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133 lines
9.4 KiB
Agda
133 lines
9.4 KiB
Agda
{-# OPTIONS --rewriting #-}
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{-# OPTIONS --allow-unsolved-metas #-}
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module Properties.DecSubtyping where
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open import Agda.Builtin.Equality using (_≡_; refl)
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open import FFI.Data.Either using (Either; Left; Right; mapLR; swapLR; cond)
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open import Luau.Subtyping using (_<:_; _≮:_; Tree; Language; ¬Language; witness; unknown; never; scalar; function; scalar-function; scalar-function-ok; scalar-function-err; scalar-scalar; function-scalar; function-ok; function-ok₁; function-ok₂; function-err; left; right; _,_)
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open import Luau.Type using (Type; Scalar; nil; number; string; boolean; never; unknown; _⇒_; _∪_; _∩_)
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open import Luau.TypeNormalization using (_∪ⁿ_; _∩ⁿ_)
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open import Properties.Contradiction using (CONTRADICTION; ¬)
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open import Properties.Functions using (_∘_)
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open import Properties.Subtyping using (<:-refl; <:-trans; ≮:-trans-<:; <:-trans-≮:; <:-never; <:-unknown; <:-∪-left; <:-∪-right; <:-∪-lub; ≮:-∪-left; ≮:-∪-right; <:-∩-left; <:-∩-right; <:-∩-glb; ≮:-∩-left; ≮:-∩-right; dec-language; scalar-<:; <:-everything; <:-function; ≮:-function-left; ≮:-function-right; <:-impl-¬≮:; <:-intersect; <:-function-∩-∪; <:-function-∩; <:-union; ≮:-left-∪; ≮:-right-∪; <:-impl-⊇)
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open import Properties.TypeNormalization using (FunType; Normal; never; unknown; function; _∩_; _∪_; _⇒_; normal; <:-normalize; normalize-<:; normal-∩ⁿ; normal-∪ⁿ; ∪-<:-∪ⁿ; ∪ⁿ-<:-∪; ∩ⁿ-<:-∩; normalⁱ)
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open import Properties.Equality using (_≢_)
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-- Honest this terminates, since src and tgt reduce the depth of nested arrows
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{-# TERMINATING #-}
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dec-subtypingˢⁿ : ∀ {T U} → Scalar T → Normal U → Either (T ≮: U) (T <: U)
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dec-subtypingᶠ : ∀ {T U} → FunType T → FunType U → Either (T ≮: U) (T <: U)
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dec-subtypingᶠⁿ : ∀ {T U} → FunType T → Normal U → Either (T ≮: U) (T <: U)
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dec-subtypingⁿ : ∀ {T U} → Normal T → Normal U → Either (T ≮: U) (T <: U)
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dec-subtyping : ∀ T U → Either (T ≮: U) (T <: U)
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srcᶠ : ∀ {F} → FunType F → Type
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normal-srcᶠ : ∀ {F} → (Fᶠ : FunType F) → Normal (srcᶠ Fᶠ)
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srcᶠ-function-err : ∀ {F} → (Fᶠ : FunType F) → ∀ s → ¬Language (srcᶠ Fᶠ) s → Language F (function-err s)
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≮:-srcᶠ : ∀ {F S T} → (Fᶠ : FunType F) → (S ≮: srcᶠ Fᶠ) → (F ≮: (S ⇒ T))
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resolveᶠⁿ : ∀ {F V} → FunType F → Normal V → Type
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normal-resolveᶠⁿ : ∀ {F V} → (Fᶠ : FunType F) → (Vⁿ : Normal V) → Normal (resolveᶠⁿ Fᶠ Vⁿ)
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resolveᶠⁿ-function-ok : ∀ {F V} → (Fᶠ : FunType F) → (Vⁿ : Normal V) → (V <: srcᶠ Fᶠ) → ∀ s t → Language (srcᶠ Fᶠ) s → ¬Language (resolveᶠⁿ Fᶠ Vⁿ) t → ¬Language F (function-ok s t)
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≮:-resolveᶠⁿ : ∀ {F S T} → (Fᶠ : FunType F) → (Sᶠ : Normal S) → (S <: srcᶠ Fᶠ) → (resolveᶠⁿ Fᶠ Sᶠ ≮: T) → (F ≮: (S ⇒ T))
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<:-resolveᶠⁿ : ∀ {F S T} → (Fᶠ : FunType F) → (Sᶠ : Normal S) → (S <: srcᶠ Fᶠ) → (resolveᶠⁿ Fᶠ Sᶠ <: T) → (F <: (S ⇒ T))
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fun-function : ∀ {F} → FunType F → Language F function
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¬fun-scalar : ∀ {F S} → (s : Scalar S) → FunType F → ¬Language F (scalar s)
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srcᶠ {S ⇒ T} F = S
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srcᶠ (F ∩ G) = srcᶠ F ∪ⁿ srcᶠ G
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normal-srcᶠ function = never
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normal-srcᶠ (S ⇒ T) = normalⁱ S
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normal-srcᶠ (F ∩ G) = normal-∪ⁿ (normal-srcᶠ F) (normal-srcᶠ G)
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srcᶠ-function-err function s p = function-err p
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srcᶠ-function-err (S ⇒ T) s p = function-err p
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srcᶠ-function-err (F ∩ G) s p with <:-impl-⊇ (∪-<:-∪ⁿ (normal-srcᶠ F) (normal-srcᶠ G)) s p
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srcᶠ-function-err (F ∩ G) s p | (p₁ , p₂) = (srcᶠ-function-err F s p₁ , srcᶠ-function-err G s p₂)
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≮:-srcᶠ F (witness s p q) = witness (function-err s) (srcᶠ-function-err F s q) (function-err p)
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resolveᶠⁿ {S ⇒ T} F V = T
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resolveᶠⁿ (F ∩ G) V with dec-subtypingⁿ V (normal-srcᶠ F) | dec-subtypingⁿ V (normal-srcᶠ G)
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resolveᶠⁿ (F ∩ G) V | Left p | Left q = resolveᶠⁿ F V ∪ⁿ resolveᶠⁿ G V
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resolveᶠⁿ (F ∩ G) V | Left p | Right q = resolveᶠⁿ G V
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resolveᶠⁿ (F ∩ G) V | Right p | Left q = resolveᶠⁿ F V
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resolveᶠⁿ (F ∩ G) V | Right p | Right q = resolveᶠⁿ F V ∩ⁿ resolveᶠⁿ G V
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normal-resolveᶠⁿ function V = unknown
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normal-resolveᶠⁿ (S ⇒ T) V = T
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normal-resolveᶠⁿ (F ∩ G) V with dec-subtypingⁿ V (normal-srcᶠ F) | dec-subtypingⁿ V (normal-srcᶠ G)
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normal-resolveᶠⁿ (F ∩ G) V | Left p | Left q = normal-∪ⁿ (normal-resolveᶠⁿ F V) (normal-resolveᶠⁿ G V)
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normal-resolveᶠⁿ (F ∩ G) V | Left p | Right q = normal-resolveᶠⁿ G V
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normal-resolveᶠⁿ (F ∩ G) V | Right p | Left q = normal-resolveᶠⁿ F V
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normal-resolveᶠⁿ (F ∩ G) V | Right p | Right q = normal-∩ⁿ (normal-resolveᶠⁿ F V) (normal-resolveᶠⁿ G V)
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resolveᶠⁿ-function-ok function V p₀ s t (scalar ()) q
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resolveᶠⁿ-function-ok (S ⇒ T) V p₀ s t p₁ p₂ = function-ok p₁ p₂
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resolveᶠⁿ-function-ok (F ∩ G) V p₀ s t p₁ p₂ with dec-subtypingⁿ V (normal-srcᶠ F) | dec-subtypingⁿ V (normal-srcᶠ G) | ∪ⁿ-<:-∪ (normal-srcᶠ F) (normal-srcᶠ G) s p₁
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resolveᶠⁿ-function-ok (F ∩ G) V p₀ s t p₁ p₂ | Left q₁ | Left q₂ | q₃ with <:-impl-⊇ (∪-<:-∪ⁿ (normal-resolveᶠⁿ F V) (normal-resolveᶠⁿ G V)) t p₂
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resolveᶠⁿ-function-ok (F ∩ G) V p₀ s t p₁ p₂ | Left q₁ | Left q₂ | left q₃ | (r₁ , r₂) = left (resolveᶠⁿ-function-ok F V {!p₀!} s t q₃ r₁)
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resolveᶠⁿ-function-ok (F ∩ G) V p₀ s t p₁ p₂ | Left q₁ | Left q₂ | right q₃ | (r₁ , r₂) = right (resolveᶠⁿ-function-ok G V {!!} s t q₃ r₂)
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resolveᶠⁿ-function-ok (F ∩ G) V p₀ s t p₁ p₂ | Left q₁ | Right q₂ | q₃ = {!resolveᶠⁿ-function-ok G V s t!}
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≮:-resolveᶠⁿ F S p (witness t q r) = witness {!function-ok !} {!!} {!p!}
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<:-resolveᶠⁿ F S p q = {!!}
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fun-function F = {!!}
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¬fun-scalar s F = {!!}
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dec-subtypingˢⁿ T U with dec-language _ (scalar T)
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dec-subtypingˢⁿ T U | Left p = Left (witness (scalar T) (scalar T) p)
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dec-subtypingˢⁿ T U | Right p = Right (scalar-<: T p)
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dec-subtypingᶠ T (U ⇒ V) with dec-subtypingⁿ (normalⁱ U) (normal-srcᶠ T) | dec-subtypingⁿ (normal-resolveᶠⁿ T (normalⁱ U)) V
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dec-subtypingᶠ T (U ⇒ V) | Left p | q = Left (≮:-srcᶠ T p)
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dec-subtypingᶠ T (U ⇒ V) | Right p | Left q = Left (≮:-resolveᶠⁿ T (normalⁱ U) p q)
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dec-subtypingᶠ T (U ⇒ V) | Right p | Right q = Right (<:-resolveᶠⁿ T (normalⁱ U) p q)
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dec-subtypingᶠ T (U ∩ V) with dec-subtypingᶠ T U | dec-subtypingᶠ T V
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dec-subtypingᶠ T (U ∩ V) | Left p | q = Left (≮:-∩-left p)
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dec-subtypingᶠ T (U ∩ V) | Right p | Left q = Left (≮:-∩-right q)
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dec-subtypingᶠ T (U ∩ V) | Right p | Right q = Right (<:-∩-glb p q)
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dec-subtypingᶠⁿ T never = Left (witness function (fun-function T) never)
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dec-subtypingᶠⁿ T unknown = Right <:-unknown
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dec-subtypingᶠⁿ T (U ⇒ V) = dec-subtypingᶠ T (U ⇒ V)
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dec-subtypingᶠⁿ T (U ∩ V) = dec-subtypingᶠ T (U ∩ V)
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dec-subtypingᶠⁿ T (U ∪ V) with dec-subtypingᶠⁿ T U
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dec-subtypingᶠⁿ T (U ∪ V) | Left (witness t p q) = Left (witness t p (q , {!!}))
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dec-subtypingᶠⁿ T (U ∪ V) | Right p = Right (<:-trans p <:-∪-left)
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dec-subtypingⁿ never U = Right <:-never
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dec-subtypingⁿ unknown unknown = Right <:-refl
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dec-subtypingⁿ unknown U with dec-subtypingᶠⁿ function U
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dec-subtypingⁿ unknown U | Left p = Left (<:-trans-≮: <:-unknown p)
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dec-subtypingⁿ unknown U | Right p₁ with dec-subtypingˢⁿ number U
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dec-subtypingⁿ unknown U | Right p₁ | Left p = Left (<:-trans-≮: <:-unknown p)
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dec-subtypingⁿ unknown U | Right p₁ | Right p₂ with dec-subtypingˢⁿ string U
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dec-subtypingⁿ unknown U | Right p₁ | Right p₂ | Left p = Left (<:-trans-≮: <:-unknown p)
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dec-subtypingⁿ unknown U | Right p₁ | Right p₂ | Right p₃ with dec-subtypingˢⁿ nil U
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dec-subtypingⁿ unknown U | Right p₁ | Right p₂ | Right p₃ | Left p = Left (<:-trans-≮: <:-unknown p)
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dec-subtypingⁿ unknown U | Right p₁ | Right p₂ | Right p₃ | Right p₄ with dec-subtypingˢⁿ boolean U
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dec-subtypingⁿ unknown U | Right p₁ | Right p₂ | Right p₃ | Right p₄ | Left p = Left (<:-trans-≮: <:-unknown p)
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dec-subtypingⁿ unknown U | Right p₁ | Right p₂ | Right p₃ | Right p₄ | Right p₅ = Right (<:-trans <:-everything (<:-∪-lub p₁ (<:-∪-lub p₂ (<:-∪-lub p₃ (<:-∪-lub p₄ p₅)))))
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dec-subtypingⁿ (S ⇒ T) U = dec-subtypingᶠⁿ (S ⇒ T) U
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dec-subtypingⁿ (S ∩ T) U = dec-subtypingᶠⁿ (S ∩ T) U
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dec-subtypingⁿ (S ∪ T) U with dec-subtypingⁿ S U | dec-subtypingˢⁿ T U
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dec-subtypingⁿ (S ∪ T) U | Left p | q = Left (≮:-∪-left p)
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dec-subtypingⁿ (S ∪ T) U | Right p | Left q = Left (≮:-∪-right q)
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dec-subtypingⁿ (S ∪ T) U | Right p | Right q = Right (<:-∪-lub p q)
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dec-subtypingⁿ function function = Right <:-refl
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dec-subtypingⁿ function (T ⇒ U) = {!!}
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dec-subtypingⁿ function (T ∩ U) = {!!}
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dec-subtypingⁿ function (T ∪ U) = {!!}
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dec-subtypingⁿ function never = Left (witness function function never)
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dec-subtypingⁿ function unknown = Right <:-unknown
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dec-subtyping T U with dec-subtypingⁿ (normal T) (normal U)
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dec-subtyping T U | Left p = Left (<:-trans-≮: (normalize-<: T) (≮:-trans-<: p (<:-normalize U)))
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dec-subtyping T U | Right p = Right (<:-trans (<:-normalize T) (<:-trans p (normalize-<: U)))
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