5.11. SPARK Libraries

The units described here have their spec in SPARK (with SPARK_Mode => On specified on the spec), more rarely their body in SPARK as well.

Subprograms in these units fall into one of the following categories:

• Subprograms which should always return without error or exception if their precondition is respected.

• Procedures marked with the annotation Exceptional_Cases. This corresponds to the possibility of exception in the procedure, even when its precondition is respected.

• Functions marked with SPARK_Mode => Off which cannot be called from SPARK code.

5.11.1. SPARK Library

As part of the SPARK product, several libraries are available through the project file <spark-install>/lib/gnat/sparklib.gpr (or through the project file <spark-install>/lib/gnat/sparklib_light.gpr in an environment without units Ada.Numerics.Big_Numbers.Big_Integers and Ada.Numerics.Big_Numbers.Big_Reals). Header files of the SPARK library are available through Help ‣ SPARK ‣ SPARKlib menu item in GNAT Studio. To use this library in a program, you need to copy the project template that corresponds to your runtime (either sparklib.gpr or sparklib_light.gpr) and adapt it by providing appropriate values for the object directory (attribute Object_Dir in the project file) and the list of excluded source files (attribute Excluded_Source_Files in the project file). The simplest is just to provide a value for Object_Dir and inherit Excluded_Source_Files from the parent project:

project SPARKlib extends "sparklib_external" is
   for Object_Dir use "sparklib_obj";
   for Source_Dirs use SPARKlib_External'Source_Dirs;
   for Excluded_Source_Files use SPARKlib_External'Excluded_Source_Files;
end SPARKlib;

Then, add a corresponding dependency in your project file, for example:

with "sparklib";
project My_Project is
   ...
end My_Project;

You may need to update the environment variable GPR_PROJECT_PATH for the lemma library project to be found by GNAT compiler, as described in Installation of GNATprove.

Finally, if you instantiate in your code a generic from the SPARK library, you may also need to pass -gnateDSPARK_BODY_MODE=Off as a compilation switch for the units with these instantiations.

5.11.2. Big Numbers Library

Annotations such as preconditions, postconditions, assertions, loop invariants, are analyzed by GNATprove with the exact same meaning that they have during execution. In particular, evaluating the expressions in an annotation may raise a run-time error, in which case GNATprove will attempt to prove that this error cannot occur, and report a warning otherwise.

In SPARK, scalar types such as integer and floating point types are bounded machine types, so arithmetic computations over them can lead to overflows when the result does not fit in the bounds of the type used to hold it. In some cases, it is convenient to express properties in annotations as they would be expressed in mathematics, where quantities are unbounded, for example:

function Add (X, Y : Integer) return Integer with
  Pre  => X + Y in Integer,
  Post => Add'Result = X + Y;

The precondition of Add states that the result of adding its two parameters should fit in type Integer. Unfortunately, evaluating this expression will fail an overflow check, because the result of X + Y is stored in a temporary of type Integer.

To alleviate this issue, it is possible to use the standard library for big numbers. It contains support for:

• Unbounded integers in SPARK.Big_Integers.

• Unbounded rational numbers in SPARK.Big_Reals.

Theses libraries define representations for big numbers and basic arithmetic operations over them, as well as conversions from bounded scalar types such as floating point numbers or integer types. Conversion from an integer to a big integer is provided by:

• function To_Big_Integer in SPARK.Big_Integers for type Integer

• function To_Big_Integer in generic package Signed_Conversions in SPARK.Big_Integers for all other signed integer types

• function To_Big_Integer in generic package Unsigned_Conversions in SPARK.Big_Integers for modular integer types

Similarly, the same packages define a function From_Big_Integer to convert from a big integer to an integer. A function To_Real in SPARK.Big_Reals converts from type Integer to a big real and function To_Big_Real in the same package converts from a big integer to a big real.

Though these operations do not have postconditions, they are interpreted by GNATprove as the equivalent operations on mathematical integers and real numbers. This allows to benefit from precise support on code using them. Note that the corresponding Ada libraries Ada.Numerics.Big_Numbers.Big_Integers and Ada.Numerics.Big_Numbers.Big_Reals will be handled in the same way, but might be not available under specific runtimes. It is preferable to use the units from the SPARK library instead, or use Ada.Numerics.Big_Numbers.Big_Integer_Ghost.

Note

Some functionality of the library are not precisely supported. This includes in particular conversions to and from strings, conversions of Big_Real to fixed-point or floating-point types, and Numerator and Denominator functions.

The big number library can be used both in annotations and in actual code, as it is executable, though of course, using it in production code means incurring its runtime costs. It can be considered a good trade-off to only use it in contracts, if they are disabled in production builds. For example, we can rewrite the precondition of our Add function with big integers to avoid overflows:

function Add (X, Y : Integer) return Integer with
  Pre  => In_Range (To_Big_Integer (X) + To_Big_Integer (Y),
                    Low  => To_Big_Integer (Integer'First),
                    High => To_Big_Integer (Integer'Last)),
  Post => Add'Result = X + Y;

As a more advanced example, it is also possible to introduce a ghost model for numerical computations on floating point numbers as a mathematical real number so as to be able to express properties about rounding errors. In the following snippet, we use the ghost variable M as a model of the floating point variable Y, so we can assert that the result of our floating point calculations are not too far from the result of the same computations on real numbers.

declare
   package Float_Convs is new Float_Conversions (Num => Float);
   --  Introduce conversions to and from values of type Float

   subtype Small_Float is Float range -100.0 .. 100.0;

   function Init return Small_Float with Import;
   --  Unknown initial value of the computation

   X : constant Small_Float := Init;
   Y : Float := X;
   M : Big_Real := Float_Convs.To_Big_Real (X) with Ghost;
   --  M is used to mimic the computations done on Y on real numbers

begin
   Y := Y * 100.0;
   M := M * Float_Convs.To_Big_Real (100.0);
   Y := Y + 100.0;
   M := M + Float_Convs.To_Big_Real (100.0);

   pragma Assert
     (In_Range (Float_Convs.To_Big_Real (Y) - M,
                Low  => Float_Convs.To_Big_Real (- 0.001),
                High => Float_Convs.To_Big_Real (0.001)));
   --  The rounding errors introduced by the floating-point computations
   --  are not too big.
end;

5.11.3. Functional Containers Library

To model complex data structures, one often needs simpler, mathematical like containers. The mathematical containers provided in the SPARK library (see the SPARK Library) are unbounded and may contain indefinite elements. However, they are controlled and thus not usable in every context. So that these containers can be used safely, we have made them functional, that is, no primitives are provided which would allow modifying an existing container. Instead, their API features functions creating new containers from existing ones. As an example, functional containers provide no Insert procedure but rather a function Add which creates a new container with one more element than its parameter:

function Add (C : Container; E : Element_Type) return Container;

As a consequence, these containers are highly inefficient. Thus, they should in general be used in ghost code and annotations so that they can be removed from the final executable.

There are 5 functional containers, which are part of the SPARK library:

• SPARK.Containers.Functional.Infinite_Sequences

• SPARK.Containers.Functional.Maps

• SPARK.Containers.Functional.Multisets

• SPARK.Containers.Functional.Sets

• SPARK.Containers.Functional.Vectors

Sequences defined in Functional.Vectors are no more than ordered collections of elements. In an Ada like manner, the user can choose the range used to index the elements:

function Length (S : Sequence) return Count_Type;
function Get (S : Sequence; N : Index_Type) return Element_Type;

The sequences defined in Functional.Infinite_Sequences behave as the one of Functional.Vectors. The difference between them lies in the fact that the inifinte one are indexed by mathematical integers.

function Length (Container : Sequence) return Big_Natural;
function Get (Container : Sequence; Position  : Big_Integer) return Element_Type;

Functional sets offer standard mathematical set functionalities such as inclusion, union, and intersection. They are neither ordered nor hashed:

function Contains (S : Set; E : Element_Type) return Boolean;
function "<=" (Left : Set; Right : Set) return Boolean;

Functional maps offer a dictionary between any two types of elements:

function Has_Key (M : Map; K : Key_Type) return Boolean;
function Get (M : Map; K : Key_Type) return Element_Type;

Multisets are mathematical sets associated with a number of occurrences:

function Nb_Occurence (S : Multiset; E : Element_Type) return Big_Natural;
function Cardinality (S : Multiset) return Big_Natural;

Each functional container type supports quantification over its elements (or keys for functional maps).

These containers can easily be used to model user defined data structures. They were used to this end to annotate and verify a package of allocators (see the allocators example provided with a SPARK installation). In this example, an allocator featuring a free list implemented in an array is modeled by a record containing a set of allocated resources and a sequence of available resources:

type Status is (Available, Allocated);
type Cell is record
   Stat : Status;
   Next : Resource;
end record;
type Allocator is array (Valid_Resource) of Cell;
type Model is record
   Available : Sequence;
   Allocated : Set;
end record;

Note

Instances of container packages, both functional and formal, are subject to particular constraints which are necessary for the contracts on the instance to be correct. For example, container primitives don’t comply with the ownership policy of SPARK if element or key types are ownership types. These constraints are verified specifically each time a container package is instantiated. For some of these checks, it is possible for the user to help the proof tool by providing some lemmas at instantiation. It is the case in particular for the constraints coming from the Ada reference manual on the container packages (that “=” is an equivalence relation, or that “<” is a strict weak order in particular). These lemmas appear in the library as additional ghost generic formal parameters.

Note

Functional sets, maps and multisets operate with a user-provided equivalence relation, which might be different from the logical equality. In this case, all elements or keys of an equivalence class are removed or included together in the container. This can sometimes have surprising results. For example, Contains can return True if an equivalent (but not equal) element has been added to a set. Similarly, the quantified expression for Some E of S => Cond (E) might be proved if Cond is False for all elements that were explicitely added to the set, but True for an object equivalent to such an element.

The functional sets, maps, sequences, and vectors have child packages providing higher order functions:

• SPARK.Containers.Functional.Infinite_Sequences.Higher_Order

• SPARK.Containers.Functional.Maps.Higher_Order

• SPARK.Containers.Functional.Sets.Higher_Order

• SPARK.Containers.Functional.Vectors.Higher_Order

These functions take as parameters access-to-functions that compute some information about an element of the container and apply it to all elements in a generic way. As an example, here is the function Count for functional sets. It counts the number of elements in the set with a given property. The property is provided by its input access-to-function parameter Test:

function Count
  (S    : Set;
   Test : not null access function (E : Element_Type) return Boolean)
   return Big_Natural
--  Count the number of elements on which the input Test function returns
--  True. Count can only be used with Test functions which return the same
--  value on equivalent elements.

with
  Global   => null,
  Annotate => (GNATprove, Higher_Order_Specialization),
  Pre      => Eq_Compatible (S, Test),
  Post     => Count'Result <= Length (S);

All the higher order functions are annotated with Higher_Order_Specialization (see Annotation for Handling Specially Higher Order Functions) so they can be used even with functions which read global data as parameters.

5.11.4. Formal Containers Library

Containers are generic data structures offering a high-level view of collections of objects, while guaranteeing fast access to their content to retrieve or modify it. The most common containers are lists, vectors, sets and maps, which are defined as generic units in the Ada Standard Library. In critical software where verification objectives severely restrict the use of pointers, containers offer an attractive alternative to pointer-intensive data structures.

The Ada Standard Library defines two kinds of containers:

• The controlled containers using dynamic allocation, for example SPARK.Containers.Formal.Unbounded_Sets. They define containers as controlled tagged types, so that memory for the container is automatic reallocated during assignment and automatically freed when the container object’s scope ends.

• The bounded containers not using dynamic allocation, for example SPARK.Containers.Formal.Vectors. They define containers as discriminated tagged types, so that the memory for the container can be reserved at initialization.

Although bounded containers are better suited to critical software development, neither controlled containers nor bounded containers can be used in SPARK, because their API does not lend itself to adding suitable contracts (in particular preconditions) ensuring correct usage in client code.

The formal containers are a variation of the standard containers with API changes that allow adding suitable contracts, so that GNATprove can prove that client code manipulates containers correctly. There are 12 formal containers, which are part of the SPARK library.

Among them, 6 are bounded and definite:

• SPARK.Containers.Formal.Vectors

• SPARK.Containers.Formal.Doubly_Linked_Lists

• SPARK.Containers.Formal.Hashed_Sets

• SPARK.Containers.Formal.Ordered_Sets

• SPARK.Containers.Formal.Hashed_Maps

• SPARK.Containers.Formal.Ordered_Maps

The 6 others are unbounded and indefinite but are controlled:

• SPARK.Containers.Formal.Unbounded_Vectors

• SPARK.Containers.Formal.Unbounded_Doubly_Linked_Lists

• SPARK.Containers.Formal.Unbounded_Hashed_Sets

• SPARK.Containers.Formal.Unbounded_Ordered_Sets

• SPARK.Containers.Formal.Unbounded_Hashed_Maps

• SPARK.Containers.Formal.Unbounded_Ordered_Maps

Bounded definite formal containers can only contain definite objects (objects for which the compiler can compute the size in memory, hence not String nor T'Class). They do not use dynamic allocation. In particular, they cannot grow beyond the bound defined at object creation.

Unbounded indefinite formal containers can contain indefinite objects. They use dynamic allocation both to allocate memory for their elements, and to expand their internal block of memory when it is full.

Note

The capacity of unbounded containers is not set using a discriminant. Instead, it is implicitly set to it maximum value. All the required memory is not reserved at declaration. As all the formal containers are internally indexed by Count_Type, their maximum size is Count_Type'Last.

5.11.4.1. Modified API of Formal Containers

The visible specification of formal containers is in SPARK, with suitable contracts on subprograms to ensure correct usage, while their private part and implementation is not in SPARK. Hence, GNATprove can be used to prove correct usage of formal containers in client code, but not to prove that formal containers implement their specification.

Procedures Update_Element or Query_Element that iterate over a container are not defined on formal containers. The effect of these procedures could not be precisely described in their contracts as there is no way to refer to the contract of their access-to-procedure parameter.

Procedures and functions that query the content of a container take the container in parameter. For example, function Has_Element that queries if a container has an element at a given position is declared as follows:

function Has_Element (Container : T; Position : Cursor) return Boolean;

This is different from the API of standard containers, where it is sufficient to pass a cursor to these subprograms, as the cursor holds a reference to the underlying container:

function Has_Element (Position : Cursor) return Boolean;

Cursors of formal containers do not hold a reference to a specific container, as this would otherwise introduce aliasing between container and cursor variables, which is not supported in SPARK. See Absence of Interferences. As a result, the same cursor can be applied to multiple container objects.

For each container type, the library provides model functions that are used to annotate subprograms from the API. The different models supply different levels of abstraction of the container’s functionalities. These model functions are grouped in Ghost Packages named Formal_Model.

The higher level view of a container is usually the mathematical structure of element it represents. We use a sequence for ordered containers such as lists and vectors and a mathematical map for imperative maps. This allows us to specify the effects of a subprogram in a very high level way, not having to consider cursors nor order of elements in a map:

procedure Increment_All (L : in out List) with
  Post =>
    (for all N in 1 .. Length (L) =>
       Element (Model (L), N) = Element (Model (L)'Old, N) + 1);

procedure Increment_All (S : in out Map) with
  Post =>
    (for all K of Model (S)'Old => Has_Key (Model (S), K))
       and
    (for all K of Model (S) =>
       Has_Key (Model (S)'Old, K)
         and Get (Model (S), K) = Get (Model (S)'Old, K) + 1);

For sets and maps, there is a lower level model representing the underlying order used for iteration in the container, as well as the actual values of elements/keys. It is a sequence of elements/keys. We can use it if we want to specify in Increment_All on maps that the order and actual values of keys are preserved:

procedure Increment_All (S : in out Map) with
  Post =>
    Keys (S) = Keys (S)'Old
      and
    (for all K of Model (S) =>
       Get (Model (S), K) = Get (Model (S)'Old, K) + 1);

Finally, cursors are modeled using a functional map linking them to their position in the container. For example, we can state that the positions of cursors in a list are not modified by a call to Increment_All:

procedure Increment_All (L : in out List) with
  Post =>
    Positions (L) = Positions (L)'Old
      and
    (for all N in 1 .. Length (L) =>
       Element (Model (L), N) = Element (Model (L)'Old, N) + 1);

Switching between the different levels of model functions allows to express precise considerations when needed without polluting upper level specifications. For example, consider a variant of the List.Find function defined in the API of formal containers, which returns a cursor holding the value searched if there is one, and the special cursor No_Element otherwise:

with Element_Lists; use Element_Lists; use Element_Lists.Lists;
with Ada.Containers; use Ada.Containers; use Element_Lists.Lists.Formal_Model;

function My_Find (L : List; E : Element_Type) return Cursor with
  SPARK_Mode,
  Contract_Cases =>
    (Contains (L, E)     => Has_Element (L, My_Find'Result) and then
                            Element (L, My_Find'Result) = E,
     not Contains (L, E) => My_Find'Result = No_Element);

The ghost functions mentioned above are specially useful in Loop Invariants to refer to cursors, and positions of elements in the containers. For example, here, ghost function Positions is used in the loop invariant to query the position of the current cursor in the list, and Model is used to specify that the value searched is not contained in the part of the container already traversed (otherwise the loop would have exited):

function My_Find (L : List; E : Element_Type) return Cursor with
  SPARK_Mode
is
   Cu : Cursor := First (L);

begin
   while Has_Element (L, Cu) loop
      pragma Loop_Variant (Increases => P.Get (Positions (L), Cu));
      pragma Loop_Invariant (for all I in 1 .. P.Get (Positions (L), Cu) - 1 =>
                                Element (Model (L), I) /= E);

      if Element (L, Cu) = E then
         return Cu;
      end if;

      Next (L, Cu);
   end loop;

   return No_Element;
end My_Find;

GNATprove proves that function My_Find implements its specification:

my_find.adb:8:28: info: loop variant proved
my_find.adb:8:42: info: precondition proved
my_find.adb:9:30: info: loop invariant initialization proved
my_find.adb:9:30: info: loop invariant preservation proved
my_find.adb:9:49: info: precondition proved
my_find.adb:10:33: info: precondition proved
my_find.adb:12:10: info: precondition proved
my_find.adb:16:07: info: precondition proved
my_find.ads:4:10: info: implicit aspect Always_Terminates on "My_Find" has been proved, subprogram will terminate
my_find.ads:6:03: info: disjoint contract cases proved
my_find.ads:6:03: info: complete contract cases proved
my_find.ads:7:26: info: contract case proved
my_find.ads:8:29: info: precondition proved
my_find.ads:9:26: info: contract case proved

Note

Just like functional containers, the formal containers do not comply with the ownership policy of SPARK if element or key types are ownership types. These constraints are verified specifically each time a container package is instantiated.

5.11.4.2. Quantification over Formal Containers

Quantified Expressions can be used over the content of a formal container to express that a property holds for all elements of a container (using for all) or that a property holds for at least one element of a container (using for some).

For example, we can express that all elements of a formal list of integers are prime as follows:

(for all Cu in My_List => Is_Prime (Element (My_List, Cu)))

On this expression, the GNAT compiler generates code that iterates over My_List using the functions First, Has_Element and Next given in the Iterable aspect applying to the type of formal lists, so the quantified expression above is equivalent to:

declare
   Cu     : Cursor_Type := First (My_List);
   Result : Boolean := True;
begin
   while Result and then Has_Element (My_List, Cu) loop
      Result := Is_Prime (Element (My_List, Cu));
      Cu     := Next (My_List, Cu);
   end loop;
end;

where Result is the value of the quantified expression. See GNAT Reference Manual for details on aspect Iterable.

5.11.5. Containers and Executablity

Some features of the container library (both functional and formal) might not have executable semantics. To ensure that user code never attempts to execute them, these subprograms call the Check_Or_Fail procedure declared in SPARK.Containers. This procedure is marked as Import, so an error will occur at link time if these features are used in normal code (or in enabled ghost code or assertions).

These non-executable features include quantified expressions over functional maps, sets, and multisets and logical equality in nearly all functional and formal containers.

Note

When instantiating containers from SPARKlib in your code, and compiling with assertions enabled (e.g. because you passed the switch -gnata), you may get the following error, even if you don’t use the non-executable features mentioned above:

undefined reference to `check_or_fail'

This spurious error comes from your executable including unused functions calling in the undefined procedure Check_Or_Fail. To get rid of this error, you should compile your code with the switch -ffunction-sections and link it with the switch -Wl,--gc-sections, so that unused functions are not included in the executable.

5.11.5.1. Quantified Expressions over Functional Maps, Sets, and Multisets

Functional maps, sets, and multisets take as parameters an equivalence relation. Inclusion in the container works modulo equivalence: when Add or Remove is called, the whole equivalence class is included or excluded at once. As equivalence classes might be infinite, quantified expressions over elements of a set or multiset or keys of a map could fail to terminate.

To replace quantified expressions over a functional map, set, or multisets occurring in the code, it is possible to use a loop over the Iterable_Map, Iterable_Set, or Iterable_Multiset types instead. They use the Choose function to get an unspecified element of the container from a different equivalence class at each iteration of the loop. As an example:

B := (for some E of S => P (E));

can be replaced by:

B := False;
for C in Iterate (S) loop
   pragma Loop_Invariant (for all E of S => (if not Contains (C, E) then not P (E)));
   if P (Choose (C)) then
      B := True;
      exit;
   end if;
end loop;

5.11.5.2. Logical Equality

The specifications of most formal and functional containers use logical equality to specify that all properties of elements placed in the container are preserved (see Annotation for Accessing the Logical Equality for a Type). Using logical equality to express such properties increases provability of user code, as it is optimally precise and natively handled by the automated solvers at the background of SPARK. However, logical equality does not always correspond to Ada equality, and there are even types for which it is not possible to write a valid logical equality in Ada, due to how things are encoded in the backend of the tool. As a result, logical equality functions used in the specification of formal and functional containers are not executable.

For formal containers, as the logical equality is given as a parameter to the functional containers used as models, the model themselves are not executable. As a result, it is not possible to execute ghost code or assertions that mention these model functions.

5.11.6. SPARK Lemma Library

As part of the SPARK library (see SPARK Library), packages declaring a set of ghost null procedures with contracts (called lemmas) are distributed. Here is an example of such a lemma:

procedure Lemma_Div_Is_Monotonic
  (Val1  : Int;
   Val2  : Int;
   Denom : Pos)
with
  Global => null,
  Pre  => Val1 <= Val2,
  Post => Val1 / Denom <= Val2 / Denom;

whose body is simply a null procedure:

procedure Lemma_Div_Is_Monotonic
  (Val1  : Int;
   Val2  : Int;
   Denom : Pos)
is null;

This procedure is ghost (as part of a ghost package), which means that the procedure body and all calls to the procedure are compiled away when producing the final executable without assertions (when switch -gnata is not set). On the contrary, when compiling with assertions for testing (when switch -gnata is set) the precondition of the procedure is executed, possibly detecting invalid uses of the lemma. However, the main purpose of such a lemma is to facilitate automatic proof, by providing the prover specific properties expressed in the postcondition. In the case of Lemma_Div_Is_Monotonic, the postcondition expresses an inequality between two expressions. You may use this lemma in your program by calling it on specific expressions, for example:

R1 := X1 / Y;
R2 := X2 / Y;
Lemma_Div_Is_Monotonic (X1, X2, Y);
--  at this program point, the prover knows that R1 <= R2
--  the following assertion is proved automatically:
pragma Assert (R1 <= R2);

Note that the lemma may have a precondition, stating in which contexts the lemma holds, which you will need to prove when calling it. For example, a precondition check is generated in the code above to show that X1 <= X2. Similarly, the types of parameters in the lemma may restrict the contexts in which the lemma holds. For example, the type Pos for parameter Denom of Lemma_Div_Is_Monotonic is the type of positive integers. Hence, a range check may be generated in the code above to show that Y is positive.

All the lemmas provided in the SPARK lemma library have been proved either automatically or using Coq interactive prover. The Why3 session file recording all proofs, as well as the individual Coq proof scripts, are available as part of the SPARK product under directory <spark-install>/lib/gnat/proof. For example, the proof of lemma Lemma_Div_Is_Monotonic is a Coq proof of the mathematical property (in Coq syntax):

Currenly, the SPARK lemma library provides the following lemmas:

• Lemmas on signed integer arithmetic in file spark-lemmas-arithmetic.ads, that are instantiated for 32 bits signed integers (Integer) in file spark-lemmas-integer_arithmetic.ads and for 64 bits signed integers (Long_Integer) in file spark-lemmas-long_integer_arithmetic.ads.

• Lemmas on modular integer arithmetic in file spark-lemmas-mod_arithmetic.ads, that are instantiated for 32 bits modular integers (Interfaces.Unsigned_32) in file spark-lemmas-mod32_arithmetic.ads and for 64 bits modular integers (Interfaces.Unsigned_64) in file spark-lemmas-mod64_arithmetic.ads.

• GNAT-specific lemmas on fixed-point arithmetic in file spark-lemmas-fixed_point_arithmetic.ads, that need to be instantiated by the user for her specific fixed-point type.

• Lemmas on floating point arithmetic in file spark-lemmas-floating_point_arithmetic.ads, that are instantiated for single-precision floats (Float) in file spark-lemmas-float_arithmetic.ads and for double-precision floats (Long_Float) in file spark-lemmas-long_float_arithmetic.ads.

• Lemmas on unconstrained arrays in file spark-lemmas-unconstrained_array.ads, that need to be instantiated by the user for her specific type of index and element, and specific ordering function between elements.

To apply lemmas to signed or modular integers of different types than the ones used in the instances provided in the library, just convert the expressions passed in arguments, as follows:

R1 := X1 / Y;
R2 := X2 / Y;
Lemma_Div_Is_Monotonic (Integer(X1), Integer(X2), Integer(Y));
--  at this program point, the prover knows that R1 <= R2
--  the following assertion is proved automatically:
pragma Assert (R1 <= R2);

5.11.7. Higher Order Function Library

The SPARK product also includes a library of higher order functions for unconstrained arrays. It is available using the SPARK library (see SPARK Library).

This library consists in a set of generic entities defining usual operations on arrays. As an example, here is a generic function for the map higher-level function on arrays. It applies a given function F to each element of an array, returning an array of results in the same order.

generic
   type Index_Type is range <>;
   type Element_In is private;
   type Array_In is array (Index_Type range <>) of Element_In;

   type Element_Out is private;
   type Array_Out is array (Index_Type range <>) of Element_Out;

   with function Init_Prop (A : Element_In) return Boolean;
   --  Potential additional constraint on values of the array to allow Map

   with function F (X : Element_In) return Element_Out;
   --  Function that should be applied to elements of Array_In

function Map (A : Array_In) return Array_Out with
  Pre  => (for all I in A'Range => Init_Prop (A (I))),
  Post => Map'Result'First = A'First
    and then Map'Result'Last = A'Last
    and then (for all I in A'Range =>
                Map'Result (I) = F (A (I)));

This function can be instantiated by providing two unconstrained array types ranging over the same index type and a function F mapping a component of the first array type to a component of the second array type. Additionally, a constraint Init_Prop can be supplied for the components of the first array to be allowed to apply F. If no such constraint is needed, Init_Prop can be instantiated with an always True function.

type Nat_Array is array (Positive range <>) of Natural;

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

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

function Increment_All is new SPARK.Higher_Order.Map
  (Index_Type  => Positive,
   Element_In  => Natural,
   Array_In    => Nat_Array,
   Element_Out => Natural,
   Array_Out   => Nat_Array,
   Init_Prop   => Small_Enough,
   F           => Increment_One);

The Increment_All function above will take as an argument an array of natural numbers small enough to be incremented and will return an array containing the result of incrementing each number by one:

function Increment_All (A : Nat_Array) return Nat_Array with
  Pre  => (for all I in A'Range => Small_Enough (A (I))),
  Post => Increment_All'Result'First = A'First
  and then Increment_All'Result'Last = A'Last
  and then (for all I in A'Range =>
              Increment_All'Result (I) = Increment_One (A (I)));

Currently, the higher-order function library provides the following functions:

• Map functions over unconstrained one-dimensional arrays in file spark-higher_order.ads. These include both in place and functional map subprograms, with and without an additional position parameter.

• Fold functions over unconstrained one-dimensional and two-dimensional arrays in file spark-higher_order-fold.ads. Both left to right and right to left fold functions are available for one-dimensional arrays. For two-dimensional arrays, fold functions go on a line by line, left to right, top-to-bottom way. For ease of use, these functions have been instantiated for the most common cases. Sum and Sum_2 respectively compute the sum of all the elements of a one-dimensional or two-dimensional array, and Count and Count_2 the number of elements with a given Choose property.

Note

Unlike the SPARK Lemma Library, these generic functions are not verified once and for all as their correction depends on the functions provided at each instance. As a result, each instance should be verified by running the SPARK tools.

5.11.8. Input-Output Libraries

The following text is about Ada.Text_IO and its child packages, Ada.Text_IO.Integer_IO, Ada.Text_IO.Modular_IO, Ada.Text_IO.Float_IO, Ada.Text_IO.Fixed_IO, Ada.Text_IO.Decimal_IO and Ada.Text_IO.Enumeration_IO.

The effect of functions and procedures of input-output units is partially modelled. This means in particular:

• that SPARK functions cannot directly call procedures that do input-output. The solution is either to transform them into procedures, or to hide the effect from GNATprove (if not relevant for analysis) by wrapping the standard input-output procedures in procedures with an explicit Global => null and body with SPARK_Mode => Off.

  with Ada.Text_IO;
  
  function Foo return Integer is
  
     procedure Put_Line (Item : String) with
       Global => null;
  
     procedure Put_Line (Item : String) with
       SPARK_Mode => Off
     is
     begin
        Ada.Text_IO.Put_Line (Item);
     end Put_Line;
  
  begin
     Put_Line ("Hello, world!");
     return 0;
  end Foo;
  

• SPARK procedures that call input-output subprograms need to reflect these effects in their Global/Depends contract if they have one.

  with Ada.Text_IO;
  
  procedure Foo with
    Global => (Input  => Var,
               In_Out => Ada.Text_IO.File_System)
  is
  begin
     Ada.Text_IO.Put_Line (Var);
  end Foo;
  
  procedure Bar is
  begin
     Ada.Text_IO.Put_Line (Var);
  end Bar;
  

In the examples above, procedure Foo and Bar have the same body, but their declarations are different. Global contracts have to be complete or not present at all. In the case of Foo, it has an Input contract on Var and an In_Out contract on File_System, an abstract state from Ada.Text_IO. Without the latter contract, a high message would be raised when running GNATprove. Global contracts will be automatically generated for Bar by flow analysis if this is user code. Both declarations are accepted by SPARK.

5.11.8.1. State Abstraction and Global Contracts

The abstract state File_System is used to model the memory on the system and the file handles (Line_Length, Col, etc.). This is explained by the fact that almost every procedure in Text_IO that actually modifies attributes of the File_Type parameter has in File_Type as a parameter and not in out. This would be inconsistent with SPARK rules without the abstract state.

All functions and procedures are annoted with Global, and Pre, Post if necessary. The Global contracts are most of the time In_Out for File_System, even in Put or Get procedures that update the current column and/or line. Functions have an Input global contract. The only functions with Global => null are the functions Get and Put in the generic packages that have a similar behaviour as sprintf and sscanf.

5.11.8.2. Functions and Procedures Removed in SPARK

Some functions and procedures are removed from SPARK usage because they are not consistent with SPARK rules:

1. Aliasing

   The functions Current_Input, Current_Output, Current_Error, Standard_Input, Standard_Output and Standard_Error are turned off in SPARK_Mode because they create aliasing, by returning the corresponding file.

   Set_Input, Set_Output and Set_Error are turned off because they also create aliasing, by assigning a File_Type variable to Current_Input or the other two.

   It is still possible to use Set_Input and the 3 others to make the code clearer. This is doable by calling Set_Input in a different subprogram whose body has SPARK_Mode => Off. However, it is necessary to check that the file is open and the mode is correct, because there are no checks made on procedures that do not take a file as a parameter (i.e. implicit, so it will write to/read from the current output/input).

2. Get_Line function

   The function Get_Line is disabled in SPARK because it is a function with side effects. Even with the Volatile_Function attribute, it is not possible to model its action on the files and global variables in SPARK. The function is very convenient because it returns an unconstrained string, but a workaround is possible by constructing the string with a buffer:

with Ada.Text_IO;
with Ada.Strings.Unbounded; use Ada.Strings.Unbounded;

procedure Echo is
   Unb_Str : Unbounded_String := Null_Unbounded_String;
   Buffer  : String (1 .. 1024);
   Last    : Natural := 1024;
begin

   while Last = 1024 loop
      Ada.Text_IO.Get_Line (Buffer, Last);
      exit when Last > Natural'Last - Length (Unb_Str);
      Unb_Str := Unb_Str & Buffer (1 .. Last);
   end loop;

   declare
      Str : String := To_String (Unb_Str);
   begin
      Ada.Text_IO.Put_Line (Str);
   end;
end Echo;

5.11.8.3. Errors Handling

Status_Error (due to a file already open/not open) and Mode_Error are fully handled.

Except for Layout_Error, which is a special case of a partially handled error and explained in a few lines below, all other errors are not handled:

• Use_Error is related to the external environment.

• Name_Error would require to check availability on disk beforehand.

• End_Error is raised when a file terminator is read while running the procedure.

For an Out_File, it is possible to set a Line_Length and Page_Length. When writing in this file, the procedures will add Line markers and Page markers each Line_Length characters or Page_Length lines respectively. Layout_Error occurs when trying to set the current column or line to a value that is greater than Line_Length or Page_Length respectively. This error is handled when using Set_Col or Set_Line procedures.

However, this error is not handled when no Line_Length or Page_Length has been specified, e.g, if the lines are unbounded, it is possible to have a Col greater than Count'Last and therefore have a Layout_Error raised when calling Col.

Not only the handling is partial, but it is also impossible to prove preconditions when working with two files or more. Since Line_Length etc. attributes are stored in the File_System, it is not posible to prove that the Line_Length of File_2 has not been modified when running any procedure that do input-output on File_1.

Finally, Layout_Error may be raised when calling Put to display the value of a real number (floating-point or fixed-point) in a string output parameter, which is not reflected currently in the precondition of Put as no simple precondition can describe the required length in such a case.

5.11.9. Strings Libraries

The following text is about Ada.Strings.Maps, Ada.Strings.Fixed, Ada.Strings.Bounded and Ada.Strings.Unbounded.

Global contracts were added to non-pure packages, and pre/postconditions were added to all SPARK subprograms to partially model their effects. In particular:

• Effects of subprograms from Ada.Strings.Maps, as specified in the Ada RM (A.4.2), are fully modeled through pre- and postconditions.

• Effects of most subprograms from Ada.Strings.Fixed are fully modeled through pre- and postconditions. Preconditions protect from exceptions specified in the Ada RM (A.4.3). Some procedures are not annotated with sufficient preconditions and may raise Length_Error when called with inconsistent parameters. They are annotated with an exceptional contract.

  Under their respective preconditions, the implementation of subprograms from Ada.Strings.Fixed is proven with GNATprove to be free from run-time errors and to comply with their postcondition, except for procedure Move and those procedures based on Move: Delete, Head, Insert, Overwrite, Replace_Slice, Tail and Trim (but the corresponding functions are proved).

• Effects of subprograms from Ada.Strings.Bounded are fully modeled through pre- and postconditions. Preconditions protect from exceptions specified in the Ada RM (A.4.4).

  Under their respective preconditions, the implementation of subprograms from Ada.Strings.Bounded is proven with GNATprove to be free from run-time errors, and except for subprograms Insert, Overwrite and Replace_Slice, to comply with their postcondition.

• Effects of subprograms from Ada.Strings.Unbounded are partially modeled. Postconditions state properties on the Length of the strings only and not on their content. Preconditions protect from exceptions specified in the Ada RM (A.4.5).

• The procedure Free in Ada.Strings.Unbounded is not in SPARK as it could be wrongly called by the user on a pointer to the stack.

Inside these packages, Translation_Error (in Ada.Strings.Maps), Index_Error and Pattern_Error are fully handled.

Length_Error is fully handled in Ada.Strings.Bounded and Ada.Strings.Unbounded and in functions from Ada.Strings.Fixed.

However, in the procedure Move and the procedures based on it except for Delete and Trim (Head, Insert, Overwrite, Replace_Slice and Tail) from Ada.Strings.Fixed, Length_Error may be raised under certain conditions. This is related to the call to Move. Each call of these subprograms can be preceded with a pragma Assert to check that the actual parameters are consistent, when parameter Drop is set to Error and the Source is longer than Target.

--  From the Ada RM for Move: "The Move procedure copies characters from
--  Source to Target.
--
--  ...
--
--  If Source is longer than Target, then the effect is based on Drop.
--
--  ...
--
--  * If Drop=Error, then the effect depends on the value of the Justify
--    parameter and also on whether any characters in Source other than Pad
--    would fail to be copied:
--
--    * If Justify=Left, and if each of the rightmost
--      Source'Length-Target'Length characters in Source is Pad, then the
--      leftmost Target'Length characters of Source are copied to Target.
--
--    * If Justify=Right, and if each of the leftmost
--      Source'Length-Target'Length characters in Source is Pad, then the
--      rightmost Target'Length characters of Source are copied to Target.
--
--    * Otherwise, Length_Error is propagated.".
--
--  Here, Move will be called with Drop = Error, Justify = Left and
--  Pad = Space, so we add the following assertion before the call to Move.

pragma Assert
 (if Source'Length > Target'Length then
    (for all J in 1 .. Source'Length - Target'Length =>
       (Source (Source'Last - J + 1) = Space)));

Move (Source  => Source,
      Target  => Target,
      Drop    => Error,
      Justify => Left,
      Pad     => Space);

5.11.10. C Strings Interface

5.11.10.1. The Ada Standard Library

Interfaces.C.Strings is a library that provides an Ada interface to allocate, reference, update and free C strings.

The provided preconditions protect users from getting Dereference_Error and Update_Error. However, those preconditions do not protect against Storage_Error and in general against leaking memory allocated by New_String and New_Char_Array.

All subprograms are annotated with Global contracts. To model the effects of the subprograms on the allocated memory, an abstract state C_Memory is defined. Since chars_ptr is an access type that is hidden from SPARK (it is a private type and the private part of Interfaces.C.Strings has SPARK_Mode => Off), the user could create aliases that SPARK would not be able to see. Hence, we consider that calling Update on any chars_ptr modifies the allocated memory, C_Memory, so that the effects of potential aliases are modelled correctly.

Additionally, some subprograms are annotated with SPARK_Mode => Off:

• To_Chars_Ptr: This function creates an alias, thus it is not compatible with SPARK.

• Free: There is no way for SPARK to know whether or not it is safe to deallocate these pointers. They might not be allocated on the heap or there might be some aliases, which could lead to dangling pointers.

Finally, the two functions used to allocate memory to create chars_ptr objects are annotated with the Volatile_Function attribute. Indeed, calling those functions twice in a row with the same parameters would return different objects.

5.11.10.2. Precise C Strings Interface

The annotations on Interfaces.C.Strings do not allow for a precise handling of the content of strings as it cannot reason about potential aliases between strings. As an alternative, when precision is required, two wrappers over Interfaces.C.Strings are provided in the SPARK library (see SPARK Library).

In the package SPARK.C.Strings, C strings are handled as SPARK pointers. Absence of aliasing and memory safety is ensured through ownership (see Memory Ownership Policy). This handling allows for safe allocation and reclamation of memory as well as precise contracts on the content of strings. However, like for regular pointers, Storage_Error might occur due to memory shortage.

Like for Interfaces.C.Strings, the To_Chars_Ptr function is disallowed but Free is retained. It is left as an assumption for the user to make sure that it is only called on a pointer allocated through New_String or New_Char_Array and not on a value created in C code. Two additional conversion functions are provided to convert to and from C strings coming from Interfaces.C.Strings but they cannot be used safely in SPARK and are annotated with SPARK_Mode => Off.

As opposed to Interfaces.C.Strings, the content of each string is modeled separately instead of collapsed together in a single abstract state and all subprograms are annotated with postconditions modeling the content of the string. This allows for a precise handling of C strings in SPARK.

The package SPARK.C.Constant_Strings offers an alternative to SPARK.C.Strings which does not enforce ownership at the cost of immutability. Aliasing between C strings is allowed but the designated values cannot be modified. The New_String and New_Char_Array are annotated with SPARK_Mode => Off as they can cause memory leaks. Instead, a version of To_Chars_Ptr taking as input an access to a constant char_array is provided. The Update procedures are removed. It is left as an assumption for the user to make sure that values of this type are never mutated by the C code.

Here again, the content of each string is modeled separately and all functions are annotated with postconditions modeling the content of the string to allow for a precise handling of C strings in SPARK.

5.11.11. Addresses to Access Conversions

The run-time library System.Address_To_Access_Conversions enables the user to convert System.Address_Type values to general access-to-object types. The conversions are subject to the same rules as Unchecked_Conversion between such types (see Data Validity), that is:

• To_Pointer is allowed in SPARK and annotated with Global => null. On a call to this function, GNATprove will emit warnings to ensure that the designated data has no aliases and is initialized.

• To_Address is forbidden in SPARK because it does not handle addresses.

5.11.12. Cut Operations

The SPARK product also includes boolean cut operations that can be used to manually help the proof of complicated assertions. These operations are provided as functions in a library but are handled in a specific way by GNATprove. They can be found in the SPARK.Cut_Operations package which is available using the same project file as the SPARK Lemma Library.

This library provides two functions named By and So which are handled in the following way:

• If A and B are two boolean expressions, proving By (A, B) requires proving B, the premise, and then A assuming B, the side-condition. When By (A, B) is assumed on the other hand, GNATprove only assumes A. The proposition B is used for the proof of A, but is not visible afterward.

• If A and B are two boolean expressions, proving So (A, B) requires proving A, the premise, and then B assuming A, the side-condition. When So (A, B) is assumed both A and B are assumed to be true.

This allows to introduce intermediate assertions to help the proof of some part of an assertion by still controlling in a precise way what is added to the enclosing context. This is interesting when doing complex proofs when the size of the proof context (the amount of information known at a given program point) is an issue.

In the example below, By is used to add the intermediate property B (I) in the proof of C (I) from A (I):

with SPARK.Cut_Operations; use SPARK.Cut_Operations;

procedure Main with SPARK_Mode is
   function A (I : Integer) return Boolean with Import;
   function B (I : Integer) return Boolean with Import,
     Post => B'Result and then (if A (I) then C (I));
   function C (I : Integer) return Boolean with Import;
begin
   pragma Assume (for all I in 1 .. 100 => A (I));
   pragma Assert (for all I in 1 .. 100 => By (C (I), B (I)));
end;

To prove the assertion, GNATprove will attempt to verify both that B (I) is true for all I and that C (I) can be deduced from B (I). After the assertion, the call to B does not occur in the context anymore, it is as if the assertion (for all I in 1 .. 100 => C (I)) had been written directly. Remark that here, B is really a lemma. Its result does not matter in itself (it always returns True) but its postcondition gives additional information for the proof (see the SPARK Lemma Library for more information about lemmas).

As can be seen in the example above, By and So may not necessarily occur at top-level in an assertion. However, because of their specific treatment, they are only allowed in specific contexts. We define these supported contexts in a recursive way:

• As the expression of a pragma Assert or Assert_And_Cut;

• As an operand of a AND, OR, AND THEN, or OR ELSE operation which itself occurs in a supported context;

• As the THEN or ELSE branch of a IF expression which itself occurs in a supported context;

• As an alternative of a CASE expression which itself occurs in a supported context;

• As the condition of a quantified expression which itself occurs in a supported context;

• As a parameter to a call to either By or So which itself occurs in a supported context;

• As the body expression of a DECLARE expression which itself occurs in a supported context.