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#![deny(missing_docs)] //! # Higher Order Core //! //! This crate contains core structs and traits for programming with higher order data structures. //! //! ### Introduction to higher order data structures //! //! A higher order data structure is a generalization of an ordinary data structure. //! //! In ordinary data structures, the default way of programming is: //! //! - Use data structures for data //! - Use methods/functions for operations on data //! //! In a higher order data structure, data and functions become the same thing. //! //! The central idea of a higher order data structure, //! is that properties can be functions of the same type. //! //! For example, a `Point` has an `x`, `y` and `z` property. //! In ordinary programming, `x`, `y` and `z` might have the type `f64`. //! //! If `x`, `y` and `z` are functions from `T -> f64`, //! then the point type is `Point<T>`. //! //! A higher order `Point<T>` can be called, just like a function. //! When called as a function, `Point<T>` returns `Point`. //! //! However, unlike functions, you can still access properties of `Point<T>`. //! You can also define methods and overload operators for `Point<T>`. //! This means that in a higher order data structure, data and functions become the same thing. //! //! ### Motivation of programming with higher order data structures //! //! The major application of higher order data structures is geometry. //! //! A typical usage is e.g. to create procedurally generated content for games. //! //! Higher order data structures is about finding the right balance between //! hiding implementation details and exposing them for various generic algorithms. //! //! For example, a circle can be thought of as having the type `Point<f64>`. //! The argument can be an angle in radians, or a value in the unit interval `[0, 1]`. //! //! Another example, a line can be thought of as having the type `Point<f64>`. //! The argument is a value in the unit interval `[0, 1]`. //! When called with `0`, you get the start point of the line. //! When called with `1`, you get the end point of the line. //! //! Instead of declaring a `Circle` type, a `Line` type and so on, //! one can use `Point<f64>` to represent both of them. //! //! Higher order data structures makes easier to write generic algorithms for geometry. //! Although it seems abstract at first, it is also practically useful in unexpected cases. //! //! For example, an animated point can be thought of as having the type `Point<(&[Frame], f64)>`. //! The first argument contains the animation data and the second argument is time in seconds. //! Properties `x`, `y` and `z` of an animated point determines how the animated point is computed. //! The details of the implementation can be hidden from the algorithm that uses animated points. //! //! Sometimes you need to work with complex geometry. //! In these cases, it is easier to work with higher order data structures. //! //! For example, a planet might have a center, equator, poles, surface etc. //! A planet orbits around a star, which orbits around the center of a galaxy. //! This means that the properties of a planet, viewed from different reference frames, //! are functions of the arguments that determine the reference frame. //! You can create a "higher order planet" to reason about a planet's properties //! under various reference frames. //! //! ### Design //! //! Here is an example of how to declare a new higher order data structure: //! //! ```rust //! use ha::{Ho, Call, Arg, Fun, Func}; //! use std::sync::Arc; //! //! /// Higher order 3D point. //! #[derive(Clone)] //! pub struct Point<T = ()> where f64: Ho<T> { //! /// Function for x-coordinates. //! pub x: Fun<T, f64>, //! /// Function for y-coordinates. //! pub y: Fun<T, f64>, //! /// Function for z-coordinates. //! pub z: Fun<T, f64>, //! } //! //! // It is common to declare a type alias for functions, e.g: //! pub type PointFunc<T> = Point<Arg<T>>; //! //! // Implement `Ho<Arg<T>>` to allow higher order data structures //! // using properties `Fun<T, Point>` (`<Point as Ho<T>>::Fun`). //! impl<T: Clone> Ho<Arg<T>> for Point { //! type Fun = PointFunc<T>; //! } //! //! // Implement `Call<T>` to allow higher order calls. //! impl<T: Copy> Call<T> for Point //! where f64: Ho<Arg<T>> + Call<T> //! { //! fn call(f: &Self::Fun, val: T) -> Point { //! Point::<()> { //! x: <f64 as Call<T>>::call(&f.x, val), //! y: <f64 as Call<T>>::call(&f.y, val), //! z: <f64 as Call<T>>::call(&f.z, val), //! } //! } //! } //! //! impl<T> PointFunc<T> { //! /// Helper method for calling value. //! pub fn call(&self, val: T) -> Point where T: Copy { //! <Point as Call<T>>::call(self, val) //! } //! } //! //! // Operations are usually defined as simple traits. //! // They look exactly the same as for normal generic programming. //! /// Dot operator. //! pub trait Dot<Rhs = Self> { //! /// The output type. //! type Output; //! //! /// Returns the dot product. //! fn dot(self, other: Rhs) -> Self::Output; //! } //! //! // Implement operator once for the ordinary case. //! impl Dot for Point { //! type Output = f64; //! fn dot(self, other: Self) -> f64 { //! self.x * other.x + //! self.y * other.y + //! self.z * other.z //! } //! } //! //! // Implement operator once for the higher order case. //! impl<T: 'static + Copy> Dot for PointFunc<T> { //! type Output = Func<T, f64>; //! fn dot(self, other: Self) -> Func<T, f64> { //! let ax = self.x; //! let ay = self.y; //! let az = self.z; //! let bx = other.x; //! let by = other.y; //! let bz = other.z; //! Arc::new(move |a| ax(a) * bx(a) + ay(a) * by(a) + az(a) * bz(a)) //! } //! } //! ``` //! //! To disambiguate impls of e.g. `Point<()>` from `Point<T>`, //! an argument type `Arg<T>` is used for point functions: `Point<Arg<T>>`. //! //! For every higher order type `U` and and argument type `T`, //! there is an associated function type `T -> U`. //! //! For primitive types, e.g. `f64`, the function type is `Func<T, f64>`. //! //! For higher order structs, e.g. `X<()>`, the function type is `X<Arg<T>>`. //! //! The code for operators on higher order data structures must be written twice: //! //! - Once for the ordinary case `X<()>` //! - Once for the higher order case `X<Arg<T>>` //! //! ### Higher Order Maps //! //! Sometimes it is useful to construct arbitrary data of the kind: //! //! - Vectors of primitives //! - Vectors of vectors, etc. //! //! For example, if a higher order point maps from angles to a circle, //! then complex geometry primitives might be defined onto the circle using angles: //! //! - Edge, e.g. `[a, b]` //! - Triangle, e.g. `[a, b, c]` //! - Square, e.g. `[[a, b], [c, d]]` //! //! The `HMap::hmap` method can be used to work with such structures. //! //! For example, if `p` is a higher order point of type `Point<Arg<f64>>`, //! then the following code maps two points at the same time: //! //! ```ignore //! let q: [Point; 2] = [0.0, 1.0].hmap(&p); //! ``` //! //! For binary higher order maps of type `f : (T, T) -> U`, //! the `HPair::hpair` method can be used before using `HMap::hmap`. //! //! For example: //! //! ```ignore //! let in_between: Func<f64, f64> = Arc::new(move |(a, b)| { //! if b < a {b += 1.0}; //! (a + (b - a) * 0.5) % 1.0 //! }); //! // Pair up. //! let args: [(f64, f64); 2] = ([0.7, 0.9], [0.9, 0.1]).hpair(); //! // `[0.8, 0.0]` //! let q: [f64; 2] = args.hmap(&in_between); //! ``` use std::sync::Arc; /// Standard function type. pub type Func<T, U> = Arc<dyn Fn(T) -> U + Send + Sync>; /// Used to disambiguate impls for Rust's type checker. #[derive(Copy, Clone)] pub struct Arg<T>(pub T); /// Implemented by higher order types. /// /// A higher order type might be a concrete value, /// or it might be a function of some input type `T`. /// /// Each higher order type has an associated function type /// for any argument of type `T`. /// /// This makes it possible to e.g. associate `PointFunc<T>` with `Point`. pub trait Ho<T>: Sized { /// The function type. type Fun: Clone; } /// Implemented by higher order calls. pub trait Call<T>: Ho<Arg<T>> { /// Calls function with some value. fn call(f: &Self::Fun, val: T) -> Self; } impl<T, U> Call<T> for U where U: Ho<Arg<T>, Fun = Func<T, Self>> { fn call(f: &Self::Fun, val: T) -> Self {f(val)} } impl<T: Clone> Ho<()> for T {type Fun = T;} /// Used to declare functions in a more readable way. pub type Fun<T, U> = <U as Ho<T>>::Fun; impl<T> Ho<Arg<T>> for f64 {type Fun = Func<T, f64>;} impl<T> Ho<Arg<T>> for f32 {type Fun = Func<T, f32>;} impl<T> Ho<Arg<T>> for u8 {type Fun = Func<T, u8>;} impl<T> Ho<Arg<T>> for u16 {type Fun = Func<T, u16>;} impl<T> Ho<Arg<T>> for u32 {type Fun = Func<T, u32>;} impl<T> Ho<Arg<T>> for u64 {type Fun = Func<T, u64>;} impl<T> Ho<Arg<T>> for usize {type Fun = Func<T, usize>;} impl<T> Ho<Arg<T>> for i8 {type Fun = Func<T, i8>;} impl<T> Ho<Arg<T>> for i16 {type Fun = Func<T, i16>;} impl<T> Ho<Arg<T>> for i32 {type Fun = Func<T, i32>;} impl<T> Ho<Arg<T>> for i64 {type Fun = Func<T, i64>;} impl<T> Ho<Arg<T>> for isize {type Fun = Func<T, isize>;} /// Higher order pairing. /// /// A higher order pairing is used pair up components of a pair of data structures. /// This is used before binary higher order maps of the type `f : (T, T) -> U`. pub trait HPair { /// Output type. type Out; /// Returns the higher order transposed value. fn hpair(self) -> Self::Out; } impl HPair for (f64, f64) {type Out = Self; fn hpair(self) -> Self {self}} impl HPair for (f32, f32) {type Out = Self; fn hpair(self) -> Self {self}} impl HPair for (u8, u8) {type Out = Self; fn hpair(self) -> Self {self}} impl HPair for (u16, u16) {type Out = Self; fn hpair(self) -> Self {self}} impl HPair for (u32, u32) {type Out = Self; fn hpair(self) -> Self {self}} impl HPair for (u64, u64) {type Out = Self; fn hpair(self) -> Self {self}} impl HPair for (usize, usize) {type Out = Self; fn hpair(self) -> Self {self}} impl HPair for (i8, i8) {type Out = Self; fn hpair(self) -> Self {self}} impl HPair for (i16, i16) {type Out = Self; fn hpair(self) -> Self {self}} impl HPair for (i32, i32) {type Out = Self; fn hpair(self) -> Self {self}} impl HPair for (i64, i64) {type Out = Self; fn hpair(self) -> Self {self}} impl HPair for (isize, isize) {type Out = Self; fn hpair(self) -> Self {self}} impl<T> HPair for ([T; 2], [T; 2]) where (T, T): HPair { type Out = [<(T, T) as HPair>::Out; 2]; fn hpair(self) -> Self::Out { let ([a, b], [c, d]) = self; [(a, c).hpair(), (b, d).hpair()] } } impl<T> HPair for ([T; 3], [T; 3]) where (T, T): HPair { type Out = [<(T, T) as HPair>::Out; 3]; fn hpair(self) -> Self::Out { let ([a, b, c], [d, e, f]) = self; [(a, d).hpair(), (b, e).hpair(), (c, f).hpair()] } } impl<T> HPair for ([T; 4], [T; 4]) where (T, T): HPair { type Out = [<(T, T) as HPair>::Out; 4]; fn hpair(self) -> Self::Out { let ([a, b, c, d], [e, f, g, h]) = self; [(a, e).hpair(), (b, f).hpair(), (c, g).hpair(), (d, h).hpair()] } } impl<T> HPair for ([T; 5], [T; 5]) where (T, T): HPair { type Out = [<(T, T) as HPair>::Out; 5]; fn hpair(self) -> Self::Out { let ([a, b, c, d, e], [f, g, h, i, j]) = self; [ (a, f).hpair(), (b, g).hpair(), (c, h).hpair(), (d, i).hpair(), (e, j).hpair() ] } } impl<T> HPair for ([T; 6], [T; 6]) where (T, T): HPair { type Out = [<(T, T) as HPair>::Out; 6]; fn hpair(self) -> Self::Out { let ([a, b, c, d, e, f], [g, h, i, j, k, l]) = self; [ (a, g).hpair(), (b, h).hpair(), (c, i).hpair(), (d, j).hpair(), (e, k).hpair(), (f, l).hpair() ] } } impl<T> HPair for (Vec<T>, Vec<T>) where (T, T): HPair { type Out = Vec<<(T, T) as HPair>::Out>; fn hpair(self) -> Self::Out { let (a, b) = self; a.into_iter().zip(b.into_iter()).map(|n| n.hpair()).collect() } } /// Implemented by higher order maps. /// /// A higher order map takes common data structures such as /// vectors and lists and applies a function to every element. /// /// This is implemented recursively, hence higher order maps. pub trait HMap<Out> { /// The out type. type Fun; /// Maps structure. fn hmap(self, f: &Self::Fun) -> Out; } impl<T, U> HMap<U> for T where U: Call<T> { type Fun = U::Fun; fn hmap(self, f: &Self::Fun) -> U { <U as Call<T>>::call(f, self) } } impl<T, U> HMap<[U; 2]> for [T; 2] where T: HMap<U> { type Fun = T::Fun; fn hmap(self, f: &Self::Fun) -> [U; 2] { let [a, b] = self; [a.hmap(f), b.hmap(f)] } } impl<T, U> HMap<[U; 3]> for [T; 3] where T: HMap<U> { type Fun = T::Fun; fn hmap(self, f: &Self::Fun) -> [U; 3] { let [a, b, c] = self; [a.hmap(f), b.hmap(f), c.hmap(f)] } } impl<T, U> HMap<[U; 4]> for [T; 4] where T: HMap<U> { type Fun = T::Fun; fn hmap(self, f: &Self::Fun) -> [U; 4] { let [a, b, c, d] = self; [a.hmap(f), b.hmap(f), c.hmap(f), d.hmap(f)] } } impl<T, U> HMap<[U; 5]> for [T; 5] where T: HMap<U> { type Fun = T::Fun; fn hmap(self, f: &Self::Fun) -> [U; 5] { let [a, b, c, d, e] = self; [a.hmap(f), b.hmap(f), c.hmap(f), d.hmap(f), e.hmap(f)] } } impl<T, U> HMap<[U; 6]> for [T; 6] where T: HMap<U> { type Fun = T::Fun; fn hmap(self, fx: &Self::Fun) -> [U; 6] { let [a, b, c, d, e, f] = self; [a.hmap(fx), b.hmap(fx), c.hmap(fx), d.hmap(fx), e.hmap(fx), f.hmap(fx)] } } impl<T, U> HMap<Vec<U>> for Vec<T> where T: HMap<U> { type Fun = T::Fun; fn hmap(self, f: &Self::Fun) -> Vec<U> { self.into_iter().map(|n| n.hmap(f)).collect() } }