Data.Monoid.GCD

Class of Abelian monoids that allow the greatest common divisor to be found for any two given values. The operations must satisfy the following laws:

gcd a b == commonPrefix a b == commonSuffix a b
Just a' = a </> p && Just b' = b </> p
   where p = gcd a b

In addition, the gcd operation must satisfy the following properties:

Uniqueness

all isJust
    [ a </> c
    , b </> c
    , c </> gcd a b
    ]
==>
    (c == gcd a b)

Idempotence

gcd a a == a

Identity

gcd mempty a == mempty

gcd a mempty == mempty

Commutativity

gcd a b == gcd b a

Associativity

gcd (gcd a b) c == gcd a (gcd b c)

Class of monoids capable of finding the equivalent of greatest common divisor on the left side of two monoidal values. The following laws must be respected:

stripCommonPrefix a b == (p, a', b')
   where p = commonPrefix a b
         Just a' = stripPrefix p a
         Just b' = stripPrefix p b
p == commonPrefix a b && p <> a' == a && p <> b' == b
   where (p, a', b') = stripCommonPrefix a b

Furthermore, commonPrefix must return the unique greatest common prefix that contains, as its prefix, any other prefix x of both values:

not (x `isPrefixOf` a && x `isPrefixOf` b) || x `isPrefixOf` commonPrefix a b

and it cannot itself be a suffix of any other common prefix y of both values:

not (y `isPrefixOf` a && y `isPrefixOf` b && commonPrefix a b `isSuffixOf` y)

In addition, the commonPrefix operation must satisfy the following properties:

Idempotence

commonPrefix a a == a

Identity

commonPrefix mempty a == mempty

commonPrefix a mempty == mempty

Commutativity

commonPrefix a b == commonPrefix b a

Associativity

commonPrefix (commonPrefix a b) c
==
commonPrefix a (commonPrefix b c)

Class of monoids capable of finding the equivalent of greatest common divisor on the right side of two monoidal values. The following laws must be respected:

stripCommonSuffix a b == (a', b', s)
   where s = commonSuffix a b
         Just a' = stripSuffix p a
         Just b' = stripSuffix p b
s == commonSuffix a b && a' <> s == a && b' <> s == b
   where (a', b', s) = stripCommonSuffix a b

Furthermore, commonSuffix must return the unique greatest common suffix that contains, as its suffix, any other suffix x of both values:

not (x `isSuffixOf` a && x `isSuffixOf` b) || x `isSuffixOf` commonSuffix a b

and it cannot itself be a prefix of any other common suffix y of both values:

not (y `isSuffixOf` a && y `isSuffixOf` b && commonSuffix a b `isPrefixOf` y)

In addition, the commonSuffix operation must satisfy the following properties:

Idempotence

commonSuffix a a == a

Identity

commonSuffix mempty a == mempty

commonSuffix a mempty == mempty

Commutativity

commonSuffix a b == commonSuffix b a

Associativity

commonSuffix (commonSuffix a b) c
==
commonSuffix a (commonSuffix b c)

Class of monoids for which the greatest overlap can be found between any two values, such that

a == a' <> overlap a b
b == overlap a b <> b'

The methods must satisfy the following laws:

stripOverlap a b == (stripSuffixOverlap b a, overlap a b, stripPrefixOverlap a b)
stripSuffixOverlap b a <> overlap a b == a
overlap a b <> stripPrefixOverlap a b == b

The result of overlap a b must be the largest prefix of b and suffix of a, in the sense that it contains any other value x that satifies the property (x isPrefixOf b) && (x isSuffixOf a):

∀x. (x `isPrefixOf` b && x `isSuffixOf` a) => (x `isPrefixOf` overlap a b && x `isSuffixOf` overlap a b)

and it must be unique so there's no other value y that satisfies the same properties for every such x:

∀y. ((∀x. (x `isPrefixOf` b && x `isSuffixOf` a) => x `isPrefixOf` y && x `isSuffixOf` y) => y == overlap a b)

In addition, the overlap operation must satisfy the following properties:

Idempotence

overlap a a == a

Identity

overlap mempty a == mempty

overlap a mempty == mempty

Class of commutative GCD monoids with symmetric distributivity.

In addition to the general GCDMonoid laws, instances of this class must also satisfy the following laws:

gcd (a <> b) (a <> c) == a <> gcd b c

gcd (a <> c) (b <> c) == gcd a b <> c

Class of left GCD monoids with left-distributivity.

In addition to the general LeftGCDMonoid laws, instances of this class must also satisfy the following law:

commonPrefix (a <> b) (a <> c) == a <> commonPrefix b c

Class of right GCD monoids with right-distributivity.

In addition to the general RightGCDMonoid laws, instances of this class must also satisfy the following law:

commonSuffix (a <> c) (b <> c) == commonSuffix a b <> c