1

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 wrap-around semantics.

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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.

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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.

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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.

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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 run-time operations. The set
of values for a modular integer type are the values from 0 to one less
than the modulus, inclusive.

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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.

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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.

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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.

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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 3.4.1)
for the class rooted at *root_integer*, allowing their use with
the operations of any integer type.

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16/2

For every modular subtype
S, the following attributes are defined:

16.1/2

S'Mod

16.2/2

16.3/2

This function returns *Arg* **mod**
S'Modulus, as a value of the type of S.

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S'Modulus

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

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For a modular type, if the result of the execution
of a predefined operator (see 4.5) 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.

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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.

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

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

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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.

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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.

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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 12.5.2.

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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.

27.1/1

For a one's complement machine, implementations
may support non-binary modulus values greater than System.Max_Nonbinary_Modulus.
It is implementation defined which specific values greater than System.Max_Nonbinary_Modulus,
if any, are supported.

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An implementation should support
Long_Integer in addition to Integer if the target machine supports 32-bit
(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 B.2).

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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

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

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28 The same arithmetic operators are predefined
for all signed integer types defined by a signed_integer_type_definition
(see 4.5, “Operators
and Expression Evaluation”). For modular types, these same
operators are predefined, plus bit-wise logical operators (**and**,
**or**, **xor**, and **not**). In addition, for the unsigned
types declared in the language-defined package Interfaces (see B.2),
functions are defined that provide bit-wise shifting and rotating.

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29 Modular types match a generic_formal_parameter_declaration
of the form "**type** T **is mod** <>;"; signed
integer types match "**type** T **is range** <>;"
(see 12.5.2).

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