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//! Group
use super::{
    monoid::{Monoid, MonoidAdd, MonoidMul},
    operator::{Addition, Multiplication, Operator},
    Loop,
};
use std::ops::{Div, DivAssign, Neg, Sub, SubAssign};

/// A Group is a triple $(\mathbb{M}, \circ, e)$, composed by a set
/// $\mathbb{M}$ and a binary inner operation $\circ$ and the element $e
/// \in \mathbb{M}$
///
/// ```math
/// \circ: \mathbb{M} \times \mathbb{M} \rightarrow \mathbb{M} , (x, y) \mapsto x \circ y
/// ```
/// 1. associativity <br>
/// $\forall x, y, z \in \mathbb{M}$: $x \circ (y \circ z) = (x \circ y)
/// \circ z$ 2. $e$ neutral element(identity) <br>
/// $\forall x \in \mathbb{M}$: $x \circ e = e \circ x = x$
/// 3.
/// $x^-1 \in \mathbb{M}: x^⁻1 \circ x = x \circ x^-1 ` = e$
pub trait Group<O: Operator>: Loop<O> + Monoid<O> {}

macro_rules! impl_group(
    ($T:ty, $($S:ty),*) =>
    {
        $(
        impl Group<$T> for $S
        {
        }
        )*
    }
);

impl_group!(Addition, i8, i16, i32, i64, i128, f32, f64);
impl_group!(Multiplication, f32, f64);

pub trait GroupAdd:
    Group<Addition> + MonoidAdd + Sub<Self, Output = Self> + SubAssign<Self> + Neg<Output = Self>
{
}

macro_rules! impl_groupadd
(
    ($($T:ty),*) =>
    {
        $(
            impl GroupAdd for $T
            {

            }
        )*
    }
);

impl_groupadd!(i8, i16, i32, i64, i128, f32, f64);

pub trait GroupMul:
    Group<Multiplication> + MonoidMul + Div<Self, Output = Self> + DivAssign<Self>
{
}

macro_rules! impl_groupmul
(
    ($($T:ty),*) =>
    {
        $(
            impl GroupMul for $T
            {

            }
        )*
    }
);

impl_groupmul!(f32, f64);