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# 3.5.4 Integer Types

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.

Syntax

2. ```integer_type_definition ::=
signed_integer_type_definition | modular_type_definition
```
3. ```signed_integer_type_definition ::=
range static_simple_expression .. static_simple_expression
```
4. ```modular_type_definition ::= mod static_expression
```

Name Resolution Rules

5. 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

6. 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.
7. 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

8. 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.
9. 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.
10. 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.
11. 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.
12. Integer has two predefined subtypes, declared in the visible part of package Standard:
13. ```subtype Natural  is Integer range 0 .. Integer'Last;
subtype Positive is Integer range 1 .. Integer'Last;
```
14. 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.
15. The position number of an integer value is equal to the value.
16. For every modular subtype S, the following attribute is defined:
17. S'Modulus
```S'Modulus yields the modulus of the type of S, as a value of
the type universal_integer.
```

Dynamic Semantics

18. The elaboration of an integer_type_definition creates the integer type and its first subtype.
19. 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.
20. 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

21. In an implementation, the range of Integer shall include the range -2**15+1 .. +2**15-1.
22. If Long_Integer is predefined for an implementation, then its range shall include the range -2**31+1 .. +2**31-1.
23. System.Max_Binary_Modulus shall be at least 2**16.

Implementation Permissions

24. 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.
25. 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.
26. 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.
27. 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

28. 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 section B.2 The Package Interfaces.
29. 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

30. (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.
31. (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 bit-wise logical operators (and, or, xor, and not). In addition, for the unsigned types declared in the language-defined package Interfaces, See section B.2 The Package Interfaces, functions are defined that provide bit-wise shifting and rotating.
32. (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

33. Examples of integer types and subtypes:
34. ```type Page_Num  is range 1 .. 2_000;
type Line_Size is range 1 .. Max_Line_Size;
```
35. ```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;
```
36. ```type Byte        is mod 256; -- an unsigned byte
type Hash_Index  is mod 97;  -- modulus is prime
```

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