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Split off DecSubtyping
This commit is contained in:
ajeffrey@roblox.com
parent
2b893dba73
commit
3170472606
2 changed files with 198 additions and 187 deletions
197
prototyping/Properties/DecSubtyping.agda
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197
prototyping/Properties/DecSubtyping.agda
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{-# OPTIONS --rewriting #-}
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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-err; left; right; _,_)
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open import Luau.Type using (Type; Scalar; nil; number; string; boolean; never; unknown; _⇒_; _∪_; _∩_; src; tgt)
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open import Properties.Contradiction using (CONTRADICTION; ¬)
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open import Properties.Functions using (_∘_)
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-- ¬Language T is the complement of Language T
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language-comp : ∀ {T} t → ¬Language T t → ¬(Language T t)
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language-comp t (p₁ , p₂) (left q) = language-comp t p₁ q
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language-comp t (p₁ , p₂) (right q) = language-comp t p₂ q
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language-comp t (left p) (q₁ , q₂) = language-comp t p q₁
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language-comp t (right p) (q₁ , q₂) = language-comp t p q₂
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language-comp (scalar s) (scalar-scalar s p₁ p₂) (scalar s) = p₂ refl
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language-comp (scalar s) (function-scalar s) (scalar s) = language-comp function (scalar-function s) function
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language-comp (scalar s) never (scalar ())
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language-comp function (scalar-function ()) function
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language-comp (function-ok t) (scalar-function-ok ()) (function-ok q)
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language-comp (function-ok t) (function-ok p) (function-ok q) = language-comp t p q
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language-comp (function-err t) (function-err p) (function-err q) = language-comp t q p
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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} never = scalar-function-err nil
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function-err-src {T = T₁ ⇒ T₂} p = function-err p
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function-err-src {T = never} (scalar-scalar number () p)
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function-err-src {T = never} (scalar-function-ok ())
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function-err-src {T = unknown} never = unknown
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function-err-src {T = boolean} p = scalar-function-err boolean
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function-err-src {T = number} p = scalar-function-err number
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function-err-src {T = string} p = scalar-function-err string
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function-err-src {T = T₁ ∪ T₂} (left p) = left (function-err-src p)
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function-err-src {T = T₁ ∪ T₂} (right p) = right (function-err-src p)
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function-err-src {T = T₁ ∩ T₂} (p₁ , p₂) = function-err-src p₁ , function-err-src p₂
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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} (scalar ())
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¬function-err-src {T = T₁ ⇒ T₂} p = function-err p
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¬function-err-src {T = never} unknown = never
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¬function-err-src {T = unknown} (scalar ())
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¬function-err-src {T = boolean} (scalar ())
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¬function-err-src {T = number} (scalar ())
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¬function-err-src {T = string} (scalar ())
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¬function-err-src {T = T₁ ∪ T₂} (p₁ , p₂) = (¬function-err-src p₁ , ¬function-err-src p₂)
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¬function-err-src {T = T₁ ∩ T₂} (left p) = left (¬function-err-src p)
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¬function-err-src {T = T₁ ∩ T₂} (right p) = right (¬function-err-src p)
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src-¬function-err : ∀ {T t} → Language T (function-err t) → (¬Language (src T) t)
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src-¬function-err {T = nil} p = never
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src-¬function-err {T = T₁ ⇒ T₂} (function-err p) = p
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src-¬function-err {T = never} (scalar-function-err ())
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src-¬function-err {T = unknown} p = never
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src-¬function-err {T = boolean} p = never
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src-¬function-err {T = number} p = never
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src-¬function-err {T = string} p = never
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src-¬function-err {T = T₁ ∪ T₂} (left p) = left (src-¬function-err p)
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src-¬function-err {T = T₁ ∪ T₂} (right p) = right (src-¬function-err p)
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src-¬function-err {T = T₁ ∩ T₂} (p₁ , p₂) = (src-¬function-err p₁ , src-¬function-err p₂)
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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 tgt
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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} (scalar ())
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tgt-function-ok {T = T₁ ⇒ T₂} p = function-ok p
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tgt-function-ok {T = never} (scalar ())
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tgt-function-ok {T = unknown} p = unknown
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tgt-function-ok {T = boolean} (scalar ())
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tgt-function-ok {T = number} (scalar ())
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tgt-function-ok {T = string} (scalar ())
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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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tgt-function-ok {T = T₁ ∩ T₂} (p₁ , p₂) = (tgt-function-ok p₁ , tgt-function-ok p₂)
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function-ok-tgt : ∀ {T t} → Language T (function-ok t) → (Language (tgt T) t)
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function-ok-tgt (function-ok p) = p
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function-ok-tgt (left p) = left (function-ok-tgt p)
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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 unknown = unknown
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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 = never} p = never
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tgt-¬function-ok {T = unknown} (scalar-scalar s () p)
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tgt-¬function-ok {T = unknown} (scalar-function ())
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tgt-¬function-ok {T = unknown} (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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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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-- Language membership is decidable
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dec-language : ∀ T t → Either (¬Language T t) (Language T t)
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dec-language nil (scalar number) = Left (scalar-scalar number nil (λ ()))
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dec-language nil (scalar boolean) = Left (scalar-scalar boolean nil (λ ()))
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dec-language nil (scalar string) = Left (scalar-scalar string nil (λ ()))
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dec-language nil (scalar nil) = Right (scalar nil)
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dec-language nil function = Left (scalar-function nil)
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dec-language nil (function-ok t) = Left (scalar-function-ok nil)
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dec-language nil (function-err t) = Right (scalar-function-err nil)
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dec-language boolean (scalar number) = Left (scalar-scalar number boolean (λ ()))
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dec-language boolean (scalar boolean) = Right (scalar boolean)
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dec-language boolean (scalar string) = Left (scalar-scalar string boolean (λ ()))
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dec-language boolean (scalar nil) = Left (scalar-scalar nil boolean (λ ()))
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dec-language boolean function = Left (scalar-function boolean)
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dec-language boolean (function-ok t) = Left (scalar-function-ok boolean)
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dec-language boolean (function-err t) = Right (scalar-function-err boolean)
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dec-language number (scalar number) = Right (scalar number)
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dec-language number (scalar boolean) = Left (scalar-scalar boolean number (λ ()))
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dec-language number (scalar string) = Left (scalar-scalar string number (λ ()))
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dec-language number (scalar nil) = Left (scalar-scalar nil number (λ ()))
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dec-language number function = Left (scalar-function number)
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dec-language number (function-ok t) = Left (scalar-function-ok number)
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dec-language number (function-err t) = Right (scalar-function-err number)
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dec-language string (scalar number) = Left (scalar-scalar number string (λ ()))
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dec-language string (scalar boolean) = Left (scalar-scalar boolean string (λ ()))
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dec-language string (scalar string) = Right (scalar string)
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dec-language string (scalar nil) = Left (scalar-scalar nil string (λ ()))
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dec-language string function = Left (scalar-function string)
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dec-language string (function-ok t) = Left (scalar-function-ok string)
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dec-language string (function-err t) = Right (scalar-function-err string)
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dec-language (T₁ ⇒ T₂) (scalar s) = Left (function-scalar s)
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dec-language (T₁ ⇒ T₂) function = Right function
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dec-language (T₁ ⇒ T₂) (function-ok t) = mapLR function-ok function-ok (dec-language T₂ t)
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dec-language (T₁ ⇒ T₂) (function-err t) = mapLR function-err function-err (swapLR (dec-language T₁ t))
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dec-language never t = Left never
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dec-language unknown t = Right unknown
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dec-language (T₁ ∪ T₂) t = cond (λ p → cond (Left ∘ _,_ p) (Right ∘ right) (dec-language T₂ t)) (Right ∘ left) (dec-language T₁ t)
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dec-language (T₁ ∩ T₂) t = cond (Left ∘ left) (λ p → cond (Left ∘ right) (Right ∘ _,_ p) (dec-language T₂ t)) (dec-language T₁ t)
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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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-- 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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@ -7,63 +7,11 @@ 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-err; left; right; _,_)
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open import Luau.Type using (Type; Scalar; nil; number; string; boolean; never; unknown; _⇒_; _∪_; _∩_; src; tgt)
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open import Properties.Contradiction using (CONTRADICTION; ¬)
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open import Properties.DecSubtyping using (language-comp; dec-language; tgt-function-ok; function-ok-tgt; function-err-src; ¬function-err-src; src-¬function-err)
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open import Properties.Equality using (_≢_)
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open import Properties.Functions using (_∘_)
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open import Properties.Product using (_×_; _,_)
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-- Language membership is decidable
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dec-language : ∀ T t → Either (¬Language T t) (Language T t)
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dec-language nil (scalar number) = Left (scalar-scalar number nil (λ ()))
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dec-language nil (scalar boolean) = Left (scalar-scalar boolean nil (λ ()))
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dec-language nil (scalar string) = Left (scalar-scalar string nil (λ ()))
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dec-language nil (scalar nil) = Right (scalar nil)
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dec-language nil function = Left (scalar-function nil)
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dec-language nil (function-ok t) = Left (scalar-function-ok nil)
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dec-language nil (function-err t) = Right (scalar-function-err nil)
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dec-language boolean (scalar number) = Left (scalar-scalar number boolean (λ ()))
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dec-language boolean (scalar boolean) = Right (scalar boolean)
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dec-language boolean (scalar string) = Left (scalar-scalar string boolean (λ ()))
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dec-language boolean (scalar nil) = Left (scalar-scalar nil boolean (λ ()))
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dec-language boolean function = Left (scalar-function boolean)
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dec-language boolean (function-ok t) = Left (scalar-function-ok boolean)
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dec-language boolean (function-err t) = Right (scalar-function-err boolean)
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dec-language number (scalar number) = Right (scalar number)
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dec-language number (scalar boolean) = Left (scalar-scalar boolean number (λ ()))
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dec-language number (scalar string) = Left (scalar-scalar string number (λ ()))
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dec-language number (scalar nil) = Left (scalar-scalar nil number (λ ()))
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dec-language number function = Left (scalar-function number)
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dec-language number (function-ok t) = Left (scalar-function-ok number)
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dec-language number (function-err t) = Right (scalar-function-err number)
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dec-language string (scalar number) = Left (scalar-scalar number string (λ ()))
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dec-language string (scalar boolean) = Left (scalar-scalar boolean string (λ ()))
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dec-language string (scalar string) = Right (scalar string)
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dec-language string (scalar nil) = Left (scalar-scalar nil string (λ ()))
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dec-language string function = Left (scalar-function string)
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dec-language string (function-ok t) = Left (scalar-function-ok string)
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dec-language string (function-err t) = Right (scalar-function-err string)
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dec-language (T₁ ⇒ T₂) (scalar s) = Left (function-scalar s)
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dec-language (T₁ ⇒ T₂) function = Right function
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dec-language (T₁ ⇒ T₂) (function-ok t) = mapLR function-ok function-ok (dec-language T₂ t)
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dec-language (T₁ ⇒ T₂) (function-err t) = mapLR function-err function-err (swapLR (dec-language T₁ t))
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dec-language never t = Left never
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dec-language unknown t = Right unknown
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dec-language (T₁ ∪ T₂) t = cond (λ p → cond (Left ∘ _,_ p) (Right ∘ right) (dec-language T₂ t)) (Right ∘ left) (dec-language T₁ t)
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dec-language (T₁ ∩ T₂) t = cond (Left ∘ left) (λ p → cond (Left ∘ right) (Right ∘ _,_ p) (dec-language T₂ t)) (dec-language T₁ t)
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-- ¬Language T is the complement of Language T
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language-comp : ∀ {T} t → ¬Language T t → ¬(Language T t)
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language-comp t (p₁ , p₂) (left q) = language-comp t p₁ q
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language-comp t (p₁ , p₂) (right q) = language-comp t p₂ q
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language-comp t (left p) (q₁ , q₂) = language-comp t p q₁
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language-comp t (right p) (q₁ , q₂) = language-comp t p q₂
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language-comp (scalar s) (scalar-scalar s p₁ p₂) (scalar s) = p₂ refl
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language-comp (scalar s) (function-scalar s) (scalar s) = language-comp function (scalar-function s) function
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language-comp (scalar s) never (scalar ())
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language-comp function (scalar-function ()) function
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language-comp (function-ok t) (scalar-function-ok ()) (function-ok q)
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language-comp (function-ok t) (function-ok p) (function-ok q) = language-comp t p q
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language-comp (function-err t) (function-err p) (function-err q) = language-comp t q p
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-- ≮: is the complement of <:
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¬≮:-impl-<: : ∀ {T U} → ¬(T ≮: U) → (T <: U)
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¬≮:-impl-<: {T} {U} p t q with dec-language U t
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@ -73,12 +21,6 @@ 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)
|
||||
|
||||
-- if T <: U then ¬Language U ⊆ ¬Language T
|
||||
<:-impl-⊇ : ∀ {T U} → (T <: U) → ∀ t → ¬Language U t → ¬Language T t
|
||||
<:-impl-⊇ {T} p t ¬Ut with dec-language T t
|
||||
<:-impl-⊇ p t ¬Ut | Left ¬Tt = ¬Tt
|
||||
<:-impl-⊇ p t ¬Ut | Right Tt = CONTRADICTION (language-comp t ¬Ut (p t Tt))
|
||||
|
||||
-- reflexivity
|
||||
≮:-refl : ∀ {T} → ¬(T ≮: T)
|
||||
≮:-refl (witness t p q) = language-comp t q p
|
||||
|
|
@ -117,40 +59,6 @@ scalar-≮:-never s = witness (scalar s) (scalar s) never
|
|||
scalar-≢-impl-≮: : ∀ {T U} → (Scalar T) → (Scalar U) → (T ≢ U) → (T ≮: U)
|
||||
scalar-≢-impl-≮: s₁ s₂ p = witness (scalar s₁) (scalar s₁) (scalar-scalar s₁ s₂ p)
|
||||
|
||||
-- Properties of tgt
|
||||
tgt-function-ok : ∀ {T t} → (Language (tgt T) t) → Language T (function-ok t)
|
||||
tgt-function-ok {T = nil} (scalar ())
|
||||
tgt-function-ok {T = T₁ ⇒ T₂} p = function-ok p
|
||||
tgt-function-ok {T = never} (scalar ())
|
||||
tgt-function-ok {T = unknown} p = unknown
|
||||
tgt-function-ok {T = boolean} (scalar ())
|
||||
tgt-function-ok {T = number} (scalar ())
|
||||
tgt-function-ok {T = string} (scalar ())
|
||||
tgt-function-ok {T = T₁ ∪ T₂} (left p) = left (tgt-function-ok p)
|
||||
tgt-function-ok {T = T₁ ∪ T₂} (right p) = right (tgt-function-ok p)
|
||||
tgt-function-ok {T = T₁ ∩ T₂} (p₁ , p₂) = (tgt-function-ok p₁ , tgt-function-ok p₂)
|
||||
|
||||
function-ok-tgt : ∀ {T t} → Language T (function-ok t) → (Language (tgt T) t)
|
||||
function-ok-tgt (function-ok p) = p
|
||||
function-ok-tgt (left p) = left (function-ok-tgt p)
|
||||
function-ok-tgt (right p) = right (function-ok-tgt p)
|
||||
function-ok-tgt (p₁ , p₂) = (function-ok-tgt p₁ , function-ok-tgt p₂)
|
||||
function-ok-tgt unknown = unknown
|
||||
|
||||
tgt-¬function-ok : ∀ {T t} → (¬Language (tgt T) t) → ¬Language T (function-ok t)
|
||||
tgt-¬function-ok {T = nil} p = scalar-function-ok nil
|
||||
tgt-¬function-ok {T = T₁ ⇒ T₂} p = function-ok p
|
||||
tgt-¬function-ok {T = never} p = never
|
||||
tgt-¬function-ok {T = unknown} (scalar-scalar s () p)
|
||||
tgt-¬function-ok {T = unknown} (scalar-function ())
|
||||
tgt-¬function-ok {T = unknown} (scalar-function-ok ())
|
||||
tgt-¬function-ok {T = boolean} p = scalar-function-ok boolean
|
||||
tgt-¬function-ok {T = number} p = scalar-function-ok number
|
||||
tgt-¬function-ok {T = string} p = scalar-function-ok string
|
||||
tgt-¬function-ok {T = T₁ ∪ T₂} (p₁ , p₂) = (tgt-¬function-ok p₁ , tgt-¬function-ok p₂)
|
||||
tgt-¬function-ok {T = T₁ ∩ T₂} (left p) = left (tgt-¬function-ok p)
|
||||
tgt-¬function-ok {T = T₁ ∩ T₂} (right p) = right (tgt-¬function-ok p)
|
||||
|
||||
skalar-function-ok : ∀ {t} → (¬Language skalar (function-ok t))
|
||||
skalar-function-ok = (scalar-function-ok number , (scalar-function-ok string , (scalar-function-ok nil , scalar-function-ok boolean)))
|
||||
|
||||
|
|
@ -169,47 +77,6 @@ never-tgt-≮: (witness function p (q₁ , scalar-function ()))
|
|||
never-tgt-≮: (witness (function-ok t) p (q₁ , function-ok q₂)) = witness t (function-ok-tgt p) q₂
|
||||
never-tgt-≮: (witness (function-err (scalar s)) p (q₁ , function-err (scalar ())))
|
||||
|
||||
tgt-≮: : ∀ {T U} → (tgt T ≮: tgt U) → (T ≮: U)
|
||||
tgt-≮: (witness t p q) = witness (function-ok t) (tgt-function-ok p) (tgt-¬function-ok q)
|
||||
|
||||
-- Properties of src
|
||||
function-err-src : ∀ {T t} → (¬Language (src T) t) → Language T (function-err t)
|
||||
function-err-src {T = nil} never = scalar-function-err nil
|
||||
function-err-src {T = T₁ ⇒ T₂} p = function-err p
|
||||
function-err-src {T = never} (scalar-scalar number () p)
|
||||
function-err-src {T = never} (scalar-function-ok ())
|
||||
function-err-src {T = unknown} never = unknown
|
||||
function-err-src {T = boolean} p = scalar-function-err boolean
|
||||
function-err-src {T = number} p = scalar-function-err number
|
||||
function-err-src {T = string} p = scalar-function-err string
|
||||
function-err-src {T = T₁ ∪ T₂} (left p) = left (function-err-src p)
|
||||
function-err-src {T = T₁ ∪ T₂} (right p) = right (function-err-src p)
|
||||
function-err-src {T = T₁ ∩ T₂} (p₁ , p₂) = function-err-src p₁ , function-err-src p₂
|
||||
|
||||
¬function-err-src : ∀ {T t} → (Language (src T) t) → ¬Language T (function-err t)
|
||||
¬function-err-src {T = nil} (scalar ())
|
||||
¬function-err-src {T = T₁ ⇒ T₂} p = function-err p
|
||||
¬function-err-src {T = never} unknown = never
|
||||
¬function-err-src {T = unknown} (scalar ())
|
||||
¬function-err-src {T = boolean} (scalar ())
|
||||
¬function-err-src {T = number} (scalar ())
|
||||
¬function-err-src {T = string} (scalar ())
|
||||
¬function-err-src {T = T₁ ∪ T₂} (p₁ , p₂) = (¬function-err-src p₁ , ¬function-err-src p₂)
|
||||
¬function-err-src {T = T₁ ∩ T₂} (left p) = left (¬function-err-src p)
|
||||
¬function-err-src {T = T₁ ∩ T₂} (right p) = right (¬function-err-src p)
|
||||
|
||||
src-¬function-err : ∀ {T t} → Language T (function-err t) → (¬Language (src T) t)
|
||||
src-¬function-err {T = nil} p = never
|
||||
src-¬function-err {T = T₁ ⇒ T₂} (function-err p) = p
|
||||
src-¬function-err {T = never} (scalar-function-err ())
|
||||
src-¬function-err {T = unknown} p = never
|
||||
src-¬function-err {T = boolean} p = never
|
||||
src-¬function-err {T = number} p = never
|
||||
src-¬function-err {T = string} p = never
|
||||
src-¬function-err {T = T₁ ∪ T₂} (left p) = left (src-¬function-err p)
|
||||
src-¬function-err {T = T₁ ∪ T₂} (right p) = right (src-¬function-err p)
|
||||
src-¬function-err {T = T₁ ∩ T₂} (p₁ , p₂) = (src-¬function-err p₁ , src-¬function-err p₂)
|
||||
|
||||
src-¬scalar : ∀ {S T t} (s : Scalar S) → Language T (scalar s) → (¬Language (src T) t)
|
||||
src-¬scalar number (scalar number) = never
|
||||
src-¬scalar boolean (scalar boolean) = never
|
||||
|
|
@ -229,9 +96,6 @@ unknown-src-≮: r (witness (function-ok (scalar s)) p (function-ok (scalar-scal
|
|||
unknown-src-≮: r (witness (function-ok (function-ok _)) p (function-ok (scalar-function-ok ())))
|
||||
unknown-src-≮: r (witness (function-err t) p (function-err q)) = witness t q (src-¬function-err p)
|
||||
|
||||
src-≮: : ∀ {T U} → (src T ≮: src U) → (U ≮: T)
|
||||
src-≮: (witness t p q) = witness (function-err t) (function-err-src q) (¬function-err-src p)
|
||||
|
||||
-- Properties of unknown and never
|
||||
unknown-≮: : ∀ {T U} → (T ≮: U) → (unknown ≮: U)
|
||||
unknown-≮: (witness t p q) = witness t unknown q
|
||||
|
|
@ -245,56 +109,6 @@ unknown-≮:-never = witness (scalar nil) unknown never
|
|||
function-≮:-never : ∀ {T U} → ((T ⇒ U) ≮: never)
|
||||
function-≮:-never = witness function function never
|
||||
|
||||
-- Subtyping is decidable
|
||||
-- Honest, this terminates (because src T and tgt T decrease the depth of the type)
|
||||
|
||||
{-# TERMINATING #-}
|
||||
dec-subtyping : ∀ T U → Either (T ≮: U) (T <: U)
|
||||
dec-subtyping T U = result where
|
||||
|
||||
P : Tree → Set
|
||||
P t = Either (¬Language T t) (Language U t)
|
||||
|
||||
Q : Tree → Set
|
||||
Q t = Either (T ≮: U) (P t)
|
||||
|
||||
decQ : ∀ t → Q t
|
||||
decQ t with dec-language T t | dec-language U t
|
||||
decQ t | Left ¬Tt | _ = Right (Left ¬Tt)
|
||||
decQ t | Right Tt | Left ¬Ut = Left (witness t Tt ¬Ut)
|
||||
decQ t | Right _ | Right Ut = Right (Right Ut)
|
||||
|
||||
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)
|
||||
lemma (Left ¬Tt) boolP nilP stringP funP srcy tgty (scalar number) Tt = CONTRADICTION (language-comp (scalar number) ¬Tt Tt)
|
||||
lemma (Right Ut) boolP nilP stringP funP srcy tgty (scalar number) Tt = Ut
|
||||
lemma numP (Left ¬Tt) nilP stringP funP srcy tgty (scalar boolean) Tt = CONTRADICTION (language-comp (scalar boolean) ¬Tt Tt)
|
||||
lemma numP (Right Ut) nilP stringP funP srcy tgty (scalar boolean) Tt = Ut
|
||||
lemma numP boolP (Left ¬Tt) stringP funP srcy tgty (scalar nil) Tt = CONTRADICTION (language-comp (scalar nil) ¬Tt Tt)
|
||||
lemma numP boolP (Right Ut) stringP funP srcy tgty (scalar nil) Tt = Ut
|
||||
lemma numP boolP nilP (Left ¬Tt) funP srcy tgty (scalar string) Tt = CONTRADICTION (language-comp (scalar string) ¬Tt Tt)
|
||||
lemma numP boolP nilP (Right Ut) funP srcy tgty (scalar string) Tt = Ut
|
||||
lemma numP boolP nilP stringP (Left ¬Tt) srcy tgty function Tt = CONTRADICTION (language-comp function ¬Tt Tt)
|
||||
lemma numP boolP nilP stringP (Right Ut) srcy tgty function Tt = Ut
|
||||
lemma numP boolP nilP stringP funP srcy tgty (function-ok t) Tt = tgt-function-ok (tgty t (function-ok-tgt Tt))
|
||||
lemma numP boolP nilP stringP funP srcy tgty (function-err t) Tt = function-err-src (<:-impl-⊇ srcy t (src-¬function-err Tt))
|
||||
|
||||
result : Either (T ≮: U) (T <: U)
|
||||
result with decQ (scalar number)
|
||||
result | Left r = Left r
|
||||
result | Right numP with decQ (scalar boolean)
|
||||
result | Right numP | Left r = Left r
|
||||
result | Right numP | Right boolP with decQ (scalar nil)
|
||||
result | Right numP | Right boolP | Left r = Left r
|
||||
result | Right numP | Right boolP | Right nilP with decQ (scalar string)
|
||||
result | Right numP | Right boolP | Right nilP | Left r = Left r
|
||||
result | Right numP | Right boolP | Right nilP | Right strP with decQ (function)
|
||||
result | Right numP | Right boolP | Right nilP | Right strP | Left r = Left r
|
||||
result | Right numP | Right boolP | Right nilP | Right strP | Right funP with dec-subtyping (src U) (src T)
|
||||
result | Right numP | Right boolP | Right nilP | Right strP | Right funP | Left r = Left (src-≮: r)
|
||||
result | Right numP | Right boolP | Right nilP | Right strP | Right funP | Right srcy with dec-subtyping (tgt T) (tgt U)
|
||||
result | Right numP | Right boolP | Right nilP | Right strP | Right funP | Right srcy | Left r = Left (tgt-≮: r)
|
||||
result | Right numP | Right boolP | Right nilP | Right strP | Right funP | Right srcy | Right tgty = Right (lemma numP boolP nilP strP funP srcy tgty)
|
||||
|
||||
-- A Gentle Introduction To Semantic Subtyping (https://www.cduce.org/papers/gentle.pdf)
|
||||
-- defines a "set-theoretic" model (sec 2.5)
|
||||
-- Unfortunately we don't quite have this property, due to uninhabited types,
|
||||
|
|
|
|||
Loading…
Add table
Reference in a new issue