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67 lines
2.5 KiB
Agda
67 lines
2.5 KiB
Agda
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module Luau.TypeSaturation where
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open import Luau.Type using (Type; _⇒_; _∩_; _∪_)
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open import Luau.TypeNormalization using (_∪ⁿ_; _∩ⁿ_)
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-- So, there's a problem with overloaded functions
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-- (of the form (S_1 ⇒ T_1) ∩⋯∩ (S_n ⇒ T_n))
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-- which is that it's not good enough to compare them
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-- for subtyping by comparing all of their overloads.
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-- For example (nil → nil) is a subtype of (number? → number?) ∩ (string? → string?)
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-- but not a subtype of any of its overloads.
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-- To fix this, we adapt the semantic subtyping algorithm for
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-- function types, given in
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-- https://www.irif.fr/~gc/papers/covcon-again.pdf and
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-- https://pnwamk.github.io/sst-tutorial/
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-- A function type is *intersection-saturated* if for any overloads
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-- (S₁ ⇒ T₁) and (S₂ ⇒ T₂), there exists an overload which is a subtype
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-- of ((S₁ ∩ S₂) ⇒ (T₁ ∩ T₂)).
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-- A function type is *union-saturated* if for any overloads
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-- (S₁ ⇒ T₁) and (S₂ ⇒ T₂), there exists an overload which is a subtype
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-- of ((S₁ ∪ S₂) ⇒ (T₁ ∪ T₂)).
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-- A function type is *saturated* if it is both intersection- and
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-- union-saturated.
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-- For example (number? → number?) ∩ (string? → string?)
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-- is not saturated, but (number? → number?) ∩ (string? → string?) ∩ (nil → nil) ∩ ((number ∪ string)? → (number ∪ string)?)
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-- is.
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-- Saturated function types have the nice property that they can ber
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-- compared by just comparing their overloads: F <: G whenever for any
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-- overload of G, there is an overload os F which is a subtype of it.
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-- Forunately every function type can be saturated!
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_⋓_ : Type → Type → Type
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(S₁ ⇒ T₁) ⋓ (S₂ ⇒ T₂) = (S₁ ∪ⁿ S₂) ⇒ (T₁ ∪ⁿ T₂)
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(F₁ ∩ G₁) ⋓ F₂ = (F₁ ⋓ F₂) ∩ (G₁ ⋓ F₂)
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F₁ ⋓ (F₂ ∩ G₂) = (F₁ ⋓ F₂) ∩ (F₁ ⋓ G₂)
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F ⋓ G = F ∩ G
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_⋒_ : Type → Type → Type
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(S₁ ⇒ T₁) ⋒ (S₂ ⇒ T₂) = (S₁ ∩ⁿ S₂) ⇒ (T₁ ∩ⁿ T₂)
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(F₁ ∩ G₁) ⋒ F₂ = (F₁ ⋒ F₂) ∩ (G₁ ⋒ F₂)
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F₁ ⋒ (F₂ ∩ G₂) = (F₁ ⋒ F₂) ∩ (F₁ ⋒ G₂)
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F ⋒ G = F ∩ G
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_∩ᵘ_ : Type → Type → Type
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F ∩ᵘ G = (F ∩ G) ∩ (F ⋓ G)
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_∩ⁱ_ : Type → Type → Type
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F ∩ⁱ G = (F ∩ G) ∩ (F ⋒ G)
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∪-saturate : Type → Type
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∪-saturate (F ∩ G) = (∪-saturate F ∩ᵘ ∪-saturate G)
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∪-saturate F = F
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∩-saturate : Type → Type
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∩-saturate (F ∩ G) = (∩-saturate F ∩ⁱ ∩-saturate G)
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∩-saturate F = F
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saturate : Type → Type
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saturate F = ∪-saturate (∩-saturate F)
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