What are lower bounds for types?

New to Julia
,
1

I’m trying to understand what lower bounds are for types. I ran the following code which shows which subtypes (recursively) of number satisfies/rejects Int <: T <: Real in hope of gaining an understanding, however I’m still lost. :sleepy:

Number
├─ Complex
└─ Real
   ├─ AbstractFloat
   │  ├─ BigFloat
   │  ├─ Float16
   │  ├─ Float32
   │  └─ Float64
   ├─ AbstractIrrational
   │  └─ Irrational
   ├─ FixedPoint
   │  ├─ Fixed
   │  └─ Normed
   ├─ Integer
   │  ├─ Bool
   │  ├─ Signed
   │  │  ├─ BigInt
   │  │  ├─ Int128
   │  │  ├─ Int16
   │  │  ├─ Int32
   │  │  ├─ Int64
   │  │  └─ Int8
   │  └─ Unsigned
   │     ├─ UInt128
   │     ├─ UInt16
   │     ├─ UInt32
   │     ├─ UInt64
   │     └─ UInt8
   ├─ Rational
   └─ TestStat

function stype(input)
    arr = []
    append!(arr,subtypes(input))
    for x in arr
        z = subtypes(x)
        if (z != Type[])
            append!(arr,z)
        end
    end
    arr
end

for T in stype(Number)
    @show T, Int <: T <: Real
end

(T, Int <: T <: Real) = (Complex, false)
(T, Int <: T <: Real) = (Real, true)
(T, Int <: T <: Real) = (AbstractFloat, false)
(T, Int <: T <: Real) = (AbstractIrrational, false)
(T, Int <: T <: Real) = (FixedPointNumbers.FixedPoint, false)
(T, Int <: T <: Real) = (Integer, true)
(T, Int <: T <: Real) = (Rational, false)
(T, Int <: T <: Real) = (StatsBase.TestStat, false)
(T, Int <: T <: Real) = (BigFloat, false)
(T, Int <: T <: Real) = (Float16, false)
(T, Int <: T <: Real) = (Float32, false)
(T, Int <: T <: Real) = (Float64, false)
(T, Int <: T <: Real) = (Irrational, false)
(T, Int <: T <: Real) = (FixedPointNumbers.Fixed, false)
(T, Int <: T <: Real) = (FixedPointNumbers.Normed, false)
(T, Int <: T <: Real) = (Bool, false)
(T, Int <: T <: Real) = (Signed, true)
(T, Int <: T <: Real) = (Unsigned, false)
(T, Int <: T <: Real) = (BigInt, false)
(T, Int <: T <: Real) = (Int128, false)
(T, Int <: T <: Real) = (Int16, false)
(T, Int <: T <: Real) = (Int32, false)
(T, Int <: T <: Real) = (Int64, true)
(T, Int <: T <: Real) = (Int8, false)
(T, Int <: T <: Real) = (UInt128, false)
(T, Int <: T <: Real) = (UInt16, false)
(T, Int <: T <: Real) = (UInt32, false)
(T, Int <: T <: Real) = (UInt64, false)
(T, Int <: T <: Real) = (UInt8, false)

I don’t understand why Int <: T <: Real is only true for Real, Integer, Signed and Int64

2

Int is just an alias for Int64 on 64-bit platforms and Int32 on 32-bit platforms. Since you seem to be running a 64-bit Julia version, you get exactly the same results for Int as you get for Int64. Does that answer your question?

3

Note Integeris probably what you meant.

4

Yes, Integer is what I meant, however I still don’t understand what the lower bound means.

Integer <: Int64 <: Real

false

Why is this false? What does the lower bound mean?

5

Int64 (which is a concrete type) is a subtype of Integer (which is an abstract type). The reverse cannot be true.

6

This makes sense, thank you for the explanation, however when would you use an upper bound with a lower bound?

7

I never use them. I’ve seen them used, but cannot remember what for.

8

One example would be Missing<:T<: Integer which is a way of writing T<: Union{Missing, Integer}. In general, lower bounds aren’t widely used.

9

I don’t think that’s true:

julia> Missing <: Missing <: Integer
false

julia> Missing <: Int <: Integer
false
10

Lower bounds are sometimes useful though, for example if you want to check whether a vector can store missing values:

julia> [1, missing, 3] isa Vector{>:Missing}
true

julia> [1, 2, 3] isa Vector{>:Missing}
false