luau/prototyping/Properties/FunctionTypes.agda
ajeffrey@roblox.com 3b55f94066 WIP
2022-05-25 15:58:14 -05:00

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{-# OPTIONS --rewriting #-}
module Properties.FunctionTypes where
open import FFI.Data.Either using (Either; Left; Right; mapLR; swapLR; cond)
open import Luau.FunctionTypes using (srcⁿ; src; tgt)
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.Functions using (_∘_)
open import Properties.Subtyping using (<:-refl; ≮:-refl; <:-trans-≮:; skalar-scalar; <:-impl-⊇; skalar-function-ok; language-comp)
open import Properties.TypeNormalization using (FunType; Normal; never; unknown; _∩_; __; _⇒_; normal; <:-normalize; normalize-<:)
-- Properties of src
function-err-srcⁿ : {T t} (FunType T) (¬Language (srcⁿ T) t) Language T (function-err t)
function-err-srcⁿ (S T) p = function-err p
function-err-srcⁿ (S T) (p₁ , p₂) = (function-err-srcⁿ S p₁ , function-err-srcⁿ T p₂)
¬function-err-srcᶠ : {T t} (FunType T) (Language (srcⁿ T) t) ¬Language T (function-err t)
¬function-err-srcᶠ (S T) p = function-err p
¬function-err-srcᶠ (S T) (left p) = left (¬function-err-srcᶠ S p)
¬function-err-srcᶠ (S T) (right p) = right (¬function-err-srcᶠ T p)
¬function-err-srcⁿ : {T t} (Normal T) (Language (srcⁿ T) t) ¬Language T (function-err t)
¬function-err-srcⁿ never p = never
¬function-err-srcⁿ unknown (scalar ())
¬function-err-srcⁿ (S T) p = function-err p
¬function-err-srcⁿ (S T) (left p) = left (¬function-err-srcᶠ S p)
¬function-err-srcⁿ (S T) (right p) = right (¬function-err-srcᶠ T p)
¬function-err-srcⁿ (S T) (scalar ())
¬function-err-src : {T t} (Language (src T) t) ¬Language T (function-err t)
¬function-err-src {T = S T} p = function-err p
¬function-err-src {T = nil} p = scalar-function-err nil
¬function-err-src {T = never} p = never
¬function-err-src {T = unknown} (scalar ())
¬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 = S T} p = <:-impl-⊇ (<:-normalize (S T)) _ (¬function-err-srcⁿ (normal (S T)) p)
¬function-err-src {T = S T} p = <:-impl-⊇ (<:-normalize (S T)) _ (¬function-err-srcⁿ (normal (S T)) p)
src-¬function-errᶠ : {T t} (FunType T) Language T (function-err t) (¬Language (srcⁿ T) t)
src-¬function-errᶠ (S T) (function-err p) = p
src-¬function-errᶠ (S T) (p₁ , p₂) = (src-¬function-errᶠ S p₁ , src-¬function-errᶠ T p₂)
src-¬function-errⁿ : {T t} (Normal T) Language T (function-err t) (¬Language (srcⁿ T) t)
src-¬function-errⁿ unknown p = never
src-¬function-errⁿ (S T) (function-err p) = p
src-¬function-errⁿ (S T) (p₁ , p₂) = (src-¬function-errᶠ S p₁ , src-¬function-errᶠ T p₂)
src-¬function-errⁿ (S T) p = never
src-¬function-err : {T t} Language T (function-err t) (¬Language (src T) t)
src-¬function-err {T = S T} (function-err p) = p
src-¬function-err {T = unknown} p = never
src-¬function-err {T = S T} p = src-¬function-errⁿ (normal (S T)) (<:-normalize (S T) _ p)
src-¬function-err {T = S T} p = src-¬function-errⁿ (normal (S T)) (<:-normalize (S T) _ p)
fun-¬scalar : {S T} (s : Scalar S) FunType T ¬Language T (scalar s)
fun-¬scalar s (S T) = function-scalar s
fun-¬scalar s (S T) = left (fun-¬scalar s S)
¬fun-scalar : {S T t} (s : Scalar S) FunType T Language T t ¬Language S t
¬fun-scalar s (S T) function = scalar-function s
¬fun-scalar s (S T) (function-ok₁ p) = scalar-function-ok s
¬fun-scalar s (S T) (function-ok₂ p) = scalar-function-ok s
¬fun-scalar s (S T) (function-err p) = scalar-function-err s
¬fun-scalar s (S T) (function-tgt p) = scalar-function-tgt s
¬fun-scalar s (S T) (p₁ , p₂) = ¬fun-scalar s T p₂
fun-function : {T} FunType T Language T function
fun-function (S T) = function
fun-function (S T) = (fun-function S , fun-function T)
srcⁿ-¬scalar : {S T t} (s : Scalar S) Normal T Language T (scalar s) (¬Language (srcⁿ T) t)
srcⁿ-¬scalar s never (scalar ())
srcⁿ-¬scalar s unknown p = never
srcⁿ-¬scalar s (S T) (scalar ())
srcⁿ-¬scalar s (S T) (p₁ , p₂) = CONTRADICTION (language-comp (scalar s) (fun-¬scalar s S) p₁)
srcⁿ-¬scalar s (S T) p = never
src-¬scalar : {S T t} (s : Scalar S) Language T (scalar s) (¬Language (src T) t)
src-¬scalar {T = nil} s p = never
src-¬scalar {T = T U} s (scalar ())
src-¬scalar {T = never} s (scalar ())
src-¬scalar {T = unknown} s p = never
src-¬scalar {T = boolean} s p = never
src-¬scalar {T = number} s p = never
src-¬scalar {T = string} s p = never
src-¬scalar {T = T U} s p = srcⁿ-¬scalar s (normal (T U)) (<:-normalize (T U) (scalar s) p)
src-¬scalar {T = T U} s p = srcⁿ-¬scalar s (normal (T U)) (<:-normalize (T U) (scalar s) p)
srcⁿ-unknown-≮: : {T U} (Normal U) (T ≮: srcⁿ U) (U ≮: (T unknown))
srcⁿ-unknown-≮: never (witness t p q) = CONTRADICTION (language-comp t q unknown)
srcⁿ-unknown-≮: unknown (witness t p q) = witness (function-err t) unknown (function-err p)
srcⁿ-unknown-≮: (U V) (witness t p q) = witness (function-err t) (function-err q) (function-err p)
srcⁿ-unknown-≮: (U V) (witness t p q) = witness (function-err t) (function-err-srcⁿ (U V) q) (function-err p)
srcⁿ-unknown-≮: (U V) (witness t p q) = witness (scalar V) (right (scalar V)) (function-scalar V)
src-unknown-≮: : {T U} (T ≮: src U) (U ≮: (T unknown))
src-unknown-≮: {U = nil} (witness t p q) = witness (scalar nil) (scalar nil) (function-scalar nil)
src-unknown-≮: {U = T U} (witness t p q) = witness (function-err t) (function-err q) (function-err p)
src-unknown-≮: {U = never} (witness t p q) = CONTRADICTION (language-comp t q unknown)
src-unknown-≮: {U = unknown} (witness t p q) = witness (function-err t) unknown (function-err p)
src-unknown-≮: {U = boolean} (witness t p q) = witness (scalar boolean) (scalar boolean) (function-scalar boolean)
src-unknown-≮: {U = number} (witness t p q) = witness (scalar number) (scalar number) (function-scalar number)
src-unknown-≮: {U = string} (witness t p q) = witness (scalar string) (scalar string) (function-scalar string)
src-unknown-≮: {U = T U} p = <:-trans-≮: (normalize-<: (T U)) (srcⁿ-unknown-≮: (normal (T U)) p)
src-unknown-≮: {U = T U} p = <:-trans-≮: (normalize-<: (T U)) (srcⁿ-unknown-≮: (normal (T U)) 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 s .(scalar s₁)) p (function-ok x (scalar-scalar s₁ () x₂)))
unknown-src-≮: r (witness (function-ok s .function) p (function-ok x (scalar-function ())))
unknown-src-≮: r (witness (function-ok s .(function-ok _ _)) p (function-ok x (scalar-function-ok ())))
unknown-src-≮: r (witness (function-ok s .(function-err _)) p (function-ok x (scalar-function-err ())))
unknown-src-≮: r (witness (function-err t) p (function-err q)) = witness t q (src-¬function-err p)
unknown-src-≮: r (witness (function-tgt t) p (function-tgt (scalar-function-tgt ())))