# 7.9.2. Loop Examples¶

The examples in this section contain loops, and thus require in general that users write suitable Loop Invariants. We start by explaining the need for a loop invariant, and we continue with a description of the most common patterns of loops and their loop invariant. We summarize each pattern in a table of the following form:

Loop Pattern

Loop Over Data Structure

Proof Objective

Establish property P.

Loop Behavior

Loops over the data structure and establishes P.

Loop Invariant

Property P is established for the part of the data structure looped over so far.

The examples in this section use the types defined in package `Loop_Types`

:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 | ```
with Ada.Containers.Formal_Doubly_Linked_Lists;
with Ada.Containers.Formal_Vectors;
package Loop_Types
with SPARK_Mode
is
subtype Index_T is Positive range 1 .. 1000;
subtype Opt_Index_T is Natural range 0 .. 1000;
subtype Component_T is Natural;
type Arr_T is array (Index_T) of Component_T;
package Vectors is new Ada.Containers.Formal_Vectors (Index_T, Component_T);
subtype Vec_T is Vectors.Vector;
package Lists is new Ada.Containers.Formal_Doubly_Linked_Lists (Component_T);
subtype List_T is Lists.List;
type List_Cell;
type List_Acc is access List_Cell;
type List_Cell is record
Value : Component_T;
Next : List_Acc;
end record;
function At_End
(L : access constant List_Cell) return access constant List_Cell
is (L)
with Ghost,
Annotate => (GNATprove, At_End_Borrow);
type Property is access function (X : Component_T) return Boolean;
function For_All_List
(L : access constant List_Cell;
P : not null Property) return Boolean
is
(L = null or else (P (L.Value) and then For_All_List (L.Next, P)))
with Annotate => (GNATprove, Terminating);
pragma Annotate (GNATprove, False_Positive, "is recursive",
"The recursive call occurs on a strictly smaller list");
pragma Annotate (GNATprove, False_Positive, "call via access-to-subprogram",
"We only call For_All_List on terminating functions");
type Relation is access function (X, Y : Component_T) return Boolean;
function For_All_List
(L1, L2 : access constant List_Cell;
P : not null Relation) return Boolean
is
((L1 = null) = (L2 = null)
and then
(if L1 /= null
then P (L1.Value, L2.Value)
and then For_All_List (L1.Next, L2.Next, P)))
with Annotate => (GNATprove, Terminating);
pragma Annotate (GNATprove, False_Positive, "is recursive",
"The recursive call occurs on a strictly smaller lists");
pragma Annotate (GNATprove, False_Positive, "call via access-to-subprogram",
"We only call For_All_List on terminating functions");
end Loop_Types;
``` |

As there is no built-in way to iterate over the elements of a recursive
data structure, the first function `For_All_List`

can be used to state that
all elements of a list have a given property. The second variant of
`For_All_List`

takes two lists and states that both lists have the same number
of elements and that the corresponding elements of both lists are related by the
given relation. The function `At_End`

is used to
refer to the value of a borrowed list or a local borrower at the end of the
borrow, see Referring to a Value at the End of a Local Borrow for more
explanations.

Note

We cannot currently prove the termination of `For_All_List`

for two
reasons. First, as it is a recursive function, we would need to provide
a Subprogram_Variant to prove that the call chain is bounded. Currently,
structural variants are not supported and we have not defined a notion of
length on lists. The second is that we have no way for now to state on
the access-to-subprogram type Property that all elements of this type
must terminate. Therefore, we justify these checks, see section on
Justifying Check Messages.

## 7.9.2.1. The Need for a Loop Invariant¶

Consider a simple procedure that increments its integer parameter `X`

a
number `N`

of times:

1 2 3 4 5 6 7 8 9 10 | ```
procedure Increment_Loop (X : in out Integer; N : Natural) with
SPARK_Mode,
Pre => X <= Integer'Last - N,
Post => X = X'Old + N
is
begin
for I in 1 .. N loop
X := X + 1;
end loop;
end Increment_Loop;
``` |

The precondition of `Increment_Loop`

ensures that there is no overflow when
incrementing `X`

in the loop, and its postcondition states that `X`

has
been incremented `N`

times. This contract is a generalization of the contract
given for a single increment in Increment. GNATprove does not manage
to prove either the absence of overflow or the postcondition of
`Increment_Loop`

:

```
increment_loop.adb:4:11: medium: postcondition might fail
4 | Post => X = X'Old + N
| ^~~~~~~~~~~~~
e.g. when N = 1
and X = 1
and X'Old = 1
possible fix: loop at line 7 should mention X in a loop invariant
7 | for I in 1 .. N loop
| ^ here
increment_loop.adb:8:14: medium: overflow check might fail, cannot prove upper bound for X + 1
8 | X := X + 1;
| ~~^~~
e.g. when X = Integer'Last
reason for check: result of addition must fit in a 32-bits machine integer
possible fix: loop at line 7 should mention X in a loop invariant
7 | for I in 1 .. N loop
| ^ here
```

As described in How to Write Loop Invariants, this is because variable
`X`

is modified in the loop, hence GNATprove knows nothing about it unless
it is stated in a loop invariant. If we add such a loop invariant, as suggested
by the possible explanation in the message issued by GNATprove, that
describes precisely the value of `X`

in each iteration of the loop:

1 2 3 4 5 6 7 8 9 10 11 | ```
procedure Increment_Loop_Inv (X : in out Integer; N : Natural) with
SPARK_Mode,
Pre => X <= Integer'Last - N,
Post => X = X'Old + N
is
begin
for I in 1 .. N loop
X := X + 1;
pragma Loop_Invariant (X = X'Loop_Entry + I);
end loop;
end Increment_Loop_Inv;
``` |

then GNATprove proves both the absence of overflow and the postcondition of
`Increment_Loop_Inv`

:

```
increment_loop_inv.adb:3:29: info: overflow check proved
increment_loop_inv.adb:4:11: info: postcondition proved
increment_loop_inv.adb:4:21: info: overflow check proved
increment_loop_inv.adb:8:14: info: overflow check proved
increment_loop_inv.adb:9:30: info: loop invariant preservation proved
increment_loop_inv.adb:9:30: info: loop invariant initialization proved
increment_loop_inv.adb:9:47: info: overflow check proved
```

Fortunately, many loops fall into some broad categories for which the loop invariant is known. In the following sections, we describe these common patterns of loops and their loop invariant, which involve in general iterating over the content of a collection (either an array, a container from the Formal Containers Library, or a pointer-based linked list).

## 7.9.2.2. Initialization Loops¶

This kind of loops iterates over a collection to initialize every element of the collection to a given value:

Loop Pattern

Separate Initialization of Each Element

Proof Objective

Every element of the collection has a specific value.

Loop Behavior

Loops over the collection and initializes every element of the collection.

Loop Invariant

Every element initialized so far has its specific value.

In the simplest case, every element is assigned the same value. For example, in
procedure `Init_Arr_Zero`

below, value zero is assigned to every element of
array `A`

:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 | ```
with Loop_Types; use Loop_Types;
procedure Init_Arr_Zero (A : out Arr_T) with
SPARK_Mode,
Post => (for all J in A'Range => A(J) = 0)
is
pragma Annotate (GNATprove, False_Positive, """A"" might not be initialized",
"Entire array is initialized element-by-element in a loop");
begin
for J in A'Range loop
A(J) := 0;
pragma Loop_Invariant (for all K in A'First .. J => A(K) = 0);
pragma Annotate (GNATprove, False_Positive, """A"" might not be initialized",
"Part of array up to index J is initialized at this point");
end loop;
end Init_Arr_Zero;
``` |

The loop invariant expresses that all elements up to the current loop index
`J`

have the value zero. With this loop invariant, GNATprove is able to
prove the postcondition of `Init_Arr_Zero`

, namely that all elements of the
array have value zero:

```
init_arr_zero.adb:5:11: info: postcondition proved
init_arr_zero.adb:12:30: info: loop invariant initialization proved
init_arr_zero.adb:12:30: info: loop invariant preservation proved
init_arr_zero.adb:12:61: info: index check proved
```

In the example above, pragma Annotate is used in `Init_Arr_Zero`

to justify
a message issued by flow analysis, about the possible read of uninitialized
value `A(K)`

in the loop invariant. Indeed, flow analysis is not currently
able to infer that all elements up to the loop index `J`

have been
initialized, hence it issues a message that `"A" might not be initialized`

.
For more details, see section on Justifying Check Messages.

To verify this loop completely, it is possible to annotate `A`

with the
Relaxed_Initialization aspect to use proof to verify its correct initialization
(see Aspect Relaxed_Initialization and Attribute Initialized for more details).
In this case, the loop invariant should be extended to state that the elements
of `A`

have been initialized by the loop up to the current index:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 | ```
with Loop_Types; use Loop_Types;
procedure Init_Arr_Zero (A : out Arr_T) with
SPARK_Mode,
Relaxed_Initialization => A,
Post => A'Initialized and then (for all J in A'Range => A(J) = 0)
is
begin
for J in A'Range loop
A(J) := 0;
pragma Loop_Invariant (for all K in A'First .. J => A(K)'Initialized);
pragma Loop_Invariant (for all K in A'First .. J => A(K) = 0);
end loop;
end Init_Arr_Zero;
``` |

Remark that the postcondition of `Init_Arr_Zero`

also needs to state that
`A`

is entirely initialized by the call.

Consider now a variant of the same initialization loop over a vector:

1 2 3 4 5 6 7 8 9 10 11 12 13 | ```
with Loop_Types; use Loop_Types; use Loop_Types.Vectors;
procedure Init_Vec_Zero (V : in out Vec_T) with
SPARK_Mode,
Post => (for all J in First_Index (V) .. Last_Index (V) => Element (V, J) = 0)
is
begin
for J in First_Index (V) .. Last_Index (V) loop
Replace_Element (V, J, 0);
pragma Loop_Invariant (Last_Index (V) = Last_Index (V)'Loop_Entry);
pragma Loop_Invariant (for all K in First_Index (V) .. J => Element (V, K) = 0);
end loop;
end Init_Vec_Zero;
``` |

Like before, the loop invariant expresses that all elements up to the current
loop index `J`

have the value zero. Another loop invariant is needed here to
express that the length of the vector does not change in the loop: as variable
`V`

is modified in the loop, GNATprove does not know its length stays the
same (for example, calling procedure `Append`

or `Delete_Last`

would change
this length) unless the user says so in the loop invariant. This is different
from arrays whose length cannot change. With this loop invariant, GNATprove
is able to prove the postcondition of `Init_Vec_Zero`

, namely that all
elements of the vector have value zero:

```
init_vec_zero.adb:5:11: info: postcondition proved
init_vec_zero.adb:5:62: info: precondition proved
init_vec_zero.adb:5:74: info: range check proved
init_vec_zero.adb:9:07: info: precondition proved
init_vec_zero.adb:9:27: info: range check proved
init_vec_zero.adb:10:30: info: loop invariant initialization proved
init_vec_zero.adb:10:30: info: loop invariant preservation proved
init_vec_zero.adb:11:30: info: loop invariant initialization proved
init_vec_zero.adb:11:30: info: loop invariant preservation proved
init_vec_zero.adb:11:67: info: precondition proved
init_vec_zero.adb:11:79: info: range check proved
```

Similarly, consider a variant of the same initialization loop over a list:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 | ```
with Loop_Types; use Loop_Types; use Loop_Types.Lists;
with Ada.Containers; use Ada.Containers; use Loop_Types.Lists.Formal_Model;
procedure Init_List_Zero (L : in out List_T) with
SPARK_Mode,
Post => (for all E of L => E = 0)
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) = 0);
Replace_Element (L, Cu, 0);
Next (L, Cu);
end loop;
end Init_List_Zero;
``` |

Contrary to arrays and vectors, lists are not indexed. Instead, a cursor can be
defined to iterate over the list. The loop invariant expresses that all
elements up to the current cursor `Cu`

have the value zero. To access the
element stored at a given position in a list, we use the function `Model`

which computes the mathematical sequence of the elements stored in the list.
The position of a cursor in this sequence is retrieved using the `Positions`

function. Contrary to the
case of vectors, no loop invariant is needed to express that the length of the
list does not change in the loop, because the postcondition remains provable
here even if the length of the list changes. With this loop invariant,
GNATprove is able to prove the postcondition of `Init_List_Zero`

, namely
that all elements of the list have value zero:

```
init_list_zero.adb:6:11: info: postcondition proved
init_list_zero.adb:6:12: info: precondition proved
init_list_zero.adb:11:30: info: loop invariant initialization proved
init_list_zero.adb:11:30: info: loop invariant preservation proved
init_list_zero.adb:11:49: info: precondition proved
init_list_zero.adb:12:32: info: precondition proved
init_list_zero.adb:13:07: info: precondition proved
init_list_zero.adb:14:07: info: precondition proved
```

The case of sets and maps is similar to the case of lists.

Note

The parameter of `Init_Vec_Zero`

and `Init_List_Zero`

is an in out
parameter. This is because some components of the vector/list parameter are
preserved by the initialization procedure (in particular the component
corresponding to its length). This is different from `Init_Arr_Zero`

which
takes an out parameter, as all components of the array are initialized by
the procedure (the bounds of an array are not modifiable, hence considered
separately from the parameter mode).

Consider now a variant of the same initialization loop over a pointer-based list:

1 2 3 4 5 6 7 8 9 10 11 | ```
with Loop_Types; use Loop_Types;
package P with
SPARK_Mode
is
function Is_Zero (X : Component_T) return Boolean is
(X = 0);
procedure Init_List_Zero (L : access List_Cell) with
Post => For_All_List (L, Is_Zero'Access);
end P;
``` |

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 | ```
with Loop_Types; use Loop_Types;
package body P with
SPARK_Mode
is
procedure Init_List_Zero (L : access List_Cell) is
B : access List_Cell := L;
begin
while B /= null loop
pragma Loop_Invariant
(if For_All_List (At_End (B), Is_Zero'Access)
then For_All_List (At_End (L), Is_Zero'Access));
B.Value := 0;
B := B.Next;
end loop;
end Init_List_Zero;
end P;
``` |

Like in the other variants, the postcondition of `Init_List_Zero`

states that
the elements of the list `L`

after the call are all `0`

. It uses the
`For_All_List`

function from `Loop_Types`

to quantify over all the elements
of the list.
The loop iterates over the list `L`

using a local borrower `B`

which is
a local variable which borrows the ownership of a part of a datastructure for
the duration of its scope, see Borrowing for more details.
The loop invariant uses the `At_End`

function to express properties about the
values of `L`

and `B`

at the end of the borrow. It states that the elements
of `L`

at the end of the borrow will all be `0`

if the elements of `B`

at
the end of the borrow are all `0`

. This is provable because we know while
verifying the invariant that the already traversed elements were all set to
`0`

and that they can no longer be changed during the scope of `B`

. With
this loop invariant, GNATprove is able to prove the postcondition of
`Init_List_Zero`

:

```
p.adb:11:13: info: loop invariant initialization proved
p.adb:11:13: info: loop invariant preservation proved
p.adb:11:49: info: null exclusion check proved
p.adb:12:51: info: null exclusion check proved
p.adb:13:11: info: pointer dereference check proved
p.adb:14:16: info: pointer dereference check proved
p.ads:10:14: info: postcondition proved
p.ads:10:38: info: null exclusion check proved
```

Consider now a case where the value assigned to each element is not the
same. For example, in procedure `Init_Arr_Index`

below, each element of array
`A`

is assigned the value of its index:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 | ```
with Loop_Types; use Loop_Types;
procedure Init_Arr_Index (A : out Arr_T) with
SPARK_Mode,
Post => (for all J in A'Range => A(J) = J)
is
pragma Annotate (GNATprove, False_Positive, """A"" might not be initialized",
"Entire array is initialized element-by-element in a loop");
begin
for J in A'Range loop
A(J) := J;
pragma Loop_Invariant (for all K in A'First .. J => A(K) = K);
pragma Annotate (GNATprove, False_Positive, """A"" might not be initialized",
"Part of array up to index J is initialized at this point");
end loop;
end Init_Arr_Index;
``` |

The loop invariant expresses that all elements up to the current loop index
`J`

have the value of their index. With this loop invariant, GNATprove is
able to prove the postcondition of `Init_Arr_Index`

, namely that all elements
of the array have the value of their index:

```
init_arr_index.adb:5:11: info: postcondition proved
init_arr_index.adb:12:30: info: loop invariant initialization proved
init_arr_index.adb:12:30: info: loop invariant preservation proved
init_arr_index.adb:12:61: info: index check proved
```

As for `Init_Arr_Zero`

above, it is possible to annotate `A`

with the
Relaxed_Initialization aspect to use proof to verify its correct initialization:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 | ```
with Loop_Types; use Loop_Types;
procedure Init_Arr_Index (A : out Arr_T) with
SPARK_Mode,
Relaxed_Initialization => A,
Post => A'Initialized and then (for all J in A'Range => A(J) = J)
is
begin
for J in A'Range loop
A(J) := J;
pragma Loop_Invariant (for all K in A'First .. J => A(K)'Initialized);
pragma Loop_Invariant (for all K in A'First .. J => A(K) = K);
end loop;
end Init_Arr_Index;
``` |

Everything is proved by GNATprove:

```
init_arr_index.adb:6:11: info: postcondition proved
init_arr_index.adb:6:59: info: initialization check proved
init_arr_index.adb:11:30: info: loop invariant initialization proved
init_arr_index.adb:11:30: info: loop invariant preservation proved
init_arr_index.adb:11:61: info: index check proved
init_arr_index.adb:12:30: info: loop invariant preservation proved
init_arr_index.adb:12:30: info: loop invariant initialization proved
init_arr_index.adb:12:59: info: initialization check proved
init_arr_index.adb:12:61: info: index check proved
```

Similarly, variants of `Init_Vec_Zero`

and `Init_List_Zero`

that assign a
different value to each element of the collection would be proved by
GNATprove.

## 7.9.2.3. Mapping Loops¶

This kind of loops iterates over a collection to map every element of the collection to a new value:

Loop Pattern

Separate Modification of Each Element

Proof Objective

Every element of the collection has an updated value.

Loop Behavior

Loops over the collection and updates every element of the collection.

Loop Invariant

Every element updated so far has its specific value.

In the simplest case, every element is assigned a new value based only on its
initial value. For example, in procedure `Map_Arr_Incr`

below, every element
of array `A`

is incremented by one:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 | ```
with Loop_Types; use Loop_Types;
procedure Map_Arr_Incr (A : in out Arr_T) with
SPARK_Mode,
Pre => (for all J in A'Range => A(J) /= Component_T'Last),
Post => (for all J in A'Range => A(J) = A'Old(J) + 1)
is
begin
for J in A'Range loop
A(J) := A(J) + 1;
pragma Loop_Invariant (for all K in A'First .. J => A(K) = A'Loop_Entry(K) + 1);
-- The following loop invariant is generated automatically by GNATprove:
-- pragma Loop_Invariant (for all K in J + 1 .. A'Last => A(K) = A'Loop_Entry(K));
end loop;
end Map_Arr_Incr;
``` |

The loop invariant expresses that all elements up to the current loop index
`J`

have been incremented (using Attribute Loop_Entry). With this loop
invariant, GNATprove is able to prove the postcondition of `Map_Arr_Incr`

,
namely that all elements of the array have been incremented:

```
map_arr_incr.adb:6:11: info: postcondition proved
map_arr_incr.adb:6:52: info: overflow check proved
map_arr_incr.adb:10:20: info: overflow check proved
map_arr_incr.adb:11:30: info: loop invariant initialization proved
map_arr_incr.adb:11:30: info: loop invariant preservation proved
map_arr_incr.adb:11:61: info: index check proved
map_arr_incr.adb:11:79: info: index check proved
map_arr_incr.adb:11:82: info: overflow check proved
```

Note that the commented loop invariant expressing that other elements have not been modified is not needed, as it is an example of Automatically Generated Loop Invariants.

Consider now a variant of the same initialization loop over a vector:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 | ```
pragma Unevaluated_Use_Of_Old (Allow);
with Loop_Types; use Loop_Types; use Loop_Types.Vectors;
use Loop_Types.Vectors.Formal_Model;
procedure Map_Vec_Incr (V : in out Vec_T) with
SPARK_Mode,
Pre => (for all I in 1 .. Last_Index (V) =>
Element (V, I) /= Component_T'Last),
Post => Last_Index (V) = Last_Index (V)'Old
and then (for all I in 1 .. Last_Index (V) =>
Element (V, I) = Element (Model (V)'Old, I) + 1)
is
begin
for J in 1 .. Last_Index (V) loop
pragma Loop_Invariant (Last_Index (V) = Last_Index (V)'Loop_Entry);
pragma Loop_Invariant
(for all I in 1 .. J - 1 =>
Element (V, I) = Element (Model (V)'Loop_Entry, I) + 1);
pragma Loop_Invariant
(for all I in J .. Last_Index (V) =>
Element (V, I) = Element (Model (V)'Loop_Entry, I));
Replace_Element (V, J, Element (V, J) + 1);
end loop;
end Map_Vec_Incr;
``` |

Like before, we need an additionnal loop invariant to state that the length of
the vector is not modified by the loop. The other two invariants are direct
translations of those used for the loop over arrays: the first one expresses
that all elements up to the current loop index `J`

have been incremented, and
the second one expresses that other elements have not been modified.
Note that, as formal vectors are limited, we need to use the `Model`

function
of vectors to express the set of elements contained in the vector before the
loop (using attributes `Loop_Entry`

and `Old`

).
With this loop invariant, GNATprove is able to prove the postcondition of
`Map_Vec_Incr`

, namely that all elements of the vector have been incremented:

```
map_vec_incr.adb:8:16: info: precondition proved
map_vec_incr.adb:8:28: info: range check proved
map_vec_incr.adb:9:11: info: postcondition proved
map_vec_incr.adb:11:18: info: precondition proved
map_vec_incr.adb:11:30: info: range check proved
map_vec_incr.adb:11:35: info: precondition proved
map_vec_incr.adb:11:59: info: range check proved
map_vec_incr.adb:11:62: info: overflow check proved
map_vec_incr.adb:15:30: info: loop invariant initialization proved
map_vec_incr.adb:15:30: info: loop invariant preservation proved
map_vec_incr.adb:17:10: info: loop invariant initialization proved
map_vec_incr.adb:17:10: info: loop invariant preservation proved
map_vec_incr.adb:18:12: info: precondition proved
map_vec_incr.adb:18:24: info: range check proved
map_vec_incr.adb:18:29: info: precondition proved
map_vec_incr.adb:18:60: info: range check proved
map_vec_incr.adb:18:63: info: overflow check proved
map_vec_incr.adb:20:10: info: loop invariant preservation proved
map_vec_incr.adb:20:10: info: loop invariant initialization proved
map_vec_incr.adb:21:12: info: precondition proved
map_vec_incr.adb:21:24: info: range check proved
map_vec_incr.adb:21:29: info: precondition proved
map_vec_incr.adb:21:60: info: range check proved
map_vec_incr.adb:22:07: info: precondition proved
map_vec_incr.adb:22:27: info: range check proved
map_vec_incr.adb:22:30: info: precondition proved
map_vec_incr.adb:22:42: info: range check proved
map_vec_incr.adb:22:45: info: overflow check proved
```

Similarly, consider a variant of the same mapping loop over a list:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 | ```
with Loop_Types; use Loop_Types; use Loop_Types.Lists;
with Ada.Containers; use Ada.Containers; use Loop_Types.Lists.Formal_Model;
procedure Map_List_Incr (L : in out List_T) with
SPARK_Mode,
Pre => (for all E of L => E /= Component_T'Last),
Post => Length (L) = Length (L)'Old
and then (for all I in 1 .. Length (L) =>
Element (Model (L), I) = Element (Model (L'Old), I) + 1)
is
Cu : Cursor := First (L);
begin
while Has_Element (L, Cu) loop
pragma Loop_Invariant (Length (L) = Length (L)'Loop_Entry);
pragma Loop_Invariant
(for all I in 1 .. P.Get (Positions (L), Cu) - 1 =>
Element (Model (L), I) = Element (Model (L'Loop_Entry), I) + 1);
pragma Loop_Invariant
(for all I in P.Get (Positions (L), Cu) .. Length (L) =>
Element (Model (L), I) = Element (Model (L'Loop_Entry), I));
Replace_Element (L, Cu, Element (L, Cu) + 1);
Next (L, Cu);
end loop;
end Map_List_Incr;
``` |

Like before, we need to use a cursor to iterate over the list. The loop
invariants express that all elements up to the current loop index `J`

have
been incremented and that other elements have not been modified. Note that it is
necessary to state here that the length of the list is not modified during the
loop. It is because the length is used to bound the quantification over the
elements of the list both in the invariant and in the postcondition. With this
loop invariant, GNATprove is able to prove the postcondition of
`Map_List_Incr`

, namely that all elements of the list have been incremented:

```
map_list_incr.adb:6:12: info: precondition proved
map_list_incr.adb:7:11: info: postcondition proved
map_list_incr.adb:9:18: info: precondition proved
map_list_incr.adb:9:43: info: precondition proved
map_list_incr.adb:9:70: info: overflow check proved
map_list_incr.adb:14:30: info: loop invariant initialization proved
map_list_incr.adb:14:30: info: loop invariant preservation proved
map_list_incr.adb:16:10: info: loop invariant initialization proved
map_list_incr.adb:16:10: info: loop invariant preservation proved
map_list_incr.adb:16:29: info: precondition proved
map_list_incr.adb:17:12: info: precondition proved
map_list_incr.adb:17:37: info: precondition proved
map_list_incr.adb:17:71: info: overflow check proved
map_list_incr.adb:19:10: info: loop invariant preservation proved
map_list_incr.adb:19:10: info: loop invariant initialization proved
map_list_incr.adb:19:24: info: precondition proved
map_list_incr.adb:20:12: info: precondition proved
map_list_incr.adb:20:32: info: range check proved
map_list_incr.adb:20:37: info: precondition proved
map_list_incr.adb:20:68: info: range check proved
map_list_incr.adb:21:07: info: precondition proved
map_list_incr.adb:21:31: info: precondition proved
map_list_incr.adb:21:47: info: overflow check proved
map_list_incr.adb:22:07: info: precondition proved
```

Finally, consider a variant of the same mapping loop over a pointer-based list:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 | ```
with Loop_Types; use Loop_Types;
package P with
SPARK_Mode
is
function Small_Enough (X : Component_T) return Boolean is
(X /= Component_T'Last);
function Bigger_Than_First (X : Component_T) return Boolean is
(X /= Component_T'First);
procedure Map_List_Incr (L : access List_Cell) with
Pre => For_All_List (L, Small_Enough'Access),
Post => For_All_List (L, Bigger_Than_First'Access);
end P;
``` |

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 | ```
with Loop_Types; use Loop_Types;
package body P with
SPARK_Mode
is
procedure Map_List_Incr (L : access List_Cell) is
B : access List_Cell := L;
begin
while B /= null loop
pragma Loop_Invariant (For_All_List (B, Small_Enough'Access));
pragma Loop_Invariant
(if For_All_List (At_End (B), Bigger_Than_First'Access)
then For_All_List (At_End (L), Bigger_Than_First'Access));
B.Value := B.Value + 1;
B := B.Next;
end loop;
end Map_List_Incr;
end P;
``` |

Like in the other variants, the precondition of `Map_List_Incr`

states that
all elements of the input list `L`

are less than `Component_T'Last`

before
the call. It uses the `For_All_List`

function from `Loop_Types`

to
quantify over all the elements of the list. The postcondition is weaker than in
other variants of the loop. Indeed, referring to the value of a pointer-based
datastructure before the call is not allowed in the SPARK language.
Therefore we changed the postcondition to state instead that
all elements of the list are bigger than `Component_T'First`

after the call.

The loop iterates over the list `L`

using a local borrower `B`

which is
a local variable which borrows the ownership of a part of a datastructure for
the duration of its scope, see Borrowing for more details.
The loop invariant is made of two parts. The first one states that the initial
property still holds on the elements of `L`

accessible through `B`

. The
second uses the `At_End`

function to express properties about the values
of `L`

and `B`

at the end of the borrow. It states that the elements of
`L`

at the end of the borrow will have the `Bigger_Than_First`

property
if the elements of `B`

at the end of the borrow have this property. This is
provable because we know when verifying the invariant that the already traversed
elements currently have the `Bigger_Than_First`

property and that they can
no longer be changed during the scope of `B`

. With this
loop invariant, GNATprove is able to prove the postcondition of
`Map_List_Incr`

:

```
p.adb:10:33: info: loop invariant preservation proved
p.adb:10:33: info: loop invariant initialization proved
p.adb:10:62: info: null exclusion check proved
p.adb:12:13: info: loop invariant initialization proved
p.adb:12:13: info: loop invariant preservation proved
p.adb:12:59: info: null exclusion check proved
p.adb:13:61: info: null exclusion check proved
p.adb:14:11: info: pointer dereference check proved
p.adb:14:22: info: pointer dereference check proved
p.adb:14:29: info: overflow check proved
p.adb:15:16: info: pointer dereference check proved
p.ads:12:43: info: null exclusion check proved
p.ads:13:14: info: postcondition proved
p.ads:13:48: info: null exclusion check proved
```

If we want to retain the most precise postcondition relating the elements of
the structure before and after the loop, we need to introduce a way to store
the values of the list before the call in a separate data structure. In the
following example, it is done by declaring a `Copy`

function which returns
a copy of its input list. In its postcondition, we use the two-valued
`For_All_List`

function to state that the elements of the new structure
are equal to the elements of the input structure. An alternative could be
to store the elements in a structure not subjected to ownership like an array.

Note

The function `Copy`

is marked as `Import`

as it is not meant to be
executed. It could be implemented in SPARK by returning a deep copy of the
argument list, reallocating all cells of the list in the result.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 | ```
with Loop_Types; use Loop_Types;
package P with
SPARK_Mode
is
function Small_Enough (X : Component_T) return Boolean is
(X /= Component_T'Last);
function Equal (X, Y : Component_T) return Boolean is (X = Y);
function Is_Incr (X, Y : Component_T) return Boolean is
(X < Y and then Y = X + 1);
function Copy (L : access List_Cell) return List_Acc with
Ghost,
Import,
Post => For_All_List (L, Copy'Result, Equal'Access);
procedure Map_List_Incr (L : access List_Cell) with
Pre => For_All_List (L, Small_Enough'Access),
Post => For_All_List (Copy (L)'Old, L, Is_Incr'Access);
end P;
``` |

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 | ```
with Loop_Types; use Loop_Types;
package body P with
SPARK_Mode
is
procedure Map_List_Incr (L : access List_Cell) is
L_Old : constant List_Acc := Copy (L) with Ghost;
pragma Annotate (GNATprove, Intentional, "memory leak might occur",
"The code will be compiled with assertions disabled");
B : access List_Cell := L;
B_Old : access constant List_Cell := L_Old with Ghost;
begin
while B /= null loop
pragma Loop_Invariant (For_All_List (B, Small_Enough'Access));
pragma Loop_Invariant (For_All_List (B, B_Old, Equal'Access));
pragma Loop_Invariant
(if For_All_List (B_Old, At_End (B), Is_Incr'Access)
then For_All_List (L_Old, At_End (L), Is_Incr'Access));
B.Value := B.Value + 1;
B := B.Next;
B_Old := B_Old.Next;
end loop;
pragma Assert
(For_All_List (L_Old, At_End (L), Is_Incr'Access));
end Map_List_Incr;
end P;
``` |

The postcondition of `Map_List_Incr`

is similar to the postcondition of
`Copy`

. It uses the two-valued `For_All_List`

function to relate the
elements of `L`

before and after the call. Like in the previous variant, the
loop traverses `L`

using a local borrower `B`

. To be able to speak
about the initial value of `L`

in the invariant, we introduce a ghost
constant `L_Old`

storing a copy of this value. As we need to traverse both
lists at the same time, we declare a ghost variable `B_Old`

as a local
observer of `L_Old`

.

The loop invariant is made of three parts now. The first one is similar to the
one in the previous example. The third loop invariant is a direct adaptation of
the second loop invariant of the previous example. It states that if, at the end
of the borrow, the values accessible through `B`

are related to their
equivalent element in `B_Old`

through `Is_Incr`

, then so are all the
elements of `L`

. The loop invariant in the middle states that the elements
reachable through `B`

have not been modified by the loop. GNATprove can
verify these loop invariants as well as the postcondition of `Map_List_Incr`

:

```
p.adb:14:33: info: loop invariant preservation proved
p.adb:14:33: info: loop invariant initialization proved
p.adb:14:62: info: null exclusion check proved
p.adb:15:33: info: loop invariant initialization proved
p.adb:15:33: info: loop invariant preservation proved
p.adb:15:62: info: null exclusion check proved
p.adb:17:13: info: loop invariant preservation proved
p.adb:17:13: info: loop invariant initialization proved
p.adb:17:56: info: null exclusion check proved
p.adb:18:58: info: null exclusion check proved
p.adb:19:11: info: pointer dereference check proved
p.adb:19:22: info: pointer dereference check proved
p.adb:19:29: info: overflow check proved
p.adb:20:16: info: pointer dereference check proved
p.adb:21:24: info: pointer dereference check proved
p.adb:24:10: info: assertion proved
p.adb:24:50: info: null exclusion check proved
p.ads:12:28: info: overflow check proved
p.ads:20:43: info: null exclusion check proved
p.ads:21:14: info: postcondition proved
p.ads:21:28: medium: memory leak might occur
21 | Post => For_All_List (Copy (L)'Old, L, Is_Incr'Access);
| ^~~~~~~
possible explanation: call to allocating function inside an assertion leaks memory
p.ads:21:52: info: null exclusion check proved
```

Note

The second loop invariant does not subsume the first. Indeed, proving that,
if all elements of `L_Old`

are small enough, so are all elements of an
unknown observer `B_Old`

of `L_Old`

, is beyond the capacity of
GNATprove.

## 7.9.2.4. Validation Loops¶

This kind of loops iterates over a collection to validate that every element of the collection has a valid value. The most common pattern is to exit or return from the loop if an invalid value if encountered:

Loop Pattern

Sequence Validation with Early Exit

Proof Objective

Determine (flag) if there are any invalid elements in a given collection.

Loop Behavior

Loops over the collection and exits/returns if an invalid element is encountered.

Loop Invariant

Every element encountered so far is valid.

Consider a procedure `Validate_Arr_Zero`

that checks that all elements of an
array `A`

have value zero:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 | ```
with Loop_Types; use Loop_Types;
procedure Validate_Arr_Zero (A : Arr_T; Success : out Boolean) with
SPARK_Mode,
Post => Success = (for all J in A'Range => A(J) = 0)
is
begin
for J in A'Range loop
if A(J) /= 0 then
Success := False;
return;
end if;
pragma Loop_Invariant (for all K in A'First .. J => A(K) = 0);
end loop;
Success := True;
end Validate_Arr_Zero;
``` |

The loop invariant expresses that all elements up to the current loop index
`J`

have value zero. With this loop invariant, GNATprove is able to prove
the postcondition of `Validate_Arr_Zero`

, namely that output parameter
`Success`

is True if-and-only-if all elements of the array have value zero:

```
validate_arr_zero.adb:3:41: info: initialization of "Success" proved
validate_arr_zero.adb:5:11: info: postcondition proved
validate_arr_zero.adb:13:30: info: loop invariant initialization proved
validate_arr_zero.adb:13:30: info: loop invariant preservation proved
validate_arr_zero.adb:13:61: info: index check proved
```

Consider now a variant of the same validation loop over a vector:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 | ```
with Loop_Types; use Loop_Types; use Loop_Types.Vectors;
procedure Validate_Vec_Zero (V : Vec_T; Success : out Boolean) with
SPARK_Mode,
Post => Success = (for all J in First_Index (V) .. Last_Index (V) => Element (V, J) = 0)
is
begin
for J in First_Index (V) .. Last_Index (V) loop
if Element (V, J) /= 0 then
Success := False;
return;
end if;
pragma Loop_Invariant (for all K in First_Index (V) .. J => Element (V, K) = 0);
end loop;
Success := True;
end Validate_Vec_Zero;
``` |

Like before, the loop invariant expresses that all elements up to the current
loop index `J`

have the value zero. Since variable `V`

is not modified in
the loop, no additional loop invariant is needed here for GNATprove to know
that its length stays the same (this is different from the case of
`Init_Vec_Zero`

seen previously). With this loop invariant, GNATprove is
able to prove the postcondition of `Validate_Vec_Zero`

, namely that output
parameter `Success`

is True if-and-only-if all elements of the vector have
value zero:

```
validate_vec_zero.adb:3:41: info: initialization of "Success" proved
validate_vec_zero.adb:5:11: info: postcondition proved
validate_vec_zero.adb:5:72: info: precondition proved
validate_vec_zero.adb:5:84: info: range check proved
validate_vec_zero.adb:9:10: info: precondition proved
validate_vec_zero.adb:9:22: info: range check proved
validate_vec_zero.adb:13:30: info: loop invariant initialization proved
validate_vec_zero.adb:13:30: info: loop invariant preservation proved
validate_vec_zero.adb:13:67: info: precondition proved
validate_vec_zero.adb:13:79: info: range check proved
```

Similarly, consider a variant of the same validation loop over a list:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 | ```
with Loop_Types; use Loop_Types; use Loop_Types.Lists;
with Ada.Containers; use Ada.Containers; use Loop_Types.Lists.Formal_Model;
procedure Validate_List_Zero (L : List_T; Success : out Boolean) with
SPARK_Mode,
Post => Success = (for all E of L => E = 0)
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) = 0);
if Element (L, Cu) /= 0 then
Success := False;
return;
end if;
Next (L, Cu);
end loop;
Success := True;
end Validate_List_Zero;
``` |

Like in the case of `Init_List_Zero`

seen previously, we need to define a
cursor here to iterate over the list. The loop invariant expresses that all
elements up to the current cursor `Cu`

have the value zero. With this loop
invariant, GNATprove is able to prove the postcondition of
`Validate_List_Zero`

, namely that output parameter `Success`

is True
if-and-only-if all elements of the list have value zero:

```
validate_list_zero.adb:4:43: info: initialization of "Success" proved
validate_list_zero.adb:6:11: info: postcondition proved
validate_list_zero.adb:6:22: info: precondition proved
validate_list_zero.adb:11:30: info: loop invariant initialization proved
validate_list_zero.adb:11:30: info: loop invariant preservation proved
validate_list_zero.adb:11:49: info: precondition proved
validate_list_zero.adb:12:32: info: precondition proved
validate_list_zero.adb:13:10: info: precondition proved
validate_list_zero.adb:17:07: info: precondition proved
```

The case of sets and maps is similar to the case of lists.

Consider now a variant of the same validation loop over a pointer-based list:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 | ```
with Loop_Types; use Loop_Types;
package P with
SPARK_Mode
is
function Is_Zero (X : Component_T) return Boolean is
(X = 0);
procedure Validate_List_Zero
(L : access constant List_Cell;
Success : out Boolean)
with
Post => Success = For_All_List (L, Is_Zero'Access);
end P;
``` |

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 | ```
with Loop_Types; use Loop_Types;
package body P with
SPARK_Mode
is
procedure Validate_List_Zero
(L : access constant List_Cell;
Success : out Boolean)
is
C : access constant List_Cell := L;
begin
while C /= null loop
pragma Loop_Invariant
(For_All_List (L, Is_Zero'Access) = For_All_List (C, Is_Zero'Access));
if C.Value /= 0 then
Success := False;
return;
end if;
C := C.Next;
end loop;
Success := True;
end Validate_List_Zero;
end P;
``` |

The loop is implemented using a local observer (see Observing) which
borrows a read-only permission over a part of a datastructure until the end
of the scope of the observer. In the loop invariant, we cannot, like in
the other versions of the algorithm, speak about the value of the elements
which have already been traversed to say that they are all `0`

. Instead,
we state that the list `L`

only contains `0`

iff `C`

only contains `0`

.
This is true since the loop exits as soon as a non-zero value is encountered.
With this invariant, the postcondition can be proved by GNATprove:

```
p.adb:14:13: info: loop invariant initialization proved
p.adb:14:13: info: loop invariant preservation proved
p.adb:14:37: info: null exclusion check proved
p.adb:14:72: info: null exclusion check proved
p.adb:15:14: info: pointer dereference check proved
p.adb:19:16: info: pointer dereference check proved
p.ads:11:07: info: initialization of "Success" proved
p.ads:13:14: info: postcondition proved
p.ads:13:48: info: null exclusion check proved
```

A variant of the previous validation pattern is to continue validating elements even after an invalid value has been encountered, which allows for example logging all invalid values:

Loop Pattern

Sequence Validation that Validates Entire Collection

Proof Objective

Determine (flag) if there are any invalid elements in a given collection.

Loop Behavior

Loops over the collection. If an invalid element is encountered, flag this, but keep validating (typically logging every invalidity) for the entire collection.

Loop Invariant

If invalidity is not flagged, every element encountered so far is valid.

Consider a variant of `Validate_Arr_Zero`

that keeps validating elements of
the array after a non-zero element has been encountered:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 | ```
with Loop_Types; use Loop_Types;
procedure Validate_Full_Arr_Zero (A : Arr_T; Success : out Boolean) with
SPARK_Mode,
Post => Success = (for all J in A'Range => A(J) = 0)
is
begin
Success := True;
for J in A'Range loop
if A(J) /= 0 then
Success := False;
-- perform some logging here instead of returning
end if;
pragma Loop_Invariant (Success = (for all K in A'First .. J => A(K) = 0));
end loop;
end Validate_Full_Arr_Zero;
``` |

The loop invariant has been modified to state that all elements up to the
current loop index J have value zero if-and-only-if the output parameter
Success is True. This in turn requires to move the assignment of `Success`

before the loop. With this loop invariant, GNATprove is able to prove the
postcondition of `Validate_Full_Arr_Zero`

, which is the same as the
postcondition of `Validate_Arr_Zero`

, namely that output parameter
`Success`

is True if-and-only-if all elements of the array have value zero:

```
validate_full_arr_zero.adb:3:46: info: initialization of "Success" proved
validate_full_arr_zero.adb:5:11: info: postcondition proved
validate_full_arr_zero.adb:15:30: info: loop invariant initialization proved
validate_full_arr_zero.adb:15:30: info: loop invariant preservation proved
validate_full_arr_zero.adb:15:72: info: index check proved
```

Similarly, variants of `Validate_Vec_Zero`

and `Validate_List_Zero`

that
keep validating elements of the collection after a non-zero element has been
encountered would be proved by GNATprove.

## 7.9.2.5. Counting Loops¶

This kind of loops iterates over a collection to count the number of elements of the collection that satisfy a given criterion:

Loop Pattern

Count Elements Satisfying Criterion

Proof Objective

Count elements that satisfy a given criterion.

Loop Behavior

Loops over the collection. Increments a counter each time the value of an element satisfies the criterion.

Loop Invariant

The value of the counter is either 0 when no element encountered so far satisfies the criterion, or a positive number bounded by the current iteration of the loop otherwise.

Consider a procedure `Count_Arr_Zero`

that counts elements with value zero
in array `A`

:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 | ```
with Loop_Types; use Loop_Types;
procedure Count_Arr_Zero (A : Arr_T; Counter : out Natural) with
SPARK_Mode,
Post => (Counter in 0 .. A'Length) and then
((Counter = 0) = (for all K in A'Range => A(K) /= 0))
is
begin
Counter := 0;
for J in A'Range loop
if A(J) = 0 then
Counter := Counter + 1;
end if;
pragma Loop_Invariant (Counter in 0 .. J);
pragma Loop_Invariant ((Counter = 0) = (for all K in A'First .. J => A(K) /= 0));
end loop;
end Count_Arr_Zero;
``` |

The loop invariant expresses that the value of `Counter`

is a natural number
bounded by the current loop index `J`

, and that `Counter`

is equal to zero
exactly when all elements up to the current loop index have a non-zero value.
With this loop invariant, GNATprove is able to prove the postcondition of
`Count_Arr_Zero`

, namely that output parameter `Counter`

is a natural
number bounded by the length of the array `A`

, and that `Counter`

is equal
to zero exactly when all elements in `A`

have a non-zero value:

```
count_arr_zero.adb:3:38: info: initialization of "Counter" proved
count_arr_zero.adb:5:11: info: postcondition proved
count_arr_zero.adb:13:29: info: overflow check proved
count_arr_zero.adb:15:30: info: loop invariant initialization proved
count_arr_zero.adb:15:30: info: loop invariant preservation proved
count_arr_zero.adb:16:30: info: loop invariant preservation proved
count_arr_zero.adb:16:30: info: loop invariant initialization proved
count_arr_zero.adb:16:78: info: index check proved
```

Consider now a variant of the same counting loop over a vector:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 | ```
with Loop_Types; use Loop_Types; use Loop_Types.Vectors;
procedure Count_Vec_Zero (V : Vec_T; Counter : out Natural) with
SPARK_Mode,
Post => (Counter in 0 .. Natural (Length (V))) and then
((Counter = 0) = (for all K in First_Index (V) .. Last_Index (V) => Element (V, K) /= 0))
is
begin
Counter := 0;
for J in First_Index (V) .. Last_Index (V) loop
if Element (V, J) = 0 then
Counter := Counter + 1;
end if;
pragma Loop_Invariant (Counter in 0 .. J);
pragma Loop_Invariant ((Counter = 0) = (for all K in First_Index (V) .. J => Element (V, K) /= 0));
end loop;
end Count_Vec_Zero;
``` |

Like before, the loop invariant expresses that the value of `Counter`

is a
natural number bounded by the current loop index `J`

, and that `Counter`

is
equal to zero exactly when all elements up to the current loop index have a
non-zero value. With this loop invariant, GNATprove is able to prove the
postcondition of `Count_Vec_Zero`

, namely that output parameter `Counter`

is a natural number bounded by the length of the vector `V`

, and that
`Counter`

is equal to zero exactly when all elements in `V`

have a non-zero
value:

```
count_vec_zero.adb:3:38: info: initialization of "Counter" proved
count_vec_zero.adb:5:11: info: postcondition proved
count_vec_zero.adb:6:79: info: precondition proved
count_vec_zero.adb:6:91: info: range check proved
count_vec_zero.adb:12:10: info: precondition proved
count_vec_zero.adb:12:22: info: range check proved
count_vec_zero.adb:13:29: info: overflow check proved
count_vec_zero.adb:15:30: info: loop invariant initialization proved
count_vec_zero.adb:15:30: info: loop invariant preservation proved
count_vec_zero.adb:16:30: info: loop invariant initialization proved
count_vec_zero.adb:16:30: info: loop invariant preservation proved
count_vec_zero.adb:16:84: info: precondition proved
count_vec_zero.adb:16:96: info: range check proved
```

## 7.9.2.6. Search Loops¶

This kind of loops iterates over a collection to search an element of the collection that meets a given search criterion:

Loop Pattern

Search with Early Exit

Proof Objective

Find an element or position that meets a search criterion.

Loop Behavior

Loops over the collection. Exits when an element that meets the search criterion is found.

Loop Invariant

Every element encountered so far does not meet the search criterion.

Consider a procedure `Search_Arr_Zero`

that searches an element with value
zero in array `A`

:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 | ```
with Loop_Types; use Loop_Types;
procedure Search_Arr_Zero (A : Arr_T; Pos : out Opt_Index_T; Success : out Boolean) with
SPARK_Mode,
Post => Success = (for some J in A'Range => A(J) = 0) and then
(if Success then A (Pos) = 0)
is
begin
for J in A'Range loop
if A(J) = 0 then
Success := True;
Pos := J;
return;
end if;
pragma Loop_Invariant (for all K in A'First .. J => A(K) /= 0);
end loop;
Success := False;
Pos := 0;
end Search_Arr_Zero;
``` |

The loop invariant expresses that all elements up to the current loop index
`J`

have a non-zero value. With this loop invariant, GNATprove is able to
prove the postcondition of `Search_Arr_Zero`

, namely that output parameter
`Success`

is True if-and-only-if there is an element of the array that has
value zero, and that `Pos`

is the index of such an element:

```
search_arr_zero.adb:3:39: info: initialization of "Pos" proved
search_arr_zero.adb:3:62: info: initialization of "Success" proved
search_arr_zero.adb:5:11: info: postcondition proved
search_arr_zero.adb:6:31: info: index check proved
search_arr_zero.adb:15:30: info: loop invariant initialization proved
search_arr_zero.adb:15:30: info: loop invariant preservation proved
search_arr_zero.adb:15:61: info: index check proved
```

Consider now a variant of the same search loop over a vector:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 | ```
with Loop_Types; use Loop_Types; use Loop_Types.Vectors;
procedure Search_Vec_Zero (V : Vec_T; Pos : out Opt_Index_T; Success : out Boolean) with
SPARK_Mode,
Post => Success = (for some J in First_Index (V) .. Last_Index (V) => Element (V, J) = 0) and then
(if Success then Element (V, Pos) = 0)
is
begin
for J in First_Index (V) .. Last_Index (V) loop
if Element (V, J) = 0 then
Success := True;
Pos := J;
return;
end if;
pragma Loop_Invariant (for all K in First_Index (V) .. J => Element (V, K) /= 0);
end loop;
Success := False;
Pos := 0;
end Search_Vec_Zero;
``` |

Like before, the loop invariant expresses that all elements up to the current
loop index `J`

have a non-zero value. With this loop invariant, GNATprove
is able to prove the postcondition of `Search_Vec_Zero`

, namely that output
parameter `Success`

is True if-and-only-if there is an element of the vector
that has value zero, and that `Pos`

is the index of such an element:

```
search_vec_zero.adb:3:39: info: initialization of "Pos" proved
search_vec_zero.adb:3:62: info: initialization of "Success" proved
search_vec_zero.adb:5:11: info: postcondition proved
search_vec_zero.adb:5:73: info: precondition proved
search_vec_zero.adb:5:85: info: range check proved
search_vec_zero.adb:6:28: info: precondition proved
search_vec_zero.adb:6:40: info: range check proved
search_vec_zero.adb:10:10: info: precondition proved
search_vec_zero.adb:10:22: info: range check proved
search_vec_zero.adb:12:17: info: range check proved
search_vec_zero.adb:15:30: info: loop invariant initialization proved
search_vec_zero.adb:15:30: info: loop invariant preservation proved
search_vec_zero.adb:15:67: info: precondition proved
search_vec_zero.adb:15:79: info: range check proved
```

Similarly, consider a variant of the same search loop over a list:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 | ```
with Loop_Types; use Loop_Types; use Loop_Types.Lists;
with Ada.Containers; use Ada.Containers; use Loop_Types.Lists.Formal_Model;
procedure Search_List_Zero (L : List_T; Pos : out Cursor; Success : out Boolean) with
SPARK_Mode,
Post => Success = (for some E of L => E = 0) and then
(if Success then Element (L, Pos) = 0)
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) /= 0);
if Element (L, Cu) = 0 then
Success := True;
Pos := Cu;
return;
end if;
Next (L, Cu);
end loop;
Success := False;
Pos := No_Element;
end Search_List_Zero;
``` |

The loop invariant expresses that all elements up to the current cursor `Cu`

have a non-zero value. With this loop invariant, GNATprove is able to prove
the postcondition of `Search_List_Zero`

, namely that output parameter
`Success`

is True if-and-only-if there is an element of the list that has
value zero, and that `Pos`

is the cursor of such an element:

```
search_list_zero.adb:4:41: info: initialization of "Pos" proved
search_list_zero.adb:4:59: info: initialization of "Success" proved
search_list_zero.adb:6:11: info: postcondition proved
search_list_zero.adb:6:22: info: precondition proved
search_list_zero.adb:7:28: info: precondition proved
search_list_zero.adb:12:30: info: loop invariant initialization proved
search_list_zero.adb:12:30: info: loop invariant preservation proved
search_list_zero.adb:12:49: info: precondition proved
search_list_zero.adb:13:32: info: precondition proved
search_list_zero.adb:14:10: info: precondition proved
search_list_zero.adb:19:07: info: precondition proved
```

The case of sets and maps is similar to the case of lists.

Consider a variant of the same search loop over a pointer-based list:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 | ```
with Loop_Types; use Loop_Types;
package P with
SPARK_Mode
is
function Is_Non_Zero (X : Component_T) return Boolean is
(X /= 0);
function Search_List_Zero (L : access List_Cell) return access List_Cell with
Post =>
((Search_List_Zero'Result = null) = For_All_List (L, Is_Non_Zero'Access)
and then
(if Search_List_Zero'Result /= null then Search_List_Zero'Result.Value = 0));
end P;
``` |

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 | ```
with Loop_Types; use Loop_Types;
package body P with
SPARK_Mode
is
function Search_List_Zero (L : access List_Cell) return access List_Cell is
B : access List_Cell := L;
begin
while B /= null and then B.Value /= 0 loop
pragma Loop_Invariant
(For_All_List (L, Is_Non_Zero'Access) =
For_All_List (B, Is_Non_Zero'Access));
B := B.Next;
end loop;
return B;
end Search_List_Zero;
end P;
``` |

As our pointer-based lists do not support cursors, the result of the search
is a pointer inside the list which can be used to access or even update the
corresponding element. Storing such an object inside an OUT parameter would
break the ownership model of SPARK by creating an alias. Instead, we use a
traversal function (see Traversal Functions) to return this pointer
as a local borrower of the input list. Since we now have a function, we can
no longer have an explicit `Success`

flag to encode whether or not the value
was found. Instead, we simply return `null`

in case of failure.

The loop iterates over the input list `L`

using a local borrower `B`

. The
iteration stops when either `B`

is `null`

or `B.Value`

is zero. In the
loop invariant, we cannot speak directly about the elements of `L`

that have
been traversed to say that they are not `0`

. Instead, we write in the
invariant that `L`

contains only non-zero values iff `B`

contains only
non-zero values. Thanks to this loop invariant, GNATprove is able to verify
the postcondition of `Search_List_Zero`

:

```
p.adb:9:33: info: pointer dereference check proved
p.adb:11:13: info: loop invariant initialization proved
p.adb:11:13: info: loop invariant preservation proved
p.adb:11:41: info: null exclusion check proved
p.adb:12:45: info: null exclusion check proved
p.adb:13:16: info: pointer dereference check proved
p.adb:16:14: info: dynamic accessibility check proved
p.ads:11:08: info: postcondition proved
p.ads:11:72: info: null exclusion check proved
p.ads:13:77: info: pointer dereference check proved
```

For more complex examples of search loops, see the SPARK Tutorial as well as the section on How to Write Loop Invariants.

## 7.9.2.7. Maximize Loops¶

This kind of loops iterates over a collection to search an element of the collection that maximizes a given optimality criterion:

Loop Pattern

Search Optimum to Criterion

Proof Objective

Find an element or position that maximizes an optimality criterion.

Loop Behavior

Loops over the collection. Records maximum value of criterion so far and possibly index that maximizes this criterion.

Loop Invariant

Exactly one element encountered so far corresponds to the recorded maximum over other elements encountered so far.

Consider a procedure `Search_Arr_Max`

that searches an element maximum value
in array `A`

:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 | ```
with Loop_Types; use Loop_Types;
procedure Search_Arr_Max (A : Arr_T; Pos : out Index_T; Max : out Component_T) with
SPARK_Mode,
Post => (for all J in A'Range => A(J) <= Max) and then
(for some J in A'Range => A(J) = Max) and then
A(Pos) = Max
is
begin
Max := 0;
Pos := A'First;
for J in A'Range loop
if A(J) > Max then
Max := A(J);
Pos := J;
end if;
pragma Loop_Invariant (for all K in A'First .. J => A(K) <= Max);
pragma Loop_Invariant (for some K in A'First .. J => A(K) = Max);
pragma Loop_Invariant (A(Pos) = Max);
end loop;
end Search_Arr_Max;
``` |

The loop invariant expresses that all elements up to the current loop index
`J`

have a value less than `Max`

, and that `Max`

is the value of one of
these elements. The last loop invariant gives in fact this element, it is
`A(Pos)`

, but this part of the loop invariant may not be present if the
position `Pos`

for the optimum is not recorded. With this loop invariant,
GNATprove is able to prove the postcondition of `Search_Arr_Max`

, namely
that output parameter `Max`

is the maximum of the elements in the array, and
that `Pos`

is the index of such an element:

```
search_arr_max.adb:3:38: info: initialization of "Pos" proved
search_arr_max.adb:3:57: info: initialization of "Max" proved
search_arr_max.adb:5:11: info: postcondition proved
search_arr_max.adb:18:30: info: loop invariant initialization proved
search_arr_max.adb:18:30: info: loop invariant preservation proved
search_arr_max.adb:18:61: info: index check proved
search_arr_max.adb:19:30: info: loop invariant preservation proved
search_arr_max.adb:19:30: info: loop invariant initialization proved
search_arr_max.adb:19:62: info: index check proved
search_arr_max.adb:20:30: info: loop invariant initialization proved
search_arr_max.adb:20:30: info: loop invariant preservation proved
```

Consider now a variant of the same search loop over a vector:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 | ```
with Loop_Types; use Loop_Types; use Loop_Types.Vectors;
procedure Search_Vec_Max (V : Vec_T; Pos : out Index_T; Max : out Component_T) with
SPARK_Mode,
Pre => not Is_Empty (V),
Post => (for all J in First_Index (V) .. Last_Index (V) => Element (V, J) <= Max) and then
(for some J in First_Index (V) .. Last_Index (V) => Element (V, J) = Max) and then
Pos in First_Index (V) .. Last_Index (V) and then
Element (V, Pos) = Max
is
begin
Max := 0;
Pos := First_Index (V);
for J in First_Index (V) .. Last_Index (V) loop
if Element (V, J) > Max then
Max := Element (V, J);
Pos := J;
end if;
pragma Loop_Invariant (for all K in First_Index (V) .. J => Element (V, K) <= Max);
pragma Loop_Invariant (for some K in First_Index (V) .. J => Element (V, K) = Max);
pragma Loop_Invariant (Pos in First_Index (V) .. J);
pragma Loop_Invariant (Element (V, Pos) = Max);
end loop;
end Search_Vec_Max;
``` |

Like before, the loop invariant expresses that all elements up to the current
loop index `J`

have a value less than `Max`

, and that `Max`

is the value
of one of these elements, most precisely the value of `Element (V, Pos)`

if
the position `Pos`

for the optimum is recorded. An additional loop invariant
is needed here compared to the case of arrays to state that `Pos`

remains
within the bounds of the vector. With this loop invariant, GNATprove is able
to prove the postcondition of `Search_Vec_Max`

, namely that output parameter
`Max`

is the maximum of the elements in the vector, and that `Pos`

is the
index of such an element:

```
search_vec_max.adb:3:38: info: initialization of "Pos" proved
search_vec_max.adb:3:57: info: initialization of "Max" proved
search_vec_max.adb:6:11: info: postcondition proved
search_vec_max.adb:6:62: info: precondition proved
search_vec_max.adb:6:74: info: range check proved
search_vec_max.adb:7:63: info: precondition proved
search_vec_max.adb:7:75: info: range check proved
search_vec_max.adb:9:11: info: precondition proved
search_vec_max.adb:16:10: info: precondition proved
search_vec_max.adb:16:22: info: range check proved
search_vec_max.adb:17:17: info: precondition proved
search_vec_max.adb:17:29: info: range check proved
search_vec_max.adb:18:17: info: range check proved
search_vec_max.adb:20:30: info: loop invariant preservation proved
search_vec_max.adb:20:30: info: loop invariant initialization proved
search_vec_max.adb:20:67: info: precondition proved
search_vec_max.adb:20:79: info: range check proved
search_vec_max.adb:21:30: info: loop invariant initialization proved
search_vec_max.adb:21:30: info: loop invariant preservation proved
search_vec_max.adb:21:68: info: precondition proved
search_vec_max.adb:21:80: info: range check proved
search_vec_max.adb:22:30: info: loop invariant preservation proved
search_vec_max.adb:22:30: info: loop invariant initialization proved
search_vec_max.adb:23:30: info: precondition proved
search_vec_max.adb:23:30: info: loop invariant initialization proved
search_vec_max.adb:23:30: info: loop invariant preservation proved
```

Similarly, consider a variant of the same search loop over a list:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 | ```
with Loop_Types; use Loop_Types; use Loop_Types.Lists;
with Ada.Containers; use Ada.Containers; use Loop_Types.Lists.Formal_Model;
procedure Search_List_Max (L : List_T; Pos : out Cursor; Max : out Component_T) with
SPARK_Mode,
Pre => not Is_Empty (L),
Post => (for all E of L => E <= Max) and then
(for some E of L => E = Max) and then
Has_Element (L, Pos) and then
Element (L, Pos) = Max
is
Cu : Cursor := First (L);
begin
Max := 0;
Pos := Cu;
while Has_Element (L, Cu) loop
pragma Loop_Invariant (for all I in 1 .. P.Get (Positions (L), Cu) - 1 =>
Element (Model (L), I) <= Max);
pragma Loop_Invariant (Has_Element (L, Pos));
pragma Loop_Invariant (Max = 0 or else Element (L, Pos) = Max);
if Element (L, Cu) > Max then
Max := Element (L, Cu);
Pos := Cu;
end if;
Next (L, Cu);
end loop;
end Search_List_Max;
``` |

The loop invariant expresses that all elements up to the current cursor `Cu`

have a value less than `Max`

, and that `Max`

is the value of one of these
elements, most precisely the value of `Element (L, Pos)`

if the cursor
`Pos`

for the optimum is recorded. Like for vectors, an additional loop
invariant is needed here compared to the case of arrays to state that cursor
`Pos`

is a valid cursor of the list. A minor difference is that a loop
invariant now starts with `Max = 0 or else ..`

because the loop invariant is
stated at the start of the loop (for convenience with the use of
`First_To_Previous`

) which requires this modification. With this loop
invariant, GNATprove is able to prove the postcondition of
`Search_List_Max`

, namely that output parameter `Max`

is the maximum of the
elements in the list, and that `Pos`

is the cursor of such an element:

```
search_list_max.adb:4:40: info: initialization of "Pos" proved
search_list_max.adb:4:58: info: initialization of "Max" proved
search_list_max.adb:7:11: info: postcondition proved
search_list_max.adb:7:12: info: precondition proved
search_list_max.adb:8:12: info: precondition proved
search_list_max.adb:10:11: info: precondition proved
search_list_max.adb:18:30: info: loop invariant preservation proved
search_list_max.adb:18:30: info: loop invariant initialization proved
search_list_max.adb:18:49: info: precondition proved
search_list_max.adb:19:32: info: precondition proved
search_list_max.adb:20:30: info: loop invariant initialization proved
search_list_max.adb:20:30: info: loop invariant preservation proved
search_list_max.adb:21:30: info: loop invariant initialization proved
search_list_max.adb:21:30: info: loop invariant preservation proved
search_list_max.adb:21:46: info: precondition proved
search_list_max.adb:23:10: info: precondition proved
search_list_max.adb:24:17: info: precondition proved
search_list_max.adb:27:07: info: precondition proved
```

The case of sets and maps is similar to the case of lists.

Consider a variant of the same search loop over a pointer-based list:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 | ```
with Loop_Types; use Loop_Types;
package P with
SPARK_Mode
is
function All_Smaller_Than_Max
(L : access constant List_Cell; Max : Component_T) return Boolean
is (L = null or else
(L.Value <= Max and then All_Smaller_Than_Max (L.Next, Max)))
with Annotate => (GNATprove, Terminating);
pragma Annotate (GNATprove, False_Positive, "is recursive",
"The recursive call occurs on a strictly smaller list");
function Search_List_Max
(L : not null access List_Cell) return not null access List_Cell
with
Post => All_Smaller_Than_Max (L, Search_List_Max'Result.Value);
end P;
``` |

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 | ```
with Loop_Types; use Loop_Types;
package body P with
SPARK_Mode
is
function Search_List_Max
(L : not null access List_Cell) return not null access List_Cell
is
B : access List_Cell := L;
begin
loop
pragma Loop_Invariant (B /= null);
pragma Loop_Invariant
(for all M in B.Value .. Component_T'Last =>
(if All_Smaller_Than_Max (B, M)
then All_Smaller_Than_Max (L, M)));
declare
Prec : access List_Cell := B;
Max : constant Component_T := B.Value;
begin
while B /= null and then B.Value <= Max loop
pragma Loop_Invariant
(for all M in Max .. Component_T'Last =>
(if All_Smaller_Than_Max (B, M)
then All_Smaller_Than_Max (L, M)));
B := B.Next;
end loop;
if B = null then
return Prec;
end if;
end;
end loop;
end Search_List_Max;
end P;
``` |

As our pointer-based lists do not support cursors, the result of the search
is a pointer inside the list which can be used to access or even update the
corresponding element. Storing such an object inside an OUT parameter would
break the ownership model of SPARK by creating an alias. Instead, we use a
traversal function (see Traversal Functions) to return this pointer
as a local borrower of the input list. Since we now have a function, we can
no longer explicitely return the value of the maximum. It is not a problem,
as it can be accessed easily as the `Value`

component of the returned
pointer. In the postcondition of `Search_List_Max`

, we cannot use
`For_All_List`

to express that the returned pointer designates the maximum
value in the list. Indeed, the property depends on the value of this maximum.
Instead, we create a specific recursive function taking the maximum as an
additional parameter.

The iteration over the input list `L`

uses a local borrower `B`

. It is
expressed as two nested loops. The inner loop declares a local borrower `Prec`

to register the current value of the maximum. Then it iterates through the loop
using `B`

until a value bigger than the current maximum is found. The outer
loop repeats this step as many times as necessary. This split into two loops
is necessary as the SPARK language prevents borrowers from jumping into a
different part of the data structure. As `B`

is not syntactically a path
rooted at `Prec`

, `Prec`

cannot be assigned the current value of `B`

when
a new maximal value is found. We therefore need to create a new variable
to hold the current maximum each time it changes.

In the loop invariant of the outer loop, we cannot speak directly about the
elements of `L`

that have been traversed to say that they are smaller than
the current maximum. Instead, we write in the invariant that the all values of
`L`

are smaller than any given value bigger than the current maximum iff the
values of `B`

are. A similar invariant is necessary on the inner loop.
Thanks to these loop invariants, GNATprove is able to verify
the postcondition of `Search_List_Max`

:

```
p.adb:12:33: info: loop invariant initialization proved
p.adb:12:33: info: loop invariant preservation proved
p.adb:14:13: info: loop invariant preservation proved
p.adb:14:13: info: loop invariant initialization proved
p.adb:14:27: info: pointer dereference check proved
p.adb:15:44: info: range check proved
p.adb:16:46: info: range check proved
p.adb:19:45: info: pointer dereference check proved
p.adb:21:39: info: pointer dereference check proved
p.adb:23:19: info: loop invariant initialization proved
p.adb:23:19: info: loop invariant preservation proved
p.adb:24:50: info: range check proved
p.adb:25:52: info: range check proved
p.adb:26:22: info: pointer dereference check proved
p.adb:29:23: info: dynamic accessibility check proved
p.adb:29:23: info: null exclusion check proved
p.ads:9:12: info: pointer dereference check proved
p.ads:9:58: info: pointer dereference check proved
p.ads:17:14: info: postcondition proved
p.ads:17:61: info: pointer dereference check proved
```

For more complex examples of search loops, see the SPARK Tutorial as well as the section on How to Write Loop Invariants.

## 7.9.2.8. Update Loops¶

This kind of loops iterates over a collection to update individual elements based either on their value or on their position. The first pattern we consider is the one that updates elements based on their value:

Loop Pattern

Modification of Elements Based on Value

Proof Objective

Elements of the collection are updated based on their value.

Loop Behavior

Loops over a collection and assigns the elements whose value satisfies a given modification criterion.

Loop Invariant

Every element encountered so far has been assigned according to its value.

Consider a procedure `Update_Arr_Zero`

that sets to zero all elements in
array `A`

that have a value smaller than a given `Threshold`

:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 | ```
with Loop_Types; use Loop_Types;
procedure Update_Arr_Zero (A : in out Arr_T; Threshold : Component_T) with
SPARK_Mode,
Post => (for all J in A'Range => A(J) = (if A'Old(J) <= Threshold then 0 else A'Old(J)))
is
begin
for J in A'Range loop
if A(J) <= Threshold then
A(J) := 0;
end if;
pragma Loop_Invariant (for all K in A'First .. J => A(K) = (if A'Loop_Entry(K) <= Threshold then 0 else A'Loop_Entry(K)));
-- The following loop invariant is generated automatically by GNATprove:
-- pragma Loop_Invariant (for all K in J + 1 .. A'Last => A(K) = A'Loop_Entry(K));
end loop;
end Update_Arr_Zero;
``` |

The loop invariant expresses that all elements up to the current loop index
`J`

have been zeroed out if initially smaller than `Threshold`

(using
Attribute Loop_Entry). With this loop invariant, GNATprove is able to
prove the postcondition of `Update_Arr_Zero`

, namely that all elements
initially smaller than `Threshold`

have been zeroed out, and that other
elements have not been modified:

```
update_arr_zero.adb:5:11: info: postcondition proved
update_arr_zero.adb:12:30: info: loop invariant initialization proved
update_arr_zero.adb:12:30: info: loop invariant preservation proved
update_arr_zero.adb:12:61: info: index check proved
update_arr_zero.adb:12:83: info: index check proved
update_arr_zero.adb:12:124: info: index check proved
```

Note that the commented loop invariant expressing that other elements have not been modified is not needed, as it is an example of Automatically Generated Loop Invariants.

Consider now a variant of the same update loop over a vector:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 | ```
pragma Unevaluated_Use_Of_Old (Allow);
with Loop_Types; use Loop_Types; use Loop_Types.Vectors;
use Loop_Types.Vectors.Formal_Model;
procedure Update_Vec_Zero (V : in out Vec_T; Threshold : Component_T) with
SPARK_Mode,
Post => Last_Index (V) = Last_Index (V)'Old
and (for all I in 1 .. Last_Index (V) =>
Element (V, I) =
(if Element (Model (V)'Old, I) <= Threshold then 0
else Element (Model (V)'Old, I)))
is
begin
for J in First_Index (V) .. Last_Index (V) loop
pragma Loop_Invariant (Last_Index (V) = Last_Index (V)'Loop_Entry);
pragma Loop_Invariant
(for all I in 1 .. J - 1 =>
Element (V, I) =
(if Element (Model (V)'Loop_Entry, I) <= Threshold then 0
else Element (Model (V)'Loop_Entry, I)));
pragma Loop_Invariant
(for all I in J .. Last_Index (V) =>
Element (V, I) = Element (Model (V)'Loop_Entry, I));
if Element (V, J) <= Threshold then
Replace_Element (V, J, 0);
end if;
end loop;
end Update_Vec_Zero;
``` |

Like for `Map_Vec_Incr`

, we need to use the `Model`

function over arrays to
access elements of the vector before the loop as the vector type is limited. The
loop invariant expresses that all elements up to the current loop index `J`

have been zeroed out if initially smaller than `Threshold`

, that elements that
follow the current loop index have not been modified, and that the length of
`V`

is not modified (like in `Init_Vec_Zero`

). With this loop invariant,
GNATprove is able to prove the postcondition of `Update_Vec_Zero`

:

```
update_vec_zero.adb:7:11: info: postcondition proved
update_vec_zero.adb:9:13: info: precondition proved
update_vec_zero.adb:9:25: info: range check proved
update_vec_zero.adb:10:18: info: precondition proved
update_vec_zero.adb:10:42: info: range check proved
update_vec_zero.adb:11:20: info: precondition proved
update_vec_zero.adb:11:44: info: range check proved
update_vec_zero.adb:15:30: info: loop invariant initialization proved
update_vec_zero.adb:15:30: info: loop invariant preservation proved
update_vec_zero.adb:17:10: info: loop invariant initialization proved
update_vec_zero.adb:17:10: info: loop invariant preservation proved
update_vec_zero.adb:17:30: info: overflow check proved
update_vec_zero.adb:18:14: info: precondition proved
update_vec_zero.adb:18:26: info: range check proved
update_vec_zero.adb:19:19: info: precondition proved
update_vec_zero.adb:19:50: info: range check proved
update_vec_zero.adb:20:21: info: precondition proved
update_vec_zero.adb:20:52: info: range check proved
update_vec_zero.adb:22:10: info: loop invariant preservation proved
update_vec_zero.adb:22:10: info: loop invariant initialization proved
update_vec_zero.adb:23:14: info: precondition proved
update_vec_zero.adb:23:26: info: range check proved
update_vec_zero.adb:23:31: info: precondition proved
update_vec_zero.adb:23:62: info: range check proved
update_vec_zero.adb:24:10: info: precondition proved
update_vec_zero.adb:24:22: info: range check proved
update_vec_zero.adb:25:10: info: precondition proved
update_vec_zero.adb:25:30: info: range check proved
```

Similarly, consider a variant of the same update loop over a list:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 | ```
with Loop_Types; use Loop_Types; use Loop_Types.Lists;
with Ada.Containers; use Ada.Containers; use Loop_Types.Lists.Formal_Model;
procedure Update_List_Zero (L : in out List_T; Threshold : Component_T) with
SPARK_Mode,
Post => Length (L) = Length (L)'Old
and (for all I in 1 .. Length (L) =>
Element (Model (L), I) =
(if Element (Model (L'Old), I) <= Threshold then 0
else Element (Model (L'Old), I)))
is
Cu : Cursor := First (L);
begin
while Has_Element (L, Cu) loop
pragma Loop_Invariant (Length (L) = Length (L)'Loop_Entry);
pragma Loop_Invariant
(for all I in 1 .. P.Get (Positions (L), Cu) - 1 =>
Element (Model (L), I) =
(if Element (Model (L'Loop_Entry), I) <= Threshold then 0
else Element (Model (L'Loop_Entry), I)));
pragma Loop_Invariant
(for all I in P.Get (Positions (L), Cu) .. Length (L) =>
Element (Model (L), I) = Element (Model (L'Loop_Entry), I));
if Element (L, Cu) <= Threshold then
Replace_Element (L, Cu, 0);
end if;
Next (L, Cu);
end loop;
end Update_List_Zero;
``` |

The loop invariant expresses that all elements up to the current cursor `Cu`

have been zeroed out if initially smaller than `Threshold`

(using function
`Model`

to access the element stored at a given position in the list and
function `Positions`

to query the position of the current cursor), and that
elements that follow the current loop index have not been
modified. Note that it is
necessary to state here that the length of the list is not modified during the
loop. It is because the length is used to bound the quantification over the
elements of the list both in the invariant and in the postcondition.

With this loop invariant, GNATprove is able to prove the postcondition of
`Update_List_Zero`

, namely that all elements initially smaller than
`Threshold`

have been zeroed out, and that other elements have not been
modified:

```
update_list_zero.adb:6:11: info: postcondition proved
update_list_zero.adb:8:13: info: precondition proved
update_list_zero.adb:9:18: info: precondition proved
update_list_zero.adb:10:20: info: precondition proved
update_list_zero.adb:15:30: info: loop invariant initialization proved
update_list_zero.adb:15:30: info: loop invariant preservation proved
update_list_zero.adb:17:10: info: loop invariant preservation proved
update_list_zero.adb:17:10: info: loop invariant initialization proved
update_list_zero.adb:17:29: info: precondition proved
update_list_zero.adb:18:13: info: precondition proved
update_list_zero.adb:19:18: info: precondition proved
update_list_zero.adb:20:20: info: precondition proved
update_list_zero.adb:22:10: info: loop invariant preservation proved
update_list_zero.adb:22:10: info: loop invariant initialization proved
update_list_zero.adb:22:24: info: precondition proved
update_list_zero.adb:23:13: info: precondition proved
update_list_zero.adb:23:33: info: range check proved
update_list_zero.adb:23:38: info: precondition proved
update_list_zero.adb:23:69: info: range check proved
update_list_zero.adb:24:10: info: precondition proved
update_list_zero.adb:25:10: info: precondition proved
update_list_zero.adb:27:07: info: precondition proved
```

The case of sets and maps is similar to the case of lists.

Consider now a variant of the same update loop over a pointer-based list.
To express the postcondition relating the elements of the structure before and
after the loop, we need to introduce a way to store the values of the list
before the call in a separate data structure. Indeed, the `Old`

attribute
cannot be used on `L`

directly has it would introduce an alias. In this
example, it is done by declaring a `Copy`

function which returns
a copy of its input list. In its postcondition, we use the two-valued
`For_All_List`

function to state that the elements of the new structure
are equal to the elements of its input structure. An alternative could be
to store the elements in a structure not subjected to ownership like an array.

Note

The function `Copy`

is marked as `Import`

as it is not meant to be
executed. It could be implemented in SPARK by returning a deep copy of the
argument list, reallocating all cells of the list in the result.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 | ```
with Loop_Types; use Loop_Types;
package P with
SPARK_Mode
is
function Equal (X, Y : Component_T) return Boolean is (X = Y);
function Copy (L : access List_Cell) return List_Acc with
Ghost,
Import,
Post => For_All_List (L, Copy'Result, Equal'Access);
function Updated_If_Less_Than_Threshold
(L1, L2 : access constant List_Cell;
Threshold : Component_T) return Boolean
is
((L1 = null) = (L2 = null)
and then
(if L1 /= null then
(if L1.Value <= Threshold then L2.Value = 0
else L2.Value = L1.Value)
and then Updated_If_Less_Than_Threshold (L1.Next, L2.Next, Threshold)))
with Annotate => (GNATprove, Terminating);
pragma Annotate (GNATprove, False_Positive, "is recursive",
"The recursive call occurs on a strictly smaller lists");
procedure Update_List_Zero (L : access List_Cell; Threshold : Component_T) with
Post => Updated_If_Less_Than_Threshold (Copy (L)'Old, L, Threshold);
end P;
``` |

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 | ```
with Loop_Types; use Loop_Types;
package body P with
SPARK_Mode
is
procedure Update_List_Zero (L : access List_Cell; Threshold : Component_T) is
L_Old : constant List_Acc := Copy (L) with Ghost;
pragma Annotate (GNATprove, Intentional, "memory leak might occur",
"The code will be compiled with assertions disabled");
B : access List_Cell := L;
B_Old : access constant List_Cell := L_Old with Ghost;
begin
while B /= null loop
pragma Loop_Invariant (For_All_List (B, B_Old, Equal'Access));
pragma Loop_Invariant
(if Updated_If_Less_Than_Threshold (B_Old, At_End (B), Threshold)
then Updated_If_Less_Than_Threshold (L_Old, At_End (L), Threshold));
if B.Value <= Threshold then
B.Value := 0;
end if;
B := B.Next;
B_Old := B_Old.Next;
end loop;
pragma Assert
(Updated_If_Less_Than_Threshold (L_Old, At_End (L), Threshold));
end Update_List_Zero;
end P;
``` |

In the postcondition of `Update_List_Zero`

, we cannot use `For_All_List`

to
express the relation between the values of the list before and after the call.
Indeed, the relation depends on the value of the input `Threshold`

.
Instead, we create a specific recursive function taking the threshold as an
additional parameter.

The loop traverses `L`

using a local borrower `B`

. To be able to speak
about the initial value of `L`

in the invariant, we introduce a ghost
constant `L_Old`

storing a copy of this value. As we need to traverse both
lists at the same time, we declare a ghost variable `B_Old`

as a local
observer of `L_Old`

.

The loop invariant is made of two parts. The first one states that the elements
reachable through `B`

have not been modified by the loop. In the second loop
invariant, we want to use `Updated_If_Less_Than_Threshold`

to relate the
elements of `L`

that were already traversed to the elements of `L_Old`

.
As we cannot speak specifically about the traversed elements of `L`

, the
invariant states that, if at the end of the borrow the values accessible
through `B`

are related to their equivalent element in `B_Old`

through
`Updated_If_Less_Than_Threshold`

, then so are all the elements of `L`

.
GNATprove can verify these invariants as well as the postcondition of
`Update_List_Zero`

:

```
p.adb:14:33: info: loop invariant preservation proved
p.adb:14:33: info: loop invariant initialization proved
p.adb:14:62: info: null exclusion check proved
p.adb:16:13: info: loop invariant initialization proved
p.adb:16:13: info: loop invariant preservation proved
p.adb:18:14: info: pointer dereference check proved
p.adb:19:14: info: pointer dereference check proved
p.adb:21:16: info: pointer dereference check proved
p.adb:22:24: info: pointer dereference check proved
p.adb:25:10: info: assertion proved
p.ads:20:20: info: pointer dereference check proved
p.ads:20:47: info: pointer dereference check proved
p.ads:21:22: info: pointer dereference check proved
p.ads:21:33: info: pointer dereference check proved
p.ads:22:53: info: pointer dereference check proved
p.ads:22:62: info: pointer dereference check proved
p.ads:28:14: info: postcondition proved
p.ads:28:46: medium: memory leak might occur
28 | Post => Updated_If_Less_Than_Threshold (Copy (L)'Old, L, Threshold);
| ^~~~~~~
possible explanation: call to allocating function inside an assertion leaks memory
```

The second pattern of update loops that we consider now is the one that updates elements based on their position:

Loop Pattern

Modification of Elements Based on Position

Proof Objective

Elements of the collection are updated based on their position.

Loop Behavior

Loops over a collection and assigns the elements whose position satisfies a given modification criterion.

Loop Invariant

Every element encountered so far has been assigned according to its position.

Consider a procedure `Update_Range_Arr_Zero`

that sets to zero all elements
in array `A`

between indexes `First`

and `Last`

:

1 2 3 4 5 6 7 8 9 10 11 12 | ```
with Loop_Types; use Loop_Types;
procedure Update_Range_Arr_Zero (A : in out Arr_T; First, Last : Index_T) with
SPARK_Mode,
Post => A = (A'Old with delta First .. Last => 0)
is
begin
for J in First .. Last loop
A(J) := 0;
pragma Loop_Invariant (A = (A'Loop_Entry with delta First .. J => 0));
end loop;
end Update_Range_Arr_Zero;
``` |

The loop invariant expresses that all elements between `First`

and the
current loop index `J`

have been zeroed out, and that other elements have not
been modified (using a combination of Attribute Loop_Entry and
Delta Aggregates to express this concisely). With this loop invariant,
GNATprove is able to prove the postcondition of `Update_Range_Arr_Zero`

,
namely that all elements between `First`

and `Last`

have been zeroed out,
and that other elements have not been modified:

```
update_range_arr_zero.adb:5:11: info: postcondition proved
update_range_arr_zero.adb:10:30: info: loop invariant initialization proved
update_range_arr_zero.adb:10:30: info: loop invariant preservation proved
```

Consider now a variant of the same update loop over a vector:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 | ```
pragma Unevaluated_Use_Of_Old (Allow);
with Loop_Types; use Loop_Types; use Loop_Types.Vectors;
use Loop_Types.Vectors.Formal_Model;
procedure Update_Range_Vec_Zero (V : in out Vec_T; First, Last : Index_T) with
SPARK_Mode,
Pre => Last <= Last_Index (V),
Post => (for all J in 1 .. Last_Index (V) =>
(if J in First .. Last then Element (V, J) = 0
else Element (V, J) = Element (Model (V)'Old, J)))
is
begin
for J in First .. Last loop
Replace_Element (V, J, 0);
pragma Loop_Invariant (Last_Index (V) = Last_Index (V)'Loop_Entry);
pragma Loop_Invariant
(for all I in 1 .. Last_Index (V) =>
(if I in First .. J then Element (V, I) = 0
else Element (V, I) = Element (Model (V)'Loop_Entry, I)));
end loop;
end Update_Range_Vec_Zero;
``` |

Like for `Map_Vec_Incr`

, we need to use the `Model`

function over arrays to
access elements of the vector before the loop as the vector type is limited. The
loop invariant expresses that all elements between `First`

and current loop
index `J`

have been zeroed, and that other elements have not been modified.
With this loop invariant, GNATprove is able to prove the
postcondition of `Update_Range_Vec_Zero`

:

```
update_range_vec_zero.adb:8:11: info: postcondition proved
update_range_vec_zero.adb:9:44: info: precondition proved
update_range_vec_zero.adb:9:56: info: range check proved
update_range_vec_zero.adb:10:22: info: precondition proved
update_range_vec_zero.adb:10:34: info: range check proved
update_range_vec_zero.adb:10:39: info: precondition proved
update_range_vec_zero.adb:10:63: info: range check proved
update_range_vec_zero.adb:14:07: info: precondition proved
update_range_vec_zero.adb:15:30: info: loop invariant preservation proved
update_range_vec_zero.adb:15:30: info: loop invariant initialization proved
update_range_vec_zero.adb:17:10: info: loop invariant preservation proved
update_range_vec_zero.adb:17:10: info: loop invariant initialization proved
update_range_vec_zero.adb:18:41: info: precondition proved
update_range_vec_zero.adb:18:53: info: range check proved
update_range_vec_zero.adb:19:22: info: precondition proved
update_range_vec_zero.adb:19:34: info: range check proved
update_range_vec_zero.adb:19:39: info: precondition proved
update_range_vec_zero.adb:19:70: info: range check proved
```

Similarly, consider a variant of the same update loop over a list:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 | ```
with Loop_Types; use Loop_Types; use Loop_Types.Lists;
with Ada.Containers; use Ada.Containers; use Loop_Types.Lists.Formal_Model;
procedure Update_Range_List_Zero (L : in out List_T; First, Last : Cursor) with
SPARK_Mode,
Pre => Has_Element (L, First) and then Has_Element (L, Last)
and then P.Get (Positions (L), First) <= P.Get (Positions (L), Last),
Post => Length (L) = Length (L)'Old
and Positions (L) = Positions (L)'Old
and (for all I in 1 .. Length (L) =>
(if I in P.Get (Positions (L), First) .. P.Get (Positions (L), Last) then
Element (Model (L), I) = 0
else Element (Model (L), I) = Element (Model (L'Old), I)))
is
Cu : Cursor := First;
begin
loop
pragma Loop_Invariant (Has_Element (L, Cu));
pragma Loop_Invariant (P.Get (Positions (L), Cu) in P.Get (Positions (L), First) .. P.Get (Positions (L), Last));
pragma Loop_Invariant (Length (L) = Length (L)'Loop_Entry);
pragma Loop_Invariant (Positions (L) = Positions (L)'Loop_Entry);
pragma Loop_Invariant (for all I in 1 .. Length (L) =>
(if I in P.Get (Positions (L), First) .. P.Get (Positions (L), Cu) - 1 then
Element (Model (L), I) = 0
else Element (Model (L), I) = Element (Model (L'Loop_Entry), I)));
Replace_Element (L, Cu, 0);
exit when Cu = Last;
Next (L, Cu);
end loop;
end Update_Range_List_Zero;
``` |

Compared to the vector example, it requires three additional invariants. As the
loop is done via a cursor, the first two loop invariants are necessary to know
that the current cursor `Cu`

stays between `First`

and `Last`

in the list.
The fourth loop invariant states that the position of cursors in `L`

is not
modified during the loop. It is necessary to know that the two cursors `First`

and
`Last`

keep designating the same range after the loop. With this loop invariant,
GNATprove is able to prove the postcondition of `Update_Range_List_Zero`

,
namely that all elements between `First`

and `Last`

have been zeroed out,
and that other elements have not been modified:

```
update_range_list_zero.adb:7:13: info: precondition proved
update_range_list_zero.adb:7:45: info: precondition proved
update_range_list_zero.adb:8:11: info: postcondition proved
update_range_list_zero.adb:11:23: info: precondition proved
update_range_list_zero.adb:11:55: info: precondition proved
update_range_list_zero.adb:12:16: info: precondition proved
update_range_list_zero.adb:13:19: info: precondition proved
update_range_list_zero.adb:13:44: info: precondition proved
update_range_list_zero.adb:18:30: info: loop invariant initialization proved
update_range_list_zero.adb:18:30: info: loop invariant preservation proved
update_range_list_zero.adb:19:30: info: loop invariant preservation proved
update_range_list_zero.adb:19:30: info: loop invariant initialization proved
update_range_list_zero.adb:19:31: info: precondition proved
update_range_list_zero.adb:19:60: info: precondition proved
update_range_list_zero.adb:19:92: info: precondition proved
update_range_list_zero.adb:20:30: info: loop invariant preservation proved
update_range_list_zero.adb:20:30: info: loop invariant initialization proved
update_range_list_zero.adb:21:30: info: loop invariant preservation proved
update_range_list_zero.adb:21:30: info: loop invariant initialization proved
update_range_list_zero.adb:22:30: info: loop invariant preservation proved
update_range_list_zero.adb:22:30: info: loop invariant initialization proved
update_range_list_zero.adb:23:42: info: precondition proved
update_range_list_zero.adb:23:74: info: precondition proved
update_range_list_zero.adb:24:36: info: precondition proved
update_range_list_zero.adb:25:38: info: precondition proved
update_range_list_zero.adb:25:63: info: precondition proved
update_range_list_zero.adb:26:07: info: precondition proved
update_range_list_zero.adb:28:07: info: precondition proved
```