Documentation

Laws for instances of LCMMonoid.

Includes the following laws:

Reductivity

isJust (lcm a b </> a)

isJust (lcm a b </> b)

Uniqueness

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

Idempotence

lcm a a == a

Identity

lcm mempty a == a

lcm a mempty == a

Commutativity

lcm a b == lcm b a

Associativity

lcm (lcm a b) c == lcm a (lcm b c)

Absorption

lcm a (gcd a b) == a

gcd a (lcm a b) == a

Note that the following superclass laws are not included:

• gcdMonoidLaws

Laws for instances of DistributiveLCMMonoid.

Includes the following laws:

Left-distributivity

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

Right-distributivity

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

Lattice distributivity

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

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

Note that the following superclass laws are not included:

• distributiveGCDMonoidLaws
• lcmMonoidLaws