A gist with the code in this post can be found here.

When planning a small library in Julia, I kept running into a similar problem across all of the type systems that I tried when looking for the proper abstraction. Briefly, I had a single parametric abstract type at the top of an arbitrary type hierarchy (with any number of abstract and concrete types below it, possibly added later by a user) and I needed to write a function that would only work on objects with the same UnionAll^{[1]} types within the hierarchy. Furthermore, I wanted this to be automatic for all of the types in the hierarchy without the user having to write any code.

Here is a basic example. Suppose I have the types:

```
abstract type A{T} end
struct B{T} <: A{T} end
struct C{T} <: A{T} end
abstract type D{T} <: A{T} end
struct E{T} <: D{T} end
struct F{T} <: D{T} end
```

Here, `D`

, `E`

, and `F`

are included to demonstrate that the type hierarchy can be arbitrarily complex and extended at any time by the user.

I want to define a function `f(x, y)`

that does something when `x`

and `y`

come from the same UnionAll type, even if they are parameterized differently, and fails otherwise. For example:

```
x = B{Int64}()
y = B{Float32}()
z = C{Int64}()
f(x, y) # -> does something
f(x, z) # -> fails
f(y, z) # -> fails
```

A simple solution to this is to just require the user to implement `f`

for their new types while providing a fall back `f(::A, ::A)`

that fails. However, the library that I am writing has a very natural interface and adding more to it would have been undesirable. Also, this struck me as something that *should* be possible programmatically.

My first attempt to do it programmatically lead to method signatures similar to (the invalid code):

`f(::X, ::Y) where {T, S, U<:A, X<:U{T}, Y<:U{S}} = ...`

I was then pointed (by a JuliaLang member on Slack, though I can't remember who) to a partial solution of determining if two objects had the same UnionAll type:

`sameunionall(::X, ::Y) where {X<:A, Y<:A} = !isabstracttype(typejoin(X, Y))`

This can be rewritten for types as:

`sameunionall(::Type{X}, ::Type{Y}) where {X<:A, Y<:A} = !isabstracttype(typejoin(X, Y))`

Combining this with 'Holy-Traits' ^{[2]} via SimpleTraits.jl leads to a nice solution:

```
using SimpleTraits
@traitdef SameUnionAll{X, Y}
@traitimpl SameUnionAll{X, Y} <- sameunionall(X, Y)
@traitfn f(::X, ::Y) where {X<:A, Y<:A; SameUnionAll{X, Y}} = "yo"
@traitfn f(::X, ::Y) where {X<:A, Y<:A; !SameUnionAll{X, Y}} = "nah"
```

This can be tested out:

```
x = B{Int64}()
y = B{Float32}()
z = C{Int64}()
f(x, y) # -> "yo"
f(x, z) # -> "nah"
f(y, z) # -> "nah"
# added later by a user
struct G{T} <: D{T} end
f(G{Int16}(), x) # -> "nah"
f(G{Int16}(), G{BigFloat}()) # -> "yo"
```

One important note is that `sameunionall`

is a pure method so `f`

does not fall back to dynamic dispatch. This can be verified by checking:

```
julia> @code_warntype f(x, y)
Body::String
1 ─ return "yo"
julia> @code_warntype f(x, z)
Body::String
1 ─ return "nah"
```

This trick will be expanded on in a future post, but to whet the appetite I will briefly introduce the exact use case that I have. Suppose I have the following structure, where more "special algebra" types can be added at will by the user:

```
abstract type AbstractAlgebraElement{T} end
struct SpecialAlgebraElement{T<:Real} <: AbstractAlgebraElement{T} end
struct DifferentAlgebraElement{T<:Number} <: AbstractAlgebraElement{T} end
```

Objects of the same algebraic type but with different parameters should be compatible (for example, a ring of real numbers represented by `Float32`

vs one represented by `Float64`

are essentially the same thing here). As such, we should be able to promote between them so that this should work:

`promote_type(SpecialAlgebraElement{Float64}, SpecialAlgebraElement{Float16}) # -> SpecialAlgebraElement{Float64}`

but this should fail:

`promote_type(SpecialAlgebraElement{Float64}, DifferentAlgebraElement{Float16})`

As a final note, more idiomatic Julia code, comments, criticisms, etc. are always welcome. Please feel free to email me.

[1] | https://docs.julialang.org/en/v1/manual/types/index.html#UnionAll-Types-1 |

[2] | https://github.com/JuliaLang/julia/issues/2345#issuecomment-54537633 |

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