Math.Core.Utils

A module of simple utility functions which are used throughout the rest of the library

Documentation

toSet :: Ord a => [a] -> [a]

sortDesc :: Ord a => [a] -> [a]

insertDesc :: Ord a => a -> [a] -> [a]

setUnionAsc :: Ord a => [a] -> [a] -> [a]

The set union of two ascending lists. If both inputs are strictly increasing, then the output is their union and is strictly increasing. The code does not check that the lists are strictly increasing.

setUnionDesc :: Ord a => [a] -> [a] -> [a]

intersectAsc :: Ord a => [a] -> [a] -> [a]

The (multi-)set intersection of two ascending lists. If both inputs are strictly increasing, then the output is the set intersection and is strictly increasing. If both inputs are weakly increasing, then the output is the multiset intersection (with multiplicity), and is weakly increasing.

multisetSumAsc :: Ord a => [a] -> [a] -> [a]

The multiset sum of two ascending lists. If xs and ys are ascending, then multisetSumAsc xs ys == sort (xs++ys). The code does not check that the lists are ascending.

multisetSumDesc :: Ord a => [a] -> [a] -> [a]

The multiset sum of two descending lists. If xs and ys are descending, then multisetSumDesc xs ys == sortDesc (xs++ys). The code does not check that the lists are descending.

diffAsc :: Ord a => [a] -> [a] -> [a]

The multiset or set difference between two ascending lists. If xs and ys are ascending, then diffAsc xs ys == xs \ ys, and diffAsc is more efficient. If xs and ys are sets (that is, have no repetitions), then diffAsc xs ys is the set difference. The code does not check that the lists are ascending.

diffDesc :: Ord a => [a] -> [a] -> [a]

The multiset or set difference between two descending lists. If xs and ys are descending, then diffDesc xs ys == xs \ ys, and diffDesc is more efficient. If xs and ys are sets (that is, have no repetitions), then diffDesc xs ys is the set difference. The code does not check that the lists are descending.

isSubsetAsc :: Ord a => [a] -> [a] -> Bool

isSubMultisetAsc :: Ord a => [a] -> [a] -> Bool

elemAsc :: Ord a => a -> [a] -> Bool

Is the element in the ascending list?

With infinite lists, this can fail to terminate. For example, elemAsc 1 [12,34,7/8..] would fail to terminate. However, with a list of Integer, this will always terminate.

notElemAsc :: Ord a => a -> [a] -> Bool

Is the element not in the ascending list? (With infinite lists, this can fail to terminate.)

picks :: [a] -> [(a, [a])]

Return all the ways to "pick one and leave the others" from a list

pairs :: [t] -> [(t, t)]

ordpair :: Ord b => b -> b -> (b, b)

foldcmpl :: (b -> b -> Bool) -> [b] -> Bool

isWeaklyIncreasing :: Ord t => [t] -> Bool

isStrictlyIncreasing :: Ord t => [t] -> Bool

isWeaklyDecreasing :: Ord t => [t] -> Bool

isStrictlyDecreasing :: Ord t => [t] -> Bool

cmpfst :: Ord a => (a, b1) -> (a, b2) -> Ordering

eqfst :: Eq a => (a, b1) -> (a, b2) -> Bool

fromBase :: (Foldable t, Num a) => a -> t a -> a

powersetdfs :: [a] -> [[a]]

Given a set xs, represented as an ordered list, powersetdfs xs returns the list of all subsets of xs, in lex order

powersetbfs :: [a] -> [[a]]

Given a set xs, represented as an ordered list, powersetbfs xs returns the list of all subsets of xs, in shortlex order

combinationsOf :: Int -> [a] -> [[a]]

Given a positive integer k, and a set xs, represented as a list, combinationsOf k xs returns all k-element subsets of xs. The result will be in lex order, relative to the order of the xs.

choose :: Integral a => a -> a -> a

choose n k is the number of ways of choosing k distinct elements from an n-set

class FinSet x where

The class of finite sets

Methods

elts :: [x]

Instances

FinSet F25

Defined in Math.Core.Field

Methods

elts :: [F25]

FinSet F16

Defined in Math.Core.Field

Methods

elts :: [F16]

FinSet F9

Defined in Math.Core.Field

Methods

elts :: [F9]

FinSet F8

Defined in Math.Core.Field

Methods

elts :: [F8]

FinSet F4

Defined in Math.Core.Field

Methods

elts :: [F4]

FinSet F23

Defined in Math.Core.Field

Methods

elts :: [F23]

FinSet F19

Defined in Math.Core.Field

Methods

elts :: [F19]

FinSet F17

Defined in Math.Core.Field

Methods

elts :: [F17]

FinSet F13

Defined in Math.Core.Field

Methods

elts :: [F13]

FinSet F11

Defined in Math.Core.Field

Methods

elts :: [F11]

FinSet F7

Defined in Math.Core.Field

Methods

elts :: [F7]

FinSet F5

Defined in Math.Core.Field

Methods

elts :: [F5]

FinSet F3

Defined in Math.Core.Field

Methods

elts :: [F3]

FinSet F2

Defined in Math.Core.Field

Methods

elts :: [F2]

IntegerAsType p => FinSet (Fp p)

Defined in Math.Algebra.Field.Base

Methods

elts :: [Fp p]

(FinSet fp, Eq fp, Num fp, PolynomialAsType fp poly) => FinSet (ExtensionField fp poly)

Defined in Math.Algebra.Field.Extension

Methods

elts :: [ExtensionField fp poly]

class HasInverses a where

A class representing algebraic structures having an inverse operation. Note that in some cases not every element has an inverse.

Methods

inverse :: a -> a

Instances

HasInverses SSymF

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

inverse :: SSymF -> SSymF

Ord a => HasInverses (Permutation a)

The HasInverses instance is what enables us to write g^-1 for the inverse of a group element.

Defined in Math.Algebra.Group.PermutationGroup

Methods

inverse :: Permutation a -> Permutation a

HasInverses (GroupAlgebra Q)

Note that the inverse of a group algebra element can only be efficiently calculated if the group generated by the non-zero terms is very small (eg <100 elements).

Defined in Math.Algebras.GroupAlgebra

Methods

inverse :: GroupAlgebra Q -> GroupAlgebra Q

(Eq k, Fractional k, Ord a, Show a) => HasInverses (Vect k (Interval a))

Defined in Math.Combinatorics.IncidenceAlgebra

Methods

inverse :: Vect k (Interval a) -> Vect k (Interval a)

(^-) :: (Num a, HasInverses a, Integral b) => a -> b -> a infix 8

A trick: x^-1 returns the inverse of x