operators.jl

# This file is a part of Julia. License is MIT: https://julialang.org/license

## types ##

"""
    <:(T1, T2)

Subtype operator, equivalent to `issubtype(T1, T2)`.

```jldoctest
julia> Float64 <: AbstractFloat
true

julia> Vector{Int} <: AbstractArray
true

julia> Matrix{Float64} <: Matrix{AbstractFloat}
false
```
"""
const (<:) = issubtype

"""
    >:(T1, T2)

Supertype operator, equivalent to `issubtype(T2, T1)`.
"""
const (>:)(a::ANY, b::ANY) = issubtype(b, a)

"""
    supertype(T::DataType)

Return the supertype of DataType `T`.

```jldoctest
julia> supertype(Int32)
Signed
```
"""
function supertype(T::DataType)
    @_pure_meta
    T.super
end

function supertype(T::UnionAll)
    @_pure_meta
    UnionAll(T.var, supertype(T.body))
end

## generic comparison ##

==(x, y) = x === y

"""
    isequal(x, y)

Similar to `==`, except treats all floating-point `NaN` values as equal to each other, and
treats `-0.0` as unequal to `0.0`. The default implementation of `isequal` calls `==`, so if
you have a type that doesn't have these floating-point subtleties then you probably only
need to define `==`.

`isequal` is the comparison function used by hash tables (`Dict`). `isequal(x,y)` must imply
that `hash(x) == hash(y)`.

This typically means that if you define your own `==` function then you must define a
corresponding `hash` (and vice versa). Collections typically implement `isequal` by calling
`isequal` recursively on all contents.

Scalar types generally do not need to implement `isequal` separate from `==`, unless they
represent floating-point numbers amenable to a more efficient implementation than that
provided as a generic fallback (based on `isnan`, `signbit`, and `==`).

```jldoctest
julia> isequal([1., NaN], [1., NaN])
true

julia> [1., NaN] == [1., NaN]
false

julia> 0.0 == -0.0
true

julia> isequal(0.0, -0.0)
false
```
"""
isequal(x, y) = x == y

signequal(x, y) = signbit(x)::Bool == signbit(y)::Bool
signless(x, y) = signbit(x)::Bool & !signbit(y)::Bool

isequal(x::AbstractFloat, y::AbstractFloat) = (isnan(x) & isnan(y)) | signequal(x, y) & (x == y)
isequal(x::Real,          y::AbstractFloat) = (isnan(x) & isnan(y)) | signequal(x, y) & (x == y)
isequal(x::AbstractFloat, y::Real         ) = (isnan(x) & isnan(y)) | signequal(x, y) & (x == y)

isless(x::AbstractFloat, y::AbstractFloat) = (!isnan(x) & isnan(y)) | signless(x, y) | (x < y)
isless(x::Real,          y::AbstractFloat) = (!isnan(x) & isnan(y)) | signless(x, y) | (x < y)
isless(x::AbstractFloat, y::Real         ) = (!isnan(x) & isnan(y)) | signless(x, y) | (x < y)


function ==(T::Type, S::Type)
    @_pure_meta
    typeseq(T, S)
end
function !=(T::Type, S::Type)
    @_pure_meta
    !(T == S)
end
==(T::TypeVar, S::Type) = false
==(T::Type, S::TypeVar) = false

## comparison fallbacks ##

"""
    !=(x, y)
    ≠(x,y)

Not-equals comparison operator. Always gives the opposite answer as `==`. New types should
generally not implement this, and rely on the fallback definition `!=(x,y) = !(x==y)` instead.

```jldoctest
julia> 3 != 2
true

julia> "foo" ≠ "foo"
false
```
"""
!=(x, y) = !(x == y)::Bool
const ≠ = !=

"""
    ===(x,y) -> Bool
    ≡(x,y) -> Bool

Determine whether `x` and `y` are identical, in the sense that no program could distinguish
them. Compares mutable objects by address in memory, and compares immutable objects (such as
numbers) by contents at the bit level. This function is sometimes called `egal`.

```jldoctest
julia> a = [1, 2]; b = [1, 2];

julia> a == b
true

julia> a === b
false

julia> a === a
true
```
"""
===
const ≡ = ===

"""
    !==(x, y)
    ≢(x,y)

Equivalent to `!(x === y)`.

```jldoctest
julia> a = [1, 2]; b = [1, 2];

julia> a ≢ b
true

julia> a ≢ a
false
```
"""
!==(x, y) = !(x === y)
const ≢ = !==

"""
    <(x, y)

Less-than comparison operator. New numeric types should implement this function for two
arguments of the new type. Because of the behavior of floating-point NaN values, `<`
implements a partial order. Types with a canonical partial order should implement `<`, and
types with a canonical total order should implement `isless`.

```jldoctest
julia> 'a' < 'b'
true

julia> "abc" < "abd"
true

julia> 5 < 3
false
```
"""
<(x, y) = isless(x, y)

"""
    >(x, y)

Greater-than comparison operator. Generally, new types should implement `<` instead of this
function, and rely on the fallback definition `>(x, y) = y < x`.

```jldoctest
julia> 'a' > 'b'
false

julia> 7 > 3 > 1
true

julia> "abc" > "abd"
false

julia> 5 > 3
true
```
"""
>(x, y) = y < x

"""
    <=(x, y)
    ≤(x,y)

Less-than-or-equals comparison operator.

```jldoctest
julia> 'a' <= 'b'
true

julia> 7 ≤ 7 ≤ 9
true

julia> "abc" ≤ "abc"
true

julia> 5 <= 3
false
```
"""
<=(x, y) = !(y < x)
const ≤ = <=

"""
    >=(x, y)
    ≥(x,y)

Greater-than-or-equals comparison operator.

```jldoctest
julia> 'a' >= 'b'
false

julia> 7 ≥ 7 ≥ 3
true

julia> "abc" ≥ "abc"
true

julia> 5 >= 3
true
```
"""
>=(x, y) = (y <= x)
const ≥ = >=

# this definition allows Number types to implement < instead of isless,
# which is more idiomatic:
isless(x::Real, y::Real) = x<y
lexcmp(x::Real, y::Real) = isless(x,y) ? -1 : ifelse(isless(y,x), 1, 0)

"""
    ifelse(condition::Bool, x, y)

Return `x` if `condition` is `true`, otherwise return `y`. This differs from `?` or `if` in
that it is an ordinary function, so all the arguments are evaluated first. In some cases,
using `ifelse` instead of an `if` statement can eliminate the branch in generated code and
provide higher performance in tight loops.

```jldoctest
julia> ifelse(1 > 2, 1, 2)
2
```
"""
ifelse(c::Bool, x, y) = select_value(c, x, y)

"""
    cmp(x,y)

Return -1, 0, or 1 depending on whether `x` is less than, equal to, or greater than `y`,
respectively. Uses the total order implemented by `isless`. For floating-point numbers, uses `<`
but throws an error for unordered arguments.

```jldoctest
julia> cmp(1, 2)
-1

julia> cmp(2, 1)
1

julia> cmp(2+im, 3-im)
ERROR: MethodError: no method matching isless(::Complex{Int64}, ::Complex{Int64})
[...]
```
"""
cmp(x, y) = isless(x, y) ? -1 : ifelse(isless(y, x), 1, 0)

"""
    lexcmp(x, y)

Compare `x` and `y` lexicographically and return -1, 0, or 1 depending on whether `x` is
less than, equal to, or greater than `y`, respectively. This function should be defined for
lexicographically comparable types, and `lexless` will call `lexcmp` by default.

```jldoctest
julia> lexcmp("abc", "abd")
-1

julia> lexcmp("abc", "abc")
0
```
"""
lexcmp(x, y) = cmp(x, y)

"""
    lexless(x, y)

Determine whether `x` is lexicographically less than `y`.

```jldoctest
julia> lexless("abc", "abd")
true
```
"""
lexless(x, y) = lexcmp(x,y) < 0

# cmp returns -1, 0, +1 indicating ordering
cmp(x::Integer, y::Integer) = ifelse(isless(x, y), -1, ifelse(isless(y, x), 1, 0))

"""
    max(x, y, ...)

Return the maximum of the arguments. See also the [`maximum`](@ref) function
to take the maximum element from a collection.

```jldoctest
julia> max(2, 5, 1)
5
```
"""
max(x, y) = ifelse(y < x, x, y)

"""
    min(x, y, ...)

Return the minimum of the arguments. See also the [`minimum`](@ref) function
to take the minimum element from a collection.

```jldoctest
julia> min(2, 5, 1)
1
```
"""
min(x,y) = ifelse(y < x, y, x)

"""
    minmax(x, y)

Return `(min(x,y), max(x,y))`. See also: [`extrema`](@ref) that returns `(minimum(x), maximum(x))`.

```jldoctest
julia> minmax('c','b')
('b', 'c')
```
"""
minmax(x,y) = y < x ? (y, x) : (x, y)

scalarmax(x,y) = max(x,y)
scalarmax(x::AbstractArray, y::AbstractArray) = throw(ArgumentError("ordering is not well-defined for arrays"))
scalarmax(x               , y::AbstractArray) = throw(ArgumentError("ordering is not well-defined for arrays"))
scalarmax(x::AbstractArray, y               ) = throw(ArgumentError("ordering is not well-defined for arrays"))

scalarmin(x,y) = min(x,y)
scalarmin(x::AbstractArray, y::AbstractArray) = throw(ArgumentError("ordering is not well-defined for arrays"))
scalarmin(x               , y::AbstractArray) = throw(ArgumentError("ordering is not well-defined for arrays"))
scalarmin(x::AbstractArray, y               ) = throw(ArgumentError("ordering is not well-defined for arrays"))

## definitions providing basic traits of arithmetic operators ##

"""
    identity(x)

The identity function. Returns its argument.

```jldoctest
julia> identity("Well, what did you expect?")
"Well, what did you expect?"
```
"""
identity(x) = x

+(x::Number) = x
*(x::Number) = x
(&)(x::Integer) = x
(|)(x::Integer) = x
xor(x::Integer) = x

const ⊻ = xor

# foldl for argument lists. expand recursively up to a point, then
# switch to a loop. this allows small cases like `a+b+c+d` to be inlined
# efficiently, without a major slowdown for `+(x...)` when `x` is big.
afoldl(op,a) = a
afoldl(op,a,b) = op(a,b)
afoldl(op,a,b,c...) = afoldl(op, op(a,b), c...)
function afoldl(op,a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,qs...)
    y = op(op(op(op(op(op(op(op(op(op(op(op(op(op(op(a,b),c),d),e),f),g),h),i),j),k),l),m),n),o),p)
    for x in qs; y = op(y,x); end
    y
end

for op in (:+, :*, :&, :|, :xor, :min, :max, :kron)
    @eval begin
        # note: these definitions must not cause a dispatch loop when +(a,b) is
        # not defined, and must only try to call 2-argument definitions, so
        # that defining +(a,b) is sufficient for full functionality.
        ($op)(a, b, c, xs...) = afoldl($op, ($op)(($op)(a,b),c), xs...)
        # a further concern is that it's easy for a type like (Int,Int...)
        # to match many definitions, so we need to keep the number of
        # definitions down to avoid losing type information.
    end
end

"""
    \\(x, y)

Left division operator: multiplication of `y` by the inverse of `x` on the left. Gives
floating-point results for integer arguments.

```jldoctest
julia> 3 \\ 6
2.0

julia> inv(3) * 6
2.0

julia> A = [1 2; 3 4]; x = [5, 6];

julia> A \\ x
2-element Array{Float64,1}:
 -4.0
  4.5

julia> inv(A) * x
2-element Array{Float64,1}:
 -4.0
  4.5
```
"""
\(x,y) = (y'/x')'

# Core <<, >>, and >>> take either Int or UInt as second arg. Signed shift
# counts can shift in either direction, and are translated here to unsigned
# counts. Integer datatypes only need to implement the unsigned version.

"""
    <<(x, n)

Left bit shift operator, `x << n`. For `n >= 0`, the result is `x` shifted left
by `n` bits, filling with `0`s. This is equivalent to `x * 2^n`. For `n < 0`,
this is equivalent to `x >> -n`.

```jldoctest
julia> Int8(3) << 2
12

julia> bits(Int8(3))
"00000011"

julia> bits(Int8(12))
"00001100"
```
See also [`>>`](@ref), [`>>>`](@ref).
"""
function <<(x::Integer, c::Integer)
    @_inline_meta
    typemin(Int) <= c <= typemax(Int) && return x << (c % Int)
    (x >= 0 || c >= 0) && return zero(x)
    oftype(x, -1)
end
<<(x::Integer, c::Unsigned) = c <= typemax(UInt) ? x << (c % UInt) : zero(x)
<<(x::Integer, c::Int) = c >= 0 ? x << unsigned(c) : x >> unsigned(-c)

"""
    >>(x, n)

Right bit shift operator, `x >> n`. For `n >= 0`, the result is `x` shifted
right by `n` bits, where `n >= 0`, filling with `0`s if `x >= 0`, `1`s if `x <
0`, preserving the sign of `x`. This is equivalent to `fld(x, 2^n)`. For `n <
0`, this is equivalent to `x << -n`.


```jldoctest
julia> Int8(13) >> 2
3

julia> bits(Int8(13))
"00001101"

julia> bits(Int8(3))
"00000011"

julia> Int8(-14) >> 2
-4

julia> bits(Int8(-14))
"11110010"

julia> bits(Int8(-4))
"11111100"
```
See also [`>>>`](@ref), [`<<`](@ref).
"""
function >>(x::Integer, c::Integer)
    @_inline_meta
    typemin(Int) <= c <= typemax(Int) && return x >> (c % Int)
    (x >= 0 || c < 0) && return zero(x)
    oftype(x, -1)
end
>>(x::Integer, c::Unsigned) = c <= typemax(UInt) ? x >> (c % UInt) : zero(x)
>>(x::Integer, c::Int) = c >= 0 ? x >> unsigned(c) : x << unsigned(-c)

"""
    >>>(x, n)

Unsigned right bit shift operator, `x >>> n`. For `n >= 0`, the result is `x`
shifted right by `n` bits, where `n >= 0`, filling with `0`s. For `n < 0`, this
is equivalent to `x << -n`.

For [`Unsigned`](@ref) integer types, this is equivalent to [`>>`](@ref). For
[`Signed`](@ref) integer types, this is equivalent to `signed(unsigned(x) >> n)`.

```jldoctest
julia> Int8(-14) >>> 2
60

julia> bits(Int8(-14))
"11110010"

julia> bits(Int8(60))
"00111100"
```

[`BigInt`](@ref)s are treated as if having infinite size, so no filling is required and this
is equivalent to [`>>`](@ref).

See also [`>>`](@ref), [`<<`](@ref).
"""
function >>>(x::Integer, c::Integer)
    @_inline_meta
    typemin(Int) <= c <= typemax(Int) ? x >>> (c % Int) : zero(x)
end
>>>(x::Integer, c::Unsigned) = c <= typemax(UInt) ? x >>> (c % UInt) : zero(x)
>>>(x::Integer, c::Int) = c >= 0 ? x >>> unsigned(c) : x << unsigned(-c)

# fallback div, fld, and cld implementations
# NOTE: C89 fmod() and x87 FPREM implicitly provide truncating float division,
# so it is used here as the basis of float div().
div{T<:Real}(x::T, y::T) = convert(T,round((x-rem(x,y))/y))

"""
    fld(x, y)

Largest integer less than or equal to `x/y`.

```jldoctest
julia> fld(7.3,5.5)
1.0
```
"""
fld{T<:Real}(x::T, y::T) = convert(T,round((x-mod(x,y))/y))

"""
    cld(x, y)

Smallest integer larger than or equal to `x/y`.
```jldoctest
julia> cld(5.5,2.2)
3.0
```
"""
cld{T<:Real}(x::T, y::T) = convert(T,round((x-modCeil(x,y))/y))
#rem{T<:Real}(x::T, y::T) = convert(T,x-y*trunc(x/y))
#mod{T<:Real}(x::T, y::T) = convert(T,x-y*floor(x/y))
modCeil{T<:Real}(x::T, y::T) = convert(T,x-y*ceil(x/y))

# operator alias

"""
    rem(x, y)
    %(x, y)

Remainder from Euclidean division, returning a value of the same sign as `x`, and smaller in
magnitude than `y`. This value is always exact.

```jldoctest
julia> x = 15; y = 4;

julia> x % y
3

julia> x == div(x, y) * y + rem(x, y)
true
```
"""
rem
const % = rem

"""
    div(x, y)
    ÷(x, y)

The quotient from Euclidean division. Computes `x/y`, truncated to an integer.

```jldoctest
julia> 9 ÷ 4
2

julia> -5 ÷ 3
-1
```
"""
div
const ÷ = div

"""
    mod1(x, y)

Modulus after flooring division, returning a value `r` such that `mod(r, y) == mod(x, y)`
in the range ``(0, y]`` for positive `y` and in the range ``[y,0)`` for negative `y`.

```jldoctest
julia> mod1(4, 2)
2

julia> mod1(4, 3)
1
```
"""
mod1{T<:Real}(x::T, y::T) = (m = mod(x, y); ifelse(m == 0, y, m))
# efficient version for integers
mod1{T<:Integer}(x::T, y::T) = (@_inline_meta; mod(x + y - T(1), y) + T(1))


"""
    fld1(x, y)

Flooring division, returning a value consistent with `mod1(x,y)`

See also: [`mod1`](@ref).

```jldoctest
julia> x = 15; y = 4;

julia> fld1(x, y)
4

julia> x == fld(x, y) * y + mod(x, y)
true

julia> x == (fld1(x, y) - 1) * y + mod1(x, y)
true
```
"""
fld1(x::T, y::T) where {T<:Real} = (m=mod(x,y); fld(x-m,y))
# efficient version for integers
fld1(x::T, y::T) where {T<:Integer} = fld(x+y-T(1),y)

"""
    fldmod1(x, y)

Return `(fld1(x,y), mod1(x,y))`.

See also: [`fld1`](@ref), [`mod1`](@ref).
"""
fldmod1(x::T, y::T) where {T<:Real} = (fld1(x,y), mod1(x,y))
# efficient version for integers
fldmod1(x::T, y::T) where {T<:Integer} = (fld1(x,y), mod1(x,y))

# transpose

"""
    ctranspose(A)

The conjugate transposition operator (`'`).

# Example

```jldoctest
julia> A =  [3+2im 9+2im; 8+7im  4+6im]
2×2 Array{Complex{Int64},2}:
 3+2im  9+2im
 8+7im  4+6im

julia> ctranspose(A)
2×2 Array{Complex{Int64},2}:
 3-2im  8-7im
 9-2im  4-6im
```
"""
ctranspose(x) = conj(transpose(x))
conj(x) = x

# transposed multiply

"""
    Ac_mul_B(A, B)

For matrices or vectors ``A`` and ``B``, calculates ``Aᴴ⋅B``.
"""
Ac_mul_B(a,b)  = ctranspose(a)*b

"""
    A_mul_Bc(A, B)

For matrices or vectors ``A`` and ``B``, calculates ``A⋅Bᴴ``.
"""
A_mul_Bc(a,b)  = a*ctranspose(b)

"""
    Ac_mul_Bc(A, B)

For matrices or vectors ``A`` and ``B``, calculates ``Aᴴ Bᴴ``.
"""
Ac_mul_Bc(a,b) = ctranspose(a)*ctranspose(b)

"""
    At_mul_B(A, B)

For matrices or vectors ``A`` and ``B``, calculates ``Aᵀ⋅B``.
"""
At_mul_B(a,b)  = transpose(a)*b

"""
    A_mul_Bt(A, B)

For matrices or vectors ``A`` and ``B``, calculates ``A⋅Bᵀ``.
"""
A_mul_Bt(a,b)  = a*transpose(b)

"""
    At_mul_Bt(A, B)

For matrices or vectors ``A`` and ``B``, calculates ``Aᵀ⋅Bᵀ``.
"""
At_mul_Bt(a,b) = transpose(a)*transpose(b)

# transposed divide

"""
    Ac_rdiv_B(A, B)

For matrices or vectors ``A`` and ``B``, calculates ``Aᴴ / B``.
"""
Ac_rdiv_B(a,b)  = ctranspose(a)/b

"""
    A_rdiv_Bc(A, B)

For matrices or vectors ``A`` and ``B``, calculates ``A / Bᴴ``.
"""
A_rdiv_Bc(a,b)  = a/ctranspose(b)

"""
    Ac_rdiv_Bc(A, B)

For matrices or vectors ``A`` and ``B``, calculates ``Aᴴ / Bᴴ``.
"""
Ac_rdiv_Bc(a,b) = ctranspose(a)/ctranspose(b)

"""
    At_rdiv_B(A, B)

For matrices or vectors ``A`` and ``B``, calculates ``Aᵀ / B``.
"""
At_rdiv_B(a,b)  = transpose(a)/b

"""
    A_rdiv_Bt(A, B)

For matrices or vectors ``A`` and ``B``, calculates ``A / Bᵀ``.
"""
A_rdiv_Bt(a,b)  = a/transpose(b)

"""
    At_rdiv_Bt(A, B)

For matrices or vectors ``A`` and ``B``, calculates ``Aᵀ / Bᵀ``.
"""
At_rdiv_Bt(a,b) = transpose(a)/transpose(b)

"""
    Ac_ldiv_B(A, B)

For matrices or vectors ``A`` and ``B``, calculates ``Aᴴ`` \\ ``B``.
"""
Ac_ldiv_B(a,b)  = ctranspose(a)\b

"""
    A_ldiv_Bc(A, B)

For matrices or vectors ``A`` and ``B``, calculates ``A`` \\ ``Bᴴ``.
"""
A_ldiv_Bc(a,b)  = a\ctranspose(b)

"""
    Ac_ldiv_Bc(A, B)

For matrices or vectors ``A`` and ``B``, calculates ``Aᴴ`` \\ ``Bᴴ``.
"""
Ac_ldiv_Bc(a,b) = ctranspose(a)\ctranspose(b)

"""
    At_ldiv_B(A, B)

For matrices or vectors ``A`` and ``B``, calculates ``Aᵀ`` \\ ``B``.
"""
At_ldiv_B(a,b)  = transpose(a)\b

"""
    A_ldiv_Bt(A, B)

For matrices or vectors ``A`` and ``B``, calculates ``A`` \\ ``Bᵀ``.
"""
A_ldiv_Bt(a,b)  = a\transpose(b)

"""
    At_ldiv_Bt(A, B)

For matrices or vectors ``A`` and ``B``, calculates ``Aᵀ`` \\ ``Bᵀ``.
"""
At_ldiv_Bt(a,b) = At_ldiv_B(a,transpose(b))

"""
    Ac_ldiv_Bt(A, B)

For matrices or vectors ``A`` and ``B``, calculates ``Aᴴ`` \\ ``Bᵀ``.
"""
Ac_ldiv_Bt(a,b) = Ac_ldiv_B(a,transpose(b))

widen(x::T) where {T<:Number} = convert(widen(T), x)

# function pipelining

"""
    |>(x, f)

Applies a function to the preceding argument. This allows for easy function chaining.

```jldoctest
julia> [1:5;] |> x->x.^2 |> sum |> inv
0.01818181818181818
```
"""
|>(x, f) = f(x)

# function composition

"""
    f ∘ g

Compose functions: i.e. `(f ∘ g)(args...)` means `f(g(args...))`. The `∘` symbol can be
entered in the Julia REPL (and most editors, appropriately configured) by typing `\\circ<tab>`.
Example:

```jldoctest
julia> map(uppercase∘hex, 250:255)
6-element Array{String,1}:
 "FA"
 "FB"
 "FC"
 "FD"
 "FE"
 "FF"
```
"""
∘(f, g) = (x...)->f(g(x...))


"""
    !f::Function

Predicate function negation: when the argument of `!` is a function, it returns a
function which computes the boolean negation of `f`. Example:

```jldoctest
julia> str = "∀ ε > 0, ∃ δ > 0: |x-y| < δ ⇒ |f(x)-f(y)| < ε"
"∀ ε > 0, ∃ δ > 0: |x-y| < δ ⇒ |f(x)-f(y)| < ε"

julia> filter(isalpha, str)
"εδxyδfxfyε"

julia> filter(!isalpha, str)
"∀  > 0, ∃  > 0: |-| <  ⇒ |()-()| < "
```
"""
!(f::Function) = (x...)->!f(x...)

# some operators not defined yet
global //, >:, <|, hcat, hvcat, ⋅, ×, ∈, ∉, ∋, ∌, ⊆, ⊈, ⊊, ∩, ∪, √, ∛

this_module = current_module()
baremodule Operators

export
    !,
    !=,
    !==,
    ===,
    xor,
    %,
    ÷,
    &,
    *,
    +,
    -,
    /,
    //,
    <,
    <:,
    >:,
    <<,
    <=,
    ==,
    >,
    >=,
    ≥,
    ≤,
    ≠,
    >>,
    >>>,
    \,
    ^,
    |,
    |>,
    <|,
    ~,
    ⋅,
    ×,
    ∈,
    ∉,
    ∋,
    ∌,
    ⊆,
    ⊈,
    ⊊,
    ∩,
    ∪,
    √,
    ∛,
    ⊻,
    ∘,
    colon,
    hcat,
    vcat,
    hvcat,
    getindex,
    setindex!,
    transpose,
    ctranspose

import ..this_module: !, !=, xor, %, ÷, &, *, +, -,
    /, //, <, <:, <<, <=, ==, >, >=, >>, >>>,
    <|, |>, \, ^, |, ~, !==, ===, >:, colon, hcat, vcat, hvcat, getindex, setindex!,
    transpose, ctranspose,
    ≥, ≤, ≠, ⋅, ×, ∈, ∉, ∋, ∌, ⊆, ⊈, ⊊, ∩, ∪, √, ∛, ⊻, ∘

end