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March 3, 2026 15:40
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| # -*- coding: utf-8 -*- | |
| # %% [markdown] | |
| # # NTuple invariance vs. tuple covariance | |
| # | |
| # Illustrates why `NTuple{N, <:AbstractFloat}` and `NTuple{N, AbstractFloat}` | |
| # mean very different things. | |
| # %% | |
| # --- helper --- | |
| check(label, val) = println(label, ": ", val) | |
| # %% [markdown] | |
| # ## 1. `NTuple{N, <:AbstractFloat}` — invariant TypeVar | |
| # | |
| # `NTuple{2, <:AbstractFloat}` expands to `NTuple{2, T} where T <: AbstractFloat`. | |
| # Julia must solve `T` to a **single** concrete type that satisfies every element, | |
| # so both elements must share exactly the same type. | |
| # %% | |
| # Both Float64 → T = Float64 ✓ | |
| check("(1.0, 2.0) <: NTuple{2,<:AbstractFloat}", | |
| (1.0, 2.0) isa NTuple{2, <:AbstractFloat}) | |
| # Both Float32 → T = Float32 ✓ | |
| check("(1f0, 2f0) <: NTuple{2,<:AbstractFloat}", | |
| (1.0f0, 2.0f0) isa NTuple{2, <:AbstractFloat}) | |
| # Float64 + Float32 → no single T satisfies both ✗ | |
| check("(1.0, 1f0) <: NTuple{2,<:AbstractFloat}", | |
| (1.0, 1.0f0) isa NTuple{2, <:AbstractFloat}) | |
| # %% [markdown] | |
| # ## 2. `NTuple{N, AbstractFloat}` — covariant tuple | |
| # | |
| # `NTuple{2, AbstractFloat}` means `Tuple{AbstractFloat, AbstractFloat}`. | |
| # Julia tuples are **covariant**: `Tuple{A,B} <: Tuple{P,Q}` holds whenever | |
| # `A <: P` and `B <: Q` are satisfied *independently* for each position. | |
| # `A` and `B` do **not** need to be the same type. | |
| # %% | |
| # Both Float64 → Float64 <: AbstractFloat ✓ | |
| check("(1.0, 2.0) <: NTuple{2,AbstractFloat}", | |
| (1.0, 2.0) isa NTuple{2, AbstractFloat}) | |
| # Float64 + Float32 → each position checked independently ✓ | |
| check("(1.0, 1f0) <: NTuple{2,AbstractFloat}", | |
| (1.0, 1.0f0) isa NTuple{2, AbstractFloat}) | |
| # Float64 + Int (not <: AbstractFloat) ✗ | |
| check("(1.0, 1) <: NTuple{2,AbstractFloat}", | |
| (1.0, 1) isa NTuple{2, AbstractFloat}) | |
| # %% [markdown] | |
| # ## 3. Type-level check (dispatch) | |
| # | |
| # The same distinction appears in method dispatch / `<:` on types. | |
| # %% | |
| # NTuple{2,<:AbstractFloat} requires a common T | |
| check("Tuple{Float64,Float64} <: NTuple{2,<:AbstractFloat}", | |
| Tuple{Float64, Float64} <: NTuple{2, <:AbstractFloat}) | |
| check("Tuple{Float64,Float32} <: NTuple{2,<:AbstractFloat}", | |
| Tuple{Float64, Float32} <: NTuple{2, <:AbstractFloat}) | |
| println() | |
| # NTuple{2,AbstractFloat} only needs each position to be <: AbstractFloat | |
| check("Tuple{Float64,Float64} <: NTuple{2,AbstractFloat}", | |
| Tuple{Float64, Float64} <: NTuple{2, AbstractFloat}) | |
| check("Tuple{Float64,Float32} <: NTuple{2,AbstractFloat}", | |
| Tuple{Float64, Float32} <: NTuple{2, AbstractFloat}) | |
| # %% [markdown] | |
| # ### Summary | |
| # | |
| # | Syntax | Constraint | | |
| # |--------|-----------| | |
| # | `NTuple{N, <:AbstractFloat}` | all N elements must be the **same** subtype T | | |
| # | `NTuple{N, AbstractFloat}` | each element only needs to satisfy `<: AbstractFloat` independently | |
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