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Add a decision procedure for subtyping to the prototype
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ajeffrey@roblox.com
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@ -75,6 +75,12 @@ language-comp (function-err t) (function-err p) (function-err q) = language-comp
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<:-impl-¬≮: : ∀ {T U} → (T <: U) → ¬(T ≮: U)
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<:-impl-¬≮: p (witness t q r) = language-comp t r (p t q)
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-- if T <: U then ¬Language U ⊆ ¬Language T
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<:-impl-⊇ : ∀ {T U} → (T <: U) → ∀ t → ¬Language U t → ¬Language T t
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<:-impl-⊇ {T} p t ¬Ut with dec-language T t
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<:-impl-⊇ p t ¬Ut | Left ¬Tt = ¬Tt
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<:-impl-⊇ p t ¬Ut | Right Tt = CONTRADICTION (language-comp t ¬Ut (p t Tt))
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-- reflexivity
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≮:-refl : ∀ {T} → ¬(T ≮: T)
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≮:-refl (witness t p q) = language-comp t q p
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@ -133,6 +139,20 @@ function-ok-tgt (right p) = right (function-ok-tgt p)
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function-ok-tgt (p₁ , p₂) = (function-ok-tgt p₁ , function-ok-tgt p₂)
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function-ok-tgt any = any
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tgt-¬function-ok : ∀ {T t} → (¬Language (tgt T) t) → ¬Language T (function-ok t)
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tgt-¬function-ok {T = nil} p = scalar-function-ok nil
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tgt-¬function-ok {T = T₁ ⇒ T₂} p = function-ok p
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tgt-¬function-ok {T = none} p = none
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tgt-¬function-ok {T = any} (scalar-scalar s () p)
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tgt-¬function-ok {T = any} (scalar-function ())
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tgt-¬function-ok {T = any} (scalar-function-ok ())
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tgt-¬function-ok {T = boolean} p = scalar-function-ok boolean
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tgt-¬function-ok {T = number} p = scalar-function-ok number
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tgt-¬function-ok {T = string} p = scalar-function-ok string
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tgt-¬function-ok {T = T₁ ∪ T₂} (p₁ , p₂) = (tgt-¬function-ok p₁ , tgt-¬function-ok p₂)
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tgt-¬function-ok {T = T₁ ∩ T₂} (left p) = left (tgt-¬function-ok p)
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tgt-¬function-ok {T = T₁ ∩ T₂} (right p) = right (tgt-¬function-ok p)
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skalar-function-ok : ∀ {t} → (¬Language skalar (function-ok t))
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skalar-function-ok = (scalar-function-ok number , (scalar-function-ok string , (scalar-function-ok nil , scalar-function-ok boolean)))
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@ -151,6 +171,9 @@ none-tgt-≮: (witness function p (q₁ , scalar-function ()))
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none-tgt-≮: (witness (function-ok t) p (q₁ , function-ok q₂)) = witness t (function-ok-tgt p) q₂
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none-tgt-≮: (witness (function-err (scalar s)) p (q₁ , function-err (scalar ())))
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tgt-≮: : ∀ {T U} → (tgt T ≮: tgt U) → (T ≮: U)
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tgt-≮: (witness t p q) = witness (function-ok t) (tgt-function-ok p) (tgt-¬function-ok q)
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-- Properties of src
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function-err-src : ∀ {T t} → (¬Language (src T) t) → Language T (function-err t)
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function-err-src {T = nil} none = scalar-function-err nil
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@ -208,6 +231,9 @@ any-src-≮: r (witness (function-ok (scalar s)) p (function-ok (scalar-scalar s
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any-src-≮: r (witness (function-ok (function-ok _)) p (function-ok (scalar-function-ok ())))
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any-src-≮: r (witness (function-err t) p (function-err q)) = witness t q (src-¬function-err p)
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src-≮: : ∀ {T U} → (src T ≮: src U) → (U ≮: T)
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src-≮: (witness t p q) = witness (function-err t) (function-err-src q) (¬function-err-src p)
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-- Properties of any and none
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any-≮: : ∀ {T U} → (T ≮: U) → (any ≮: U)
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any-≮: (witness t p q) = witness t any q
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@ -221,6 +247,56 @@ any-≮:-none = witness (scalar nil) any none
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function-≮:-none : ∀ {T U} → ((T ⇒ U) ≮: none)
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function-≮:-none = witness function function none
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-- Subtyping is decidable
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-- Honest, this terminates (because src T and tgt T decrease the depth of the type)
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{-# TERMINATING #-}
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dec-subtyping : ∀ T U → Either (T ≮: U) (T <: U)
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dec-subtyping T U = result where
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P : Tree → Set
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P t = Either (¬Language T t) (Language U t)
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Q : Tree → Set
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Q t = Either (T ≮: U) (P t)
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decQ : ∀ t → Q t
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decQ t with dec-language T t | dec-language U t
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decQ t | Left ¬Tt | _ = Right (Left ¬Tt)
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decQ t | Right Tt | Left ¬Ut = Left (witness t Tt ¬Ut)
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decQ t | Right _ | Right Ut = Right (Right Ut)
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lemma : P(scalar number) → P(scalar boolean) → P(scalar nil) → P(scalar string) → P(function) → (src U <: src T) → (tgt T <: tgt U) → (T <: U)
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lemma (Left ¬Tt) boolP nilP stringP funP srcy tgty (scalar number) Tt = CONTRADICTION (language-comp (scalar number) ¬Tt Tt)
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lemma (Right Ut) boolP nilP stringP funP srcy tgty (scalar number) Tt = Ut
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lemma numP (Left ¬Tt) nilP stringP funP srcy tgty (scalar boolean) Tt = CONTRADICTION (language-comp (scalar boolean) ¬Tt Tt)
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lemma numP (Right Ut) nilP stringP funP srcy tgty (scalar boolean) Tt = Ut
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lemma numP boolP (Left ¬Tt) stringP funP srcy tgty (scalar nil) Tt = CONTRADICTION (language-comp (scalar nil) ¬Tt Tt)
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lemma numP boolP (Right Ut) stringP funP srcy tgty (scalar nil) Tt = Ut
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lemma numP boolP nilP (Left ¬Tt) funP srcy tgty (scalar string) Tt = CONTRADICTION (language-comp (scalar string) ¬Tt Tt)
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lemma numP boolP nilP (Right Ut) funP srcy tgty (scalar string) Tt = Ut
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lemma numP boolP nilP stringP (Left ¬Tt) srcy tgty function Tt = CONTRADICTION (language-comp function ¬Tt Tt)
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lemma numP boolP nilP stringP (Right Ut) srcy tgty function Tt = Ut
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lemma numP boolP nilP stringP funP srcy tgty (function-ok t) Tt = tgt-function-ok (tgty t (function-ok-tgt Tt))
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lemma numP boolP nilP stringP funP srcy tgty (function-err t) Tt = function-err-src (<:-impl-⊇ srcy t (src-¬function-err Tt))
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result : Either (T ≮: U) (T <: U)
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result with decQ (scalar number)
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result | Left r = Left r
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result | Right numP with decQ (scalar boolean)
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result | Right numP | Left r = Left r
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result | Right numP | Right boolP with decQ (scalar nil)
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result | Right numP | Right boolP | Left r = Left r
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result | Right numP | Right boolP | Right nilP with decQ (scalar string)
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result | Right numP | Right boolP | Right nilP | Left r = Left r
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result | Right numP | Right boolP | Right nilP | Right strP with decQ (function)
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result | Right numP | Right boolP | Right nilP | Right strP | Left r = Left r
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result | Right numP | Right boolP | Right nilP | Right strP | Right funP with dec-subtyping (src U) (src T)
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result | Right numP | Right boolP | Right nilP | Right strP | Right funP | Left r = Left (src-≮: r)
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result | Right numP | Right boolP | Right nilP | Right strP | Right funP | Right srcy with dec-subtyping (tgt T) (tgt U)
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result | Right numP | Right boolP | Right nilP | Right strP | Right funP | Right srcy | Left r = Left (tgt-≮: r)
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result | Right numP | Right boolP | Right nilP | Right strP | Right funP | Right srcy | Right tgty = Right (lemma numP boolP nilP strP funP srcy tgty)
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-- A Gentle Introduction To Semantic Subtyping (https://www.cduce.org/papers/gentle.pdf)
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-- defines a "set-theoretic" model (sec 2.5)
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-- Unfortunately we don't quite have this property, due to uninhabited types,
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