Copyright	(c) Erich Gut
License	BSD3
Maintainer	zerich.gut@gmail.com
Safe Haskell	Safe-Inferred
Language	Haskell2010

OAlg.Structure.Multiplicative.Proposition

Contents

Multiplicative
X
Oriented Direction
- Total
- Orientation

Description

Propositions on Multiplicative structures.

Synopsis

prpMlt :: Multiplicative c => XMlt c -> Statement
data XMlt c = XMlt {
- xMltN :: X N
- xMltPoint :: X (Point c)
- xMltFactor :: X c
- xMltEndo :: X (Endo c)
- xMltMltp2 :: X (Mltp2 c)
- xMltMltp3 :: X (Mltp3 c)
}
newtype Endo q = Endo q
data Mltp2 c = Mltp2 c c
data Mltp3 c = Mltp3 c c c
prpMlt1 :: Multiplicative c => p c -> X (Point c) -> Statement
prpMlt2 :: Multiplicative c => X (c, c) -> Statement
prpMlt2_1 :: Multiplicative c => Mltp2 c -> Statement
prpMlt2_2 :: Multiplicative c => c -> c -> Statement
prpMlt3 :: Multiplicative c => X c -> Statement
prpMlt4 :: Multiplicative c => X (Mltp3 c) -> Statement
prpMlt5 :: Multiplicative c => X N -> X c -> X (Endo c) -> Statement
class XStandardMlt c where
- xStandardMlt :: XMlt c
xMlt :: Multiplicative c => XOrtSite d c -> X N -> X (Endo c) -> XMlt c
xMltp2 :: Multiplicative c => XOrtSite d c -> X (Mltp2 c)
xMltp3 :: Multiplicative c => XOrtSite d c -> X (Mltp3 c)
xoMlt :: Multiplicative c => X N -> XOrtOrientation c -> XMlt c
xMltTtl :: Singleton (Point c) => X N -> X c -> XMlt c
xMltOrnt :: Entity p => X N -> X p -> XMlt (Orientation p)

Multiplicative

prpMlt :: Multiplicative c => XMlt c -> Statement Source #

validity of the Multiplicative structure of c.

data XMlt c Source #

random variable for Multiplicative structures.

Note As the multiplication could by costly, it is recommended to use a bounded random variable for xMltN which serves to validate npower.

Constructors

XMlt
Fields xMltN :: X N xMltPoint :: X (Point c) xMltFactor :: X c xMltEndo :: X (Endo c) xMltMltp2 :: X (Mltp2 c) xMltMltp3 :: X (Mltp3 c)

Instances

Instances details

Oriented c => Validable (XMlt c) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Proposition Methods valid :: XMlt c -> Statement Source #
type Dual (XMlt c :: Type) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Proposition type Dual (XMlt c :: Type) = XMlt (Op c)

newtype Endo q Source #

predicate for endos.

Constructors

Endo q

Instances

Instances details

Show q => Show (Endo q) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Proposition Methods showsPrec :: Int -> Endo q -> ShowS # show :: Endo q -> String # showList :: [Endo q] -> ShowS #
Eq q => Eq (Endo q) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Proposition Methods (==) :: Endo q -> Endo q -> Bool # (/=) :: Endo q -> Endo q -> Bool #
Ord q => Ord (Endo q) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Proposition Methods compare :: Endo q -> Endo q -> Ordering # (<) :: Endo q -> Endo q -> Bool # (<=) :: Endo q -> Endo q -> Bool # (>) :: Endo q -> Endo q -> Bool # (>=) :: Endo q -> Endo q -> Bool # max :: Endo q -> Endo q -> Endo q # min :: Endo q -> Endo q -> Endo q #
Oriented q => Validable (Endo q) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Proposition Methods valid :: Endo q -> Statement Source #

data Mltp2 c Source #

predicate for two multiplicable factors.

Constructors

Mltp2 c c

Instances

Instances details

Show c => Show (Mltp2 c) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Proposition Methods showsPrec :: Int -> Mltp2 c -> ShowS # show :: Mltp2 c -> String # showList :: [Mltp2 c] -> ShowS #
Eq c => Eq (Mltp2 c) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Proposition Methods (==) :: Mltp2 c -> Mltp2 c -> Bool # (/=) :: Mltp2 c -> Mltp2 c -> Bool #
Ord c => Ord (Mltp2 c) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Proposition Methods compare :: Mltp2 c -> Mltp2 c -> Ordering # (<) :: Mltp2 c -> Mltp2 c -> Bool # (<=) :: Mltp2 c -> Mltp2 c -> Bool # (>) :: Mltp2 c -> Mltp2 c -> Bool # (>=) :: Mltp2 c -> Mltp2 c -> Bool # max :: Mltp2 c -> Mltp2 c -> Mltp2 c # min :: Mltp2 c -> Mltp2 c -> Mltp2 c #
Oriented c => Dualisable (Mltp2 c) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Proposition Methods toDual :: Mltp2 c -> Dual (Mltp2 c) Source # fromDual :: Dual (Mltp2 c) -> Mltp2 c Source #
Oriented c => Validable (Mltp2 c) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Proposition Methods valid :: Mltp2 c -> Statement Source #
type Dual (Mltp2 c :: Type) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Proposition type Dual (Mltp2 c :: Type) = Mltp2 (Op c)

data Mltp3 c Source #

predicate for three multiplicable factors.

Constructors

Mltp3 c c c

Instances

Instances details

Show c => Show (Mltp3 c) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Proposition Methods showsPrec :: Int -> Mltp3 c -> ShowS # show :: Mltp3 c -> String # showList :: [Mltp3 c] -> ShowS #
Eq c => Eq (Mltp3 c) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Proposition Methods (==) :: Mltp3 c -> Mltp3 c -> Bool # (/=) :: Mltp3 c -> Mltp3 c -> Bool #
Ord c => Ord (Mltp3 c) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Proposition Methods compare :: Mltp3 c -> Mltp3 c -> Ordering # (<) :: Mltp3 c -> Mltp3 c -> Bool # (<=) :: Mltp3 c -> Mltp3 c -> Bool # (>) :: Mltp3 c -> Mltp3 c -> Bool # (>=) :: Mltp3 c -> Mltp3 c -> Bool # max :: Mltp3 c -> Mltp3 c -> Mltp3 c # min :: Mltp3 c -> Mltp3 c -> Mltp3 c #
Oriented c => Dualisable (Mltp3 c) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Proposition Methods toDual :: Mltp3 c -> Dual (Mltp3 c) Source # fromDual :: Dual (Mltp3 c) -> Mltp3 c Source #
Oriented c => Validable (Mltp3 c) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Proposition Methods valid :: Mltp3 c -> Statement Source #
type Dual (Mltp3 c :: Type) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Proposition type Dual (Mltp3 c :: Type) = Mltp3 (Op c)

prpMlt1 :: Multiplicative c => p c -> X (Point c) -> Statement Source #

validity of one according to OAlg.Structure.Multiplicative.Definition.

prpMlt2 :: Multiplicative c => X (c, c) -> Statement Source #

validity of * according to OAlg.Structure.Multiplicative.Definition.

prpMlt2_1 :: Multiplicative c => Mltp2 c -> Statement Source #

validity of * for two multiplicable factors according to OAlg.Structure.Multiplicative.Definition.

prpMlt2_2 :: Multiplicative c => c -> c -> Statement Source #

validity of * for two not multiplicable factors according to OAlg.Structure.Multiplicative.Definition.

prpMlt3 :: Multiplicative c => X c -> Statement Source #

validity according to OAlg.Structure.Multiplicative.Definition.

prpMlt4 :: Multiplicative c => X (Mltp3 c) -> Statement Source #

validity according to OAlg.Structure.Multiplicative.Definition.

prpMlt5 :: Multiplicative c => X N -> X c -> X (Endo c) -> Statement Source #

validity of npower according to OAlg.Structure.Multiplicative.Definition.

Note As the multiplication can by very costly the random variable for X N - which serves to check OAlg.Structure.Multiplicative.Definition - has to be chosen carefully.

X

class XStandardMlt c where Source #

standard random variable for Multiplicative structures.

Methods

xStandardMlt :: XMlt c Source #

Instances

Instances details

XStandardMlt N Source #
Instance details Defined in OAlg.Structure.Multiplicative.Proposition Methods xStandardMlt :: XMlt N Source #
XStandardMlt Q Source #
Instance details Defined in OAlg.Structure.Multiplicative.Proposition Methods xStandardMlt :: XMlt Q Source #
XStandardMlt Z Source #
Instance details Defined in OAlg.Structure.Multiplicative.Proposition Methods xStandardMlt :: XMlt Z Source #
(Entity p, XStandard p) => XStandardMlt (Orientation p) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Proposition Methods xStandardMlt :: XMlt (Orientation p) Source #

xMlt :: Multiplicative c => XOrtSite d c -> X N -> X (Endo c) -> XMlt c Source #

random variable for Multiplicative structures.

xMltp2 :: Multiplicative c => XOrtSite d c -> X (Mltp2 c) Source #

random variable of two multiplicable factors.

xMltp3 :: Multiplicative c => XOrtSite d c -> X (Mltp3 c) Source #

random variable of three multiplicable factors.

xoMlt :: Multiplicative c => X N -> XOrtOrientation c -> XMlt c Source #

the induced random variable for multiplicable structures.

Oriented Direction

Total

xMltTtl :: Singleton (Point c) => X N -> X c -> XMlt c Source #

random variable for total Multiplicative structures.

Orientation

xMltOrnt :: Entity p => X N -> X p -> XMlt (Orientation p) Source #

random variable for the Multiplicative structure of Orientation p.