dsalgo 0.3.10

//! binary function

pub trait BinaryFunc {
    type L;

    type R;

    type Cod;

    fn f(
        _: Self::L,
        _: Self::R,
    ) -> Self::Cod;
}

/// external binary operation

pub trait ExtBinaryOp: BinaryFunc {}

impl<T: BinaryFunc> ExtBinaryOp for T {}

/// binary operation on a set.

pub trait BinaryOp {
    type S;

    fn op(
        _: Self::S,
        _: Self::S,
    ) -> Self::S;
}

fn is_left_identity<S, F>(
    f: &F,
    e: S,
    x: S,
) -> bool
where
    F: Fn(S, S) -> S,
    S: Clone + PartialEq,
{
    f(e, x.clone()) == x
}

fn is_right_identity<S, F>(
    f: &F,
    e: S,
    x: S,
) -> bool
where
    F: Fn(S, S) -> S,
    S: Clone + PartialEq,
{
    f(x.clone(), e) == x
}

fn is_identity<S, F>(
    f: &F,
    e: S,
    x: S,
) -> bool
where
    F: Fn(S, S) -> S,
    S: Clone + PartialEq,
{
    is_left_identity(f, e.clone(), x.clone()) && is_right_identity(f, e, x)
}

/// identity element

pub trait Identity: BinaryOp {
    fn e() -> Self::S;

    fn assert(x: Self::S)
    where
        Self::S: Clone + PartialEq,
    {
        assert!(is_identity(&Self::op, Self::e(), x));
    }
}

fn is_invertible<F, G, X>(
    op: &F,
    inv: &G,
    e: X,
    x: X,
) -> bool
where
    F: Fn(X, X) -> X,
    G: Fn(X) -> X,
    X: Clone + PartialEq,
{
    op(x.clone(), inv(x.clone())) == e.clone()
        && op(inv(x.clone()), x.clone()) == e
}

/// inverse element

pub trait Inverse: Identity {
    fn inv(_: Self::S) -> Self::S;

    fn assert(x: Self::S)
    where
        Self::S: Clone + PartialEq,
    {
        assert!(is_invertible(&Self::op, &Self::inv, Self::e(), x));
    }
}

fn is_commutative<F, X, Y>(
    f: &F,
    a: X,
    b: X,
) -> bool
where
    F: Fn(X, X) -> Y,
    X: Clone,
    Y: PartialEq,
{
    f(a.clone(), b.clone()) == f(b, a)
}

/// commutative property

pub trait Commutative: BinaryOp {
    fn assert(
        a: Self::S,
        b: Self::S,
    ) where
        Self::S: Clone + PartialEq,
    {
        assert!(is_commutative(&Self::op, a, b));
    }
}

fn is_associative<F, X>(
    f: &F,
    a: X,
    b: X,
    c: X,
) -> bool
where
    F: Fn(X, X) -> X,
    X: Clone + PartialEq,
{
    f(f(a.clone(), b.clone()), c.clone()) == f(a, f(b, c))
}

/// associative property

pub trait Associative: BinaryOp {
    fn assert(
        x: Self::S,
        y: Self::S,
        z: Self::S,
    ) where
        Self::S: Clone + PartialEq,
    {
        assert!(is_associative(&Self::op, x, y, z));
    }
}

fn is_idempotent<F, X>(
    f: &F,
    x: X,
) -> bool
where
    F: Fn(X, X) -> X,
    X: Clone + PartialEq,
{
    f(x.clone(), x.clone()) == x
}

pub trait Idempotence: BinaryOp {
    fn assert(x: Self::S)
    where
        Self::S: Clone + PartialEq,
    {
        assert!(is_idempotent(&Self::op, x));
    }
}

/// latin square property

pub trait LatinSquare: BinaryOp {}

impl<T: Inverse> LatinSquare for T {}

// TODO: assertion latin square
fn is_left_absorbing<F, X>(
    f: &F,
    z: X,
    x: X,
) -> bool
where
    F: Fn(X, X) -> X,
    X: Clone + PartialEq,
{
    f(z.clone(), x) == z
}

fn is_right_absorbing<F, X>(
    f: &F,
    z: X,
    x: X,
) -> bool
where
    F: Fn(X, X) -> X,
    X: Clone + PartialEq,
{
    f(x, z.clone()) == z
}

fn is_absorbing<F, X>(
    f: &F,
    z: X,
    x: X,
) -> bool
where
    F: Fn(X, X) -> X,
    X: Clone + PartialEq,
{
    is_left_absorbing(f, z.clone(), x.clone()) && is_right_absorbing(f, z, x)
}

/// absorbing element

pub trait Absorbing: BinaryOp {
    fn z() -> Self::S;

    fn assert(x: Self::S)
    where
        Self::S: Clone + PartialEq,
    {
        assert!(is_absorbing(&Self::op, Self::z(), x));
    }
}

pub trait Add {
    type S;

    fn add(
        _: Self::S,
        _: Self::S,
    ) -> Self::S;
}

pub trait Mul: Add {
    fn mul(
        _: Self::S,
        _: Self::S,
    ) -> Self::S;
}

fn iz_zero<F, X>(
    mul: &F,
    z: X,
    x: X,
) -> bool
where
    F: Fn(X, X) -> X,
    X: Clone + PartialEq,
{
    is_absorbing(mul, z, x)
}

pub trait Zero: Mul {
    fn zero() -> Self::S;

    fn assert(x: Self::S)
    where
        Self::S: Clone + PartialEq,
    {
        assert!(iz_zero(&Self::mul, Self::zero(), x));
    }
}

pub trait One: Mul {
    fn one() -> Self::S;
}

pub trait AddInv: Add {
    fn add_inv(_: Self::S) -> Self::S;
}

pub trait MulInv: Mul {
    fn mul_inv(_: Self::S) -> Self::S;
}

fn is_left_distributive<Add, Mul, X>(
    add: &Add,
    mul: &Mul,
    x: X,
    y: X,
    z: X,
) -> bool
where
    Add: Fn(X, X) -> X,
    Mul: Fn(X, X) -> X,
    X: Clone + PartialEq,
{
    mul(x.clone(), add(y.clone(), z.clone()))
        == add(mul(x.clone(), y), mul(x, z))
}

fn is_right_distributive<Add, Mul, X>(
    add: &Add,
    mul: &Mul,
    y: X,
    z: X,
    x: X,
) -> bool
where
    Add: Fn(X, X) -> X,
    Mul: Fn(X, X) -> X,
    X: Clone + PartialEq,
{
    mul(add(y.clone(), z.clone()), x.clone())
        == add(mul(y, x.clone()), mul(z, x))
}

fn is_distributive<Add, Mul, X>(
    add: &Add,
    mul: &Mul,
    x: X,
    y: X,
    z: X,
) -> bool
where
    Add: Fn(X, X) -> X,
    Mul: Fn(X, X) -> X,
    X: Clone + PartialEq,
{
    is_left_distributive(add, mul, x.clone(), y.clone(), z.clone())
        && is_right_distributive(add, mul, y, z, x)
}

pub trait Distributive: Mul {
    fn assert(
        x: Self::S,
        y: Self::S,
        z: Self::S,
    ) where
        Self::S: Clone + PartialEq,
    {
        assert!(is_distributive(&Self::mul, &Self::add, x, y, z));
    }
}

pub mod dynamic {

    pub trait BinaryOp {
        type S;

        fn op(
            &self,
            _: Self::S,
            _: Self::S,
        ) -> Self::S;
    }
}

pub mod itself {

    pub trait Id {}

    impl<T> Id for T {}

    pub trait BinaryOp<I: Id> {
        fn op(
            _: Self,
            _: Self,
        ) -> Self;
    }

    pub trait Commutative<I: Id> {}

    pub trait Associative<I: Id> {}

    pub trait Idempotence<I: Id> {}

    pub trait Identity<I: Id> {
        fn e() -> Self;
    }

    pub trait Inverse<I: Id> {
        fn inv(_: Self) -> Self;
    }
}