/**
 * A collection of typeclasses that add additional laws to existing fp-ts typeclasses.
 *
 * @since 1.0.0
 */
import * as Grp from 'fp-ts/Group'
import * as Rng from 'fp-ts/Ring'

// #############
// ### Model ###
// #############

/**
 * An `AbelianGroup` is a `Group` that abides the following laws:
 *
 * - Commutativity: `a * b = b * a`
 *
 * @since 1.0.0
 * @category Typeclasses
 */
export interface AbelianGroup<A> extends Grp.Group<A> {}

/**
 * Adapted from:
 * https://pursuit.purescript.org/packages/purescript-prelude/6.0.0/docs/Data.DivisionRing
 *
 * A ring structure with division
 *
 * - One != zero
 * - Nonzero multiplicative inverse
 *
 * @since 1.0.0
 * @category Typeclasses
 */
export interface DivisionRing<A> extends Rng.Ring<A> {
  readonly recip: (x: A) => A
}

/**
 * Adapted from:
 * https://pursuit.purescript.org/packages/purescript-prelude/6.0.0/docs/Data.CommutativeRing
 *
 * A Ring structure with commutativity of multiplcation
 *
 * - Commutativity: `a * b = b * a`
 *
 * @since 1.0.0
 * @category Typeclasses
 */
export interface CommutativeRing<A> extends Rng.Ring<A> {}

/**
 * Adapted from:
 * https://pursuit.purescript.org/packages/purescript-prelude/6.0.0/docs/Data.EuclideanRing
 *
 * @since 1.0.0
 * @category Typeclasses
 */
export interface EuclidianRing<A> extends CommutativeRing<A> {
  readonly degree: (a: A) => number
  readonly div: (x: A, y: A) => A
  readonly mod: (x: A, y: A) => A
}

/**
 * A `Module` over a Ring R extends an Abelian Group A follows all the laws for an Abelian
 * Group and the following:
 *
 * - Distributivity over the Abelian Group: `r * (x + y) = r * x + r * y`
 * - Distributivity over the Ring R: `(r + s) * x = r * x + s * x`
 * - Associativity over the Ring R: `(r * s) * x = r * (s * x)`
 * - Unital element over the Ring R: `1 * x = x`
 *
 * @since 1.0.0
 * @category Typeclasses
 */
export interface LeftModule<A, L> extends AbelianGroup<A> {
  readonly leftScalarMul: (r: L, x: A) => A
}

/**
 * See `LeftModule` for Module laws
 *
 * @since 1.0.0
 * @category Typeclasses
 */
export interface RightModule<A, R> extends AbelianGroup<A> {
  readonly rightScalarMul: (x: A, r: R) => A
}

/**
 * @since 1.0.0
 * @category Typeclasses
 */
export interface Bimodule<A, L, R = L> extends LeftModule<A, L>, RightModule<A, R> {}