This chapter describes the Common Dylan implementation of arithmetic functions, especially integer arithmetic. It describes a number of extensions to the Dylan language, which are available from the Dylan library. It also describes a generic arithmetic facility that, through the use of other libraries, allows you to extend arithmetic to special number types, such as “big” (64-bit) integers.
Throughout this chapter, arguments are instances of the class specified by the argument name (ignoring any numeric suffixes), unless otherwise noted. Thus, the arguments integer, integer1, and integer2 would all be instances of the class <integer>.
The goals of the extensions to the Dylan language described in this chapter are as follows:
Provide arithmetic operations that are closed over small integers.
This allows type inference to propagate small integer declarations more widely, because there is no possibility of automatic coercion into some more general format.
Make the arithmetic operations that are closed over small integers easily accessible to programmers.
Allow the Dylan library to be described in such a way that only small integers are present by default, moving support for infinite precision integer arithmetic to the Big-Integers library, which must be explicitly used.
Support infinite precision integer arithmetic through the Big-Integers library.
Note
Using that library in another library does not have a negative effect on the correctness or performance of other libraries in the same application that do not use it.
Maintain compatibility with the DRM specification.
In particular, the extensions support the production of efficient code for programs written to be portable with respect to the DRM specification. Use of implementation-specific types or operations in order to get reasonable efficiency is not required. This precludes relegating the <integer> class and limited-<integer> types to inefficient implementations.
Note
When there are several distinct interfaces with the same name but in different modules, the notation interface # module is used in this chapter to remove ambiguity.
Specify that the class <integer> has a finite, implementation-dependent range, bounded by the constants $minimum-integer and $maximum-integer.
The representation for integers must be at least 28 bits, including the sign. That is, the minimum conforming value for $maximum-integer is 2*27* -1 and the maximum conforming value for $minimum-integer is -2*27*.
Rationale: Restricting <integer> in this way allows the programmer to stay in the efficient range without requiring exact knowledge of what that range might be. The full generality of extended precision integers is provided by the Big-Integers library, for programmers who actually need that functionality.
Define the type <machine-number> to be the type union of <float> and <integer>.
The Dylan library provides implementations of the generic functions and functions described in this chapter. If the result of one of these operations is specified to be an instance of <integer> and the mathematically correct result cannot be represented as an <integer> then an error is signaled. This removes fully generic arithmetic from the Dylan library. In particular, it removes extended integers, ratios, and rectangular complex numbers.
This section describes the extensions to the Dylan library that provide the arithmetic operations available as standard to your applications. You do not have to explicitly use any additional libraries to have access to any of the functionality described in this section. Note that this section only describes extensions to the Dylan library; for complete descriptions, you should also refer to the Dylan Reference Manual.
Note that the Common-Dylan library also has these extensions because it uses the Dylan library.
The initialization arguments for <range> must all be instances of <machine-number> rather than <real>.
The following specific constructors are available for use with the class <integer>.
Summary
Defines a new type that represents a subset of the class <integer>.
Arguments
Signature
limited integer-class #key min max => limited-type
Description
The integer-class argument is the class <integer>, and all other arguments are instances of <integer>. The range of <integer> is bounded by default.
Summary
This function is used to specify ranges of numbers.
Arguments
Signature
range (#key from:, to:, above:, below:, by:, size:) => <range>
Description
All of the supplied arguments must be instances of <machine-number>.
The = function compares two objects and returns #t if the values of the two objects are equal to each other, that is of the same magnitude.
=
~
Summary
Tests its arguments to see if they are of the same magnitude.
Signature
Value
<boolean>
Other available methods are described in the Dylan Reference Manual.
The Dylan library provides the following interfaces for testing the magnitude of two numbers:
<
~
Summary
Returns #t if its first argument is less than its second argument.
Signature
< *object1* *object2* => *boolean* (*Generic function*)*
< *complex1* *complex2* (*Sealed domain*)
< *machine-number1* *machine-number2* => *boolean* (*G.f. method*)
Other available methods are described in the Dylan Reference Manual.
Various number properties can be tested using the following predicates in the Dylan library:
Summary
Tests whether the argument supplied represents an odd value.
Signature
odd? *object* => *boolean* (*Open generic function*)
odd? *complex* => *boolean* (*Sealed domain*)
odd? *integer* => *boolean* (*G.f. method*)
Summary
Tests whether the argument supplied represents an even value
Signature
even? *object* => *boolean* (*Open generic function*)
even? *complex* *=>* *boolean* (*Sealed domain*)
even? *integer* => *boolean* (*G.f. method*)
zero?
Open generic function
zero? object => boolean
zero?
Sealed domain
zero? complex
zero?
G.f. method
zero? machine-number => boolean
Tests whether the argument supplied represents a zero value.
positive?
Open generic function
positive? object => boolean
positive?
Sealed domain
positive? complex
positive?
G.f. method
positive? machine-number => boolean
Tests whether the argument supplied represents a positive value.
negative?
Open generic function
negative? object => boolean
negative?
Sealed domain
negative? complex
negative?
G.f. method
negative? machine-number => boolean
Tests whether the argument supplied represents a negative value.
integral?
Open generic function
integral? object => boolean
integral?
Sealed domain
integral? complex
integral?
G.f. method
integral? machine-number => boolean
Tests whether the argument supplied represents an integral value.
The following arithmetic operations are available in the Dylan library:
+
Open generic function
+ object1 object2 => #rest object
+
Sealed domain
+ complex1 complex 2
+
G.f. method
+ integer1 integer 2 => integer
+
G.f. method
+ machine-number1 machine-number2 => machine-number
Returns the sum of the two supplied arguments. The actual type of the value is determined by the contagion rules when applied to the arguments.
-
Open generic function
- object1 object2 => #rest object
-
Sealed domain
- complex1 complex2
-
G.f. method
- integer1 integer2 => integer
-
G.f. method
- machine-number1 machine-number2 => machine-number
Returns the result of subtracting the second argument from the first. The actual type of the value is determined by the contagion rules when applied to the arguments.
*
Open generic function
* object1 object2 => #rest object
*
Sealed domain
* complex1 complex2
*
G.f. method
* integer1 integer 2 => integer
*
G.f. method
* machine-number1 machine-number2 => machine-number
Returns the result of multiplying the two arguments. The actual type of the value is determined by the contagion rules when applied to the arguments.
/
Open generic function
/ object1 object2 => #rest object
/
Sealed domain
/ complex1 complex2
/
G.f. method
/ float1 float 2 => float
Returns the result of dividing the first argument by the second. The actual type of the value is determined by the contagion rules when applied to the arguments.
negative
Open generic function
negative object => #rest negative-object
negative
Sealed domain
negative complex
negative
G.f. method
negative integer => negative-integer
negative
G.f. method
negative float => negative-float
Negates the supplied argument. The returned value is of the same float format as the supplied argument.
floor
Function
Truncates a number toward negative infinity. The integer part is returned as integer, the remainder is of the same float format as the argument.
ceiling
Function
Truncates a number toward positive infinity. The integer part is returned as integer, the remainder is of the same float format as the argument.
round
Function
Rounds a number toward the nearest mathematical integer. The integer part is returned as integer, the remainder is of the same float format as the argument. If the argument is exactly between two integers, then the result integer will be a multiple of two.
truncate
Function
Truncates a number toward zero. The integer part is returned as integer, the remainder is of the same float format as the argument.
floor/
Function
floor/ *machine-number1* *machine-number2* => *integer* *machine-number*
floor/ *integer1* *integer2* => *integer* *integer*
floor/ *machine-number1* *machine-number2* => *integer* *machine-number*
Divides the first argument into the second and truncates the result toward negative infinity. The integer part is returned as integer, the type of the remainder is determined by the contagion rules when applied to the arguments.
ceiling/
Function
ceiling/ *machine-number1* *machine-number2* => *integer* *machine-number*
ceiling/ *integer1* *integer2* => *integer* *integer*
ceiling/ *machine-number1* *machine-number2* => *integer* *machine-number*
Divides the first argument into the second and truncates the result toward positive infinity. The integer part is returned as integer, the type of the remainder is determined by the contagion rules when applied to the arguments.
round/
Function
round/ *machine-number1* *machine-number2* => *integer* *machine-number*
round/ *integer1* *integer2* => *integer* *integer*
round/ *machine-number1* *machine-number2* => *integer* *machine-number*
Divides the first argument into the second and rounds the result toward the nearest mathematical integer. The integer part is returned as integer, the type of the remainder is determined by the contagion rules when applied to the arguments.
truncate/
Function
truncate/ *machine-number1* *machine-number2* => *integer* *machine-number*
truncate/ *integer1* *integer* 2 => *integer* *integer*
truncate/ *machine-number1* *machine-number2* => *integer* *machine-number*
Divides the first argument into the second and truncates the result toward zero. The integer part is returned as integer, the type of the remainder is determined by the contagion rules when applied to the arguments.
modulo
Function
modulo *machine-number1* *machine-number2* => *machine-number*
modulo *integer1* *integer2* => *integer*
modulo *machine-number1* *machine-number2* => *machine-number*
Returns the second value of floor/ ( arg1 , arg2 ). The actual type of the second value is determined by the contagion rules when applied to the arguments.
remainder
Function
remainder *machine-number1* *machine-number2* => *machine-number*
remainder *integer1* *integer2* => *integer*
remainder *machine-number1* *machine-number2* => *machine-number*
Returns the second value of truncate/ ( arg1 , arg2 ).The actual type of the second value is determined by the contagion rules when applied to the arguments.
^
Open generic function
^ object1 object2 => #rest object
^
Sealed domain
^ complex1 complex 2
^
G.f. method
^ integer1 integer2 => integer
^
G.f. method
^ float1 integer2 => float
Returns the first argument raised to the power of the second argument. The value is of the same float format as the first argument. An error is signalled if both arguments are 0.
abs
Open generic function
abs object => #rest object
abs
Sealed domain
abs complex
abs
G.f. method
abs integer => integer
abs
G.f. method
abs float => float
Returns the absolute value of the argument. The value is of the same float format as the argument.
logior
Function
logior #rest integers => integer
Returns the bitwise inclusive OR of its integer arguments.
logxor
Function
logxor #rest integers => integer
Returns the bitwise exclusive OR of its integer arguments.
logand
Function
logand #rest integers => integer
Returns the bitwise AND of its integer arguments.
lognot
Function
lognot integer1 => integer2
Returns the bitwise NOT of its integer arguments.
logbit?
Function
logbit? index integer => boolean
Tests the value of a particular bit in its integer argument. The index argument is an instance of <integer>.
ash
Function
ash integer1 count => integer
Performs an arithmetic shift on its first argument.
lcm
Function
lcm integer1 integer2 => integer
Returns the least common multiple of its two arguments.
gcd
Function
gcd integer1 integer2 => integer
Returns the greatest common divisor of its two arguments.
The keys for sequences are always instances of <integer>. This means that certain kinds of collections cannot be sequences; very large (or unbounded) sparse arrays are an example.
The following functions in the Dylan library are extended. Note that the hash IDs for tables are always instances of <integer>.
merge-hash-codes
Function
Returns a hash code created from the merging of two argument hash codes. The id arguments are hash IDs, and the state arguments are hash states (instances of <object>). The ordered? argument is an instance of <boolean>. The returned merged values are instances of <integer> and <object>, as determined by the name of each argument.
object-hash
Function
object-hash object => hash-id hash-state
The hash function for the equivalence predicate ==. The return values are of the same types as the return values of merge-hash-codes.
for
Statement macro
The start, bound, and increment expressions in a numeric clause must evaluate to instances of <machine-number> for this macro.
The Generic-Arithmetic library exports the functions described in this section from an exported module called generic-arithmetic.
The Generic-Arithmetic library provides a fully extensible version of all arithmetic operations. If an application only uses Generic-Arithmetic, these versions of the operators reduce themselves to be equivalent to those in the Dylan library. But when you use additional implementation libraries, the arithmetic operators are extended.
The Big-Integers library is one such implementation library. It provides a 64-bit implementation of <integer>.
The standard integer implementation in the Dylan library is actually part of the following class hierarchy:
<abstract-integer>
<integer>
<big-integer>
<double-integer>
(The classes <big-integer> and <double-integer> are implementation classes. You do not need to use them.)
The modules in the Generic-Arithmetic library export <abstract-integer> with the name <integer>. They also export a full set of arithmetic operators that use instances of <abstract-integer> rather than instances of <integer> (in the Dylan library naming scheme). However, those operators just fall back to the Dylan library operators until you include an implementation library, such as Big-Integers, in your application.
When you use the Big-Integers library, the arithmetic operators exported by Generic-Arithmetic are enhanced to extend their results to 64-bit integers. If a result is small enough to fit in a Dylan library <integer>, it will be fitted into one.
Note that the Generic-Arithmetic library uses the same naming conventions for arithmetic operators as used by the Dylan library. This means that some renaming is required in modules that require access to both the basic Dylan interfaces and the interfaces supplied by the Generic-Arithmetic library. As described earlier, the notation interface # module is used to denote different interfaces of the same name, where interface is the name of the interface, and module is the name of the module it is exported from.
See Using special arithmetic features for an example of how to use an implementation library with Generic-Arithmetic.
The Generic-Arithmetic library defines the class <range>, which is in most respects functionally equivalent to <range>#Dylan, but uses generic arithmetic operations in its implementation so that the initialization arguments can be instances of <real>, rather than being restricted to <machine-number>.
The class <abstract-integer> is imported and re-exported under the name <integer>#generic-arithmetic.
range
Function
range #key from to above below by size => range
This function is identical to the function range#Dylan, except that all of the supplied arguments must be instances of <real>.
The following functions all apply function #Dylan to the arguments and return the results, where function is the appropriate function name. See Arithmetic operations for descriptions of each function as implemented in the Dylan library.
* object1 object2 => #rest object
/ object1 object2 => #rest object
negative object => #rest negative-object
floor real1 => abstract-integer real
ceiling real1 => abstract-integer real
round real1 => abstract-integer real
truncate real1 => abstract-integer real
floor/ real1 real2 => abstract-integer real
ceiling/ real1 real2 => abstract-integer real
round/ real1 real2 => abstract-integer real
truncate/ real1 real2 => abstract-integer real
modulo real1 real2 => real
remainder real1 real2 => real
^ object1 object2 => #rest object
abs object1 => #rest object
logior #rest abstract-integer1 => abstract-integer
logxor #rest abstract-integer1 => abstract-integer
logand #rest abstract-integer1 => abstract-integer
lognot abstract-integer1 => abstract-integer
logbit? integer abstract-integer => boolean
ash abstract-integer1 integer => abstract-integer
lcm abstract-integer1 abstract-integer2 => abstract-integer
gcd abstract-integer1 abstract-integer2 => abstract-integer
While a programmer could make use of generic arithmetic in a for loop by using explicit-step clauses, this approach leads to a loss of clarity. The definition of the for macro is complex, so a version that uses generic arithmetic in numeric clauses is provided, rather than requiring programmers who want that feature to reconstruct it.
for
Statement macro
The start, bound, and increment expressions in a numeric clause must evaluate to instances of <machine-number> for this macro. Otherwise, this macro is similar to for#Dylan.
The Generic-Arithmetic library exports several modules that are provided for the convenience of programmers who wish to create additional modules based on the dylan module plus various combinations of the arithmetic models.
The Dylan-Excluding-Arithmetic module imports and re-exports all of the interfaces exported by the dylan module from the Dylan library, except for the following excluded interfaces:
<integer>
range
negative
floor ceiling round truncate
floor/ ceiling/ round/ truncate/
modulo remainder
^
abs
logior logxor logand lognot
logbit?
ash
lcm gcd
for
The Dylan-Arithmetic module imports and re-exports all of the interfaces exported by the dylan module from the Dylan library which are excluded by the dylan-excluding-arithmetic module.
The Generic-Arithmetic-Dylan module imports and reexports all of the interfaces exported by the dylan-excluding-arithmetic module and the generic-arithmetic module.
The dylan-excluding-arithmetic, dylan-arithmetic, and generic-arithmetic modules provide convenient building blocks for programmers to build the particular set of global name bindings they wish to work with. The purpose of the generic-arithmetic-dylan module is to provide a standard environment in which generic arithmetic is the norm, for those programmers who might want that.
As noted in The Generic-Arithmetic library, the Generic-Arithmetic library provides an extensible protocol for adding specialized arithmetic functionality to your applications. By using the Generic-Arithmetic library alongside a special implementation library, you can make the standard arithmetic operations support number types such as big (64-bit) integers, or complex numbers.
This section provides an example of extending the basic Dylan arithmetic features using the Generic-Arithmetic library and the Big-Integers implementation library.
To use special arithmetic features, an a library’s define library declaration must use at least the following libraries:
common-dylan
generic-arithmetic
special-arithmetic-implementation-library
So for Big-Integers you would write:
define library foo
use common-dylan;
use generic-arithmetic;
use big-integers;
…
end library foo;
Next you have to declare a module. There are three ways of using big-integer arithmetic that we can arrange with a suitable module declaration:
Replace all integer arithmetic with the big-integer arithmetic
Use both, with normal arithmetic remaining the default
Use both, with the big-integer arithmetic becoming the default
To get one of the three different effects described above, you need to arrange the define module declaration accordingly. To replace all integer arithmetic with big-integer arithmetic, include the following in your define module declaration:
use generic-arithmetic-common-dylan;
(Note that the module definition should not use the Big-Integers module. The Big-Integers library is used as a side-effects library only, that is, it is referenced in the library definition so that it will be loaded. Its definitions extend the Generic-Arithmetic library.)
If you replace all integer arithmetic with big-integer arithmetic in this way, there will be performance hits. For instance, loop indices will have to be checked at run-time to see whether a normal or big integer representation is being used, and a choice must be made about the representation for an incremented value.
You can take a different approach that reduces the cost of big-integer arithmetic. Under this approach you leave normal integer arithmetic unchanged, and get access to big-integer arithmetic when you need it. To do this, use the same libraries but instead of using the common-dylan-generic-arithmetic module, include the following in your define module declaration:
use common-dylan;
use generic-arithmetic, prefix: “ga/”; // use any prefix you like
This imports the big-integer arithmetic binding names, but gives them a prefix ga/, using the standard renaming mechanism available in module declarations. Thus you gain access to big arithmetic using renamed classes and operations like:
ga/<integer>
ga/+
ga/-
ga/*
…
The operations take either instances of <integer> or ga/<integer> (a subclass of <integer>) and return instances of ga/<integer>.
Note that having imported the big-integer operations under new names, you have to use prefix rather than infix syntax when calling them. For example:
ga/+ (5, 4);
not:
5 ga/+ 4;
The existing functions like + and - will only accept <integer> instances and ga/<integer> instances small enough to be represented as <integer> instances.
Under this renaming scheme, reduced performance will be confined to the ga/ operations. Other operations, such as loop index increments and decrements, will retain their efficiency.
Finally, you can make big-integer arithmetic the default but keep normal arithmetic around for when you need it. Your define module declaration should contain:
use generic-arithmetic-common-dylan;
use dylan-arithmetic, prefix: “dylan/”; //use any prefix you like
The Big-Integers library exports a module called big-integers, which imports and re-exports all of the interfaces exported by the generic-arithmetic module of the Generic-Arithmetic library.
The Big-Integers library modifies the behavior of functions provided by the Dylan library as described in this section.
The Big-Integers library extends the functionality of specific constructors in the Dylan library as follows:
limited
G.f. method
limited abstract-integer-class #key min max => limited-type
Returns a limited integer type, which is a subtype of <abstract-integer>, whose instances are integers greater than or equal to min (if specified) and less than or equal to max (if specified). If no keyword arguments are specified, the result type is equivalent to <abstract-integer>. The argument abstract-integer-class is the class <abstract-integer>.
If both min and max are supplied, and both are instances of <integer>, then the result type is equivalent to calling limited on <integer> with those same bounds.
The Limited Integer Type Protocol is extended to account for limited <abstract-integer> types.
Instances and subtypes in the Big-Integers library
:: todo Fix header style here—
This is true if and only if …
… all these clauses are true
(w >= y)
(x <= z)
Type-equivalence in the Big-Integers library :: todo Fix header style here—
This is type equivalent to …
… this, if and only if …
… this is true
y and z are both instances of <integer>.
y is an instance of <integer>.
z is an instance of <integer>.
Type disjointness is modified as follows to account for limited <abstract-integer> types.
A limited integer type is disjoint from a class if their base types are disjoint or the class is <integer> and the range of the limited integer type is disjoint from the range of <integer> (that is, from $minimum-integer to $maximum-integer).
The behavior of equality comparisons in the Dylan library is modified by the Big-Integers library as follows:
= *abstract-integer1* *abstract-integer2* => *boolean*
= *abstract-integer* *float* => *boolean*
= *float* *abstract-integer* => *boolean*
The behavior of magnitude comparisons in the Dylan library is modified by the Big-Integers library as follows:
< *abstract-integer1* *abstract-integer2* => *boolean
< *abstract-integer* *float* => *boolean*
< *float* *abstract-integer* => *boolean*
The behavior of number property tests in the Dylan library is modified by the Big-Integers library as follows:
odd? *abstract-integer* => *boolean*
even? *abstract-integer* => *boolean*
zero? *abstract-integer* => *boolean*
positive? *abstract-integer* => *boolean*
negative? *abstract-integer* => *boolean*
integral? *abstract-integer* => *boolean*
The Big-Integers library modifies the behavior of the functions provided by the Generic-Arithmetic library as described below.
The actual type of the return value for all the following interfaces is determined by the contagion rules when applied to the arguments.
+ *abstract-integer1* *abstract-integer2* => *abstract-integer*
+ *abstract-integer* *float1* => *float*
+ *float1* *abstract-integer* => *float*
- *abstract-integer1* *abstract-integer2* => *abstract-integer*
- *abstract-integer* *float1* => *float*
- *float1* *abstract-integer* => *float*
\* *abstract-integer1* *abstract-integer2* => *abstract-integer*
\* *abstract-integer* *float1* => *float*
\* *float1* *abstract-integer* => *float*
The return value of the following interface is of the same float format as the argument:
negative *abstract-integer* => *negative-abstract-integer*
The second return value of all the following interfaces is of the same float format as the argument:
floor *abstract-integer* => *abstract-integer1* *abstract-integer2*
floor *float1* => *abstract-integer* *float*
ceiling *abstract-integer* => *abstract-integer1* *abstract-integer2*
ceiling *float1* => *abstract-integer* *float*
round *abstract-integer* => *abstract-integer1* *abstract-integer2*
round *float1* => *abstract-integer* *float*
truncate *abstract-integer* => *abstract-integer1* *abstract-integer2*
truncate *float1* => *abstract-integer* *float*
The second return value of all the following interfaces is of the same float format as the first argument:
floor/ *abstract-integer1* *abstract-integer2* => *abstract-integer3* *abstract-integer4*
floor/ *float1* *abstract-integer1* => *abstract-integer2* *float2*
ceiling/ *abstract-integer1* *abstract-integer2* => *abstract-integer3* *abstract-integer4*
ceiling/ *float1* *abstract-integer1* => *abstract-integer2* *float2*
round/ *abstract-integer1* *abstract-integer2* => *abstract-integer3* *abstract-integer4*
round/ *float1* *abstract-integer1* => *abstract-integer2* *float2*
truncate/ *abstract-integer1* *abstract-integer2* => *abstract-integer3* *abstract-integer4
truncate/ *float1* *abstract-integer1* => *abstract-integer2* *float2*
The second return value of the following interfaces is of the same float format as the second argument:
floor/ *abstract-integer1* *float1* => *abstract-integer2* *float2*
ceiling/ *abstract-integer1* *float1* => *abstract-integer2* *float2*
round/ *abstract-integer1* *float1* => *abstract-integer2* *float2*
truncate/ *abstract-integer1* *float1* => *abstract-integer2* *float2*
The return value of the following interfaces is of the same float format as the first argument:
modulo *float1* *abstract-integer* => *float*
remainder *float1* *abstract-integer* => *float*
The return value of the following interfaces is of the same float format as the second argument:
modulo *abstract-integer1* *abstract-integer2* => *abstract-integer*
modulo *abstract-integer* *float1* => *float*
remainder *abstract-integer1* *abstract-integer2* => *abstract-integer*
remainder *abstract-integer* *float1* => *float*
The behavior of the following miscellaneous interfaces is also modified by the Big-Integers library:
^ *abstract-integer1* *integer* => *abstract-integer
abs *abstract-integer1* => *abstract-integer*
logior #rest *abstract-integer1* => *abstract-integer*
logxor #rest *abstract-integer1* => *abstract-integer*
logand #rest *abstract-integer1* => *abstract-integer*
lognot *abstract-integer1* => *abstract-integer*
logbit? *integer* *abstract-integer* => *boolean*
ash *abstract-integer1* *integer* => *abstract-integer*
lcm *abstract-integer1* *abstract-integer2* => *abstract-integer*
gcd *abstract-integer1* *abstract-integer2* => *abstract-integer*