A data kind distinguishing links in a chain of Galois connections of length 2 or 3.

• L-tagged types are increasing (e.g. ceiling, maximize)
• R-tagged types are decreasing (e.g. floor, minimize)

If a connection is existentialized over this value (i.e. has type forall k. Cast k a b) then it can provide either of two functions f, h :: a -> b.

This is useful because it enables rounding, truncation, medians, etc.

A chain of Galois connections of length 2 or 3.

Connections have many nice properties wrt numerical conversion:

>>> upper ratf32 (1 / 8)    -- eighths are exactly representable in a float
1 % 8
>>> upper ratf32 (1 / 7)    -- sevenths are not
9586981 % 67108864
>>> floor ratf32 &&& ceiling ratf32 $ 1 % 8
(0.125,0.125)
>>> floor ratf32 &&& ceiling ratf32 $ 1 % 7
(0.14285713,0.14285715)

Another example avoiding loss-of-precision:

>>> f x y = (x + y) - x
>>> maxOdd32 = 1.6777215e7
>>> f maxOdd32 2.0 :: Float
1.0
>>> round2 f64f32 f maxOdd32 2.0
2.0

Left-to-right composition

Right-to-left composition

Lift a Cast into a functor.

Caution: This function will result in an invalid connection if the functor alters the internal preorder (e.g. Down).

Lift two connections into a connection on the coproduct order

(choice id) (ab >>> cd) = (choice id) ab >>> (choice id) cd
(flip choice id) (ab >>> cd) = (flip choice id) ab >>> (flip choice id) cd

Lift two connections into a connection on the coproduct order

Lift two connections into a connection on the product order

(strong id) (ab >>> cd) = (strong id) ab >>> (strong id) cd
(flip strong id) (ab >>> cd) = (flip strong id) ab >>> (flip strong id) cd

Lift two connections into a connection on the product order

The defining connections of a bounded preorder.

The defining connections of a total order.

>>> floor ordered &&& ceiling ordered $ (True, False)
(False,True)

The identity connection.

A Galois connection between two monotone functions.

A Galois connection between f and g, written \(f \dashv g \) is an adjunction in the category of preorders.

Each side of the connection may be defined in terms of the other:

\( g(x) = \sup \{y \in E \mid f(y) \leq x \} \)

\( f(x) = \inf \{y \in E \mid g(y) \geq x \} \)

Caution: CastL f g must obey \(f \dashv g \). This condition is not checked.

For further information see Property.

Witness to the mirror symmetry between CastL and CastR.

swapL . swapR = id
swapR . swapL = id

Extract the upper half of a CastL.

>>> upper ratf32 (1 / 8)    -- eighths are exactly representable in a float
1 % 8
>>> upper ratf32 (1 / 7)    -- sevenths are not
9586981 % 67108864

Map over a CastL from the right.

This is the unit of the resulting monad:

x <~ upper1 c id x

>>> compare pi $ upper1 f64f32 id pi
LT

Zip over a CastL from the right.

Extract the lower half of a CastL.

ceiling identity = id
ceiling c (x \/ y) = ceiling c x \/ ceiling c y

The latter law is the adjoint functor theorem for preorders.

>>> ceiling ratf32 (0 :% 0)
NaN
>>> ceiling ratf32 (13 :% 10)
1.3000001
>>> ceiling f64f32 pi
3.1415927

Map over a CastL from the left.

ceiling1 identity = id

This is the counit of the resulting comonad:

x >~ ceiling1 c id x

Zip over a CastL from the left.

Generalized maximum.

A Galois connection between two monotone functions.

CastR is the mirror image of CastL.

If you only require one connection there is no particular reason to use one version over the other. However many use cases (e.g. rounding) require an adjoint triple of connections that can lower into a standard connection in either of two ways.

Caution: CastR f g must obey \(f \dashv g \). This condition is not checked.

For further information see Property.

Witness to the mirror symmetry between CastL and CastR.

swapL . swapR = id
swapR . swapL = id

Extract the lower half of a CastR.

Map over a CastR from the left.

This is the counit of the resulting comonad:

x >~ lower1 c id x

>>> compare pi $ lower1 f64f32 id pi
GT

Zip over a CastR from the left.

Extract the upper half of a CastR

floor identity = id
floor c (x /\ y) = floor c x /\ floor c y

The latter law is the adjoint functor theorem for preorders.

>>> floor ratf32 (0 :% 0)
NaN
>>> floor ratf32 (13 :% 10)
1.3
>>> floor f64f32 pi
3.1415925

Map over a CastR from the right.

floor1 identity = id

This is the unit of the resulting monad:

x <~ floor1 c id x

Zip over a CastR from the right.

Generalized minimum.

An adjoint triple of Galois connections.

An adjoint triple is a chain of connections of length 3:

\(f \dashv g \dashv h \)

When applied to a CastL or CastR, the two functions of type a -> b returned will be identical.

Caution: Cast f g h must obey \(f \dashv g \dashv h\). This condition is not checked.

For detailed properties see Property.

Return the midpoint of the interval containing x.

For example, the (double-precision) error of the single-precision floating point approximation of pi is:

>>> pi - midpoint f64f32 pi
3.1786509424591713e-8

Determine which half of the interval between 2 representations of a a particular value lies.

 interval c x = pcompare (x - lower1 c id x) (upper1 c id x - x)

>>> maybe False (== EQ) $ interval f64f32 (midpoint f64f32 pi)
True

Return the nearest value to x.

round identity = id

If x lies halfway between two finite values, then return the value with the smaller absolute value (i.e. round towards from zero).

See https://en.wikipedia.org/wiki/Rounding.

Lift a unary function over an adjoint triple.

round1 identity = id

Results are rounded to the nearest value with ties away from 0.

Lift a binary function over an adjoint triple.

round2 identity = id

Results are rounded to the nearest value with ties away from 0.

For example, to avoid a loss of precision:

>>> f x y = (x + y) - x
>>> maxOdd32 = 1.6777215e7
>>> f maxOdd32 2.0 :: Float
1.0
>>> round2 ratf32 f maxOdd32 2.0
2.0

Truncate towards zero.

truncate identity = id

Lift a unary function over an adjoint triple.

truncate1 identity = id

Results are truncated towards zero.

Lift a binary function over an adjoint triple.

truncate2 identity = id

Results are truncated towards zero.

Birkoff's median operator.

median x x y = x
median x y z = median z x y
median x y z = median x z y
median (median x w y) w z = median x w (median y w z)

>>> median f32f32 1.0 9.0 7.0
7.0
>>> median f32f32 1.0 9.0 (0.0 / 0.0)
9.0

Invert a Cast.

upL . downL = downL . upL = id

Invert a Cast.

upR . downR = downR . upR = id

Invert a Cast.

>>> let counit = upper1 (downL $ bounded @Ordering) id
>>> counit (Down LT)
Down LT
>>> counit (Down GT)
Down LT

Invert a Cast.

>>> let unit = lower1 (downR $ bounded @Ordering) id
>>> unit (Down LT)
Down GT
>>> unit (Down GT)
Down GT

Obtain the principal filter in B generated by an element of A.

A subset B of a lattice is an filter if and only if it is an upper set that is closed under finite meets, i.e., it is nonempty and for all x, y in B, the element meet c x y is also in b.

filterL and filterR commute with Down:

filterL c a b <=> filterR c (Down a) (Down b)

filterL c (Down a) (Down b) <=> filterR c a b

filterL c a is upward-closed for all a:

a <= b1 && b1 <= b2 => a <= b2
a1 <= b && a2 <= b => minimize c (ceiling c a1) (ceiling c a2) <= b

See https://en.wikipedia.org/wiki/Filter_(mathematics)

Obtain the principal ideal in B generated by an element of A.

A subset B of a lattice is an ideal if and only if it is a lower set that is closed under finite joins, i.e., it is nonempty and for all x, y in B, the element join c x y is also in B.

filterR c a is downward-closed for all a:

a >= b1 && b1 >= b2 => a >= b2

a1 >= b && a2 >= b => maximize c (floor c a1) (floor c a2) >= b

See https://en.wikipedia.org/wiki/Ideal_(order_theory)

Eliminate an Extended.

Extended r is an extension of r with positive/negative infinity (±∞).