- class (C a, C a) => C a where
- extendedGCD :: a -> a -> (a, (a, a))
- gcd :: a -> a -> a
- lcm :: a -> a -> a

- euclid :: (C a, C a) => (a -> a -> a) -> a -> a -> a
- extendedEuclid :: (C a, C a) => (a -> a -> (a, a)) -> a -> a -> (a, (a, a))
- extendedGCDMulti :: C a => [a] -> (a, [a])
- diophantine :: C a => a -> a -> a -> Maybe (a, a)
- diophantineMin :: C a => a -> a -> a -> Maybe (a, a)
- diophantineMulti :: C a => a -> [a] -> Maybe [a]
- chineseRemainder :: C a => (a, a) -> (a, a) -> Maybe (a, a)
- chineseRemainderMulti :: C a => [(a, a)] -> Maybe (a, a)
- propMaximalDivisor :: C a => a -> a -> a -> Property
- propGCDDiophantine :: (Eq a, C a) => a -> a -> Bool
- propExtendedGCDMulti :: (Eq a, C a) => [a] -> Bool
- propDiophantine :: (Eq a, C a) => a -> a -> a -> a -> Bool
- propDiophantineMin :: (Eq a, C a) => a -> a -> a -> a -> Bool
- propDiophantineMulti :: (Eq a, C a) => [(a, a)] -> Bool
- propDiophantineMultiMin :: (Eq a, C a) => [(a, a)] -> Bool
- propChineseRemainder :: (Eq a, C a) => a -> a -> [a] -> Property
- propDivisibleGCD :: C a => a -> a -> Bool
- propDivisibleLCM :: C a => a -> a -> Bool
- propGCDIdentity :: (Eq a, C a) => a -> Bool
- propGCDCommutative :: (Eq a, C a) => a -> a -> Bool
- propGCDAssociative :: (Eq a, C a) => a -> a -> a -> Bool
- propGCDHomogeneous :: (Eq a, C a) => a -> a -> a -> Bool
- propGCD_LCM :: (Eq a, C a) => a -> a -> Bool

# Class

class (C a, C a) => C a whereSource

A principal ideal domain is a ring in which every ideal
(the set of multiples of some generating set of elements)
is principal:
That is,
every element can be written as the multiple of some generating element.
`gcd a b`

gives a generator for the ideal generated by `a`

and `b`

.
The algorithm above works whenever `mod x y`

is smaller
(in a suitable sense) than both `x`

and `y`

;
otherwise the algorithm may run forever.

Laws:

divides x (lcm x y) x `gcd` (y `gcd` z) == (x `gcd` y) `gcd` z gcd x y * z == gcd (x*z) (y*z) gcd x y * lcm x y == x * y

(etc: canonical)

Minimal definition:
* nothing, if the standard Euclidean algorithm work
* if `extendedGCD`

is implemented customly, `gcd`

and `lcm`

make use of it

extendedGCD :: a -> a -> (a, (a, a))Source

Compute the greatest common divisor and solve a respective Diophantine equation.

(g,(a,b)) = extendedGCD x y ==> g==a*x+b*y && g == gcd x y

TODO: This method is not appropriate for the PID class, because there are rings like the one of the multivariate polynomials, where for all x and y greatest common divisors of x and y exist, but they cannot be represented as a linear combination of x and y. TODO: The definition of extendedGCD does not return the canonical associate.

The Greatest Common Divisor is defined by:

gcd x y == gcd y x divides z x && divides z y ==> divides z (gcd x y) (specification) divides (gcd x y) x

Least common multiple

# Standard implementations for instances

extendedEuclid :: (C a, C a) => (a -> a -> (a, a)) -> a -> a -> (a, (a, a))Source

# Algorithms

extendedGCDMulti :: C a => [a] -> (a, [a])Source

Compute the greatest common divisor for multiple numbers by repeated application of the two-operand-gcd.

diophantine :: C a => a -> a -> a -> Maybe (a, a)Source

A variant with small coefficients.

`Just (a,b) = diophantine z x y`

means
`a*x+b*y = z`

.
It is required that `gcd(y,z) `

.
`divides`

x

diophantineMin :: C a => a -> a -> a -> Maybe (a, a)Source

Like `diophantine`

, but `a`

is minimal
with respect to the measure function of the Euclidean algorithm.

diophantineMulti :: C a => a -> [a] -> Maybe [a]Source

chineseRemainder :: C a => (a, a) -> (a, a) -> Maybe (a, a)Source

Not efficient enough, because GCD/LCM is computed twice.

chineseRemainderMulti :: C a => [(a, a)] -> Maybe (a, a)Source

For `Just (b,n) = chineseRemainder [(a0,m0), (a1,m1), ..., (an,mn)]`

and all `x`

with `x = b mod n`

the congruences
`x=a0 mod m0, x=a1 mod m1, ..., x=an mod mn`

are fulfilled.

# Properties

propMaximalDivisor :: C a => a -> a -> a -> PropertySource

propGCDDiophantine :: (Eq a, C a) => a -> a -> BoolSource

propExtendedGCDMulti :: (Eq a, C a) => [a] -> BoolSource

propDiophantine :: (Eq a, C a) => a -> a -> a -> a -> BoolSource

propDiophantineMin :: (Eq a, C a) => a -> a -> a -> a -> BoolSource

propDiophantineMulti :: (Eq a, C a) => [(a, a)] -> BoolSource

propDiophantineMultiMin :: (Eq a, C a) => [(a, a)] -> BoolSource

propChineseRemainder :: (Eq a, C a) => a -> a -> [a] -> PropertySource

propDivisibleGCD :: C a => a -> a -> BoolSource

propDivisibleLCM :: C a => a -> a -> BoolSource

propGCDIdentity :: (Eq a, C a) => a -> BoolSource

propGCDCommutative :: (Eq a, C a) => a -> a -> BoolSource

propGCDAssociative :: (Eq a, C a) => a -> a -> a -> BoolSource

propGCDHomogeneous :: (Eq a, C a) => a -> a -> a -> BoolSource

propGCD_LCM :: (Eq a, C a) => a -> a -> BoolSource