Documentation

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)

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

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

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).

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

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.

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.

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.

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.

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.

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.

Div function : Multiply by the reciprocal.

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.

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.

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.

Compute reciprocal (by negating exponents or equivalent).

Root operator

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

Greatest Common Divisor function

Least Common Multiple function

Modulus function

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.

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.

Totient function(s)

Totient function(s)

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

Function to check if a Canon is positive

Check if Canon is Integral

Check if Canon is Rational

Check if Canon is Irrational

Check if Canon is prime

Check if Canon is simplified, meaning completely factored

Split a Canon into the numerator and denominator.

Return the numerator

Return the denominator

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

Checks if the Canon just a Bare Integer.

Returns the status for Bare numbers.

Return the CanonValueType (Integral, etc).

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

Check if Canon is a prime tower

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

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

Compare the level and the "mantissa"

Tetration Function - Level 4

Pentation Function - Level 5

Hexation Function - Level 6

Heptation Function - Level 7

Octation Function -- Level 8

Nonation Function -- Level 9

Tetration List Function

Pentation List Function

Hexation List Function

Heptation List Function

Octation List Function

Nonation List Function

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

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

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

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

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

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

Tetration list operator

Pentation list operator

Hexation list operator

Heptation list operator

Octation list operator

Nonation list operator

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Variable reps ranging from 1 to 9 in the hyperoperation hierarchy

Variable reps ranging from 1 to 9 in the hyperoperation hierarchy

Variable reps ranging from 1 to 9 in the hyperoperation hierarchy

Variable reps ranging from 1 to 9 in the hyperoperation hierarchy

Variable reps ranging from 1 to 9 in the hyperoperation hierarchy

Variable reps ranging from 1 to 9 in the hyperoperation hierarchy

Variable reps ranging from 1 to 9 in the hyperoperation hierarchy

Variable reps ranging from 1 to 9 in the hyperoperation hierarchy

Variable reps ranging from 1 to 9 in the hyperoperation hierarchy

Return the list of canons from a hyper expression

Return the level of hyperoperation from a hyper expression.

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

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

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

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

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

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.

wrapper to create apply a hyperoperation to a list

Generalized Hyperoperation Function

Check if Canon is a hyperexpression

Check if Canon has a hyperexpression anywhere in its definition

Find the maximum hyperoperation embedded in a Canon.

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

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

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

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

Check if Canon is an exponential expression

Check if Canon has a hypersum embedded in the expression

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

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

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

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

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

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

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

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.

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

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

Used when computing "Super Log"

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

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.

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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