying the value of X, the type of X cannot be limited.

Strictly speaking, attribute Old must apply to a name in Ada syntax, for example a variable, a component selection, a call, but not an addition like X + Y. For expressions that are not names, attribute Old can be applied to their qualified version, for example:

procedure Increment_One_Of (X, Y : in out Integer) with
  Post => X + Y = Integer'(X + Y)'Old + 1;

Because the compiler unconditionally creates a copy of the expression to which attribute Old is applied at subprogram entry, there is a risk that this feature might confuse users in more complex postconditions. Take the example of a procedure Extract, which copies the value of array A at index J into parameter V, and zeroes out this value in the array, but only if J is in the bounds of A:

procedure Extract (A : in out My_Array; J : Integer; V : out Value) with
  Post => (if J in A'Range then V = A(J)'Old);  --  INCORRECT

Clearly, the value of A(J) at subprogram entry is only meaningful if J is in the bounds of A. If the code above was allowed, then a copy of A(J) would be made on entry to subprogram Extract, even when J is out of bounds, which would raise a run-time error. To avoid this common pitfall, use of attribute Old in expressions that are potentially unevaluated (like the then-part in an if-expression, or the right argument of a shortcut boolean expression - See Ada RM 6.1.1) is restricted to plain variables: A is allowed, but not A(J). The GNAT compiler issues the following error on the code above:

prefix of attribute "Old" that is potentially unevaluated must denote an entity

The correct way to specify the postcondition in the case above is to apply attribute Old to the entity prefix A:

procedure Extract (A : in out My_Array; J : Integer; V : out Value) with
  Post => (if J in A'Range then V = A'Old(J));

5.5.5.2. In Contract Cases¶

The rule for attribute Old inside Contract Cases is more permissive. Take for example the same contract as above for procedure Extract, expressed with contract cases:

procedure Extract (A : in out My_Array; J : Integer; V : out Value) with
  Contract_Cases => ((J in A'Range) => V = A(J)'Old,
                     others         => True);

Only the expressions used as prefixes of attribute Old in the currently enabled case are copied on entry to the subprogram. So if Extract is called with J out of the range of A, then the second case is enabled, so A(J) is not copied when entering procedure Extract. Hence, the above code is allowed.

It may still be the case that some contracts refer to the value of objects at subprogram entry inside potentially unevaluated expressions. For example, an incorrect variation of the above contract would be:

procedure Extract (A : in out My_Array; J : Integer; V : out Value) with
  Contract_Cases => (J >= A'First => (if J <= A'Last then V = A(J)'Old),  --  INCORRECT
                     others       => True);

For the same reason that such uses are forbidden by Ada RM inside postconditions, the SPARK RM forbids these uses inside contract cases (see SPARK RM 6.1.3(2)). The GNAT compiler issues the following error on the code above:

prefix of attribute "Old" that is potentially unevaluated must denote an entity

The correct way to specify the consequence expression in the case above is to apply attribute Old to the entity prefix A:

procedure Extract (A : in out My_Array; J : Integer; V : out Value) with
  Contract_Cases => (J >= A'First => (if J <= A'Last then V = A'Old(J)),
                     others       => True);

5.5.5.3. In a Potentially Unevaluated Expression¶

In some cases, the compiler issues the error discussed above (on attribute Old applied to a non-entity in a potentially unevaluated context) on an expression that can safely be evaluated on subprogram entry, for example:

procedure Extract (A : in out My_Array; J : Integer; V : out Value) with
  Post => (if J in A'Range then V = Get_If_In_Range(A,J)'Old);  --  ERROR

where function Get_If_In_Range returns the value A(J) when J is in the bounds of A, and a default value otherwise.

In that case, the solution is either to rewrite the postcondition using non-shortcut boolean operators, so that the expression is not potentially evaluated anymore, for example:

procedure Extract (A : in out My_Array; J : Integer; V : out Value) with
  Post => J not in A'Range or V = Get_If_In_Range(A,J)'Old;

or to rewrite the postcondition using an intermediate expression function, so that the expression is not potentially evaluated anymore, for example:

function Extract_Post (A : My_Array; J : Integer; V, Get_V : Value) return Boolean is
  (if J in A'Range then V = Get_V);

procedure Extract (A : in out My_Array; J : Integer; V : out Value) with
  Post => Extract_Post (A, J, V, Get_If_In_Range(A,J)'Old);

or to use the GNAT pragma Unevaluated_Use_Of_Old to allow such uses of attribute Old in potentially unevaluated expressions:

pragma Unevaluated_Use_Of_Old (Allow);

procedure Extract (A : in out My_Array; J : Integer; V : out Value) with
  Post => (if J in A'Range then V = Get_If_In_Range(A,J)'Old);

GNAT does not issue an error on the code above, and always evaluates the call to Get_If_In_Range on entry to procedure Extract, even if this value may not be used when executing the postcondition. Note that the formal verification tool GNATprove correctly generates all required checks to prove that this evaluation on subprogram entry does not fail a run-time check or a contract (like the precondition of Get_If_In_Range if any).

Pragma Unevaluated_Use_Of_Old applies to uses of attribute Old both inside postconditions and inside contract cases. See GNAT RM for a detailed description of this pragma.

5.5.6. Attribute `Result`¶

[Ada 2012]

Inside Postconditions of functions, attribute Result refers to the value returned by the function. For example, the postcondition of function Increment might specify that it returns the value of parameter X plus one:

function Increment (X : Integer) return Integer with
  Post => Increment'Result = X + 1;

Contrary to Attribute Old, attribute Result does not require copying the value, hence it can be applied to functions that return a limited type. Attribute Result can also be used inside consequence expressions in Contract Cases.

5.5.7. Aggregates¶

Aggregates are expressions, and as such can appear in assertions and contracts to specify the value of a composite type (record or array), without having to specify the value of each component of the object separately.

5.5.7.1. Record Aggregates¶

[Ada 83]

Since the first version, Ada has a compact syntax for expressing the value of a record type, optionally allowing to name the components. Given the following declaration of type Point:

type Point is record
   X, Y, Z : Float;
end record;

the value of the origin can be expressed with a named notation:

Origin : constant Point := (X => 0.0, Y => 0.0, Z => 0.0);

or with a positional notation, where the values for components are taken in the order in which they are declared in type Point, so the following is equivalent to the above named notation:

Origin : constant Point := (0.0, 0.0, 0.0);

With named notation, components can be given in any order:

Origin : constant Point := (Z => 0.0, Y => 0.0, X => 0.0);

Positional notation and named notation can be mixed, but, in that case, named associations should always follow positional associations, so positional notation will refer to the first components of the record, and named notation will refer to the last components of the record:

Origin : constant Point := (0.0, Y => 0.0, Z => 0.0);
Origin : constant Point := (0.0, 0.0, Z => 0.0);

Choices can be grouped with the bar symbol | to denote sets:

Origin : constant Point := (X | Y | Z => 0.0);

The choice others can be used with a value to refer to all other components, provided these components have the same type, and the others choice should come last:

Origin : constant Point := (X => 0.0, others => 0.0);
Origin : constant Point := (Z => 0.0, others => 0.0);
Origin : constant Point := (0.0, others => 0.0);  --  positional for X
Origin : constant Point := (others => 0.0);

The box notation <> can be used instead of an explicit value to denote the default value of the corresponding type:

Origin : constant Point := (X => <>, Y => 0.0, Z => <>);

In SPARK, this is only allowed if the types of the corresponding components have a default value, for example here:

type Zero_Init_Float is new Float with Default_Value => 0.0;

type Point is record
   X : Float := 0.0;
   Y : Float;
   Z : Zero_Init_Float;
end record;

Note that, when using box notation <> with an others choice, it is not required that these components have the same type.

5.5.7.2. Array Aggregates¶

[Ada 83]

Since the first version, Ada has the same compact syntax for expressing the value of an array type as for record types, optionally allowing to name the indexes. Given the following declaration of type Point:

type Dimension is (X, Y, Z);

type Point is array (Dimension) of Float;

the value of the origin can be expressed with a named notation:

Origin : constant Point := (X => 0.0, Y => 0.0, Z => 0.0);

or with a positional notation, where the values for components are taken in the order in which they are declared in type Point, so the following is equivalent to the above named notation:

Origin : constant Point := (0.0, 0.0, 0.0);

With the difference that named notation and positional notation cannot be mixed in an array aggregate, all other explanations presented for aggregates of record type Point in Record Aggregates are applicable to array aggregates here, so all the following declarations are valid:

Origin : constant Point := (Z => 0.0, Y => 0.0, X => 0.0);
Origin : constant Point := (X | Y | Z => 0.0);
Origin : constant Point := (X => 0.0, others => 0.0);
Origin : constant Point := (Z => 0.0, others => 0.0);
Origin : constant Point := (0.0, others => 0.0);  --  positional for X
Origin : constant Point := (others => 0.0);

while the use of box notation <> is only allowed in SPARK if array components have a default value, either through their type, or through aspect Default_Component_Value on the array type:

type Point is array (Dimension) of Float
  with Default_Component_Value => 0.0;

Note that in many cases, indexes take an integer value rather than an enumeration value:

type Dimension is range 1 .. 3;

type Point is array (Dimension) of Float;

In that case, choices will take an integer value too:

Origin : constant Point := (3 => 0.0, 2 => 0.0, 1 => 0.0);
Origin : constant Point := (1 | 2 | 3 => 0.0);
Origin : constant Point := (1 => 0.0, others => 0.0);
Origin : constant Point := (3 => 0.0, others => 0.0);
Origin : constant Point := (0.0, others => 0.0);  --  positional for 1
Origin : constant Point := (others => 0.0);

Note that one can also use X, Y and Z in place of literals 1, 2 and 3 with the prior definition of suitable named numbers:

X : constant := 1;
Y : constant := 2;
Z : constant := 3;

Note that allocators are allowed inside expressions, and that values in aggregates are evaluated for each corresponding choice, so it is possible to write the following without violating the Memory Ownership Policy of SPARK:

type Ptr is access Integer;
type Data is array (1 .. 10) of Ptr;

Database : Data := (others => new Integer'(0));

This would be also possible in a record aggregate, but it is more common in array aggregates.

5.5.7.3. Iterated Component Associations¶

[Ada 2022]

It is possible to have the value of an association depending on the choice, with the feature called iterated component associations. Here is how we can express that Ident is the identity mapping from values in Index to themselves:

type Index is range 1 .. 100;
type Mapping is array (Index) of Index;

Ident : constant Mapping := (for J in Index => J);

Such an iterated component association can appear next to other associations in an array aggregate using named notation. Here is how we can express that Saturation is the identity mapping between 10 and 90, and saturates outside of this range:

Saturation : constant Mapping :=
  (1 .. 10 => 10, for J in 11 .. 89 => J, 90 .. 100 => 90);

5.5.7.4. Delta Aggregates¶

[Ada 2022]

It is quite common in Postconditions to relate the input and output values of parameters. While this can be as easy as X = X'Old + 1 in the case of scalar parameters, it is more complex to express for array and record parameters. Delta aggregates are useful in that case, to denote the updated value of a composite variable. For example, we can express more clearly that procedure Zero_Range zeroes out the elements of its array parameter X between From and To by using a delta aggregate:

procedure Zero_Range (X : in out Integer_Array; From, To : Positive) with
  Post => X = (X'Old with delta From .. To => 0);

than with an equivalent postcondition using Quantified Expressions and Conditional Expressions:

procedure Zero_Range (X : in out Integer_Array; From, To : Positive) with
  Post => (for all J in X'Range =>
             (if J in From .. To then X(J) = 0 else X(J) = X'Old(J)));

Delta aggregates allow to specify a list of associations between indexes (for arrays) or components (for records) and values. Components can only be mentioned once, with the semantics that all values are evaluated before any update. Array indexes may be mentioned more than once, with the semantics that updates are applied in left-to-right order. For example, the postcondition of procedure Swap expresses that the values at indexes J and K in array X have been swapped:

procedure Swap (X : in out Integer_Array; J, K : Positive) with
  Post => X = (X'Old with delta J => X'Old(K), K => X'Old(J));

and the postcondition of procedure Rotate_Clockwize_Z expresses that the point P given in parameter has been rotated 90 degrees clockwise around the Z axis (thus component Z is preserved while components X and Y are modified):

procedure Rotate_Clockwize_Z (P : in out Point_3D) with
  Post => P = (P'Old with delta X => P.Y'Old, Y => - P.X'Old);

Similarly to their use in combination with attribute Old in postconditions, delta aggregates are useful in combination with Attribute Loop_Entry inside Loop Invariants. For example, we can express the property that, after iteration J in the main loop in procedure Zero_Range, the value of parameter array X at all indexes already seen is equal to zero:

procedure Zero_Range (X : in out Integer_Array; From, To : Positive) is
begin
   for J in From .. To loop
      X(J) := 0;
      pragma Loop_Invariant (X = (X'Loop_Entry with delta From .. J => 0));
   end loop;
end Zero_Range;

Delta aggregates can also be used outside of assertions. They are particularly useful in expression functions. For example, the functionality in procedure Rotate_Clockwize_Z could be expressed equivalently as an expression function:

function Rotate_Clockwize_Z (P : Point_3D) return Point_3D is
  (P with delta X => P.Y, Y => - P.X);

Because it requires copying the value of P, the type of P cannot be limited.

Note

In SPARK versions up to SPARK 21, delta aggregates are not supported and an equivalent attribute named Update can be used instead.

5.5.8. Conditional Expressions¶

[Ada 2012]

A conditional expression is a way to express alternative possibilities in an expression. It is like the ternary conditional expression cond ? expr1 : expr2 in C or Java, except more powerful. There are two kinds of conditional expressions in Ada:

if-expressions are the counterpart of if-statements in expressions
case-expressions are the counterpart of case-statements in expressions

For example, consider the variant of procedure Add_To_Total seen in Contract Cases, which saturates at a given threshold. Its postcondition can be expressed with an if-expression as follows:

procedure Add_To_Total (Incr : in Integer) with
  Post => (if Total'Old + Incr < Threshold  then
             Total = Total'Old + Incr
           else
             Total = Threshold);

Each branch of an if-expression (there may be one, two or more branches when elsif is used) can be seen as a logical implication, which explains why the above postcondition can also be written:

procedure Add_To_Total (Incr : in Integer) with
  Post => (if Total'Old + Incr < Threshold then Total = Total'Old + Incr) and
          (if Total'Old + Incr >= Threshold then Total = Threshold);

or equivalently (as the absence of else branch above is implicitly the same as else True):

procedure Add_To_Total (Incr : in Integer) with
  Post => (if Total'Old + Incr < Threshold then Total = Total'Old + Incr else True) and
          (if Total'Old + Incr >= Threshold then Total = Threshold else True);

If-expressions are not necessarily of boolean type, in which case they must have an else branch that gives the value of the expression for cases not covered in previous conditions (as there is no implicit else True in such a case). For example, here is a postcondition equivalent to the above, that uses an if-expression of Integer type:

procedure Add_To_Total (Incr : in Integer) with
  Post => Total = (if Total'Old + Incr < Threshold then Total'Old + Incr else Threshold);

Although case-expressions can be used to cover cases of any scalar type, they are mostly used with enumerations, and the compiler checks that all cases are disjoint and that together they cover all possible cases. For example, consider a variant of procedure Add_To_Total which takes an additional Mode global input of enumeration value Single, Double, Negate or Ignore, with the intuitive corresponding leverage effect on the addition. The postcondition of this variant can be expressed using a case-expression as follows:

procedure Add_To_Total (Incr : in Integer) with
  Post => (case Mode is
             when Single => Total = Total'Old + Incr,
             when Double => Total = Total'Old + 2 * Incr,
             when Ignore => Total = Total'Old,
             when Negate => Total = Total'Old - Incr);

Like if-expressions, case-expressions are not necessarily of boolean type. For example, here is a postcondition equivalent to the above, that uses a case-expression of Integer type:

procedure Add_To_Total (Incr : in Integer) with
  Post => Total = Total'Old + (case Mode is
                                 when Single => Incr,
                                 when Double => 2 * Incr,
                                 when Ignore => 0,
                                 when Negate => - Incr);

A last case of others can be used to denote all cases not covered by previous conditions. If-expressions and case-expressions should always be parenthesized.

5.5.9. Declare Expressions¶

[Ada 2022]

Declare expressions are used to factorize parts of an expression. They allow to declare constants and renamings which are local to the expression. A declare expression is made of two parts:

A list of declarations of local constants and renamings
An expression using the names introduced in these declarations.

To match the syntax of declare blocks, the first part is introduced by declare and the second by begin. The scope is delimited by enclosing parentheses, without end to close the scope.

As an example, we introduce a Find_First_Zero function which finds the index of the first occurrence of 0 in an array of integers and a procedure Set_Range_To_Zero which zeros out all elements located between the first and second occurrence of 0 in the array:

function Has_Zero (A : My_Array) return Boolean is
  (for some E of A => E = 0);

function Has_Two_Zeros (A : My_Array) return Boolean is
  (for some I in A'Range => A (I) = 0 and
     (for some J in A'Range => A (J) = 0 and I /= J));

function Find_First_Zero (A : My_Array) return Natural with
  Pre  => Has_Zero (A),
  Post => Find_First_Zero'Result in A'Range
    and A (Find_First_Zero'Result) = 0
    and not Has_Zero (A (A'First .. Find_First_Zero'Result - 1));

procedure Set_Range_To_Zero (A : in out My_Array) with
  Pre  => Has_Two_Zeros (A),
  Post =>
     A = (A'Old with delta
            Find_First_Zero (A'Old) ..
              Find_First_Zero
                (A'Old (Find_First_Zero (A'Old) + 1 .. A'Last)) => 0);

In the contract of Set_Range_To_Zero, we use Delta Aggregates to state that elements of A located in the range between the first and the second occurrence of 0 in A have been set to 0 by the procedure. The second occurrence is found by calling Find_First_Zero on the slice of A starting just after the first occurrence of 0.

To make the contract of Set_Range_To_Zero more readable, we can use a declare expression to introduce constants for the first and second occurrence of 0 in the array. The explicit names make it easier to understand what the bounds of the updated slice are supposed to be. It also avoids repeating the call to Find_First_Zero on A in the computation of the second bound:

procedure Set_Range_To_Zero (A : in out My_Array) with
  Pre  => Has_Two_Zeros (A),
  Post =>
    (declare
       Fst_Zero : constant Positive := Find_First_Zero (A'Old);
       Snd_Zero : constant Positive := Find_First_Zero
          (A'Old (Fst_Zero + 1 .. A'Last));
     begin
       A = (A'Old with delta Fst_Zero .. Snd_Zero => 0));

5.5.10. Expression Functions¶

[Ada 2012]

An expression function is a function whose implementation is given by a single expression. For example, the function Increment can be defined as an expression function as follows:

function Increment (X : Integer) return Integer is (X + 1);

For compilation and execution, this definition is equivalent to:

function Increment (X : Integer) return Integer is
begin
   return X + 1;
end Increment;

For GNATprove, this definition as expression function is equivalent to the same function body as above, plus a postcondition:

function Increment (X : Integer) return Integer with
  Post => Increment'Result = X + 1
is
begin
   return X + 1;
end Increment;

Thus, a user does not need in general to add a postcondition to an expression function, as the implicit postcondition generated by GNATprove is the most precise one. If a user adds a postcondition to an expression function, GNATprove uses this postcondition to analyze the function’s callers as well as the most precise implicit postcondition.

On the contrary, it may be useful in general to add a precondition to an expression function, to constrain the contexts in which it can be called. For example, parameter X passed to function Increment should be less than the maximal integer value, otherwise an overflow would occur. We can specify this property in Increment’s precondition as follows:

function Increment (X : Integer) return Integer is (X + 1) with
  Pre => X < Integer'Last;

Note that the contract of an expression function follows its expression.

Expression functions can be defined in package declarations, hence they are well suited for factoring out common properties that are referred to in contracts. For example, consider the procedure Increment_Array that increments each element of its array parameter X by one. Its precondition can be expressed using expression functions as follows:

package Increment_Utils is

   function Not_Max (X : Integer) return Boolean is (X < Integer'Last);

   function None_Max (X : Integer_Array) return Boolean is
     (for all J in X'Range => Not_Max (X(J)));

   procedure Increment_Array (X : in out Integer_Array) with
     Pre => None_Max (X);

end Increment_Utils;

Expression functions can be defined over private types, and still be used in the contracts of publicly visible subprograms of the package, by declaring the function publicly and defining it in the private part. For example:

package Increment_Utils is

   type Integer_Array is private;

   function None_Max (X : Integer_Array) return Boolean;

   procedure Increment_Array (X : in out Integer_Array) with
     Pre => None_Max (X);

private

   type Integer_Array is array (Positive range <>) of Integer;

   function Not_Max (X : Integer) return Boolean is (X < Integer'Last);

   function None_Max (X : Integer_Array) return Boolean is
     (for all J in X'Range => Not_Max (X(J)));

end Increment_Utils;

If an expression function is defined in a unit spec, GNATprove can use its implicit postcondition at every call. If an expression function is defined in a unit body, GNATprove can use its implicit postcondition at every call in the same unit, but not at calls inside other units. This is true even if the expression function is declared in the unit spec and defined in the unit body.

5.5.11. Ghost Code¶

[SPARK]

Sometimes, the variables and functions that are present in a program are not sufficient to specify intended properties and to verify these properties with GNATprove. In such a case, it is possible in SPARK to insert in the program additional code useful for specification and verification, specially identified with the aspect Ghost so that it can be discarded during compilation. So-called ghost code in SPARK are these parts of the code that are only meant for specification and verification, and have no effect on the functional behavior of the program.

Various kinds of ghost code are useful in different situations:

Ghost functions are typically used to express properties used in contracts.
Global ghost variables are typically used to keep track of the current state of a program, or to maintain a log of past events of some type. This information can then be referred to in contracts.
Local ghost variables are typically used to hold intermediate values during computation, which can then be referred to in assertion pragmas like loop invariants.
Ghost types are those types only useful for defining ghost variables.
Ghost procedures can be used to factor out common treatments on ghost variables. Ghost procedures should not have non-ghost outputs, either output parameters or global outputs.
Ghost packages provide a means to encapsulate all types and operations for a specific kind of ghost code.
Imported ghost subprograms are used to provide placeholders for properties that are defined in a logical language, when using manual proof.

When the program is compiled with assertions (for example with switch -gnata in GNAT), ghost code is executed like normal code. Ghost code can also be selectively enabled by setting pragma Assertion_Policy as follows:

pragma Assertion_Policy (Ghost => Check);

GNATprove checks that ghost code cannot have an effect on the behavior of the program. GNAT compiler also performs some of these checks, although not all of them. Apart from these checks, GNATprove treats ghost code like normal code during its analyses.

5.5.11.1. Ghost Functions¶

Ghost functions are useful to express properties only used in contracts, and to factor out common expressions used in contracts. For example, function Get_Total introduced in State Abstraction and Functional Contracts to retrieve the value of variable Total in the contract of Add_To_Total could be marked as a ghost function as follows:

function Get_Total return Integer with Ghost;

and still be used exactly as seen in State Abstraction and Functional Contracts:

procedure Add_To_Total (Incr : in Integer) with
  Pre  => Incr >= 0 and then Get_Total in 0 .. Integer'Last - Incr,
  Post => Get_Total = Get_Total'Old + Incr;

The definition of Get_Total would be also the same:

Total : Integer;

function Get_Total return Integer is (Total);

Although it is more common to define ghost functions as Expression Functions, a regular function might be used too:

function Get_Total return Integer is
begin
   return Total;
end Get_Total;

In that case, GNATprove uses only the contract of Get_Total (either user-specified or the default one) when analyzing its callers, like for a non-ghost regular function. (The same exception applies as for regular functions, when GNATprove can analyze a subprogram in the context of its callers, as described in Contextual Analysis of Subprograms Without Contracts.)

All functions which are only used in specification can be marked as ghost, but most don’t need to. However, there are cases where marking a specification-only function as ghost really brings something. First, as ghost entities are not allowed to interfere with normal code, marking a function as ghost avoids having to break state abstraction for the purpose of specification. For example, marking Get_Total as ghost will prevent users of the package Account from accessing the value of Total from non-ghost code.

Then, in the usual context where ghost code is not kept in the final executable, the user is given more freedom to use in ghost code constructs that are less efficient than in normal code, which may be useful to express rich properties. For example, the ghost functions defined in the Formal Containers Library in GNAT typically copy the entire content of the argument container, which would not be acceptable for non-ghost functions.

5.5.11.2. Ghost Variables¶

Ghost variables are useful to keep track of local or global information during the computation, which can then be referred to in contracts or assertion pragmas.

Case 1: Keeping Intermediate Values¶

Local ghost variables are commonly used to keep intermediate values. For example, we can define a local ghost variable Init_Total to hold the initial value of variable Total in procedure Add_To_Total, which allows checking the relation between the initial and final values of Total in an assertion:

procedure Add_To_Total (Incr : in Integer) is
   Init_Total : Integer := Total with Ghost;
begin
   Total := Total + Incr;
   pragma Assert (Total = Init_Total + Incr);
end Add_To_Total;

Case 2: Keeping Memory of Previous State¶

Global ghost variables are commonly used to memorize the value of a previous state. For example, we can define a global ghost variable Last_Incr to hold the previous value passed in argument when calling procedure Add_To_Total, which allows checking in its precondition that the sequence of values passed in argument is non-decreasing:

Last_Incr : Integer := Integer'First with Ghost;

procedure Add_To_Total (Incr : in Integer) with
  Pre => Incr >= Last_Incr;

procedure Add_To_Total (Incr : in Integer) is
begin
   Total := Total + Incr;
   Last_Incr := Incr;
end Add_To_Total;

Case 3: Logging Previous Events¶

Going beyond the previous case, global ghost variables can be used to store a complete log of events. For example, we can define global ghost variables Log and Log_Size to hold the sequence of values passed in argument to procedure Add_To_Total, as in State Abstraction:

Log      : Integer_Array with Ghost;
Log_Size : Natural with Ghost;

procedure Add_To_Total (Incr : in Integer) with
  Post => Log_Size = Log_Size'Old + 1 and Log = (Log'Old with delta Log_Size => Incr);

procedure Add_To_Total (Incr : in Integer) is
begin
   Total := Total + Incr;
   Log_Size := Log_Size + 1;
   Log (Log_Size) := Incr;
end Add_To_Total;

The postcondition of Add_To_Total above expresses that Log_Size is incremented by one at each call, and that the current value of parameter Incr is appended to Log at each call (using Attribute Old and Delta Aggregates).

Case 4: Expressing Existentially Quantified Properties¶

In SPARK, universal quantification is only allowed in restricted cases (over integer ranges and over the content of a container). To express the existence of a particular object, it is sometimes easier to simply provide it. This can be done using a global ghost variable. This can be used in particular to split the specification of a complex procedure into smaller parts:

X_Interm : T with Ghost;

procedure Do_Two_Thing (X : in out T) with
  Post => First_Thing_Done (X'Old, X_Interm) and then
          Second_Thing_Done (X_Interm, X)
is
  X_Init : constant T := X with Ghost;
begin
  Do_Something (X);
  pragma Assert (First_Thing_Done (X_Init, X));
  X_Interm := X;

  Do_Something_Else (X);
  pragma Assert (Second_Thing_Done (X_Interm, X));
end Do_Two_Things;

More complicated uses can also be envisioned, up to constructing ghost data structures reflecting complex properties. For example, we can express that two arrays are a permutation of each other by constructing a permutation from one to the other:

Perm : Permutation with Ghost;

procedure Permutation_Sort (A : Nat_Array) with
  Post => A = Apply_Perm (Perm, A'Old)
is
begin
  --  Initalize Perm with the identity
  Perm := Identity_Perm;

  for Current in A'First .. A'Last - 1 loop
    Smallest := Index_Of_Minimum_Value (A, Current, A'Last);
    if Smallest /= Current then
      Swap (A, Current, Smallest);

      --  Update Perm each time we permute two elements in A
      Permute (Perm, Current, Smallest);
    end if;
   end loop;
 end Permutation_Sort;

5.5.11.3. Ghost Types¶

Ghost types can only be used to define ghost variables. For example, we can define ghost types Log_Type and Log_Size_Type that specialize the types Integer_Array and Natural for ghost variables:

subtype Log_Type is Integer_Array with Ghost;
subtype Log_Size_Type is Natural with Ghost;

Log      : Log_Type with Ghost;
Log_Size : Log_Size_Type with Ghost;

5.5.11.4. Ghost Procedures¶

Ghost procedures are useful to factor out common treatments on ghost variables. For example, we can define a ghost procedure Append_To_Log to append a value to the log as seen previously.

Log      : Integer_Array with Ghost;
Log_Size : Natural with Ghost;

procedure Append_To_Log (Incr : in Integer) with
  Ghost,
  Post => Log_Size = Log_Size'Old + 1 and Log = (Log'Old with delta Log_Size => Incr);

procedure Append_To_Log (Incr : in Integer) is
begin
   Log_Size := Log_Size + 1;
   Log (Log_Size) := Incr;
end Append_To_Log;

Then, this procedure can be called in Add_To_Total as follows:

procedure Add_To_Total (Incr : in Integer) is
begin
   Total := Total + Incr;
   Append_To_Log (Incr);
end Add_To_Total;

5.5.11.5. Ghost Packages¶

Ghost packages are useful to encapsulate all types and operations for a specific kind of ghost code. For example, we can define a ghost package Logging to deal with all logging operations on package Account:

package Logging with
  Ghost
is
   Log      : Integer_Array;
   Log_Size : Natural;

   procedure Append_To_Log (Incr : in Integer) with
     Post => Log_Size = Log_Size'Old + 1 and Log = (Log'Old with delta Log_Size => Incr);

   ...

end Logging;

The implementation of package Logging is the same as if it was not a ghost package. In particular, a Ghost aspect is implicitly added to all declarations in Logging, so it is not necessary to specify it explicitly. Logging can be defined either as a local ghost package or as a separate unit. In the latter case, unit Account needs to reference unit Logging in a with-clause like for a non-ghost unit:

with Logging;

package Account is
   ...
end Account;

5.5.11.6. Imported Ghost Subprograms¶

When using manual proof (see GNATprove and Manual Proof), it may be more convenient to define some properties in the logical language of the prover rather than in SPARK. In that case, ghost functions might be marked as imported, so that no implementation is needed. For example, the ghost procedure Append_To_Log seen previously may be defined equivalently as a ghost imported function as follows:

function Append_To_Log (Log : Log_type; Incr : in Integer) return Log_Type with
  Ghost,
  Import;

where Log_Type is an Ada type used also as placeholder for a type in the logical language of the prover. To avoid any inconsistency between the interpretations of Log_Type in GNATprove and in the manual prover, it is preferable in such a case to mark the definition of Log_Type as not in SPARK, so that GNATprove does not make any assumptions on its content. This can be achieved by defining Log_Type as a private type and marking the private part of the enclosing package as not in SPARK:

package Logging with
  SPARK_Mode,
  Ghost
is
   type Log_Type is private;

   function Append_To_Log (Log : Log_type; Incr : in Integer) return Log_Type with
     Import;

   ...

private
   pragma SPARK_Mode (Off);

   type Log_Type is new Integer;  --  Any definition is fine here
end Logging;

A ghost imported subprogram cannot be executed, so calls to Append_To_Log above should not be enabled during compilation, otherwise a compilation error is issued. Note also that GNATprove will not attempt proving the contract of a ghost imported subprogram, as it does not have its body.

5.5.11.7. Ghost Models¶

When specifying a program, it is common to use a model, that is, an alternative, simpler view of a part of the program. As they are only used in annotations, models can be computed using ghost code.

Models of Control Flow¶

Global variables can be used to enforce properties over call cahains in the program. For example, we may want to express that Total cannot be incremented twice in a row without registering the transaction in between. This can be done by introducing a ghost global variable Last_Transaction_Registered, used to encode whether Register_Transaction was called since the last call to Add_To_Total:

Last_Transaction_Registered : Boolean := True with Ghost;

procedure Add_To_Total (Incr : Integer) with
  Pre  => Last_Transaction_Registered,
  Post => not Last_Transaction_Registered;

procedure Register_Transaction with
  Post => Last_Transaction_Registered;

The value of Last_Transaction_Registered should also be updated in the body of Add_To_Total and Register_Transaction to reflect their contracts:

procedure Add_To_Total (Incr : in Integer) is
begin
   Total := Total + Incr;
   Last_Transaction_Registered := False;
end Add_To_Total;

More generally, the expected control flow of a program can be modeled using an automaton. We can take as an example a mailbox containing only one message. The expected way Receive and Send should be interleaved can be expressed as a two state automaton. The mailbox can either be full, in which case Receive can be called but not Send, or it can be empty, in which case it is Send that can be called and not Receive. To express this property, we can define a ghost global variable of a ghost enumeration type to hold the state of the automaton:

type Mailbox_Status_Kind is (Empty, Full) with Ghost;
Mailbox_Status : Mailbox_Status_Kind := Empty with Ghost;

procedure Receive (X : out Message) with
  Pre  => Mailbox_Status = Full,
  Post => Mailbox_Status = Empty;

procedure Send (X : Message) with
  Pre  => Mailbox_Status = Empty,
  Post => Mailbox_Status = Full;

Like before, Receive and Send should update Mailbox_Status in their bodies. Note that all the transitions of the automaton need not be specified, only the part which are relevant to the properties we want to express.

If the program also has some regular state, an invariant can be used to link the value of this state to the value of the ghost state of the automaton. For example, in our mailbox, we may have a regular variable Message_Content holding the content of the current message, which is only known to be valid after a call to Send. We can introduce a ghost function linking the value of Message_Content to the value of Mailbox_Status, so that we can ensure that Message_Content is always valid when accessed from Receive:

function Invariant return Boolean is
  (if Mailbox_Status = Full then Valid (Message_Content))
with Ghost;

procedure Receive (X : out Message) with
  Pre  => Invariant and then Mailbox_Status = Full,
  Post => Invariant and then Mailbox_Status = Empty
      and then Valid (X)
is
  X := Message_Content;
end Receive;

Models of Data Structures¶

For specifying programs that use complex data structures (doubly-linked lists, maps…), it can be useful to supply a model for the data structure. A model is an alternative, simpler view of the data-structure which allows to write properties more easily. For example, a ring buffer, or a doubly-linked list, can be modeled using an array containing the elements from the buffer or the list in the right order. Typically, though simpler to reason with, the model is less efficient than the regular data structure. For example, inserting an element at the beginning of a doubly-linked list or at the beginning of a ring buffer can be done in constant time whereas inserting an element at the beginning of an array requires to slide all the elements to the right. As a result, models of data structures are usually supplied using ghost code. As an example, the package Ring_Buffer offers an implementation of a single instance ring buffer. A ghost variable Buffer_Model is used to write the specification of the Enqueue procedure:

package Ring_Buffer is
  function Get_Model return Nat_Array with Ghost;

  procedure Enqueue (E : Natural) with
    Post => Get_Model = E & Get_Model'Old (1 .. Max – 1);
private
  Buffer_Content : Nat_Array;
  Buffer_Top     : Natural;
  Buffer_Model   : Nat_Array with Ghost;

  function Get_Model return Nat_Array is (Buffer_Model);
end Ring_Buffer;

Then, just like for models of control flow, an invariant should be supplied to link the regular data structure to its model:

package Ring_Buffer is
  function Get_Model return Nat_Array with Ghost;
  function Invariant return Boolean with Ghost;

  procedure Enqueue (E : Natural) with
    Pre  => Invariant,
    Post => Invariant and then Get_Model = E & Get_Model'Old (1 .. Max – 1);
private
  Buffer_Content : Nat_Array;
  Buffer_Top     : Natural;
  Buffer_Model   : Nat_Array with Ghost;

  function Get_Model return Nat_Array is (Buffer_Model);
  function Invariant return Boolean is
    (Buffer_Model = Buffer_Content (Buffer_Top .. Max)
                  & Buffer_Content (1 .. Buffer_Top - 1));
end Ring_Buffer;

If a data structure type is defined, a ghost function can be provided to compute a model for objects of the data structure type, and the invariant can be stated as a postcondition of this function:

package Ring_Buffer is
  type Buffer_Type is private;
  subtype Model_Type is Nat_Array with Ghost;

  function Invariant (X : Buffer_Type; M : Model_Type) return Boolean with
    Ghost;
  function Get_Model (X : Buffer_Type) return Model_Type with
    Ghost,
    Post => Invariant (X, Get_Model'Result);

  procedure Enqueue (X : in out Buffer_Type; E : Natural) with
    Post => Get_Model (X) = E & Get_Model (X)'Old (1 .. Max – 1);
private
  type Buffer_Type is record
    Content : Nat_Array;
    Top     : Natural;
  end record;
end Ring_Buffer;

More complex examples of models of data structure can be found in the Formal Containers Library.

5.5.11.8. Removal of Ghost Code¶

By default, GNAT completely discards ghost code during compilation, so that no ghost code is present in the object code or the executable. This ensures that, even if parts of the ghost could have side-effects when executed (writing to variables, performing system calls, raising exceptions, etc.), by default the compiler ensures that it cannot have any effect on the behavior of the program.

This is also essential in domains submitted to certification where all instructions in the object code should be traceable to source code and requirements, and where testing should ensure coverage of the object code. As ghost code is not present in the object code, there is no additional cost for maintaining its traceability and ensuring its coverage by tests.

GNAT provides an easy means to check that no ignored ghost code is present in a given object code or executable, which relies on the property that, by definition, each ghost declaration or ghost statement mentions at least one ghost entity. GNAT prefixes all names of such ignored ghost entities in the object code with the string ___ghost_ (except for names of ghost compilation units). The initial triple underscore ensures that this substring cannot appear anywhere in the name of non-ghost entities or ghost entities that are not ignored. Thus, one only needs to check that the substring ___ghost_ does not appear in the list of names from the object code or executable.

On Unix-like platforms, this can done by checking that the following command does not output anything:

nm <object files or executable> | grep ___ghost_

The same can be done to check that a ghost compilation unit called my_unit (whatever the capitalization) is not included at all (entities in that unit would have been detected by the previous check) in the object code or executable. For example on Unix-like platforms:

nm <object files or executable> | grep my_unit

5.5.12. Quantified Expressions¶

[Ada 2012]

A quantified expression is a way to express a property over a collection, either an array or a container (see Formal Containers Library):

a universally quantified expression using for all expresses a property that holds for all elements of a collection
an existentially quantified expression using for some expresses a property that holds for at least one element of a collection

Quantified expressions should always be parenthesized.

5.5.12.1. Iteration Over Content vs. Over Positions¶

Iteration can be expressed either directly over the content of the collection, or over the range of positions of elements in the collection. The former is preferred when the property involved does not refer to the position of elements in the collection or to the previous value of the element at the same position in the collection (e.g. in a postcondition). Otherwise, the latter is needed. For example, consider the procedure Nullify_Array that sets each element of its array parameter X to zero. Its postcondition can be expressed using a universally quantified expression iterating over the content of the array as follows:

procedure Nullify_Array (X : out Integer_Array) with
  Post => (for all E in X => E = 0);

or using a universally quantified expression iterating over the range of the array as follows:

procedure Nullify_Array (X : out Integer_Array) with
  Post => (for all J in X'Range => X(J) = 0);

Quantification over formal containers can similarly iterate over their content, using the syntax for .. of, or their positions, using the syntax for .. in, see examples in Loop Examples.

Iteration over positions is needed when the property refers to the position of elements in the collection. For example, consider the procedure Initialize_Array that sets each element of its array parameter X to its position. Its postcondition can be expressed using a universally quantified expression as follows:

procedure Initialize_Array (X : out Integer_Array) with
  Post => (for all J in X'Range => X(J) = J);

Iteration over positions is also needed when the property refers to the previous value of the element at the same position in the collection. For example, consider the procedure Increment_Array that increments each element of its array parameter X by one. Its postcondition can be expressed using a universally quantified expression as follows:

procedure Increment_Array (X : in out Integer_Array) with
  Post => (for all J in X'Range => X(J) = X'Old(J) + 1);

The negation of a universal property being an existential property (the opposite is true too), the postcondition above can be expressed also using an existentially quantified expression as follows:

procedure Increment_Array (X : in out Integer_Array) with
  Post => not (for some J in X'Range => X(J) /= X'Old(J) + 1);

5.5.12.2. Execution vs. Proof¶

At run time, a quantified expression is executed like a loop, which exits as soon as the value of the expression is known: if the property does not hold (resp. holds) for a given element of a universally (resp. existentially) quantified expression, execution of the loop does not proceed with remaining elements and returns the value False (resp. True) for the expression.

When a quantified expression is analyzed with GNATprove, it uses the logical counterpart of the quantified expression. GNATprove also checks that the expression is free from run-time errors. For this checking, GNATprove checks that the enclosed expression is free from run-time errors over the entire range of the quantification, not only at points that would actually be reached at run time. As an example, consider the following expression:

(for all I in 1 .. 10 => 1 / (I - 3) > 0)

This quantified expression cannot raise a run-time error, because the enclosed expression 1 / (I - 3) > 0 is false for the first value of the range I = 1, so the execution of the loop exits immediately with the value False for the quantified expression. GNATprove is stricter and requires the enclosed expression 1 / (I - 3) > 0 to be free from run-time errors over the entire range I in 1 .. 10 (including I = 3) so it issues a check message for a possible division by zero in this case.

5.5.12.3. Iterator Filters¶

The set of values or positions over which iteration is performed can be filtered with an iterator filter introduced by the keyword when. For example, we can express a property for all prime numbers in a given range as follows:

(for all N in 1 .. 1000 when Is_Prime (N) => ...)

5.5. Specification Features¶

5.5.1. Aspect Constant_After_Elaboration¶

5.5.2. Aspect No_Caching¶

5.5.3. Aspect Relaxed_Initialization and Attribute Initialized¶

5.5.3.1. Aspect Relaxed_Initialization¶

5.5.3.2. Attribute Initialized¶

5.5.4. Attribute Loop_Entry¶

5.5.5. Attribute Old¶

5.5.5.1. In a Postcondition¶

5.5.5.2. In Contract Cases¶

5.5.5.3. In a Potentially Unevaluated Expression¶

5.5.6. Attribute Result¶

5.5.7. Aggregates¶

5.5.7.1. Record Aggregates¶

5.5.7.2. Array Aggregates¶

5.5.7.3. Iterated Component Associations¶

5.5.7.4. Delta Aggregates¶

5.5.8. Conditional Expressions¶

5.5.9. Declare Expressions¶

5.5.10. Expression Functions¶

5.5.11. Ghost Code¶

5.5.11.1. Ghost Functions¶

5.5.11.2. Ghost Variables¶

Case 1: Keeping Intermediate Values¶

Case 2: Keeping Memory of Previous State¶

Case 3: Logging Previous Events¶

Case 4: Expressing Existentially Quantified Properties¶

5.5.11.3. Ghost Types¶

5.5.11.4. Ghost Procedures¶

5.5.11.5. Ghost Packages¶

5.5.11.6. Imported Ghost Subprograms¶

5.5.11.7. Ghost Models¶

Models of Control Flow¶

Models of Data Structures¶

5.5.11.8. Removal of Ghost Code¶

5.5.12. Quantified Expressions¶

5.5.12.1. Iteration Over Content vs. Over Positions¶

5.5.12.2. Execution vs. Proof¶

5.5.12.3. Iterator Filters¶

5.5.1. Aspect `Constant_After_Elaboration`¶

5.5.2. Aspect `No_Caching`¶

5.5.3. Aspect `Relaxed_Initialization` and Attribute `Initialized`¶

5.5.3.1. Aspect `Relaxed_Initialization`¶

5.5.3.2. Attribute `Initialized`¶

5.5.4. Attribute `Loop_Entry`¶

5.5.5. Attribute `Old`¶

5.5.6. Attribute `Result`¶