luau-lang/luau
1
0
Fork 0
mirror of https://github.com/luau-lang/luau.git synced 2024-12-13 05:20:38 +00:00
Code Issues Projects Releases Packages Wiki Activity Actions
luau/prototyping/Properties/Subtyping.agda
Alan Jeffrey 948f678f93
Prototyping function overload resolution (#508)
2022-06-14 20:03:43 -07:00

481 lines
25 KiB
Agda
Raw Blame History

This file contains ambiguous Unicode characters

This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

{-# OPTIONS --rewriting #-}
module Properties.Subtyping where
open import Agda.Builtin.Equality using (_≡_; refl)
open import FFI.Data.Either using (Either; Left; Right; mapLR; swapLR; cond)
open import FFI.Data.Maybe using (Maybe; just; nothing)
open import Luau.Subtyping using (_<:_; _≮:_; Tree; Language; ¬Language; witness; unknown; never; scalar; function; scalar-function; scalar-function-ok; scalar-function-err; scalar-function-tgt; scalar-scalar; function-scalar; function-ok; function-ok₁; function-ok₂; function-err; function-tgt; left; right; _,_)
open import Luau.Type using (Type; Scalar; nil; number; string; boolean; never; unknown; _⇒_; _∪_; _∩_; skalar)
open import Properties.Contradiction using (CONTRADICTION; ¬; ⊥)
open import Properties.Equality using (_≢_)
open import Properties.Functions using (_∘_)
open import Properties.Product using (_×_; _,_)
-- Language membership is decidable
dec-language : ∀ T t → Either (¬Language T t) (Language T t)
dec-language nil (scalar number) = Left (scalar-scalar number nil (λ ()))
dec-language nil (scalar boolean) = Left (scalar-scalar boolean nil (λ ()))
dec-language nil (scalar string) = Left (scalar-scalar string nil (λ ()))
dec-language nil (scalar nil) = Right (scalar nil)
dec-language nil function = Left (scalar-function nil)
dec-language nil (function-ok s t) = Left (scalar-function-ok nil)
dec-language nil (function-err t) = Left (scalar-function-err nil)
dec-language boolean (scalar number) = Left (scalar-scalar number boolean (λ ()))
dec-language boolean (scalar boolean) = Right (scalar boolean)
dec-language boolean (scalar string) = Left (scalar-scalar string boolean (λ ()))
dec-language boolean (scalar nil) = Left (scalar-scalar nil boolean (λ ()))
dec-language boolean function = Left (scalar-function boolean)
dec-language boolean (function-ok s t) = Left (scalar-function-ok boolean)
dec-language boolean (function-err t) = Left (scalar-function-err boolean)
dec-language number (scalar number) = Right (scalar number)
dec-language number (scalar boolean) = Left (scalar-scalar boolean number (λ ()))
dec-language number (scalar string) = Left (scalar-scalar string number (λ ()))
dec-language number (scalar nil) = Left (scalar-scalar nil number (λ ()))
dec-language number function = Left (scalar-function number)
dec-language number (function-ok s t) = Left (scalar-function-ok number)
dec-language number (function-err t) = Left (scalar-function-err number)
dec-language string (scalar number) = Left (scalar-scalar number string (λ ()))
dec-language string (scalar boolean) = Left (scalar-scalar boolean string (λ ()))
dec-language string (scalar string) = Right (scalar string)
dec-language string (scalar nil) = Left (scalar-scalar nil string (λ ()))
dec-language string function = Left (scalar-function string)
dec-language string (function-ok s t) = Left (scalar-function-ok string)
dec-language string (function-err t) = Left (scalar-function-err string)
dec-language (T₁ ⇒ T₂) (scalar s) = Left (function-scalar s)
dec-language (T₁ ⇒ T₂) function = Right function
dec-language (T₁ ⇒ T₂) (function-ok s t) = cond (Right ∘ function-ok₁) (λ p → mapLR (function-ok p) function-ok₂ (dec-language T₂ t)) (dec-language T₁ s)
dec-language (T₁ ⇒ T₂) (function-err t) = mapLR function-err function-err (swapLR (dec-language T₁ t))
dec-language never t = Left never
dec-language unknown t = Right unknown
dec-language (T₁ ∪ T₂) t = cond (λ p → cond (Left ∘ _,_ p) (Right ∘ right) (dec-language T₂ t)) (Right ∘ left) (dec-language T₁ t)
dec-language (T₁ ∩ T₂) t = cond (Left ∘ left) (λ p → cond (Left ∘ right) (Right ∘ _,_ p) (dec-language T₂ t)) (dec-language T₁ t)
dec-language nil (function-tgt t) = Left (scalar-function-tgt nil)
dec-language (T₁ ⇒ T₂) (function-tgt t) = mapLR function-tgt function-tgt (dec-language T₂ t)
dec-language boolean (function-tgt t) = Left (scalar-function-tgt boolean)
dec-language number (function-tgt t) = Left (scalar-function-tgt number)
dec-language string (function-tgt t) = Left (scalar-function-tgt string)
-- ¬Language T is the complement of Language T
language-comp : ∀ {T} t → ¬Language T t → ¬(Language T t)
language-comp t (p₁ , p₂) (left q) = language-comp t p₁ q
language-comp t (p₁ , p₂) (right q) = language-comp t p₂ q
language-comp t (left p) (q₁ , q₂) = language-comp t p q₁
language-comp t (right p) (q₁ , q₂) = language-comp t p q₂
language-comp (scalar s) (scalar-scalar s p₁ p₂) (scalar s) = p₂ refl
language-comp (scalar s) (function-scalar s) (scalar s) = language-comp function (scalar-function s) function
language-comp (scalar s) never (scalar ())
language-comp function (scalar-function ()) function
language-comp (function-ok s t) (scalar-function-ok ()) (function-ok₁ p)
language-comp (function-ok s t) (function-ok p₁ p₂) (function-ok₁ q) = language-comp s q p₁
language-comp (function-ok s t) (function-ok p₁ p₂) (function-ok₂ q) = language-comp t p₂ q
language-comp (function-err t) (function-err p) (function-err q) = language-comp t q p
language-comp (function-tgt t) (scalar-function-tgt ()) (function-tgt q)
language-comp (function-tgt t) (function-tgt p) (function-tgt q) = language-comp t p q
-- ≮: is the complement of <:
¬≮:-impl-<: : ∀ {T U} → ¬(T ≮: U) → (T <: U)
¬≮:-impl-<: {T} {U} p t q with dec-language U t
¬≮:-impl-<: {T} {U} p t q | Left r = CONTRADICTION (p (witness t q r))
¬≮:-impl-<: {T} {U} p t q | Right r = r
<:-impl-¬≮: : ∀ {T U} → (T <: U) → ¬(T ≮: U)
<:-impl-¬≮: p (witness t q r) = language-comp t r (p t q)
<:-impl-⊇ : ∀ {T U} → (T <: U) → ∀ t → ¬Language U t → ¬Language T t
<:-impl-⊇ {T} p t q with dec-language T t
<:-impl-⊇ {_} p t q | Left r = r
<:-impl-⊇ {_} p t q | Right r = CONTRADICTION (language-comp t q (p t r))
-- reflexivity
≮:-refl : ∀ {T} → ¬(T ≮: T)
≮:-refl (witness t p q) = language-comp t q p
<:-refl : ∀ {T} → (T <: T)
<:-refl = ¬≮:-impl-<: ≮:-refl
-- transititivity
≮:-trans-≡ : ∀ {S T U} → (S ≮: T) → (T ≡ U) → (S ≮: U)
≮:-trans-≡ p refl = p
<:-trans-≡ : ∀ {S T U} → (S <: T) → (T ≡ U) → (S <: U)
<:-trans-≡ p refl = p
≡-impl-<: : ∀ {T U} → (T ≡ U) → (T <: U)
≡-impl-<: refl = <:-refl
≡-trans-≮: : ∀ {S T U} → (S ≡ T) → (T ≮: U) → (S ≮: U)
≡-trans-≮: refl p = p
≡-trans-<: : ∀ {S T U} → (S ≡ T) → (T <: U) → (S <: U)
≡-trans-<: refl p = p
≮:-trans : ∀ {S T U} → (S ≮: U) → Either (S ≮: T) (T ≮: U)
≮:-trans {T = T} (witness t p q) = mapLR (witness t p) (λ z → witness t z q) (dec-language T t)
<:-trans : ∀ {S T U} → (S <: T) → (T <: U) → (S <: U)
<:-trans p q t r = q t (p t r)
<:-trans-≮: : ∀ {S T U} → (S <: T) → (S ≮: U) → (T ≮: U)
<:-trans-≮: p (witness t q r) = witness t (p t q) r
≮:-trans-<: : ∀ {S T U} → (S ≮: U) → (T <: U) → (S ≮: T)
≮:-trans-<: (witness t p q) r = witness t p (<:-impl-⊇ r t q)
-- Properties of union
<:-union : ∀ {R S T U} → (R <: T) → (S <: U) → ((R ∪ S) <: (T ∪ U))
<:-union p q t (left r) = left (p t r)
<:-union p q t (right r) = right (q t r)
<:-∪-left : ∀ {S T} → S <: (S ∪ T)
<:-∪-left t p = left p
<:-∪-right : ∀ {S T} → T <: (S ∪ T)
<:-∪-right t p = right p
<:-∪-lub : ∀ {S T U} → (S <: U) → (T <: U) → ((S ∪ T) <: U)
<:-∪-lub p q t (left r) = p t r
<:-∪-lub p q t (right r) = q t r
<:-∪-symm : ∀ {T U} → (T ∪ U) <: (U ∪ T)
<:-∪-symm t (left p) = right p
<:-∪-symm t (right p) = left p
<:-∪-assocl : ∀ {S T U} → (S ∪ (T ∪ U)) <: ((S ∪ T) ∪ U)
<:-∪-assocl t (left p) = left (left p)
<:-∪-assocl t (right (left p)) = left (right p)
<:-∪-assocl t (right (right p)) = right p
<:-∪-assocr : ∀ {S T U} → ((S ∪ T) ∪ U) <: (S ∪ (T ∪ U))
<:-∪-assocr t (left (left p)) = left p
<:-∪-assocr t (left (right p)) = right (left p)
<:-∪-assocr t (right p) = right (right p)
≮:-∪-left : ∀ {S T U} → (S ≮: U) → ((S ∪ T) ≮: U)
≮:-∪-left (witness t p q) = witness t (left p) q
≮:-∪-right : ∀ {S T U} → (T ≮: U) → ((S ∪ T) ≮: U)
≮:-∪-right (witness t p q) = witness t (right p) q
≮:-left-∪ : ∀ {S T U} → (S ≮: (T ∪ U)) → (S ≮: T)
≮:-left-∪ (witness t p (q₁ , q₂)) = witness t p q₁
≮:-right-∪ : ∀ {S T U} → (S ≮: (T ∪ U)) → (S ≮: U)
≮:-right-∪ (witness t p (q₁ , q₂)) = witness t p q₂
-- Properties of intersection
<:-intersect : ∀ {R S T U} → (R <: T) → (S <: U) → ((R ∩ S) <: (T ∩ U))
<:-intersect p q t (r₁ , r₂) = (p t r₁ , q t r₂)
<:-∩-left : ∀ {S T} → (S ∩ T) <: S
<:-∩-left t (p , _) = p
<:-∩-right : ∀ {S T} → (S ∩ T) <: T
<:-∩-right t (_ , p) = p
<:-∩-glb : ∀ {S T U} → (S <: T) → (S <: U) → (S <: (T ∩ U))
<:-∩-glb p q t r = (p t r , q t r)
<:-∩-symm : ∀ {T U} → (T ∩ U) <: (U ∩ T)
<:-∩-symm t (p₁ , p₂) = (p₂ , p₁)
<:-∩-assocl : ∀ {S T U} → (S ∩ (T ∩ U)) <: ((S ∩ T) ∩ U)
<:-∩-assocl t (p , (p₁ , p₂)) = (p , p₁) , p₂
<:-∩-assocr : ∀ {S T U} → ((S ∩ T) ∩ U) <: (S ∩ (T ∩ U))
<:-∩-assocr t ((p , p₁) , p₂) = p , (p₁ , p₂)
≮:-∩-left : ∀ {S T U} → (S ≮: T) → (S ≮: (T ∩ U))
≮:-∩-left (witness t p q) = witness t p (left q)
≮:-∩-right : ∀ {S T U} → (S ≮: U) → (S ≮: (T ∩ U))
≮:-∩-right (witness t p q) = witness t p (right q)
-- Distribution properties
<:-∩-distl-∪ : ∀ {S T U} → (S ∩ (T ∪ U)) <: ((S ∩ T) ∪ (S ∩ U))
<:-∩-distl-∪ t (p₁ , left p₂) = left (p₁ , p₂)
<:-∩-distl-∪ t (p₁ , right p₂) = right (p₁ , p₂)
∩-distl-∪-<: : ∀ {S T U} → ((S ∩ T) ∪ (S ∩ U)) <: (S ∩ (T ∪ U))
∩-distl-∪-<: t (left (p₁ , p₂)) = (p₁ , left p₂)
∩-distl-∪-<: t (right (p₁ , p₂)) = (p₁ , right p₂)
<:-∩-distr-∪ : ∀ {S T U} → ((S ∪ T) ∩ U) <: ((S ∩ U) ∪ (T ∩ U))
<:-∩-distr-∪ t (left p₁ , p₂) = left (p₁ , p₂)
<:-∩-distr-∪ t (right p₁ , p₂) = right (p₁ , p₂)
∩-distr-∪-<: : ∀ {S T U} → ((S ∩ U) ∪ (T ∩ U)) <: ((S ∪ T) ∩ U)
∩-distr-∪-<: t (left (p₁ , p₂)) = (left p₁ , p₂)
∩-distr-∪-<: t (right (p₁ , p₂)) = (right p₁ , p₂)
<:-∪-distl-∩ : ∀ {S T U} → (S ∪ (T ∩ U)) <: ((S ∪ T) ∩ (S ∪ U))
<:-∪-distl-∩ t (left p) = (left p , left p)
<:-∪-distl-∩ t (right (p₁ , p₂)) = (right p₁ , right p₂)
∪-distl-∩-<: : ∀ {S T U} → ((S ∪ T) ∩ (S ∪ U)) <: (S ∪ (T ∩ U))
∪-distl-∩-<: t (left p₁ , p₂) = left p₁
∪-distl-∩-<: t (right p₁ , left p₂) = left p₂
∪-distl-∩-<: t (right p₁ , right p₂) = right (p₁ , p₂)
<:-∪-distr-∩ : ∀ {S T U} → ((S ∩ T) ∪ U) <: ((S ∪ U) ∩ (T ∪ U))
<:-∪-distr-∩ t (left (p₁ , p₂)) = left p₁ , left p₂
<:-∪-distr-∩ t (right p) = (right p , right p)
∪-distr-∩-<: : ∀ {S T U} → ((S ∪ U) ∩ (T ∪ U)) <: ((S ∩ T) ∪ U)
∪-distr-∩-<: t (left p₁ , left p₂) = left (p₁ , p₂)
∪-distr-∩-<: t (left p₁ , right p₂) = right p₂
∪-distr-∩-<: t (right p₁ , p₂) = right p₁
∩-<:-∪ : ∀ {S T} → (S ∩ T) <: (S ∪ T)
∩-<:-∪ t (p , _) = left p
-- Properties of functions
<:-function : ∀ {R S T U} → (R <: S) → (T <: U) → (S ⇒ T) <: (R ⇒ U)
<:-function p q function function = function
<:-function p q (function-ok s t) (function-ok₁ r) = function-ok₁ (<:-impl-⊇ p s r)
<:-function p q (function-ok s t) (function-ok₂ r) = function-ok₂ (q t r)
<:-function p q (function-err s) (function-err r) = function-err (<:-impl-⊇ p s r)
<:-function p q (function-tgt t) (function-tgt r) = function-tgt (q t r)
<:-function-∩-∩ : ∀ {R S T U} → ((R ⇒ T) ∩ (S ⇒ U)) <: ((R ∩ S) ⇒ (T ∩ U))
<:-function-∩-∩ function (function , function) = function
<:-function-∩-∩ (function-ok s t) (function-ok₁ p , q) = function-ok₁ (left p)
<:-function-∩-∩ (function-ok s t) (function-ok₂ p , function-ok₁ q) = function-ok₁ (right q)
<:-function-∩-∩ (function-ok s t) (function-ok₂ p , function-ok₂ q) = function-ok₂ (p , q)
<:-function-∩-∩ (function-err s) (function-err p , q) = function-err (left p)
<:-function-∩-∩ (function-tgt s) (function-tgt p , function-tgt q) = function-tgt (p , q)
<:-function-∩-∪ : ∀ {R S T U} → ((R ⇒ T) ∩ (S ⇒ U)) <: ((R ∪ S) ⇒ (T ∪ U))
<:-function-∩-∪ function (function , function) = function
<:-function-∩-∪ (function-ok s t) (function-ok₁ p₁ , function-ok₁ p₂) = function-ok₁ (p₁ , p₂)
<:-function-∩-∪ (function-ok s t) (p₁ , function-ok₂ p₂) = function-ok₂ (right p₂)
<:-function-∩-∪ (function-ok s t) (function-ok₂ p₁ , p₂) = function-ok₂ (left p₁)
<:-function-∩-∪ (function-err s) (function-err p₁ , function-err q₂) = function-err (p₁ , q₂)
<:-function-∩-∪ (function-tgt t) (function-tgt p , q) = function-tgt (left p)
<:-function-∩ : ∀ {S T U} → ((S ⇒ T) ∩ (S ⇒ U)) <: (S ⇒ (T ∩ U))
<:-function-∩ function (function , function) = function
<:-function-∩ (function-ok s t) (p₁ , function-ok₁ p₂) = function-ok₁ p₂
<:-function-∩ (function-ok s t) (function-ok₁ p₁ , p₂) = function-ok₁ p₁
<:-function-∩ (function-ok s t) (function-ok₂ p₁ , function-ok₂ p₂) = function-ok₂ (p₁ , p₂)
<:-function-∩ (function-err s) (function-err p₁ , function-err p₂) = function-err p₂
<:-function-∩ (function-tgt t) (function-tgt p₁ , function-tgt p₂) = function-tgt (p₁ , p₂)
<:-function-∪ : ∀ {R S T U} → ((R ⇒ S) ∪ (T ⇒ U)) <: ((R ∩ T) ⇒ (S ∪ U))
<:-function-∪ function (left function) = function
<:-function-∪ (function-ok s t) (left (function-ok₁ p)) = function-ok₁ (left p)
<:-function-∪ (function-ok s t) (left (function-ok₂ p)) = function-ok₂ (left p)
<:-function-∪ (function-err s) (left (function-err p)) = function-err (left p)
<:-function-∪ (scalar s) (left (scalar ()))
<:-function-∪ function (right function) = function
<:-function-∪ (function-ok s t) (right (function-ok₁ p)) = function-ok₁ (right p)
<:-function-∪ (function-ok s t) (right (function-ok₂ p)) = function-ok₂ (right p)
<:-function-∪ (function-err s) (right (function-err x)) = function-err (right x)
<:-function-∪ (scalar s) (right (scalar ()))
<:-function-∪ (function-tgt t) (left (function-tgt p)) = function-tgt (left p)
<:-function-∪ (function-tgt t) (right (function-tgt p)) = function-tgt (right p)
<:-function-∪-∩ : ∀ {R S T U} → ((R ∩ S) ⇒ (T ∪ U)) <: ((R ⇒ T) ∪ (S ⇒ U))
<:-function-∪-∩ function function = left function
<:-function-∪-∩ (function-ok s t) (function-ok₁ (left p)) = left (function-ok₁ p)
<:-function-∪-∩ (function-ok s t) (function-ok₂ (left p)) = left (function-ok₂ p)
<:-function-∪-∩ (function-ok s t) (function-ok₁ (right p)) = right (function-ok₁ p)
<:-function-∪-∩ (function-ok s t) (function-ok₂ (right p)) = right (function-ok₂ p)
<:-function-∪-∩ (function-err s) (function-err (left p)) = left (function-err p)
<:-function-∪-∩ (function-err s) (function-err (right p)) = right (function-err p)
<:-function-∪-∩ (function-tgt t) (function-tgt (left p)) = left (function-tgt p)
<:-function-∪-∩ (function-tgt t) (function-tgt (right p)) = right (function-tgt p)
<:-function-left : ∀ {R S T U} → (S ⇒ T) <: (R ⇒ U) → (R <: S)
<:-function-left {R} {S} p s Rs with dec-language S s
<:-function-left p s Rs | Right Ss = Ss
<:-function-left p s Rs | Left ¬Ss with p (function-err s) (function-err ¬Ss)
<:-function-left p s Rs | Left ¬Ss | function-err ¬Rs = CONTRADICTION (language-comp s ¬Rs Rs)
<:-function-right : ∀ {R S T U} → (S ⇒ T) <: (R ⇒ U) → (T <: U)
<:-function-right p t Tt with p (function-tgt t) (function-tgt Tt)
<:-function-right p t Tt | function-tgt St = St
≮:-function-left : ∀ {R S T U} → (R ≮: S) → (S ⇒ T) ≮: (R ⇒ U)
≮:-function-left (witness t p q) = witness (function-err t) (function-err q) (function-err p)
≮:-function-right : ∀ {R S T U} → (T ≮: U) → (S ⇒ T) ≮: (R ⇒ U)
≮:-function-right (witness t p q) = witness (function-tgt t) (function-tgt p) (function-tgt q)
-- Properties of scalars
skalar-function-ok : ∀ {s t} → (¬Language skalar (function-ok s t))
skalar-function-ok = (scalar-function-ok number , (scalar-function-ok string , (scalar-function-ok nil , scalar-function-ok boolean)))
scalar-<: : ∀ {S T} → (s : Scalar S) → Language T (scalar s) → (S <: T)
scalar-<: number p (scalar number) (scalar number) = p
scalar-<: boolean p (scalar boolean) (scalar boolean) = p
scalar-<: string p (scalar string) (scalar string) = p
scalar-<: nil p (scalar nil) (scalar nil) = p
scalar-∩-function-<:-never : ∀ {S T U} → (Scalar S) → ((T ⇒ U) ∩ S) <: never
scalar-∩-function-<:-never number .(scalar number) (() , scalar number)
scalar-∩-function-<:-never boolean .(scalar boolean) (() , scalar boolean)
scalar-∩-function-<:-never string .(scalar string) (() , scalar string)
scalar-∩-function-<:-never nil .(scalar nil) (() , scalar nil)
function-≮:-scalar : ∀ {S T U} → (Scalar U) → ((S ⇒ T) ≮: U)
function-≮:-scalar s = witness function function (scalar-function s)
scalar-≮:-function : ∀ {S T U} → (Scalar U) → (U ≮: (S ⇒ T))
scalar-≮:-function s = witness (scalar s) (scalar s) (function-scalar s)
unknown-≮:-scalar : ∀ {U} → (Scalar U) → (unknown ≮: U)
unknown-≮:-scalar s = witness function unknown (scalar-function s)
scalar-≮:-never : ∀ {U} → (Scalar U) → (U ≮: never)
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)
scalar-≢-∩-<:-never : ∀ {T U V} → (Scalar T) → (Scalar U) → (T ≢ U) → (T ∩ U) <: V
scalar-≢-∩-<:-never s t p u (scalar s₁ , scalar s₂) = CONTRADICTION (p refl)
skalar-scalar : ∀ {T} (s : Scalar T) → (Language skalar (scalar s))
skalar-scalar number = left (scalar number)
skalar-scalar boolean = right (right (right (scalar boolean)))
skalar-scalar string = right (left (scalar string))
skalar-scalar nil = right (right (left (scalar nil)))
-- Properties of unknown and never
unknown-≮: : ∀ {T U} → (T ≮: U) → (unknown ≮: U)
unknown-≮: (witness t p q) = witness t unknown q
never-≮: : ∀ {T U} → (T ≮: U) → (T ≮: never)
never-≮: (witness t p q) = witness t p never
unknown-≮:-never : (unknown ≮: never)
unknown-≮:-never = witness (scalar nil) unknown never
unknown-≮:-function : ∀ {S T} → (unknown ≮: (S ⇒ T))
unknown-≮:-function = witness (scalar nil) unknown (function-scalar nil)
function-≮:-never : ∀ {T U} → ((T ⇒ U) ≮: never)
function-≮:-never = witness function function never
<:-never : ∀ {T} → (never <: T)
<:-never t (scalar ())
≮:-never-left : ∀ {S T U} → (S <: (T ∪ U)) → (S ≮: T) → (S ∩ U) ≮: never
≮:-never-left p (witness t q₁ q₂) with p t q₁
≮:-never-left p (witness t q₁ q₂) | left r = CONTRADICTION (language-comp t q₂ r)
≮:-never-left p (witness t q₁ q₂) | right r = witness t (q₁ , r) never
≮:-never-right : ∀ {S T U} → (S <: (T ∪ U)) → (S ≮: U) → (S ∩ T) ≮: never
≮:-never-right p (witness t q₁ q₂) with p t q₁
≮:-never-right p (witness t q₁ q₂) | left r = witness t (q₁ , r) never
≮:-never-right p (witness t q₁ q₂) | right r = CONTRADICTION (language-comp t q₂ r)
<:-unknown : ∀ {T} → (T <: unknown)
<:-unknown t p = unknown
<:-everything : unknown <: ((never ⇒ unknown) ∪ skalar)
<:-everything (scalar s) p = right (skalar-scalar s)
<:-everything function p = left function
<:-everything (function-ok s t) p = left (function-ok₁ never)
<:-everything (function-err s) p = left (function-err never)
<:-everything (function-tgt t) p = left (function-tgt unknown)
-- 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,
-- for example (never -> T) is equivalent to (never -> U)
-- when types are interpreted as sets of syntactic values.
_⊆_ : ∀ {A : Set} → (A → Set) → (A → Set) → Set
(P ⊆ Q) = ∀ a → (P a) → (Q a)
_⊗_ : ∀ {A B : Set} → (A → Set) → (B → Set) → ((A × B) → Set)
(P ⊗ Q) (a , b) = (P a) × (Q b)
Comp : ∀ {A : Set} → (A → Set) → (A → Set)
Comp P a = ¬(P a)
Lift : ∀ {A : Set} → (A → Set) → (Maybe A → Set)
Lift P nothing = ⊥
Lift P (just a) = P a
set-theoretic-if : ∀ {S₁ T₁ S₂ T₂} →
-- This is the "if" part of being a set-theoretic model
-- though it uses the definition from Frisch's thesis
-- rather than from the Gentle Introduction. The difference
-- being the presence of Lift, (written D_Ω in Defn 4.2 of
-- https://www.cduce.org/papers/frisch_phd.pdf).
(Language (S₁ ⇒ T₁) ⊆ Language (S₂ ⇒ T₂)) →
(∀ Q → Q ⊆ Comp((Language S₁) ⊗ Comp(Lift(Language T₁))) → Q ⊆ Comp((Language S₂) ⊗ Comp(Lift(Language T₂))))
set-theoretic-if {S₁} {T₁} {S₂} {T₂} p Q q (t , just u) Qtu (S₂t , ¬T₂u) = q (t , just u) Qtu (S₁t , ¬T₁u) where
S₁t : Language S₁ t
S₁t with dec-language S₁ t
S₁t | Left ¬S₁t with p (function-err t) (function-err ¬S₁t)
S₁t | Left ¬S₁t | function-err ¬S₂t = CONTRADICTION (language-comp t ¬S₂t S₂t)
S₁t | Right r = r
¬T₁u : ¬(Language T₁ u)
¬T₁u T₁u with p (function-ok t u) (function-ok₂ T₁u)
¬T₁u T₁u | function-ok₁ ¬S₂t = language-comp t ¬S₂t S₂t
¬T₁u T₁u | function-ok₂ T₂u = ¬T₂u T₂u
set-theoretic-if {S₁} {T₁} {S₂} {T₂} p Q q (t , nothing) Qt- (S₂t , _) = q (t , nothing) Qt- (S₁t , λ ()) where
S₁t : Language S₁ t
S₁t with dec-language S₁ t
S₁t | Left ¬S₁t with p (function-err t) (function-err ¬S₁t)
S₁t | Left ¬S₁t | function-err ¬S₂t = CONTRADICTION (language-comp t ¬S₂t S₂t)
S₁t | Right r = r
not-quite-set-theoretic-only-if : ∀ {S₁ T₁ S₂ T₂} →
-- We don't quite have that this is a set-theoretic model
-- it's only true when Language S₂ is inhabited
-- in particular it's not true when S₂ is never,
∀ s₂ → Language S₂ s₂ →
-- This is the "only if" part of being a set-theoretic model
(∀ Q → Q ⊆ Comp((Language S₁) ⊗ Comp(Lift(Language T₁))) → Q ⊆ Comp((Language S₂) ⊗ Comp(Lift(Language T₂)))) →
(Language (S₁ ⇒ T₁) ⊆ Language (S₂ ⇒ T₂))
not-quite-set-theoretic-only-if {S₁} {T₁} {S₂} {T₂} s₂ S₂s₂ p = r where
Q : (Tree × Maybe Tree) → Set
Q (t , just u) = Either (¬Language S₁ t) (Language T₁ u)
Q (t , nothing) = ¬Language S₁ t
q : Q ⊆ Comp(Language S₁ ⊗ Comp(Lift(Language T₁)))
q (t , just u) (Left ¬S₁t) (S₁t , ¬T₁u) = language-comp t ¬S₁t S₁t
q (t , just u) (Right T₂u) (S₁t , ¬T₁u) = ¬T₁u T₂u
q (t , nothing) ¬S₁t (S₁t , _) = language-comp t ¬S₁t S₁t
r : Language (S₁ ⇒ T₁) ⊆ Language (S₂ ⇒ T₂)
r function function = function
r (function-err s) (function-err ¬S₁s) with dec-language S₂ s
r (function-err s) (function-err ¬S₁s) | Left ¬S₂s = function-err ¬S₂s
r (function-err s) (function-err ¬S₁s) | Right S₂s = CONTRADICTION (p Q q (s , nothing) ¬S₁s (S₂s , λ ()))
r (function-ok s t) (function-ok₁ ¬S₁s) with dec-language S₂ s
r (function-ok s t) (function-ok₁ ¬S₁s) | Left ¬S₂s = function-ok₁ ¬S₂s
r (function-ok s t) (function-ok₁ ¬S₁s) | Right S₂s = CONTRADICTION (p Q q (s , nothing) ¬S₁s (S₂s , λ ()))
r (function-ok s t) (function-ok₂ T₁t) with dec-language T₂ t
r (function-ok s t) (function-ok₂ T₁t) | Left ¬T₂t with dec-language S₂ s
r (function-ok s t) (function-ok₂ T₁t) | Left ¬T₂t | Left ¬S₂s = function-ok₁ ¬S₂s
r (function-ok s t) (function-ok₂ T₁t) | Left ¬T₂t | Right S₂s = CONTRADICTION (p Q q (s , just t) (Right T₁t) (S₂s , language-comp t ¬T₂t))
r (function-ok s t) (function-ok₂ T₁t) | Right T₂t = function-ok₂ T₂t
r (function-tgt t) (function-tgt T₁t) with dec-language T₂ t
r (function-tgt t) (function-tgt T₁t) | Left ¬T₂t = CONTRADICTION (p Q q (s₂ , just t) (Right T₁t) (S₂s₂ , language-comp t ¬T₂t))
r (function-tgt t) (function-tgt T₁t) | Right T₂t = function-tgt T₂t
-- A counterexample when the argument type is empty.
set-theoretic-counterexample-one : (∀ Q → Q ⊆ Comp((Language never) ⊗ Comp(Lift(Language number))) → Q ⊆ Comp((Language never) ⊗ Comp(Lift(Language string))))
set-theoretic-counterexample-one Q q ((scalar s) , u) Qtu (scalar () , p)
set-theoretic-counterexample-two : (never ⇒ number) ≮: (never ⇒ string)
set-theoretic-counterexample-two = witness (function-tgt (scalar number)) (function-tgt (scalar number)) (function-tgt (scalar-scalar number string (λ ())))
Reference in a new issue View git blame Copy permalink