/*!
 * Numerical type definitions.
 *
 * `Numeric` together with `where for<'a> &'a T: NumericRef<T>`
 * expresses the operations in [`NumericByValue`] for
 * all 4 combinations of by value and by reference. [`Numeric`]
 * additionally adds some additional constraints only needed by value on an implementing
 * type such as `PartialOrd`, [`ZeroOne`] and
 * [`FromUsize`].
 */

use std::cmp::PartialOrd;
use std::iter::Sum;
use std::marker::Sized;
use std::num::Wrapping;
use std::ops::Add;
use std::ops::Div;
use std::ops::Mul;
use std::ops::Neg;
use std::ops::Sub;

/**
 * A trait defining what a numeric type is in terms of by value
 * numerical operations matrices need their types to support for
 * math operations.
 *
 * The requirements are Add, Sub, Mul, Div, Neg and Sized. Note that
 * unsigned integers do not implement Neg unless they are wrapped by
 * [Wrapping](std::num::Wrapping).
 */
pub trait NumericByValue<Rhs = Self, Output = Self>:
    Add<Rhs, Output = Output>
    + Sub<Rhs, Output = Output>
    + Mul<Rhs, Output = Output>
    + Div<Rhs, Output = Output>
    + Neg<Output = Output>
    + Sized
{
}

/**
 * Anything which implements all the super traits will automatically implement this trait too.
 * This covers primitives such as f32, f64, signed integers and
 * [Wrapped unsigned integers](std::num::Wrapping)
 * as well as [Traces](super::differentiation::Trace) and
 * [Records](super::differentiation::Record) of those types.
 *
 * It will not include Matrix because Matrix does not implement Div.
 * Similarly, unwrapped unsigned integers do not implement Neg so are not included.
 */
impl<T, Rhs, Output> NumericByValue<Rhs, Output> for T where
    // Div is first here because Matrix does not implement it.
    // if Add, Sub or Mul are first the rust compiler gets stuck
    // in an infinite loop considering arbitarily nested matrix
    // types, even though any level of nested Matrix types will
    // never implement Div so shouldn't be considered for
    // implementing NumericByValue
    T: Div<Rhs, Output = Output>
        + Add<Rhs, Output = Output>
        + Sub<Rhs, Output = Output>
        + Mul<Rhs, Output = Output>
        + Neg<Output = Output>
        + Sized
{
}

/**
 * The trait to define `&T op T` and `&T op &T` versions for NumericByValue
 * based off the MIT/Apache 2.0 licensed code from num-traits 0.2.10:
 *
 * **This trait is not ever used directly for users of this library. You
 * don't need to deal with it unless
 * [implementing custom numeric types](super::using_custom_types)
 * and even then it will be implemented automatically.**
 *
 * - [http://opensource.org/licenses/MIT](http://opensource.org/licenses/MIT)
 * - [https://docs.rs/num-traits/0.2.10/src/num_traits/lib.rs.html#112](https://docs.rs/num-traits/0.2.10/src/num_traits/lib.rs.html#112)
 *
 * The trick is that all types implementing this trait will be references,
 * so the first constraint expresses some &T which can be operated on with
 * some right hand side type T to yield a value of type T.
 *
 * In a similar way the second constraint expresses `&T op &T -> T` operations
 */
pub trait NumericRef<T>:
    // &T op T -> T
    NumericByValue<T, T>
    // &T op &T -> T
    + for<'a> NumericByValue<&'a T, T> {}

/**
 * Anything which implements all the super traits will automatically implement this trait too.
 * This covers primitives such as `&f32`, `&f64`, ie a type like `&u8` is `NumericRef<u8>`,
 * as well as [Traces](super::differentiation::Trace) and
 * [Records](super::differentiation::Record) of those types.
 */
impl<RefT, T> NumericRef<T> for RefT where
    RefT: NumericByValue<T, T> + for<'a> NumericByValue<&'a T, T>
{
}

/**
 * A general purpose numeric trait that defines all the behaviour numerical
 * matrices need their types to support for math operations.
 *
 * This trait extends the constraints in [NumericByValue]
 * to types which also support the operations with a right hand side type
 * by reference, and adds some additional constraints needed only
 * by value on types.
 *
 * When used together with [NumericRef] this
 * expresses all 4 by value and by reference combinations for the
 * operations using the following syntax:
 *
 * ```ignore
 * fn function_name<T: Numeric>()
 * where for<'a> &'a T: NumericRef<T> {
 *
 * }
 * ```
 *
 * This pair of constraints is used nearly everywhere some numeric
 * type is needed, so although this trait does not require reference
 * type methods by itself, in practise you won't be able to call many
 * functions in this library with a numeric type that doesn't.
 */
pub trait Numeric:
    // T op T -> T
    NumericByValue
    // T op &T -> T
    + for<'a> NumericByValue<&'a Self>
    + Clone
    + ZeroOne
    + FromUsize
    + Sum
    + PartialOrd {}

/**
 * All types implemeting the operations in NumericByValue with a right hand
 * side type by reference are Numeric.
 *
 * This covers primitives such as f32, f64, signed integers and
 * [Wrapped unsigned integers](std::num::Wrapping),
 * as well as [Traces](super::differentiation::Trace) and
 * [Records](super::differentiation::Record) of those types.
 */
impl<T> Numeric for T where
    T: NumericByValue
        + for<'a> NumericByValue<&'a T>
        + Clone
        + ZeroOne
        + FromUsize
        + Sum
        + PartialOrd
{
}

/**
 * A trait defining how to obtain 0 and 1 for every implementing type.
 *
 * The boilerplate implementations for primitives is performed with a macro.
 * If a primitive type is missing from this list, please open an issue to add it in.
 */
pub trait ZeroOne: Sized {
    fn zero() -> Self;
    fn one() -> Self;
}

impl<T: ZeroOne> ZeroOne for Wrapping<T> {
    #[inline]
    fn zero() -> Wrapping<T> {
        Wrapping(T::zero())
    }
    #[inline]
    fn one() -> Wrapping<T> {
        Wrapping(T::one())
    }
}

macro_rules! zero_one_integral {
    ($T:ty) => {
        impl ZeroOne for $T {
            #[inline]
            fn zero() -> $T {
                0
            }
            #[inline]
            fn one() -> $T {
                1
            }
        }
    };
}

macro_rules! zero_one_float {
    ($T:ty) => {
        impl ZeroOne for $T {
            #[inline]
            fn zero() -> $T {
                0.0
            }
            #[inline]
            fn one() -> $T {
                1.0
            }
        }
    };
}

zero_one_integral!(u8);
zero_one_integral!(i8);
zero_one_integral!(u16);
zero_one_integral!(i16);
zero_one_integral!(u32);
zero_one_integral!(i32);
zero_one_integral!(u64);
zero_one_integral!(i64);
zero_one_integral!(u128);
zero_one_integral!(i128);
zero_one_float!(f32);
zero_one_float!(f64);
zero_one_integral!(usize);
zero_one_integral!(isize);

/**
 * Specifies how to obtain an instance of this numeric type
 * equal to the usize primitive. If the number is too large to
 * represent in this type, `None` should be returned instead.
 *
 * The boilerplate implementations for primitives is performed with a macro.
 * If a primitive type is missing from this list, please open an issue to add it in.
 */
pub trait FromUsize: Sized {
    fn from_usize(n: usize) -> Option<Self>;
}

impl<T: FromUsize> FromUsize for Wrapping<T> {
    fn from_usize(n: usize) -> Option<Wrapping<T>> {
        Some(Wrapping(T::from_usize(n)?))
    }
}

macro_rules! from_usize_integral {
    ($T:ty) => {
        impl FromUsize for $T {
            #[inline]
            fn from_usize(n: usize) -> Option<$T> {
                if n <= (<$T>::max_value() as usize) {
                    Some(n as $T)
                } else {
                    None
                }
            }
        }
    };
}

macro_rules! from_usize_float {
    ($T:ty) => {
        impl FromUsize for $T {
            #[inline]
            fn from_usize(n: usize) -> Option<$T> {
                Some(n as $T)
            }
        }
    };
}

from_usize_integral!(u8);
from_usize_integral!(i8);
from_usize_integral!(u16);
from_usize_integral!(i16);
from_usize_integral!(u32);
from_usize_integral!(i32);
from_usize_integral!(u64);
from_usize_integral!(i64);
from_usize_integral!(u128);
from_usize_integral!(i128);
from_usize_float!(f32);
from_usize_float!(f64);
from_usize_integral!(usize);
from_usize_integral!(isize);

/**
 * Additional traits for more complex numerical operations on real numbers.
 */
pub mod extra {

    /**
     * A type which can be square rooted.
     *
     * This is implemented by `f32` and `f64` by value and by reference, as well as
     * [Traces](super::super::differentiation::Trace)
     * and [Records](super::super::differentiation::Record) of these.
     */
    pub trait Sqrt {
        type Output;
        fn sqrt(self) -> Self::Output;
    }

    macro_rules! sqrt_float {
        ($T:ty) => {
            impl Sqrt for $T {
                type Output = $T;
                #[inline]
                fn sqrt(self) -> Self::Output {
                    self.sqrt()
                }
            }
            impl Sqrt for &$T {
                type Output = $T;
                #[inline]
                fn sqrt(self) -> Self::Output {
                    self.clone().sqrt()
                }
            }
        };
    }

    sqrt_float!(f32);
    sqrt_float!(f64);

    /**
     * A type which can compute e^self.
     *
     * This is implemented by `f32` and `f64` by value and by reference, as well as
     * [Traces](super::super::differentiation::Trace)
     * and [Records](super::super::differentiation::Record) of these.
     */
    pub trait Exp {
        type Output;
        fn exp(self) -> Self::Output;
    }

    macro_rules! exp_float {
        ($T:ty) => {
            impl Exp for $T {
                type Output = $T;
                #[inline]
                fn exp(self) -> Self::Output {
                    self.exp()
                }
            }
            impl Exp for &$T {
                type Output = $T;
                #[inline]
                fn exp(self) -> Self::Output {
                    self.clone().exp()
                }
            }
        };
    }

    exp_float!(f32);
    exp_float!(f64);

    /**
     * A type which can compute self^rhs.
     *
     * This is implemented by `f32` and `f64` for all combinations of
     * by value and by reference, as well as
     * [Traces](super::super::differentiation::Trace)
     * and [Records](super::super::differentiation::Record) of these.
     *
     * The Trace and Record implementations also implement versions with the other
     * argument being a raw `f32` or `f64`, for convenience.
     */
    pub trait Pow<Rhs = Self> {
        type Output;
        fn pow(self, rhs: Rhs) -> Self::Output;
    }

    macro_rules! pow_float {
        ($T:ty) => {
            // T ^ T
            impl Pow<$T> for $T {
                type Output = $T;
                #[inline]
                fn pow(self, rhs: Self) -> Self::Output {
                    self.powf(rhs)
                }
            }
            // T ^ &T
            impl<'a> Pow<&'a $T> for $T {
                type Output = $T;
                #[inline]
                fn pow(self, rhs: &Self) -> Self::Output {
                    self.powf(rhs.clone())
                }
            }
            // &T ^ T
            impl<'a> Pow<$T> for &'a $T {
                type Output = $T;
                #[inline]
                fn pow(self, rhs: $T) -> Self::Output {
                    self.powf(rhs)
                }
            }
            // &T ^ &T
            impl<'a, 'b> Pow<&'b $T> for &'a $T {
                type Output = $T;
                #[inline]
                fn pow(self, rhs: &$T) -> Self::Output {
                    self.powf(rhs.clone())
                }
            }
        };
    }

    pow_float!(f32);
    pow_float!(f64);

    /**
     * A type which can represent Pi.
     */
    pub trait Pi {
        fn pi() -> Self;
    }

    impl Pi for f32 {
        fn pi() -> f32 {
            std::f32::consts::PI
        }
    }

    impl Pi for f64 {
        fn pi() -> f64 {
            std::f64::consts::PI
        }
    }

    /**
     * A type which can compute the natural logarithm of itself: ln(self).
     *
     * This is implemented by `f32` and `f64` by value and by reference, as well as
     * [Traces](super::super::differentiation::Trace)
     * and [Records](super::super::differentiation::Record) of these.
     */
    pub trait Ln {
        type Output;
        fn ln(self) -> Self::Output;
    }

    macro_rules! ln_float {
        ($T:ty) => {
            impl Ln for $T {
                type Output = $T;
                #[inline]
                fn ln(self) -> Self::Output {
                    self.ln()
                }
            }
            impl Ln for &$T {
                type Output = $T;
                #[inline]
                fn ln(self) -> Self::Output {
                    self.clone().ln()
                }
            }
        };
    }

    ln_float!(f32);
    ln_float!(f64);

    /**
     * A type which can compute the sine of itself: sin(self)
     *
     * This is implemented by `f32` and `f64` by value and by reference, as well as
     * [Traces](super::super::differentiation::Trace)
     * and [Records](super::super::differentiation::Record) of these.
     */
    pub trait Sin {
        type Output;
        fn sin(self) -> Self::Output;
    }

    macro_rules! sin_float {
        ($T:ty) => {
            impl Sin for $T {
                type Output = $T;
                #[inline]
                fn sin(self) -> Self::Output {
                    self.sin()
                }
            }
            impl Sin for &$T {
                type Output = $T;
                #[inline]
                fn sin(self) -> Self::Output {
                    self.clone().sin()
                }
            }
        };
    }

    sin_float!(f32);
    sin_float!(f64);

    /**
     * A type which can compute the cosine of itself: cos(self)
     *
     * This is implemented by `f32` and `f64` by value and by reference, as well as
     * [Traces](super::super::differentiation::Trace)
     * and [Records](super::super::differentiation::Record) of these.
     */
    pub trait Cos {
        type Output;
        fn cos(self) -> Self::Output;
    }

    macro_rules! cos_float {
        ($T:ty) => {
            impl Cos for $T {
                type Output = $T;
                #[inline]
                fn cos(self) -> Self::Output {
                    self.cos()
                }
            }
            impl Cos for &$T {
                type Output = $T;
                #[inline]
                fn cos(self) -> Self::Output {
                    self.clone().cos()
                }
            }
        };
    }

    cos_float!(f32);
    cos_float!(f64);

    /**
     * A trait defining what a real number type is in terms of by value
     * numerical operations needed on top of operations defined by Numeric
     * for some functions.
     *
     * The requirements are Sqrt, Exp, Pow, Ln, Sin, Cos and Sized.
     */
    pub trait RealByValue<Rhs = Self, Output = Self>:
        Sqrt<Output = Output>
        + Exp<Output = Output>
        + Pow<Rhs, Output = Output>
        + Ln<Output = Output>
        + Sin<Output = Output>
        + Cos<Output = Output>
        + Sized
    {
    }

    /**
     * Anything which implements all the super traits will automatically implement this trait too.
     * This covers primitives such as f32 & f64 as well as
     * [Traces](super::super::differentiation::Trace) and
     * [Records](super::super::differentiation::Record) of those types.
     */
    impl<T, Rhs, Output> RealByValue<Rhs, Output> for T where
        T: Sqrt<Output = Output>
            + Exp<Output = Output>
            + Pow<Rhs, Output = Output>
            + Ln<Output = Output>
            + Sin<Output = Output>
            + Cos<Output = Output>
            + Sized
    {
    }

    /**
     * The trait to define `&T op T` and `&T op &T` versions for RealByValue
     * based off the MIT/Apache 2.0 licensed code from num-traits 0.2.10:
     *
     * **This trait is not ever used directly for users of this library. You
     * don't need to deal with it unless
     * [implementing custom numeric types](super::super::using_custom_types)
     * and even then it will be implemented automatically.**
     *
     * - [http://opensource.org/licenses/MIT](http://opensource.org/licenses/MIT)
     * - [https://docs.rs/num-traits/0.2.10/src/num_traits/lib.rs.html#112](https://docs.rs/num-traits/0.2.10/src/num_traits/lib.rs.html#112)
     *
     * The trick is that all types implementing this trait will be references,
     * so the first constraint expresses some &T which can be operated on with
     * some right hand side type T to yield a value of type T.
     *
     * In a similar way the second constraint expresses `&T op &T -> T` operations
     */
    pub trait RealRef<T>:
    // &T op T -> T
    RealByValue<T, T>
    // &T op &T -> T
    + for<'a> RealByValue<&'a T, T> {}

    /**
     * Anything which implements all the super traits will automatically implement this trait too.
     * This covers primitives such as `&f32` & `&f64`, ie a type like `&f64` is `RealRef<&f64>`
     * as well as [Traces](super::super::differentiation::Trace) and
     * [Records](super::super::differentiation::Record) of those types.
     */
    impl<RefT, T> RealRef<T> for RefT where RefT: RealByValue<T, T> + for<'a> RealByValue<&'a T, T> {}

    /**
     * A general purpose extension to the numeric trait that adds many operations needed
     * for more complex math operations.
     *
     * This trait extends the constraints in [RealByValue]
     * to types which also support the operations with a right hand side type
     * by reference, and adds some additional constraints needed only
     * by value on types.
     *
     * When used together with [RealRef] this
     * expresses all 4 by value and by reference combinations for the
     * operations using the following syntax:
     *
     * ```ignore
     * fn function_name<T: Numeric + Real>()
     * where for<'a> &'a T: NumericRef<T> + RealRef<T> {
     *
     * }
     * ```
     *
     * This pair of constraints is used where any real number type is needed, so although
     * this trait does not require reference type methods by itself, or re-require
     * what is in Numeric, in practise you won't be able to call many
     * functions in this library with a real type that doesn't.
     */
    pub trait Real:
    // T op T -> T
    RealByValue
    // T op &T -> T
    + for<'a> RealByValue<&'a Self>
    + Pi {}

    /**
     * All types implemeting the operations in RealByValue with a right hand
     * side type by reference are Real.
     *
     * This covers primitives such as f32 & f64 as well as
     * [Traces](super::super::differentiation::Trace) and
     * [Records](super::super::differentiation::Record) of those types.
     */
    impl<T> Real for T where T: RealByValue + for<'a> RealByValue<&'a T> + Pi {}
}