# OrderedCompletion R¶

complet.spad line 1 [edit on github]

R: SetCategory

Adjunction of two real infinites quantities to a set. Date Created: 4 Oct 1989

+: (%, %) -> % if R has AbelianMonoid

-: % -> % if R has AbelianGroup

- <=: (%, %) -> Boolean if R has OrderedSet
from PartialOrder

- <: (%, %) -> Boolean if R has OrderedSet
from PartialOrder

- >=: (%, %) -> Boolean if R has OrderedSet
from PartialOrder

- >: (%, %) -> Boolean if R has OrderedSet
from PartialOrder

- coerce: % -> OutputForm
from CoercibleTo OutputForm

- coerce: Fraction Integer -> % if R has RetractableTo Fraction Integer
from CoercibleFrom Fraction Integer

- coerce: Integer -> % if R has RetractableTo Integer
from CoercibleFrom Integer

- coerce: R -> %
from CoercibleFrom R

- convert: % -> InputForm if R has ConvertibleTo InputForm
from ConvertibleTo InputForm

- finite?: % -> Boolean
`finite?(x)`

tests if`x`

is finite.

- infinite?: % -> Boolean
`infinite?(x)`

tests if`x`

is +infinity or -infinity,

- latex: % -> String
from SetCategory

- max: (%, %) -> % if R has OrderedSet
from OrderedSet

- min: (%, %) -> % if R has OrderedSet
from OrderedSet

- minusInfinity: () -> %
`minusInfinity()`

returns -infinity.

- plusInfinity: () -> %
`plusInfinity()`

returns +infinity.

- rational?: % -> Boolean if R has IntegerNumberSystem
`rational?(x)`

tests if`x`

is a finite rational number.

- rational: % -> Fraction Integer if R has IntegerNumberSystem
`rational(x)`

returns`x`

as a finite rational number. Error: if`x`

cannot be so converted.

- rationalIfCan: % -> Union(Fraction Integer, failed) if R has IntegerNumberSystem
`rationalIfCan(x)`

returns`x`

as a finite rational number if it is one and “failed” otherwise.

- retract: % -> Fraction Integer if R has RetractableTo Fraction Integer
from RetractableTo Fraction Integer

- retract: % -> Integer if R has RetractableTo Integer
from RetractableTo Integer

- retract: % -> R
from RetractableTo R

- retractIfCan: % -> Union(Fraction Integer, failed) if R has RetractableTo Fraction Integer
from RetractableTo Fraction Integer

- retractIfCan: % -> Union(Integer, failed) if R has RetractableTo Integer
from RetractableTo Integer

- retractIfCan: % -> Union(R, failed)
from RetractableTo R

- smaller?: (%, %) -> Boolean if R has OrderedSet
from Comparable

- whatInfinity: % -> SingleInteger
`whatInfinity(x)`

returns 0 if`x`

is finite, 1 if`x`

is +infinity, and`-1`

if`x`

is -infinity.

CoercibleFrom Fraction Integer if R has RetractableTo Fraction Integer

CoercibleFrom Integer if R has RetractableTo Integer

Comparable if R has OrderedSet

ConvertibleTo InputForm if R has ConvertibleTo InputForm

OrderedSet if R has OrderedSet

PartialOrder if R has OrderedSet

RetractableTo Fraction Integer if R has RetractableTo Fraction Integer

RetractableTo Integer if R has RetractableTo Integer