mirrors/luau
1
0
Fork 0
mirror of https://github.com/luau-lang/luau.git synced 2024-11-15 22:35:43 +08:00
Code Issues Actions Packages Projects Releases Wiki Activity
d37d0c857b
luau/prototyping/Properties/Subtyping.agda
Alan Jeffrey d37d0c857b
Prototype: Renamed any/none to unknown/never (#447) 
* Renamed any/none to unknown/never
* Pin hackage version
* Update Agda version
2022-04-09 00:07:08 -05:00

312 lines
17 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 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; _,_)
open import Luau.Type using (Type; Scalar; strict; nil; number; string; boolean; never; unknown; _⇒_; _∪_; _∩_; tgt)
open import Properties.Contradiction using (CONTRADICTION; ¬)
open import Properties.Equality using (_≢_)
open import Properties.Functions using (_∘_)
open import Properties.Product using (_×_; _,_)
src = Luau.Type.src strict
-- 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 t) = Left (scalar-function-ok nil)
dec-language nil (function-err t) = Right (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 t) = Left (scalar-function-ok boolean)
dec-language boolean (function-err t) = Right (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 t) = Left (scalar-function-ok number)
dec-language number (function-err t) = Right (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 t) = Left (scalar-function-ok string)
dec-language string (function-err t) = Right (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 t) = mapLR function-ok function-ok (dec-language T₂ t)
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)
-- ¬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 t) (scalar-function-ok ()) (function-ok q)
language-comp (function-ok t) (function-ok p) (function-ok q) = language-comp t p q
language-comp (function-err t) (function-err p) (function-err q) = language-comp t q p
-- ≮: 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)
-- 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-≮: 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 = ¬≮:-impl-<: (cond (<:-impl-¬≮: p) (<:-impl-¬≮: q) ∘ ≮:-trans)
-- Properties of scalars
skalar = number ∪ (string ∪ (nil ∪ boolean))
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-ok (scalar s)) unknown (scalar-function-ok 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)
-- 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
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)))
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)))
tgt-never-≮: : ∀ {T U} → (tgt T ≮: U) → (T ≮: (skalar ∪ (never ⇒ U)))
tgt-never-≮: (witness t p q) = witness (function-ok t) (tgt-function-ok p) (skalar-function-ok , function-ok q)
never-tgt-≮: : ∀ {T U} → (T ≮: (skalar ∪ (never ⇒ U))) → (tgt T ≮: U)
never-tgt-≮: (witness (scalar s) p (q₁ , q₂)) = CONTRADICTION (≮:-refl (witness (scalar s) (skalar-scalar s) q₁))
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 ())))
-- 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
src-¬scalar string (scalar string) = never
src-¬scalar nil (scalar nil) = never
src-¬scalar s (left p) = left (src-¬scalar s p)
src-¬scalar s (right p) = right (src-¬scalar s p)
src-¬scalar s (p₁ , p₂) = (src-¬scalar s p₁ , src-¬scalar s p₂)
src-¬scalar s unknown = never
src-unknown-≮: : ∀ {T U} → (T ≮: src U) → (U ≮: (T ⇒ unknown))
src-unknown-≮: (witness t p q) = witness (function-err t) (function-err-src q) (¬function-err-src p)
unknown-src-≮: : ∀ {S T U} → (U ≮: S) → (T ≮: (U ⇒ unknown)) → (U ≮: src T)
unknown-src-≮: (witness t x x₁) (witness (scalar s) p (function-scalar s)) = witness t x (src-¬scalar s p)
unknown-src-≮: r (witness (function-ok (scalar s)) p (function-ok (scalar-scalar s () q)))
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)
-- 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
function-≮:-never : ∀ {T U} → ((T ⇒ U) ≮: never)
function-≮:-never = witness function function never
-- 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)
set-theoretic-if : ∀ {S₁ T₁ S₂ T₂} →
-- This is the "if" part of being a set-theoretic model
(Language (S₁ ⇒ T₁) ⊆ Language (S₂ ⇒ T₂)) →
(∀ Q → Q ⊆ Comp((Language S₁) ⊗ Comp(Language T₁)) → Q ⊆ Comp((Language S₂) ⊗ Comp(Language T₂)))
set-theoretic-if {S₁} {T₁} {S₂} {T₂} p Q q (t , u) Qtu (S₂t , ¬T₂u) = q (t , 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 u) (function-ok T₁u)
¬T₁u T₁u | function-ok T₂u = ¬T₂u T₂u
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 T₁ and ¬Language T₂ t₂ are inhabited
-- in particular it's not true when T₁ is never, or T₂ is unknown.
∀ s₂ t₂ → Language S₂ s₂ → ¬Language T₂ t₂ →
-- This is the "only if" part of being a set-theoretic model
(∀ Q → Q ⊆ Comp((Language S₁) ⊗ Comp(Language T₁)) → Q ⊆ Comp((Language S₂) ⊗ Comp(Language T₂))) →
(Language (S₁ ⇒ T₁) ⊆ Language (S₂ ⇒ T₂))
not-quite-set-theoretic-only-if {S₁} {T₁} {S₂} {T₂} s₂ t₂ S₂s₂ ¬T₂t₂ p = r where
Q : (Tree × Tree) → Set
Q (t , u) = Either (¬Language S₁ t) (Language T₁ u)
q : Q ⊆ Comp((Language S₁) ⊗ Comp(Language T₁))
q (t , u) (Left ¬S₁t) (S₁t , ¬T₁u) = language-comp t ¬S₁t S₁t
q (t , u) (Right T₂u) (S₁t , ¬T₁u) = ¬T₁u T₂u
r : Language (S₁ ⇒ T₁) ⊆ Language (S₂ ⇒ T₂)
r function function = function
r (function-err t) (function-err ¬S₁t) with dec-language S₂ t
r (function-err t) (function-err ¬S₁t) | Left ¬S₂t = function-err ¬S₂t
r (function-err t) (function-err ¬S₁t) | Right S₂t = CONTRADICTION (p Q q (t , t₂) (Left ¬S₁t) (S₂t , language-comp t₂ ¬T₂t₂))
r (function-ok t) (function-ok T₁t) with dec-language T₂ t
r (function-ok t) (function-ok T₁t) | Left ¬T₂t = CONTRADICTION (p Q q (s₂ , t) (Right T₁t) (S₂s₂ , language-comp t ¬T₂t))
r (function-ok t) (function-ok T₁t) | Right T₂t = function-ok T₂t
-- A counterexample when the argument type is empty.
set-theoretic-counterexample-one : (∀ Q → Q ⊆ Comp((Language never) ⊗ Comp(Language number)) → Q ⊆ Comp((Language never) ⊗ Comp(Language string)))
set-theoretic-counterexample-one Q q ((scalar s) , u) Qtu (scalar () , p)
set-theoretic-counterexample-one Q q ((function-err t) , u) Qtu (scalar-function-err () , p)
set-theoretic-counterexample-two : (never ⇒ number) ≮: (never ⇒ string)
set-theoretic-counterexample-two = witness
(function-ok (scalar number)) (function-ok (scalar number))
(function-ok (scalar-scalar number string (λ ())))
-- At some point we may deal with overloaded function resolution, which should fix this problem...
-- The reason why this is connected to overloaded functions is that currently we have that the type of
-- f(x) is (tgt T) where f:T. Really we should have the type depend on the type of x, that is use (tgt T U),
-- where U is the type of x. In particular (tgt (S => T) (U & V)) should be the same as (tgt ((S&U) => T) V)
-- and tgt(never => T) should be unknown. For example
--
-- tgt((number => string) & (string => bool))(number)
-- is tgt(number => string)(number) & tgt(string => bool)(number)
-- is tgt(number => string)(number) & tgt(string => bool)(number&unknown)
-- is tgt(number => string)(number) & tgt(string&number => bool)(unknown)
-- is tgt(number => string)(number) & tgt(never => bool)(unknown)
-- is string & unknown
-- is string
--
-- there's some discussion of this in the Gentle Introduction paper.
Reference in New Issue View Git Blame Copy Permalink