
An integer_type_definition defines an integer type; it defines either a
signed integer type, or a modular integer type. The base range of a
signed integer type includes at least the values of the specified range.
A modular type is an integer type with all arithmetic modulo a specified
positive modulus; such a type corresponds to an unsigned type with
wraparound semantics.
Syntax

integer_type_definition ::=
signed_integer_type_definition  modular_type_definition

signed_integer_type_definition ::=
range static_simple_expression .. static_simple_expression

modular_type_definition ::= mod static_expression
Name Resolution Rules

Each simple_expression in a signed_integer_type_definition is expected
to be of any integer type; they need not be of the same type. The
expression in a modular_type_definition is likewise expected to be of
any integer type.
Legality Rules

The simple_expressions of a signed_integer_type_definition shall be
static, and their values shall be in the range System.Min_Int ..
System.Max_Int.

The expression of a modular_type_definition shall be static, and its
value (the modulus) shall be positive, and shall be no greater than
System.Max_Binary_Modulus if a power of 2, or no greater than
System.Max_Nonbinary_Modulus if not.
Static Semantics

The set of values for a signed integer type is the (infinite) set of
mathematical integers, though only values of the base range of the type
are fully supported for runtime operations. The set of values for a
modular integer type are the values from 0 to one less than the modulus,
inclusive.

A signed_integer_type_definition defines an integer type whose base
range includes at least the values of the simple_expressions and is
symmetric about zero, excepting possibly an extra negative value. A
signed_integer_type_definition also defines a constrained first subtype
of the type, with a range whose bounds are given by the values of the
simple_expressions, converted to the type being defined.

A modular_type_definition defines a modular type whose base range is
from zero to one less than the given modulus. A modular_type_definition
also defines a constrained first subtype of the type with a range that
is the same as the base range of the type.

There is a predefined signed integer subtype named Integer, declared in
the visible part of package Standard. It is constrained to the base
range of its type.

Integer has two predefined subtypes, declared in the visible part of
package Standard:

subtype Natural is Integer range 0 .. Integer'Last;
subtype Positive is Integer range 1 .. Integer'Last;

A type defined by an integer_type_definition is implicitly derived from
root_integer, an anonymous predefined (specific) integer type, whose
base range is System.Min_Int .. System.Max_Int. However, the base range
of the new type is not inherited from root_integer, but is instead
determined by the range or modulus specified by the
integer_type_definition. Integer literals are all of the type
universal_integer, the universal type, See section 3.4.1 Derivation Classes, for the class
rooted at root_integer, allowing their use with the operations of any
integer type.

The position number of an integer value is equal to the value.

For every modular subtype S, the following attribute is defined:

S'Modulus
S'Modulus yields the modulus of the type of S, as a value of
the type universal_integer.
Dynamic Semantics

The elaboration of an integer_type_definition creates the integer type
and its first subtype.

For a modular type, if the result of the execution of a predefined
operator, See section 4.5 Operators and Expression Evaluation is outside the base range of the type, the result
is reduced modulo the modulus of the type to a value that is within the
base range of the type.

For a signed integer type, the exception Constraint_Error is raised by
the execution of an operation that cannot deliver the correct result
because it is outside the base range of the type. For any integer type,
Constraint_Error is raised by the operators "/", "rem", and "mod" if the
right operand is zero.
Implementation Requirements

In an implementation, the range of Integer shall include the range
2**15+1 .. +2**151.

If Long_Integer is predefined for an implementation, then its range
shall include the range 2**31+1 .. +2**311.

System.Max_Binary_Modulus shall be at least 2**16.
Implementation Permissions

For the execution of a predefined operation of a signed integer type,
the implementation need not raise Constraint_Error if the result is
outside the base range of the type, so long as the correct result is
produced.

An implementation may provide additional predefined signed integer
types, declared in the visible part of Standard, whose first subtypes
have names of the form Short_Integer, Long_Integer, Short_Short_Integer,
Long_Long_Integer, etc. Different predefined integer types are allowed
to have the same base range. However, the range of Integer should be no
wider than that of Long_Integer. Similarly, the range of Short_Integer
(if provided) should be no wider than Integer. Corresponding
recommendations apply to any other predefined integer types. There need
not be a named integer type corresponding to each distinct base range
supported by an implementation. The range of each first subtype should
be the base range of its type.

An implementation may provide nonstandard integer types, descendants of
root_integer that are declared outside of the specification of package
Standard, which need not have all the standard characteristics of a type
defined by an integer_type_definition. For example, a nonstandard
integer type might have an asymmetric base range or it might not be
allowed as an array or loop index (a very long integer). Any type
descended from a nonstandard integer type is also nonstandard. An
implementation may place arbitrary restrictions on the use of such
types; it is implementation defined whether operators that are
predefined for "any integer type" are defined for a particular
nonstandard integer type. In any case, such types are not permitted as
explicit_generic_actual_parameters for formal scalar types 
See section 12.5.2 Formal Scalar Types.

For a one's complement machine, the high bound of the base range of a
modular type whose modulus is one less than a power of 2 may be equal to
the modulus, rather than one less than the modulus. It is implementation
defined for which powers of 2, if any, this permission is exercised.
Implementation Advice

An implementation should support Long_Integer in addition to Integer if
the target machine supports 32bit (or longer) arithmetic. No other
named integer subtypes are recommended for package Standard. Instead,
appropriate named integer subtypes should be provided in the library
package Interfaces, See section B.2 The Package Interfaces.

An implementation for a two's complement machine should support modular
types with a binary modulus up to System.Max_Int*2+2. An implementation
should support a nonbinary modulus up to Integer'Last.
NOTES

(25) Integer literals are of the anonymous predefined integer type
universal_integer. Other integer types have no literals. However, the
overload resolution rules, See section 8.6 The Context of Overload Resolution, allow expressions of the type universal_integer whenever
an integer type is expected.

(26) The same arithmetic operators are predefined for all signed integer
types defined by a signed_integer_type_definition, See section 4.5 Operators and Expression Evaluation. For modular types, these same
operators are predefined, plus bitwise logical operators (and, or, xor,
and not). In addition, for the unsigned types declared in the
languagedefined package Interfaces, See section B.2 The Package Interfaces, functions are defined
that provide bitwise shifting and rotating.

(27) Modular types match a generic_formal_parameter_declaration of the
form "type T is mod <>;"; signed integer types match "type T is range
<>;", See section 12.5.2 Formal Scalar Types.
Examples

Examples of integer types and subtypes:

type Page_Num is range 1 .. 2_000;
type Line_Size is range 1 .. Max_Line_Size;

subtype Small_Int is Integer range 10 .. 10;
subtype Column_Ptr is Line_Size range 1 .. 10;
subtype Buffer_Size is Integer range 0 .. Max;

type Byte is mod 256;  an unsigned byte
type Hash_Index is mod 97;  modulus is prime