Section2.2

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## 2.2 Writing Types and Modes

We have already seen in the last section ugTypesBasic several examples of types. Most of these examples had either no arguments (for example, Integer) or one argument (for example, Polynomial (Integer)). In this section we give details about writing arbitrary types. We then define modes and discuss how to write them. We conclude the section with a discussion on constructor abbreviations.

When might you need to write a type or mode? You need to do so when you declare variables.

a : PositiveInteger
Type: Void

You need to do so when you declare functions (See Section ugTypesDeclare ),

f : Integer -> String
Type: Void

You need to do so when you convert an object from one type to another (See Section ugTypesConvert ).

factor(2 :: Complex(Integer))

-i(1+i)2

Type: Factored Complex Integer
(2 = 3) $Integer

false

Type: Boolean

You need to do so when you give computation target type information (See Section ugTypesPkgCall ).

(2 = 3)@Boolean

false

Type: Boolean

### 2.2.1 Types with No Arguments

A constructor with no arguments can be written either type:using parentheses with or without parentheses:using with types trailing opening and closing parentheses ``()''.

Boolean() is the same as Boolean
Integer() is the same as Integer
String() is the same as String
Void() is the same as Void

It is customary to omit the parentheses.

### 2.2.2 Types with One Argument

A constructor with one argument can frequently be type:using parentheses written with no parentheses:using with types parentheses. Types nest from right to left so that Complex Fraction Polynomial Integer is the same as Complex (Fraction (Polynomial (Integer))). You need to use parentheses to force the application of a constructor to the correct argument, but you need not use any more than is necessary to remove ambiguities.

Here are some guidelines for using parentheses (they are possibly slightly more restrictive than they need to be).

If the argument is an expression like 2+3 then you must enclose the argument in parentheses.

e : PrimeField(2 + 3)
Type: Void

If the type is to be used with package calling then you must enclose the argument in parentheses.

content(2) $Polynomial(Integer)

2

Type: Integer

Alternatively, you can write the type without parentheses then enclose the whole type expression with parentheses.

content(2) $(Polynomial Complex Fraction Integer)

2

Type: Complex Fraction Integer

If you supply computation target type information (See Section ugTypesPkgCall ) then you should enclose the argument in parentheses.

(2/3)@Fraction(Polynomial(Integer))

2 3

Type: Fraction Polynomial Integer

If the type itself has parentheses around it and we are not in the case of the first example above, then the parentheses can usually be omitted.

(2/3)@Fraction(Polynomial Integer)

2 3

Type: Fraction Polynomial Integer

If the type is used in a declaration and the argument is a single-word type, integer or symbol, then the parentheses can usually be omitted.

(d,f,g) : Complex Polynomial Integer
Type: Void

### 2.2.3 Types with More Than One Argument

If a constructor type:using parentheses has more than parentheses:using with types one argument, you must use parentheses. Some examples are

UnivariatePolynomial(x, Float)
MultivariatePolynomial([z,w,r], Complex Float)
SquareMatrix(3, Integer)
FactoredFunctions2(Integer,Fraction Integer)

### 2.2.4 Modes

A mode is a type that possibly is a question mark (?) or contains one in an argument position. For example, the following are all modes.

?

Polynomial ?

Matrix Polynomial ?

SquareMatrix(3,?)

Integer

OneDimensionalArray(Float)

As is evident from these examples, a mode is a type with a part that is not specified (indicated by a question mark). Only one ``?'' is allowed per mode and it must appear in the most deeply nested argument that is a type. Thus ?(Integer), Matrix(? (Polynomial)), SquareMatrix(?, Integer) (it requires a numeric argument) and SquareMatrix(?, ?) are all invalid. The question mark must take the place of a domain, not data. This rules out, for example, the two SquareMatrix expressions.

Modes can be used for declarations (See Section ugTypesDeclare ) and conversions (Section ugTypesConvert ). However, you cannot use a mode for package calling or giving target type information.

### 2.2.5 Abbreviations

Every constructor has an abbreviation that abbreviation:constructor you can freely constructor:abbreviation substitute for the constructor name. In some cases, the abbreviation is nothing more than the capitalized version of the constructor name.

Aside from allowing types to be written more concisely, abbreviations are used by Axiom to name various system files for constructors (such as library filenames, test input files and example files). Here are some common abbreviations.

COMPLEX abbreviates Complex | DFLOAT abbreviates DoubleFloat
EXPR abbreviates Expression | FLOAT abbreviates Float
FRAC abbreviates Fraction | INT abbreviates Integer
MATRIX abbreviates Matrix | NNI abbreviates NonNegativeInteger
PI abbreviates PositiveInteger | POLY abbreviates Polynomial
STRING abbreviates String | UP abbreviates UnivariatePolynomial

You can combine both full constructor names and abbreviations in a type expression. Here are some types using abbreviations.

POLY INT | is the same as | Polynomial(INT)
POLY(Integer) | is the same as | Polynomial(Integer)
POLY(Integer) | is the same as | Polynomial(INT)
FRAC(COMPLEX(INT)) | is the same as | Fraction Complex Integer
FRAC(COMPLEX(INT)) | is the same as | FRAC(Complex Integer)

There are several ways of finding the names of constructors and their abbreviations. For a specific constructor, use )abbreviation query. abbreviation You can also use the )what system command to see the names and abbreviations of constructors. what For more information about )what, see ugSysCmdwhat .

)abbreviation query can be abbreviated (no pun intended) to )abb q.

)abb q Integer
INT abbreviates domain Integer

The )abbreviation query command lists the constructor name if you give the abbreviation. Issue )abb q if you want to see the names and abbreviations of all Axiom constructors.

)abb q DMP

DMP abbreviates domain DistributedMultivariatePolynomial

Issue this to see all packages whose names contain the string ``ode''. what packages

)what packages ode
---------------------- Packages -----------------------
Packages with names matching patterns:
ode
EXPRODE ExpressionSpaceODESolver
FCPAK1 FortranCodePackage1
GRAY GrayCode
LODEEF ElementaryFunctionLODESolver
NODE1 NonLinearFirstOrderODESolver
ODECONST ConstantLODE
ODEEF ElementaryFunctionODESolver
ODEINT ODEIntegration
ODEPAL PureAlgebraicLODE
ODERAT RationalLODE
ODERED ReduceLODE
ODESYS SystemODESolver
ODETOOLS ODETools
UTSODE UnivariateTaylorSeriesODESolver
UTSODETL UTSodetools
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