# 5.10. SPARK Libraries¶

## 5.10.1. Functional Containers Library¶

To model complex data structures, one often needs simpler, mathematical like
containers. The mathematical containers provided in the SPARK library are
unbounded and may contain indefinite elements. Furthermore, to be usable in
every context, they are neither controlled nor limited. 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. They are also memory consuming as the allocated memory is not reclaimed when the container is no longer referenced. Thus, they should in general be used in ghost code and annotations so that they can be removed from the final executable.

There are 3 functional containers, which are part of the GNAT standard library:

`Ada.Containers.Functional_Maps`

`Ada.Containers.Functional_Sets`

`Ada.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;
```

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;
```

Each functional container type supports iteration as appropriate, so that its elements can easily be quantified over.

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 allocators example in the Examples in the Toolset Distribution). 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

Functional sets and maps represent elements modulo equivalence. For proof, the range of quantification over their content includes all elements that are equivalent to elements included in the container. On the other hand, for execution, the iteration is only done on elements which have actually been included in the container. This difference may make interaction between test and proof tricky when the equivalence relation is not the equality.

## 5.10.2. 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
`Ada.Containers.Vectors`

. 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
`Ada.Containers.Bounded_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 bounded containers with API changes that allow adding suitable contracts, so that GNATprove can prove that client code manipulates containers correctly. There are 7 formal containers, which are part of the GNAT standard library:

`Ada.Containers.Formal_Vectors`

`Ada.Containers.Formal_Indefinite_Vectors`

`Ada.Containers.Formal_Doubly_Linked_Lists`

`Ada.Containers.Formal_Hashed_Sets`

`Ada.Containers.Formal_Ordered_Sets`

`Ada.Containers.Formal_Hashed_Maps`

`Ada.Containers.Formal_Ordered_Maps`

Lists, sets and maps can only be used with definite objects (objects for which
the compiler can compute the size in memory, hence not `String`

nor
`T'Class`

). Vectors come in two flavors for definite objects
(`Formal_Vectors`

) and indefinite objects (`Formal_Indefinite_Vectors`

).

Lists, sets, maps, and definite vectors are always bounded. Indefinite vectors
can be bounded or unbounded
depending on the value of the formal parameter `Bounded`

when instantiating
the generic unit. Bounded containers do not use dynamic allocation. Unbounded
vectors use dynamic allocation to expand their internal block of memory.

### 5.10.2.1. 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. Specification and analysis of
such procedures that take an access-to-procedure in parameter is beyond the
capabilities of SPARK and GNATprove. See Excluded Ada Features.

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 controlled containers and bounded 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:

1 2 3 4 5 6 7 8 9 | ```
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):

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 | ```
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_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:7:26: info: initialization of "Cu.Node" proved
my_find.adb:8:30: info: loop invariant initialization proved
my_find.adb:8:30: info: loop invariant preservation proved
my_find.adb:8:49: info: precondition proved
my_find.adb:8:70: info: initialization of "Cu.Node" proved
my_find.adb:9:33: info: precondition proved
my_find.adb:11:10: info: precondition proved
my_find.adb:11:22: info: initialization of "Cu.Node" proved
my_find.adb:12:17: info: initialization of "Cu.Node" proved
my_find.adb:15:07: info: precondition proved
my_find.adb:15:16: info: initialization of "Cu.Node" proved
my_find.ads:6:03: info: complete contract cases proved
my_find.ads:6:03: info: disjoint 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
```

### 5.10.2.2. 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`

.

## 5.10.3. SPARK Lemma Library¶

As part of the SPARK product, a library of lemmas is available through the
project file `<spark-install>/lib/gnat/spark_lemmas.gpr`

. To use this
library in a program, you need to add a corresponding dependency in your
project file, for example:

```
with "spark_lemmas";
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.

You also need to set the environment variable `SPARK_LEMMAS_OBJECT_DIR`

to
the absolute path of the object directory where you want compilation and
verification artefacts for the lemma library to be created. This should be an
absolute path (not a relative one) otherwise these artefacts will be created
inside you SPARK install.

Finally, if you instantiate in your code a generic from the lemma library, you
also need to pass `-gnateDSPARK_BODY_MODE=Off`

as a compilation switch for
these generic units.

This library consists in a set of ghost null procedures with contracts (called lemmas). 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-arithmetic_lemmas.ads`

, that are instantiated for 32 bits signed integers (`Integer`

) in file`spark-integer_arithmetic_lemmas.ads`

and for 64 bits signed integers (`Long_Integer`

) in file`spark-long_integer_arithmetic_lemmas.ads`

. - Lemmas on modular integer arithmetic in file
`spark-mod_arithmetic_lemmas.ads`

, that are instantiated for 32 bits modular integers (`Interfaces.Unsigned_32`

) in file`spark-mod32_arithmetic_lemmas.ads`

and for 64 bits modular integers (`Interfaces.Unsigned_64`

) in file`spark-mod64_arithmetic_lemmas.ads`

. - GNAT-specific lemmas on fixed-point arithmetic in file
`spark-fixed_point_arithmetic_lemmas.ads`

, that need to be instantiated by the user for her specific fixed-point type. - Lemmas on floating point arithmetic in file
`spark-floating_point_arithmetic_lemmas.ads`

, that are instantiated for single-precision floats (`Float`

) in file`spark-float_arithmetic_lemmas.ads`

and for double-precision floats (`Long_Float`

) in file`spark-long_float_arithmetic_lemmas.ads`

. - Lemmas on unconstrained arrays in file
`spark-unconstrained_array_lemmas.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);
```

## 5.10.4. Higher Order Function Library¶

The SPARK product also includes a library of higher order functions for unconstrained arrays. It is available using the same project file as the SPARK Lemma 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.