Math.NumberTheory.Canon

A Canon is exponentation-based representation for arbitrarily massive numbers, including prime towers and hyper-expressions.

Documentation

data Canon

This is a GADT representing either a Bare (Integer), some variation of a Can(onical) form or a HX (Hyper eXpression) consisting of a level and list of Canons (see CanonValueType)

Instances

Enum Canon

Defined in Math.NumberTheory.Canon

Methods

succ :: Canon -> Canon

pred :: Canon -> Canon

toEnum :: Int -> Canon

fromEnum :: Canon -> Int

enumFrom :: Canon -> [Canon]

enumFromThen :: Canon -> Canon -> [Canon]

enumFromTo :: Canon -> Canon -> [Canon]

enumFromThenTo :: Canon -> Canon -> Canon -> [Canon]

Eq Canon

Instances for Canon

Defined in Math.NumberTheory.Canon

Methods

(==) :: Canon -> Canon -> Bool

(/=) :: Canon -> Canon -> Bool

Fractional Canon

Defined in Math.NumberTheory.Canon

Methods

(/) :: Canon -> Canon -> Canon

recip :: Canon -> Canon

fromRational :: Rational -> Canon

Integral Canon

Defined in Math.NumberTheory.Canon

Methods

quot :: Canon -> Canon -> Canon

rem :: Canon -> Canon -> Canon

div :: Canon -> Canon -> Canon

mod :: Canon -> Canon -> Canon

quotRem :: Canon -> Canon -> (Canon, Canon)

divMod :: Canon -> Canon -> (Canon, Canon)

toInteger :: Canon -> Integer

Num Canon

Defined in Math.NumberTheory.Canon

Methods

(+) :: Canon -> Canon -> Canon

(-) :: Canon -> Canon -> Canon

(*) :: Canon -> Canon -> Canon

negate :: Canon -> Canon

abs :: Canon -> Canon

signum :: Canon -> Canon

fromInteger :: Integer -> Canon

Ord Canon

Defined in Math.NumberTheory.Canon

Methods

compare :: Canon -> Canon -> Ordering

(<) :: Canon -> Canon -> Bool

(<=) :: Canon -> Canon -> Bool

(>) :: Canon -> Canon -> Bool

(>=) :: Canon -> Canon -> Bool

max :: Canon -> Canon -> Canon

min :: Canon -> Canon -> Canon

Real Canon

Defined in Math.NumberTheory.Canon

Methods

toRational :: Canon -> Rational

Show Canon

Defined in Math.NumberTheory.Canon

Methods

showsPrec :: Int -> Canon -> ShowS

show :: Canon -> String

showList :: [Canon] -> ShowS

CanonConv Canon

Instance of CanonConv class

Defined in Math.NumberTheory.Canon

Methods

toSC :: Canon -> SimpleCanon

toRC :: Canon -> RationalSimpleCanon

makeCanon :: Integer -> Canon

Create a Canon from an Integer. This may involve expensive factorization.

makeCanon' :: Integer -> (Canon, Bool)

Create a Canon from an Integer. Returns the Canon and flag indicating full factorization or not.

data BareStatus

BareStatus: A "Bare Simp" number means a prime number, +/-1 or 0. The code must set the flag properly. A "Bare NSim" number is an Integer that has not been checked (to see if it can be factored).

Constructors

Simp

NSim

Instances

Eq BareStatus

Defined in Math.NumberTheory.Canon

Methods

(==) :: BareStatus -> BareStatus -> Bool

(/=) :: BareStatus -> BareStatus -> Bool

Ord BareStatus

Defined in Math.NumberTheory.Canon

Methods

compare :: BareStatus -> BareStatus -> Ordering

(<) :: BareStatus -> BareStatus -> Bool

(<=) :: BareStatus -> BareStatus -> Bool

(>) :: BareStatus -> BareStatus -> Bool

(>=) :: BareStatus -> BareStatus -> Bool

max :: BareStatus -> BareStatus -> BareStatus

min :: BareStatus -> BareStatus -> BareStatus

Show BareStatus

Defined in Math.NumberTheory.Canon

Methods

showsPrec :: Int -> BareStatus -> ShowS

show :: BareStatus -> String

showList :: [BareStatus] -> ShowS

data CanonValueType

CanonValueType: 3 possibilities for this GADT (integral, non-integral rational, irrational). Imaginary/complex numbers are not supported

Constructors

IntC

NirC

IrrC

Instances

Eq CanonValueType

Defined in Math.NumberTheory.Canon

Methods

(==) :: CanonValueType -> CanonValueType -> Bool

(/=) :: CanonValueType -> CanonValueType -> Bool

Ord CanonValueType

Defined in Math.NumberTheory.Canon

Methods

compare :: CanonValueType -> CanonValueType -> Ordering

(<) :: CanonValueType -> CanonValueType -> Bool

(<=) :: CanonValueType -> CanonValueType -> Bool

(>) :: CanonValueType -> CanonValueType -> Bool

(>=) :: CanonValueType -> CanonValueType -> Bool

max :: CanonValueType -> CanonValueType -> CanonValueType

min :: CanonValueType -> CanonValueType -> CanonValueType

Show CanonValueType

Defined in Math.NumberTheory.Canon

Methods

showsPrec :: Int -> CanonValueType -> ShowS

show :: CanonValueType -> String

showList :: [CanonValueType] -> ShowS

cShowFull :: Canon -> String

Various show functions: cShowFull - fully expand large primes and composites in Canon expression. Unf in name means don't factor unless it's too big too display AsCode in name means you can copy and paste the results and execute them.

cShowFullAsCode :: Canon -> String

Various show functions: cShowFull - fully expand large primes and composites in Canon expression. Unf in name means don't factor unless it's too big too display AsCode in name means you can copy and paste the results and execute them.

cShowAsCode :: Canon -> String

Various show functions: cShowFull - fully expand large primes and composites in Canon expression. Unf in name means don't factor unless it's too big too display AsCode in name means you can copy and paste the results and execute them.

cShowAsCodeUnf :: Canon -> String

Various show functions: cShowFull - fully expand large primes and composites in Canon expression. Unf in name means don't factor unless it's too big too display AsCode in name means you can copy and paste the results and execute them.

cShowUnf :: Canon -> String

Various show functions: cShowFull - fully expand large primes and composites in Canon expression. Unf in name means don't factor unless it's too big too display AsCode in name means you can copy and paste the results and execute them.

cMult :: Canon -> Canon -> CycloMap -> (Canon, CycloMap)

Multiply Function: Generally speaking, this will be much cheaper than addition/subtraction which requires factoring. You are usually just merging lists of prime, exponent pairs and adding exponents where common primes are found. This notion is the crux of the library.

Note: This can be used instead of the * operator if you want to maintain a CycloMap for performance reasons.

cDiv :: Canon -> Canon -> CycloMap -> (Canon, CycloMap)

Div function : Multiply by the reciprocal.

cAdd :: Canon -> Canon -> CycloMap -> (Canon, CycloMap)

Addition and subtraction is generally much more expensive because it requires refactorization. There is logic to look for algebraic forms which can greatly reduce simplify factorization. Note: This can be used instead of the +/- operators if you want to maintain a CycloMap for performance reasons.

cSubtract :: Canon -> Canon -> CycloMap -> (Canon, CycloMap)

Addition and subtraction is generally much more expensive because it requires refactorization. There is logic to look for algebraic forms which can greatly reduce simplify factorization. Note: This can be used instead of the +/- operators if you want to maintain a CycloMap for performance reasons.

cExp :: Canon -> Canon -> Bool -> CycloMap -> (Canon, CycloMap)

Exponentiation: This does allow for negative exponentiation if the Bool flag is True. Note: This can be used instead of the exponentiation operator if you want to maintain a CycloMap for performance reasons.

cReciprocal :: Canon -> Canon

Compute reciprocal (by negating exponents or equivalent).

(>^) :: Canon -> Canon -> Canon infixr 9

Root operator

(<^) :: Canon -> Canon -> Canon infixr 9

Special exponentiation operator for Canon (One MUST use this instead of the standard exp. operator: ^)

cGCD :: Canon -> Canon -> Canon

Greatest Common Divisor function

cLCM :: Canon -> Canon -> Canon

Least Common Multiple function

cMod :: Canon -> Canon -> Canon

Modulus function

cOdd :: Canon -> Bool

Check if a Canon is integral and odd/even, respectively. Note: Return False for both if the Canon is not integral. See CanonValueType for possible cases.

cEven :: Canon -> Bool

Check if a Canon is integral and odd/even, respectively. Note: Return False for both if the Canon is not integral. See CanonValueType for possible cases.

cTotient :: Canon -> CycloMap -> (Canon, CycloMap)

Totient function(s)

cPhi :: Canon -> CycloMap -> (Canon, CycloMap)

Totient function(s)

cNegative :: Canon -> Bool

Function to check if a Canon is negative cNegative c | trace ("cNegative: (l=" ++ show c ++ "))") False = undefined

cPositive :: Canon -> Bool

Function to check if a Canon is positive

cIntegral :: Canon -> Bool

Check if Canon is Integral

cRational :: Canon -> Bool

Check if Canon is Rational

cIrrational :: Canon -> Bool

Check if Canon is Irrational

cPrime :: Canon -> Bool

Check if Canon is prime

cSimplified :: Canon -> Bool

Check if Canon is simplified, meaning completely factored

cSplit :: Canon -> (Canon, Canon)

Split a Canon into the numerator and denominator.

cNumerator :: Canon -> Canon

Return the numerator

cDenominator :: Canon -> Canon

Return the denominator

cCanonical :: Canon -> Bool

Checks if the Canon is Canonical, a more complex expression.

cBare :: Canon -> Bool

Checks if the Canon just a Bare Integer.

cBareStatus :: Canon -> BareStatus

Returns the status for Bare numbers.

cValueType :: Canon -> CanonValueType

Return the CanonValueType (Integral, etc).

cDelve :: Canon -> [Int] -> Canon

Take a Canon and a list of indexes and delve into the Canon. This operates on the internal hyper lists.

cIsPrimeTower :: Canon -> Bool

Check if Canon is a prime tower

cPrimeTowerLevel :: Canon -> Canon

This is used for tetration, etc. It defaults to zero for non-integral reps.

cSuperLog :: Canon -> (SuperPairC, Int)

This is the super log function. Similar to the iterated log function. A level and mantissa is returned along with the number's sign.

cSuperLogCmp :: SuperPairC -> SuperPairC -> Ordering

Compare the level and the "mantissa"

cTetration :: Canon -> Canon -> Canon

Tetration Function - Level 4

cPentation :: Canon -> Canon -> Canon

Pentation Function - Level 5

cHexation :: Canon -> Canon -> Canon

Hexation Function - Level 6

cHeptation :: Canon -> Canon -> Canon

Heptation Function - Level 7

cOctation :: Canon -> Canon -> Canon

Octation Function -- Level 8

cNonation :: Canon -> Canon -> Canon

Nonation Function -- Level 9

cTetrationL :: Canon -> [Canon] -> CycloMap -> (Canon, CycloMap)

Tetration List Function

cPentationL :: Canon -> [Canon] -> CycloMap -> (Canon, CycloMap)

Pentation List Function

cHexationL :: Canon -> [Canon] -> CycloMap -> (Canon, CycloMap)

Hexation List Function

cHeptationL :: Canon -> [Canon] -> CycloMap -> (Canon, CycloMap)

Heptation List Function

cOctationL :: Canon -> [Canon] -> CycloMap -> (Canon, CycloMap)

Octation List Function

cNonationL :: Canon -> [Canon] -> CycloMap -> (Canon, CycloMap)

Nonation List Function

(<^>) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(<<^>>) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(<<<^>>>) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(<<<<^>>>>) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(<<<<<^>>>>>) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(|<^>|) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(<^^>) :: Canon -> [Canon] -> Canon infixr 9

Tetration list operator

(<<^^>>) :: Canon -> [Canon] -> Canon infixr 9

Pentation list operator

(<<<^^>>>) :: Canon -> [Canon] -> Canon infixr 9

Hexation list operator

(<<<<^^>>>>) :: Canon -> [Canon] -> Canon infixr 9

Heptation list operator

(<<<<<^^>>>>>) :: Canon -> [Canon] -> Canon infixr 9

Octation list operator

(|<^^>|) :: Canon -> [Canon] -> Canon infixr 9

Nonation list operator

(~^~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~<^>~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~<<^>>~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~<<<^>>>~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~<<<<^>>>>~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~|^|~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~|<^>|~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~|<<^>>|~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~|<<<^>>>|~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~|<<<<^>>>>|~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~~^~~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~~<^>~~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~~<<^>>~~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~~<<<^>>>~~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~~<<<<^>>>>~~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~~|^|~~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~~|<^>|~~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~~|<<^>>|~~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~~|<<<^>>>|~~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~~|<<<<^>>>>|~~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~~~^~~~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~~~<^>~~~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~~~<<^>>~~~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~~~<<<^>>>~~~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~~~<<<<^>>>>~~~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~~~|^|~~~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~~~|<^>|~~~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~~~|<<^>>|~~~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~~~|<<<^>>>|~~~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~~~|<<<<^>>>>|~~~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~~~~^~~~~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~~~~<^>~~~~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~~~~<<^>>~~~~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~~~~<<<^>>>~~~~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~~~~<<<<^>>>>~~~~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~~~~|^|~~~~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~~~~|<^>|~~~~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~~~~|<<^>>|~~~~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~~~~|<<<^>>>|~~~~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~~~~|<<<<^>>>>|~~~~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

(~~~~~^~~~~~) :: Canon -> Canon -> Canon infixr 9

Defined specialty hyper operators range from level 4 / tetration (<^>) to level 50 (~~~~~^~~~~~). See source or .odp file for the operator naming convention.

cAddOpLevel :: Canon

Variable reps ranging from 1 to 9 in the hyperoperation hierarchy

cMultOpLevel :: Canon

Variable reps ranging from 1 to 9 in the hyperoperation hierarchy

cExpOpLevel :: Canon

Variable reps ranging from 1 to 9 in the hyperoperation hierarchy

cTetrOpLevel :: Canon

Variable reps ranging from 1 to 9 in the hyperoperation hierarchy

cPentOpLevel :: Canon

Variable reps ranging from 1 to 9 in the hyperoperation hierarchy

cHexOpLevel :: Canon

Variable reps ranging from 1 to 9 in the hyperoperation hierarchy

cHeptOpLevel :: Canon

Variable reps ranging from 1 to 9 in the hyperoperation hierarchy

cOctOpLevel :: Canon

Variable reps ranging from 1 to 9 in the hyperoperation hierarchy

cNonOpLevel :: Canon

Variable reps ranging from 1 to 9 in the hyperoperation hierarchy

cGetHyperList :: Canon -> [Canon]

Return the list of canons from a hyper expression

cGetHyperOp :: Canon -> Canon

Return the level of hyperoperation from a hyper expression.

maxHyperOpDispLevel :: Integer

Internal value that corresponds with ~~~~~^~~~~~ (level 50 hyperoperation)

maxHyperOpDelveLevel :: Canon

Max hyper operaton level when converting to canonical form (for the sake of combining and reducing terms)

cFactorSum :: Canon -> Bool -> Canon

cFactor : Factor simple terms from a sum. If the flag is true, only factor by the gcd if the gcd is a hyper expression

cConvertToSum :: Canon -> (Canon, Bool)

Convert a hyperexpression to a sum if possible. Useful in comparison. Will expand polynomials to a limited degree.

cMaxExpoToExpand :: Canon

Maximum exponent (of a polynomial) to distribute into a sum of terms.

cFactorHorizon :: Canon -> Canon

This is good for polynomial-like expressions like: (1 + 3<^>4 + 3<^>5) <^ 3 - 1, where there's a mixture of "canonical" and hE exponents.

cApplyHy :: Canon -> [Canon] -> Bool -> Canon

wrapper to create apply a hyperoperation to a list

cHyperOp :: Canon -> [Canon] -> CycloMap -> (Canon, CycloMap)

Generalized Hyperoperation Function

cHyperExpr :: Canon -> Bool

Check if Canon is a hyperexpression

cHyperExprAny :: Canon -> Bool

Check if Canon has a hyperexpression anywhere in its definition

cMaxHyperOp :: Canon -> Canon

Find the maximum hyperoperation embedded in a Canon.

cMinHyperOp :: Canon -> Canon

Find the minimum hyperoperation embedded in a Canon. (If it's not a hyper expression, return zero.)

cMaxHyperOpForQC :: Canon -> Canon

Used when checking if one can quasi-canonize a hyper expression (should be compared against the cutoff)

cHyperSum :: Canon -> Bool

Check if Canon is a positive sum (or negative one times a positive sum). Internally, all level 1 expressions are positive.

cHyperProd :: Canon -> Bool

Check if Canon is a product (and specificlly not a negative sum)

cHyperExpo :: Canon -> Bool

Check if Canon is an exponential expression

cHyperSumAny :: Canon -> Bool

Check if Canon has a hypersum embedded in the expression

cHyperize :: Canon -> Canon

cHyperize will take a Canon in quasi-canonized form and try to clean it up in a tidier expression

>>> cShowUnf $ cHyperize $  7 <^ ( 1 + 2 * (49 <^> 7))
7 * 49 <^> 8

cQuasiCanonize :: Canon -> Canon

This is akin to canonical form except you may have sums in the bases. It converts expression up to a hyperoperational cutoff.

cQuasiCanonized :: Canon -> Bool

This checks if the (hyper)expression is in quasi-canonical form.

cCleanup :: Canon -> Canon

Try to make expression more elegant by creating exponential tails and then hyperizing them

cGetAddends :: Canon -> [Canon]

If the Canon is a product, return the factors. Otherwise, return the Canon itself.

cGetFactors :: Canon -> [Canon]

If the Canon is a sum, return the addends. Otherwise, return the Canon itself.

cCleanupAsNumDenPair :: Canon -> (Canon, Canon)

Split the hyperoperation into a cleaned-up numerator and denominator pair (if denom is 1). This still represents an integral value. e.g. 3 ^ 7 / 3 ^ 4

type CanonElement = (Canon, Canon)

This element is a base, exponent pair. The base is an integer and is generally prime or 0, -1. The exponent is also a Canon (allowing for arbitrary nesting) A Canon conceptually consists of a list of these elements. The first member of the pair will be a Canon raised to the first power. By doing this, we allow for further generality in the definition of a Canon.

cGetBase :: CanonElement -> Canon

Return the base b from a CanonElement (equivalent to b^e).

cGetExponent :: CanonElement -> Canon

Return the exponent e from a CanonElement (equivalent to b^e).

type SuperPairC = (Canon, Double)

Used when computing "Super Log"

cGetBases :: Canon -> [Canon]

Return the list of bases from a Canon (conceptually of the form [b^e]).

cGetBasesDeep :: Canon -> [Canon]

This is similar to cGetBases except that it will do trial factoring of any hyper sums. i So, for obvious reasons, this is not a complete factorization.

cGetExponents :: Canon -> [Canon]

Return the list of exponents from a Canon (conceptually of the form [b^e]).

cGetElements :: Canon -> [CanonElement]

Return the list of CanonElements from a Canon (conceptually of the form [b^e]).

cNumDivisors :: Canon -> Canon

Note: Divisor count function(s) should be called with integral Canons. Restricted to positive divisors. Returns Either String Canon

cTau :: Canon -> Canon

Note: Divisor count function(s) should be called with integral Canons. Restricted to positive divisors. Returns Either String Canon

cDivisors :: Canon -> Either String [Canon]

Efficiently compute all of the divisors based on the canonical representation. Returns either an error message or a list of Canons.

cNthDivisor :: Canon -> Canon -> Either String Canon

Compute the nth divisor of a Canon. It operates on the absolute value of the Canon and is zero based. Note: This is deterministic but it's not ordered by the value of the divisor.

cWhichDivisor :: Canon -> Canon -> Either String Canon

Consider this to be the inverse of the cNthDivisor function. This function ignores signs but both parameters must be integral.

cRelativelyPrime :: Canon -> Canon -> Maybe Bool

This will determine if two arbitary expressions are relatively prime or not (if possible). Goes deep.

cGetFirstNDivisors :: Int -> Canon -> Either String [Canon]

Return the first N divisors of a hyper expression (if possible)

cN1 :: Canon

Define some small canons for convenience from (-1 to 9)

c0 :: Canon

Define some small canons for convenience from (-1 to 9)

c1 :: Canon

Define some small canons for convenience from (-1 to 9)

c2 :: Canon

Define some small canons for convenience from (-1 to 9)

c3 :: Canon

Define some small canons for convenience from (-1 to 9)

c4 :: Canon

Define some small canons for convenience from (-1 to 9)

c5 :: Canon

Define some small canons for convenience from (-1 to 9)

c6 :: Canon

Define some small canons for convenience from (-1 to 9)

c7 :: Canon

Define some small canons for convenience from (-1 to 9)

c8 :: Canon

Define some small canons for convenience from (-1 to 9)

c9 :: Canon

Define some small canons for convenience from (-1 to 9)

data CycloMap

CycloMap is a newtype hiding the details of a map of CR_ to pairs of integers and corresponding cyclotomic polynomials.

Instances

Eq CycloMap

Defined in Math.NumberTheory.Canon.AurifCyclo

Methods

(==) :: CycloMap -> CycloMap -> Bool

(/=) :: CycloMap -> CycloMap -> Bool

Show CycloMap

Defined in Math.NumberTheory.Canon.AurifCyclo

Methods

showsPrec :: Int -> CycloMap -> ShowS

show :: CycloMap -> String

showList :: [CycloMap] -> ShowS

getIntegerBasedCycloMap :: CycloMap -> Map Integer CycloPair

Unwrap the CycloMap and convert the internal canon rep keys to Integers, returning a "raw" map

crCycloInitMap :: CycloMap

This is an initial map with the cyclotomic polynomials for 1.