Idris2Doc : Data.Nat.Factor


CommonFactor : Nat -> Nat -> Nat -> Type
CommonFactor n m p is a witness that p is a factor of both n and m.
Totality: total
CommonFactorExists : (p : Nat) -> Factorpa -> Factorpb -> CommonFactorpab
DecFactor : Nat -> Nat -> Type
DecFactor n p is a result of the process which decides
whether or not p is a factor on n.
Totality: total
ItIsFactor : Factorpn -> DecFactorpn
ItIsNotFactor : NotFactorpn -> DecFactorpn
Factor : Nat -> Nat -> Type
Factor n p is a witness that p is indeed a factor of n,
i.e. there exists a q such that p * q = n.
Totality: total
CofactorExists : (q : Nat) -> n = p*q -> Factorpn
GCD : Nat -> Nat -> Nat -> Type
GCD n m p is a witness that p is THE greatest common factor of both n and m.
The second argument to the constructor is a function which for all q being
a factor of both n and m, proves that q is a factor of p.

This is equivalent to a more straightforward definition, stating that for
all q being a factor of both n and m, q is less than or equal to p, but more
powerful and therefore more useful for further proofs. See below for a proof
that if q is a factor of p then q must be less than or equal to p.
Totality: total
MkGCD : NotBothZeroab => CommonFactorpab -> ((q : Nat) -> CommonFactorqab -> Factorqp) -> GCDpab
NotFactor : Nat -> Nat -> Type
NotFactor n p is a witness that p is NOT a factor of n,
i.e. there exist a q and an r, greater than 0 but smaller than p,
such that p * q + r = n.
Totality: total
ZeroNotFactorS : (n : Nat) -> NotFactorZ (Sn)
ProperRemExists : (q : Nat) -> (r : Fin (predp)) -> n = (p*q) +S (finToNatr) -> NotFactorpn
anythingFactorZero : (a : Nat) -> FactoraZ
Anything is a factor of 0.
Totality: total
cofactor : Factorpn -> DPairNat (\q => Factorqn)
Given a statement that p is factor of n, return its cofactor.
Totality: total
commonFactorAlsoFactorOfGCD : Factorpa -> Factorpb -> GCDqab -> Factorpq
If p is a common factor of a and b, then it is also a factor of their GCD.
This actually follows directly from the definition of GCD.
Totality: total
commonFactorSym : CommonFactorpab -> CommonFactorpba
The relation of common factor is symmetric, that is if p is a common factor
of n and m, then it is also a common factor if m and n.
Totality: total
decFactor : (n : Nat) -> (d : Nat) -> DecFactordn
A decision procedure for whether of not p is a factor of n.
Totality: total
divByGcdGcdOne : GCDa (a*b) (a*c) -> GCD 1 bc
For every two natural numbers, if we divide both of them by their GCD,
the GCD of resulting numbers will always be 1.
Totality: total
factNotSuccFact : GTp 1 -> Factorpn -> NotFactorp (Sn)
For all p greater than 1, if p is a factor of n, then it is NOT a factor
of (n + 1).
Totality: total
factorAntisymmetric : Factorab -> Factorba -> a = b
If a is a factor of b and b is a factor of a, then we can conclude a = b.
Totality: total
factorLteNumber : Factorpn -> LTE 1 n => LTEpn
If n > 0 then any factor of n must be less than or equal to n.
Totality: total
factorNotFactorAbsurd : Factorpn -> NotFactorpn -> Void
No number can simultaneously be and not be a factor of another number.
Totality: total
factorReflexive : (n : Nat) -> Factornn
Every natural number is factor of itself.
Totality: total
factorTransitive : (a : Nat) -> (b : Nat) -> (c : Nat) -> Factorab -> Factorbc -> Factorac
Factor relation is transitive. If b is factor of a and c is b factor of c
is also a factor of a.
Totality: total
gcd : (a : Nat) -> (b : Nat) -> NotBothZeroab => DPairNat (\f => GCDfab)
An implementation of Euclidean Algorithm for computing greatest common
divisors. It is proven correct and total; returns a proof that computed
number actually IS the GCD. Unfortunately it's very slow, so improvements
in terms of efficiency would be welcome.
Totality: total
gcdSym : GCDpab -> GCDpba
The relation of greates common divisor is symmetric.
Totality: total
gcdUnique : GCDpab -> GCDqab -> p = q
For every two natural numbers there is a unique greatest common divisor.
Totality: total
minusFactor : Factorp (a+b) -> Factorpa -> Factorpb
If p is a factor of a sum (n + m) and a factor of n, then it is also
a factor of m. This could be expressed more naturally with minus, but
it would be more difficult to prove, since minus lacks certain properties
that one would expect from decent subtraction.
Totality: total
multFactor : (p : Nat) -> (q : Nat) -> Factorp (p*q)
For all natural numbers p and q, p is a factor of (p * q).
Totality: total
multNAlsoFactor : Factorpn -> (a : Nat) -> LTE 1 a => Factorp (n*a)
If p is a factor of n, then it is also a factor of any multiply of n.
Totality: total
oneCommonFactor : (a : Nat) -> (b : Nat) -> CommonFactor 1 ab
1 is a common factor of any pair of natural numbers.
Totality: total
oneIsFactor : (n : Nat) -> Factor 1 n
1 is a factor of any natural number.
Totality: total
oneSoleFactorOfOne : (a : Nat) -> Factora 1 -> a = 1
1 is the only factor of itself
Totality: total
plusDivisorAlsoFactor : Factorpn -> Factorp (n+p)
If p is a factor of n, then it is also a factor of (n + p).
Totality: total
plusDivisorNeitherFactor : NotFactorpn -> NotFactorp (n+p)
If p is NOT a factor of n, then it also is NOT a factor of (n + p).
Totality: total
plusFactor : Factorpn -> Factorpm -> Factorp (n+m)
If p is a factor of both n and m, then it is also a factor of their sum.
Totality: total
selfIsCommonFactor : (a : Nat) -> LTE 1 a => CommonFactoraaa
Any natural number is a common factor of itself and itself.
Totality: total