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//! //!Traits for sets with a single binary operation and various properties of that operation //! //!Currently, the group operation is interpreted as being either the [`Add`] or [`Mul`] operation, //!and each of the group properties in this module have both an additive and multiplicative variant. //! //!As it stands currently, there is no real difference between the two, so it is ultimately up //!to the implementor's preference which one (or both) to use. Obviously though, addition and multiplication //!carry difference connotations in different contexts, so for clarity and consistency it is //!suggested to try to follow the general mathematical or programming conventions whenever possible. //!In particular: //!* Try to use multiplication for structures with a single operation //!except when convention dictates otherwise (such as the case of string concatenation). //!* While the option does exist, avoid implementing a non-commutative or especially a non-associative //!addition operation unless convention dictates otherwise. //!* Avoid implementing both an addition and multiplication where the multiplication *doesn't* distrubute //!or where the addition distributes instead. //! //!# Implementation //! //!To implement a struct as a group-like structure, the various group-like trait aliases will be usable //!according to which and how the following properties are implemented: //!* An additive or multiplicative binary operation: //! * Has some function taking any pair of elements from `Self` and outputing any other member of `Self` //! * Represented with either [`Add`] and [`AddAssign`] or [`Mul`] and [`MulAssign`] from [`core::ops`] //! * (Note that for the auto-implementing categorization traits to work, the corresponding //! "Assign" traits must be implemented.) //!* An identity element: //! * Contains a unique element `0` or `1` such that `0+x=x` and `x+0=x` or //! `1*x=x`,`x*1=x` for all `x` //! * Represented with either [`Zero`] or [`One`] from [`num_traits`] //!* Invertibility: //! * For every `x` in the set, there exists some other `y` in the struct such that //! `x*y=1` and `y*x=1` (or `x+y=0` and `y+x=0` if additive), and there exists //! a corresponding inverse operation. //! * Represented with either [`Neg`], [`Sub`], and [`SubAssign`] or [`Inv`], [`Div`], and [`DivAssign`] //! from [`core::ops`] and [`num_traits`] //! * Note, again, that the "Assign" variants are required //!* Commutative: //! * If the operation is order invariant, ie `x+y=y+x` or `x*y=y*x` for all `x` and `y`. //! * Represented with [`AddCommutative`] or [`MulCommutative`] //!* Associative: //! * If operation sequences are _evaluation_ order invariant, ie `x+(y+z)=(x+y)+z` or `x*(y*z)=(x*y)*z` //! for all `x`, `y`, and `z`. //! * Represented with [`AddAssociative`] or [`MulAssociative`] //! //!# Exponentiation //! //!In addition to these traits, it may be desirable to implement a [multiplication](Mul) or //![exponentiation](num_traits::Pow) operation with particular [integer](crate::algebra::Integer) //!or [natural](crate::algebra::Natural) type. See [`MulN`], [`MulZ`], [`PowN`], and [`PowZ`] for more details. //! //!# Usage //! //!Structs with the above properties implemented will be automatically usable under a number of trait //!aliases for particular group-like structures. These structures follow standard mathematical //!convention and roughly correspond to a heirarchy varying levels of "niceness" of each //!binary operation. //! //!These structures can be diagrammed as follows: //!```text //! ---Magma--- //! | | //! | Semigroup //! Loop | //! | Monoid //! | | //! ---Group--- //! | //! Abelian Group //!``` //!where: //!* A [Magma](MulMagma) is a set with any binary operation //!* A [Semigroup](MulSemigroup) is an [associative](MulAssociative) Magma //!* A [Monoid](MulMonoid) is a Semigroup with an [identity](One) element //!* A [Loop](MulLoop) is a Magma with an [identity](One) element and [inverses](Invertable) //!* A [Group](MulGroup) is a Monoid with [inverses](Invertable), or alternatively, an [associative](MulAssociative) Loop //!* An [Abelian Group](MulAbelianGroup) is a [commutative](MulCommutative) Group //! //!For more information, see Wikipedia's article on //![algebraic structures](https://en.wikipedia.org/wiki/Outline_of_algebraic_structures) //! //!# Additional Notes //! //!It is worth noting that this particular system is certainly non-exhaustive and is missing a number //!of group-like structures. In particular, it is missing the Category-like structures and Quasigroups //! //!In the case of categories, this is because as it stands currently, there are few uses for partial //!operations while including them would add noticeable complexity. //! //!In the case of quasigroups however, the reason is because with the way the primitive types are //!designed, the only way to determine if an addition or subtraction operation is *truely* invertible //!is to check against the [Neg] or [Inv] traits, as the unsigned integers implement **both** division //!and subtraction even though technically those operations are not valid on all natural numbers. //!So seeing as quasigroups aren't particularly useful anyway, the decision was made to omit them. //! //! pub use self::additive::*; pub use self::multiplicative::*; ///Traits for group-like structures using addition pub mod additive { pub use core::ops::{Add, Sub, Neg, AddAssign, SubAssign}; pub use num_traits::{Zero}; use core::convert::{From}; use core::ops::{Mul}; use super::{repeated_doubling, repeated_doubling_neg}; use crate::algebra::{Natural, IntegerSubset, Semiring, Ring}; #[allow(unused_imports)] use crate::algebra::Integer; /// ///A marker trait for stucts whose addition operation is evaluation order independent, ///ie `x+(y+z)=(x+y)+z` for all `x`, `y`, and `z`. /// ///This is an extremely common property, and _most_ commonly used algebraic systems have it. ///Nonetheless, there are some algebraic constructions like loop concatenation, the cross product, ///lie algebras, and octonions that do not have this property, so the option to _not_ implement it exists. /// ///Note however, it is _highly_ recommended to implement non-associative structs as multiplicative ///to be consistent with convention. /// pub trait AddAssociative {} /// ///A marker trait for stucts whose addition operation is order independent, ///ie `x+y=y+x` for all `x`, `y`, and `z`. /// ///This is an extremely common property, and _most_ commonly used algebraic systems have it. ///Nonetheless, there are also a fairly number of algebraic constructions do not, such as ///matrix multiplication, most finite groups, and in particular, string concatenation. /// ///Note however, it is _highly_ recommended to implement non-commutative structs ///(except string concatentation) as multiplicative to be consistent with convention. /// pub trait AddCommutative {} /// ///An auto-implemented trait for multiplication by [natural numbers](Natural) with ///[associative](AddAssociative) types using repeated addition /// ///This is intended as a simple and easy way to compute object multiples in abstract algebraic ///algorithms without resorting to explicitly applying addition repeatedly. For this reason, the ///trait is automatically implemented for any relevant associative algebraic structure and ///the supplied function is generic over the [`Natural`] type. /// ///``` ///# use maths_traits::algebra::*; /// /// assert_eq!(2.5f32.mul_n(4u8), 10.0); /// assert_eq!(2.5f32.mul_n(4u16), 10.0); /// assert_eq!(2.5f32.mul_n(4u128), 10.0); /// assert_eq!(2.5f64.mul_n(4u8), 10.0); /// assert_eq!(2.5f64.mul_n(4u16), 10.0); /// assert_eq!(2.5f64.mul_n(4u128), 10.0); /// ///``` /// ///# Usage Notes /// ///This trait is intended for use with *small* integers and can make no guarrantee that the ///mathematical output will actually fit in the valid range for the output type. As such, ///it is possible for the method to panic, overflow, or return an NaN or error value, so if this ///is an expected possibility, it is recommended to use [TryInto](core::convert::TryInto) or ///a different to perform the operation. /// ///It is worth noting that this particular design was chosen over returning a [Result] ///or [Option] since this general behavior is already the default for primitive types ///despite the relative ease with which it can happen. /// ///# Implementation Notes /// ///Note that while multiplication by natural numbers is very simply defined using ///repeated addition, in order to add flexibility in implementation and the possibility for ///proper optimization, the automatic implmentation of this trait will first try to use other ///traits as a base before defaulting to the general [repeated_doubling] algorithm /// ///Specifically, for a given [Natural] type `N`, the auto-impl will first attempt to use ///[`Mul<N>`](Mul), if implemented. If that fails, it will then try to convert using [`From<N>`](From) ///and multiplying if if it implemented and the struct is a [Semiring]. ///Finally, in the general case, it will use the [repeated_doubling] function. /// pub trait MulN: AddSemigroup + Zero { #[inline] fn mul_n<N:Natural>(self, n:N) -> Self { trait Helper1<Z:Natural>: AddSemigroup + Zero { fn _mul1(self, n:Z) -> Self; } impl<H: AddSemigroup + Zero, Z:Natural> Helper1<Z> for H { #[inline] default fn _mul1(self, n:Z) -> Self {self._mul2(n)} } impl<H: AddSemigroup + Zero + Mul<Z,Output=H>, Z:Natural> Helper1<Z> for H { #[inline] fn _mul1(self, n:Z) -> Self {self * n} } trait Helper2<Z:Natural>: AddSemigroup + Zero { fn _mul2(self, n:Z) -> Self; } impl<H: AddSemigroup + Zero, Z:Natural> Helper2<Z> for H { #[inline] default fn _mul2(self, n:Z) -> Self {repeated_doubling(self, n)} } impl<H: Semiring + From<Z>, Z:Natural> Helper2<Z> for H { #[inline] fn _mul2(self, n:Z) -> Self {H::from(n) * self} } self._mul1(n) } } impl<G:AddSemigroup + Zero> MulN for G {} /// ///An auto-implemented trait for multiplication by [integers](Integer) with ///[associative](AddAssociative) and [negatable](Negatable) types using ///negation and repeated addition /// ///This is intended as a simple and easy way to compute object multiples in abstract algebraic ///algorithms without resorting to explicitly applying addition repeatedly. For this reason, the ///trait is automatically implemented for any relevant associative and negatable algebraic structure and ///the supplied function is generic over the [`Integer`] type. /// ///``` ///# use maths_traits::algebra::*; /// /// assert_eq!(2.5f32.mul_z(5u8), 12.5); /// assert_eq!(2.5f32.mul_z(5u128), 12.5); /// assert_eq!(2.5f64.mul_z(5u8), 12.5); /// assert_eq!(2.5f64.mul_z(5u128), 12.5); /// assert_eq!(2.5f32.mul_z(-5i8), -12.5); /// assert_eq!(2.5f32.mul_z(-5i64), -12.5); /// ///``` /// ///# Usage Notes /// ///This trait is intended for use with *small* integers and can make no guarrantee that the ///mathematical output will actually fit in the valid range for the output type. As such, ///it is possible for the method to panic, overflow, or return an NaN or error value, so if this ///is an expected possibility, it is recommended to use [TryInto](core::convert::TryInto) or ///a different to perform the operation. /// ///It is worth noting that this particular design was chosen over returning a [Result] ///or [Option] since this general behavior is already the default for primitive types ///despite the relative ease with which it can happen. /// ///# Implementation Notes /// ///Note that while multiplication by integers is very simply defined using ///repeated addition and subtraction, in order to add flexibility in implementation and the possibility for ///proper optimization, the automatic implmentation of this trait will first try to use other ///traits as a base before defaulting to the general [repeated_doubling_neg] algorithm /// ///Specifically, for a given [Integer] type `Z`, the auto-impl will first attempt to use ///[`Mul<Z>`](Mul), if implemented. If that fails, it will then try to convert using [`From<Z>`](From) ///and multiplying if if it implemented and the struct is a [Ring]. ///Finally, in the general case, it will use the [repeated_doubling_neg] function. /// pub trait MulZ: AddMonoid + Negatable { #[inline] fn mul_z<N:IntegerSubset>(self, n:N) -> Self { trait Helper1<Z:IntegerSubset>: AddMonoid + Negatable { fn _mul1(self, n:Z) -> Self; } impl<H: AddMonoid + Negatable, Z:IntegerSubset> Helper1<Z> for H { #[inline] default fn _mul1(self, n:Z) -> Self {self._mul2(n)} } impl<H: AddMonoid + Negatable + Mul<Z,Output=H>, Z:IntegerSubset> Helper1<Z> for H { #[inline] fn _mul1(self, n:Z) -> Self {self * n} } trait Helper2<Z:IntegerSubset>: AddSemigroup + Zero { fn _mul2(self, n:Z) -> Self; } impl<H: AddMonoid + Negatable, Z:IntegerSubset> Helper2<Z> for H { #[inline] default fn _mul2(self, n:Z) -> Self {repeated_doubling_neg(self, n)} } impl<H: Ring + From<Z>, Z:IntegerSubset> Helper2<Z> for H { #[inline] fn _mul2(self, n:Z) -> Self {H::from(n) * self} } self._mul1(n) } } impl<G:AddMonoid + Negatable> MulZ for G {} ///A set with an fully described additive inverse pub trait Negatable = Sized + Clone + Neg<Output=Self> + Sub<Self, Output=Self> + SubAssign<Self>; ///A set with an addition operation pub trait AddMagma = Sized + Clone + Add<Self,Output=Self> + AddAssign<Self>; ///An associative additive magma pub trait AddSemigroup = AddMagma + AddAssociative; ///An additive semigroup with an identity element pub trait AddMonoid = AddSemigroup + Zero + MulN; ///An additive magma with an inverse operation and identity pub trait AddLoop = AddMagma + Negatable + Zero; ///An additive monoid with an inverse operation pub trait AddGroup = AddMagma + AddAssociative + Negatable + Zero + MulZ; ///A commutative additive group pub trait AddAbelianGroup = AddGroup + AddCommutative; } ///Traits for group-like structures using Multiplication pub mod multiplicative { pub use core::ops::{Mul, Div, MulAssign, DivAssign}; pub use num_traits::{Inv, One}; use num_traits::Pow; use super::{repeated_squaring, repeated_squaring_inv}; use crate::algebra::{Natural, IntegerSubset}; #[allow(unused_imports)] use crate::algebra::Integer; /// ///A marker trait for stucts whose multiplication operation is evaluation order independent, ///ie `x*(y*z)=(x*y)*z` for all `x`, `y`, and `z`. /// ///This is an extremely common property, and _most_ commonly used algebraic systems have it. ///Nonetheless, there are some algebraic constructions like loop concatenation, the cross product, ///lie algebras, and octonions that do not have this property, so the option to _not_ implement it exists. /// pub trait MulAssociative: {} /// ///A marker trait for stucts whose addition operation is order independent, ///ie `x+y=y+x` for all `x`, `y`, and `z`. /// ///This is an extremely common property, and _most_ commonly used algebraic systems have it. ///Nonetheless, there are also a fairly number of algebraic constructions do not, such as ///matrix multiplication and most finite groups. /// pub trait MulCommutative {} /// ///An auto-implemented trait for exponentiation by [natural numbers](Natural) with ///[associative](MulAssociative) types using repeated multiplication /// ///This is intended as a simple and easy way to compute object powers in abstract algebraic ///algorithms without resorting to explicitly applying multiplication repeatedly. For this reason, the ///trait is automatically implemented for any relevant associative algebraic structure and ///the supplied function is generic over the [`Natural`] type. /// ///``` ///# use maths_traits::algebra::*; /// /// assert_eq!(2.0f32.pow_n(4u8), 16.0); /// assert_eq!(2.0f32.pow_n(4u16), 16.0); /// assert_eq!(2.0f32.pow_n(4u128), 16.0); /// assert_eq!(2.0f64.pow_n(4u8), 16.0); /// assert_eq!(2.0f64.pow_n(4u16), 16.0); /// assert_eq!(2.0f64.pow_n(4u128), 16.0); /// ///``` ///# Usage Notes /// ///This trait is intended for use with *small* integers and can make no guarrantee that the ///mathematical output will actually fit in the valid range for the output type. As such, ///it is possible for the method to panic, overflow, or return an NaN or error value, so if this ///is an expected possibility, it is recommended to use [TryInto](core::convert::TryInto) or ///a different to perform the operation. /// ///It is worth noting that this particular design was chosen over returning a [Result] ///or [Option] since this general behavior is already the default for primitive types ///despite the relative ease with which it can happen. /// ///# Implementation Notes /// ///Note that while exponentiation by natural numbers is very simply defined using ///repeated multiplication, in order to add flexibility in implementation and the possibility for ///proper optimization, the automatic implmentation of this trait will first try to use other ///traits as a base before defaulting to the general [repeated_squaring] algorithm /// ///Specifically, for a given [Natural] type `N`, the auto-impl will first attempt to use ///[`Pow<N>`](Pow), if implemented, then if that fails, it will use the general ///[repeated_squaring] algorithm /// pub trait PowN: MulSemigroup + One { #[inline] fn pow_n<N:Natural>(self, n:N) -> Self { trait Helper<Z:Natural>: MulSemigroup + One { fn _pow_n(self, n:Z) -> Self; } impl<G:MulSemigroup+One, Z:Natural> Helper<Z> for G { #[inline] default fn _pow_n(self, n:Z) -> Self {repeated_squaring(self, n)} } impl<G:MulSemigroup+One+Pow<Z,Output=Self>, Z:Natural> Helper<Z> for G { #[inline] fn _pow_n(self, n:Z) -> Self {self.pow(n)} } self._pow_n(n) } } impl<G:MulSemigroup+One> PowN for G {} /// ///An auto-implemented trait for exponentiation by [integers](Integer) with ///[associative](MulAssociative) and [invertable](Invertable) types using ///inversion and repeated multiplication /// ///This is intended as a simple and easy way to compute object powers in abstract algebraic ///algorithms without resorting to explicitly applying multiplication repeatedly. For this reason, the ///trait is automatically implemented for any relevant associative and invertable algebraic structure and ///the supplied function is generic over the [`Integer`] type. /// ///``` ///# use maths_traits::algebra::*; /// /// assert_eq!(2.0f32.pow_z(3u8), 8.0); /// assert_eq!(2.0f32.pow_z(3u128), 8.0); /// assert_eq!(2.0f64.pow_z(3u8), 8.0); /// assert_eq!(2.0f64.pow_z(3u128), 8.0); /// assert_eq!(2.0f32.pow_z(-3i8), 0.125); /// assert_eq!(2.0f32.pow_z(-3i64), 0.125); /// ///``` ///# Usage Notes /// ///This trait is intended for use with *small* integers and can make no guarrantee that the ///mathematical output will actually fit in the valid range for the output type. As such, ///it is possible for the method to panic, overflow, or return an NaN or error value, so if this ///is an expected possibility, it is recommended to use [TryInto](core::convert::TryInto) or ///a different to perform the operation. /// ///It is worth noting that this particular design was chosen over returning a [Result] ///or [Option] since this general behavior is already the default for primitive types ///despite the relative ease with which it can happen. /// ///# Implementation Notes /// ///Note that while exponentiation by integers is very simply defined using ///repeated multiplication and inversion, in order to add flexibility in implementation and the possibility for ///proper optimization, the automatic implmentation of this trait will first try to use other ///traits as a base before defaulting to the general [repeated_squaring_inv] algorithm /// ///Specifically, for a given [Natural] type `N`, the auto-impl will first attempt to use ///[`Pow<N>`](Pow), if implemented, then if that fails, it will use the general ///[repeated_squaring_inv] algorithm /// pub trait PowZ: MulMonoid + Invertable { #[inline] fn pow_z<Z:IntegerSubset>(self, n:Z) -> Self { trait Helper<N:IntegerSubset>: MulMonoid + Invertable { fn _pow_z(self, n:N) -> Self; } impl<G:MulMonoid+Invertable, N:IntegerSubset> Helper<N> for G { #[inline] default fn _pow_z(self, n:N) -> Self {repeated_squaring_inv(self, n)} } impl<G:MulMonoid+Invertable+Pow<N,Output=Self>, N:IntegerSubset> Helper<N> for G { #[inline] fn _pow_z(self, n:N) -> Self {self.pow(n)} } self._pow_z(n) } } impl<G:MulMonoid+Invertable> PowZ for G {} ///A set with an fully described multiplicative inverse pub trait Invertable = Sized + Clone + Inv<Output=Self> + Div<Self, Output=Self> + DivAssign<Self>; ///A set with a multiplication operation pub trait MulMagma = Sized + Clone + Mul<Self, Output=Self> + MulAssign<Self>; ///An associative multiplicative magma pub trait MulSemigroup = MulMagma + MulAssociative; ///A multiplicative semigroup with an identity element pub trait MulMonoid = MulSemigroup + One + PowN; ///A multiplicative magma with an inverse operation and identity pub trait MulLoop = MulMagma + Invertable + One; ///A multiplicative monoid with an inverse operation pub trait MulGroup = MulMagma + MulAssociative + Invertable + One + PowZ; ///A commutative multiplicative group pub trait MulAbelianGroup = MulGroup + MulCommutative; } use crate::algebra::{Natural, IntegerSubset}; trait IsZero:Sized { fn _is_zero(&self) -> bool; } impl<Z> IsZero for Z { #[inline(always)] default fn _is_zero(&self) -> bool {false} } impl<Z:Zero> IsZero for Z { #[inline(always)] fn _is_zero(&self) -> bool {self.is_zero()} } fn mul_pow_helper<E:Natural, R:Clone, Op: Fn(R,R) -> R>(mut b: R, mut p: E, op: Op) -> R { //repeated squaring/doubling let mut res = b.clone(); p -= E::one(); while !p.is_zero() { if p.even() { //if the exponent (or multiple) is even, we can factor out the two //ie b^2p = (b^2)^p or 2p*b = p*2b b = op(b.clone(), b); p = p.div_two(); } else { //else b^(p+1)=(b^p)*b or (p+1)*b = p*b + b res = op(res, b.clone()); p -= E::one(); } } res } /// ///Raises a [monoid](MulMonoid) element to a integral power using inversion and repeated squaring /// ///# Panics ///If both the base and power are `0` /// #[inline] pub fn repeated_squaring_inv<E:IntegerSubset, R:MulGroup+Clone>(b: R, p: E) -> R { if p.negative() { repeated_squaring(b, p.abs_unsigned()).inv() } else { repeated_squaring(b, p.as_unsigned()) } } /// ///Raises a [monoid](MulMonoid) element to a positive integer power using the repeated squaring algorithm /// ///# Panics ///If both the base and power are `0` /// #[inline] pub fn repeated_squaring<E:Natural, R:MulMonoid+Clone>(b: R, p: E) -> R { if p.is_zero() { if b._is_zero() { panic!("Attempted to raise 0^0") } R::one() } else { mul_pow_helper(b, p, R::mul) } } ///Multiplies a [monoid](AddMonoid) by a positive integer using negation and repeated doublings #[inline] pub fn repeated_doubling_neg<E:IntegerSubset, R:AddGroup>(b: R, p: E) -> R { if p.negative() { -repeated_doubling(b, p.abs_unsigned()) } else { repeated_doubling(b, p.as_unsigned()) } } ///Multiplies a [monoid](AddMonoid) by a positive integer using repeated doublings #[inline] pub fn repeated_doubling<E:Natural, R:AddMonoid>(b: R, p: E) -> R { if p.is_zero() { R::zero() } else { mul_pow_helper(b, p, R::add) } } macro_rules! impl_props { ($($z:ty)*; $($f:ty)*) => { $(impl_props!(@int $z);)* $(impl_props!(@float $f);)* }; (@int $z:ty) => { impl_props!(@props $z); impl_props!(@props core::num::Wrapping<$z>); }; (@float $f:ty) => {impl_props!(@props $f);}; (@props $t:ty) => { impl AddAssociative for $t {} impl AddCommutative for $t {} impl MulAssociative for $t {} impl MulCommutative for $t {} }; } impl_props!{ usize u8 u16 u32 u64 u128 isize i8 i16 i32 i64 i128; f32 f64 } #[cfg(std)] impl<'a> AddAssociative for ::std::borrow::Cow<'a,str> {}