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Prototyping: add semantic subtyping (#424) 

Adds subtyping to strict mode.
This commit is contained in:
Alan Jeffrey 2022-03-23 15:02:57 -05:00 committed by GitHub
parent 362428f8b4
commit 7721955ba5
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14 changed files with 623 additions and 414 deletions
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4
prototyping/FFI/Data/Aeson.agda
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@ -10,7 +10,7 @@ open import Agda.Builtin.String using (String)
open import FFI.Data.ByteString using (ByteString)
open import FFI.Data.HaskellString using (HaskellString; pack)
open import FFI.Data.Maybe using (Maybe; just; nothing)
open import FFI.Data.Either using (Either; mapLeft)
open import FFI.Data.Either using (Either; mapL)
open import FFI.Data.Scientific using (Scientific)
open import FFI.Data.Vector using (Vector)
@ -73,5 +73,5 @@ postulate
{-# COMPILE GHC eitherHDecode = Data.Aeson.eitherDecodeStrict #-}
eitherDecode : ByteString → Either String Value
eitherDecode bytes = mapLeft pack (eitherHDecode bytes)
eitherDecode bytes = mapL pack (eitherHDecode bytes)

24
prototyping/FFI/Data/Either.agda
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@ -7,10 +7,22 @@ data Either (A B : Set) : Set where
Right : B → Either A B
{-# COMPILE GHC Either = data Data.Either.Either (Data.Either.Left|Data.Either.Right) #-}
mapLeft : ∀ {A B C} → (A → B) → (Either A C) → (Either B C)
mapLeft f (Left x) = Left (f x)
mapLeft f (Right x) = Right x
swapLR : ∀ {A B} → Either A B → Either B A
swapLR (Left x) = Right x
swapLR (Right x) = Left x
mapRight : ∀ {A B C} → (B → C) → (Either A B) → (Either A C)
mapRight f (Left x) = Left x
mapRight f (Right x) = Right (f x)
mapL : ∀ {A B C} → (A → B) → Either A C → Either B C
mapL f (Left x) = Left (f x)
mapL f (Right x) = Right x
mapR : ∀ {A B C} → (B → C) → Either A B → Either A C
mapR f (Left x) = Left x
mapR f (Right x) = Right (f x)
mapLR : ∀ {A B C D} → (A → B) → (C → D) → Either A C → Either B D
mapLR f g (Left x) = Left (f x)
mapLR f g (Right x) = Right (g x)
cond : ∀ {A B C : Set} → (A → C) → (B → C) → (Either A B) → C
cond f g (Left x) = f x
cond f g (Right x) = g x

2
prototyping/Luau/RuntimeError.agda
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@ -25,7 +25,7 @@ data RuntimeErrorᴮ {a} (H : Heap a) : Block a → Set
data RuntimeErrorᴱ {a} (H : Heap a) : Expr a → Set
data RuntimeErrorᴱ H where
FunctionMismatch : ∀ v w → (function ≢ valueType v) → RuntimeErrorᴱ H (val v $ val w)
FunctionMismatch : ∀ v w → (valueType v ≢ function) → RuntimeErrorᴱ H (val v $ val w)
BinOpMismatch₁ : ∀ v w {op} → (BinOpError op (valueType v)) → RuntimeErrorᴱ H (binexp (val v) op (val w))
BinOpMismatch₂ : ∀ v w {op} → (BinOpError op (valueType w)) → RuntimeErrorᴱ H (binexp (val v) op (val w))
UnboundVariable : ∀ {x} → RuntimeErrorᴱ H (var x)

34
prototyping/Luau/StrictMode.agda
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@ -5,28 +5,18 @@ module Luau.StrictMode where
open import Agda.Builtin.Equality using (_≡_)
open import FFI.Data.Maybe using (just; nothing)
open import Luau.Syntax using (Expr; Stat; Block; BinaryOperator; yes; nil; addr; var; binexp; var_∈_; _⟨_⟩∈_; function_is_end; _$_; block_is_end; local_←_; _∙_; done; return; name; +; -; *; /; <; >; <=; >=; ··)
open import Luau.Type using (Type; strict; nil; number; string; _⇒_; tgt)
open import Luau.Type using (Type; strict; nil; number; string; boolean; none; any; _⇒_; _∪_; _∩_; tgt)
open import Luau.Subtyping using (_≮:_)
open import Luau.Heap using (Heap; function_is_end) renaming (_[_] to _[_]ᴴ)
open import Luau.VarCtxt using (VarCtxt; ∅; _⋒_; _↦_; _⊕_↦_; _⊝_) renaming (_[_] to _[_]ⱽ)
open import Luau.TypeCheck(strict) using (_⊢ᴮ_∈_; _⊢ᴱ_∈_; ⊢ᴴ_; ⊢ᴼ_; _⊢ᴴᴱ_▷_∈_; _⊢ᴴᴮ_▷_∈_; var; addr; app; binexp; block; return; local; function)
open import Properties.Equality using (_≢_)
open import Luau.TypeCheck(strict) using (_⊢ᴮ_∈_; _⊢ᴱ_∈_; ⊢ᴴ_; ⊢ᴼ_; _⊢ᴴᴱ_▷_∈_; _⊢ᴴᴮ_▷_∈_; var; addr; app; binexp; block; return; local; function; srcBinOp)
open import Properties.Contradiction using (¬)
open import Properties.TypeCheck(strict) using (typeCheckᴮ)
open import Properties.Product using (_,_)
src : Type → Type
src = Luau.Type.src strict
data BinOpWarning : BinaryOperator → Type → Set where
+ : ∀ {T} → (T ≢ number) → BinOpWarning + T
- : ∀ {T} → (T ≢ number) → BinOpWarning - T
* : ∀ {T} → (T ≢ number) → BinOpWarning * T
/ : ∀ {T} → (T ≢ number) → BinOpWarning / T
< : ∀ {T} → (T ≢ number) → BinOpWarning < T
> : ∀ {T} → (T ≢ number) → BinOpWarning > T
<= : ∀ {T} → (T ≢ number) → BinOpWarning <= T
>= : ∀ {T} → (T ≢ number) → BinOpWarning >= T
·· : ∀ {T} → (T ≢ string) → BinOpWarning ·· T
data Warningᴱ (H : Heap yes) {Γ} : ∀ {M T} → (Γ ⊢ᴱ M ∈ T) → Set
data Warningᴮ (H : Heap yes) {Γ} : ∀ {B T} → (Γ ⊢ᴮ B ∈ T) → Set
@ -46,7 +36,7 @@ data Warningᴱ H {Γ} where
FunctionCallMismatch : ∀ {M N T U} {D₁ : Γ ⊢ᴱ M ∈ T} {D₂ : Γ ⊢ᴱ N ∈ U} →
(src T ≢ U) →
(U ≮: src T) →
-----------------
Warningᴱ H (app D₁ D₂)
@ -64,13 +54,13 @@ data Warningᴱ H {Γ} where
BinOpMismatch₁ : ∀ {op M N T U} {D₁ : Γ ⊢ᴱ M ∈ T} {D₂ : Γ ⊢ᴱ N ∈ U} →
BinOpWarning op T →
(T ≮: srcBinOp op) →
------------------------------
Warningᴱ H (binexp {op} D₁ D₂)
BinOpMismatch₂ : ∀ {op M N T U} {D₁ : Γ ⊢ᴱ M ∈ T} {D₂ : Γ ⊢ᴱ N ∈ U} →
BinOpWarning op U →
(U ≮: srcBinOp op) →
------------------------------
Warningᴱ H (binexp {op} D₁ D₂)
@ -88,7 +78,7 @@ data Warningᴱ H {Γ} where
FunctionDefnMismatch : ∀ {f x B T U V} {D : (Γ ⊕ x ↦ T) ⊢ᴮ B ∈ V} →
(U ≢ V) →
(V ≮: U) →
-------------------------
Warningᴱ H (function {f} {U = U} D)
@ -100,7 +90,7 @@ data Warningᴱ H {Γ} where
BlockMismatch : ∀ {b B T U} {D : Γ ⊢ᴮ B ∈ U} →
(T ≢ U) →
(U ≮: T) →
------------------------------
Warningᴱ H (block {b} {T = T} D)
@ -120,7 +110,7 @@ data Warningᴮ H {Γ} where
LocalVarMismatch : ∀ {x M B T U V} {D₁ : Γ ⊢ᴱ M ∈ U} {D₂ : (Γ ⊕ x ↦ T) ⊢ᴮ B ∈ V} →
(T ≢ U) →
(U ≮: T) →
--------------------
Warningᴮ H (local D₁ D₂)
@ -138,7 +128,7 @@ data Warningᴮ H {Γ} where
FunctionDefnMismatch : ∀ {f x B C T U V W} {D₁ : (Γ ⊕ x ↦ T) ⊢ᴮ C ∈ V} {D₂ : (Γ ⊕ f ↦ (T ⇒ U)) ⊢ᴮ B ∈ W} →
(U ≢ V) →
(V ≮: U) →
-------------------------------------
Warningᴮ H (function D₁ D₂)
@ -158,7 +148,7 @@ data Warningᴼ (H : Heap yes) : ∀ {V} → (⊢ᴼ V) → Set where
FunctionDefnMismatch : ∀ {f x B T U V} {D : (x ↦ T) ⊢ᴮ B ∈ V} →
(U ≢ V) →
(V ≮: U) →
---------------------------------
Warningᴼ H (function {f} {U = U} D)

39
prototyping/Luau/StrictMode/ToString.agda
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@ -2,37 +2,58 @@
module Luau.StrictMode.ToString where
open import Agda.Builtin.Nat using (Nat; suc)
open import FFI.Data.String using (String; _++_)
open import Luau.Subtyping using (_≮:_; Tree; witness; scalar; function; function-ok; function-err)
open import Luau.StrictMode using (Warningᴱ; Warningᴮ; UnallocatedAddress; UnboundVariable; FunctionCallMismatch; FunctionDefnMismatch; BlockMismatch; app₁; app₂; BinOpMismatch₁; BinOpMismatch₂; bin₁; bin₂; block₁; return; LocalVarMismatch; local₁; local₂; function₁; function₂; heap; expr; block; addr)
open import Luau.Syntax using (Expr; val; yes; var; var_∈_; _⟨_⟩∈_; _$_; addr; number; binexp; nil; function_is_end; block_is_end; done; return; local_←_; _∙_; fun; arg; name)
open import Luau.Type using (strict)
open import Luau.Type using (strict; number; boolean; string; nil)
open import Luau.TypeCheck(strict) using (_⊢ᴮ_∈_; _⊢ᴱ_∈_)
open import Luau.Addr.ToString using (addrToString)
open import Luau.Var.ToString using (varToString)
open import Luau.Type.ToString using (typeToString)
open import Luau.Syntax.ToString using (binOpToString)
tmp : Nat → String
tmp 0 = "w"
tmp 1 = "x"
tmp 2 = "y"
tmp 3 = "z"
tmp (suc (suc (suc n))) = tmp n ++ "'"
treeToString : Tree → Nat → String → String
treeToString (scalar number) n v = v ++ " is a number"
treeToString (scalar boolean) n v = v ++ " is a boolean"
treeToString (scalar string) n v = v ++ " is a string"
treeToString (scalar nil) n v = v ++ " is nil"
treeToString function n v = v ++ " is a function"
treeToString (function-ok t) n v = treeToString t n (v ++ "()")
treeToString (function-err t) n v = v ++ "(" ++ w ++ ") can error when\n " ++ treeToString t (suc n) w where w = tmp n
subtypeWarningToString : ∀ {T U} → (T ≮: U) → String
subtypeWarningToString (witness t p q) = "\n because provided type contains v, where " ++ treeToString t 0 "v"
warningToStringᴱ : ∀ {H Γ T} M → {D : Γ ⊢ᴱ M ∈ T} → Warningᴱ H D → String
warningToStringᴮ : ∀ {H Γ T} B → {D : Γ ⊢ᴮ B ∈ T} → Warningᴮ H D → String
warningToStringᴱ (var x) (UnboundVariable p) = "Unbound variable " ++ varToString x
warningToStringᴱ (val (addr a)) (UnallocatedAddress p) = "Unallocated adress " ++ addrToString a
warningToStringᴱ (M $ N) (FunctionCallMismatch {T = T} {U = U} p) = "Function has type " ++ typeToString T ++ " but argument has type " ++ typeToString U
warningToStringᴱ (val (addr a)) (UnallocatedAddress p) = "Unallocated address " ++ addrToString a
warningToStringᴱ (M $ N) (FunctionCallMismatch {T = T} {U = U} p) = "Function has type " ++ typeToString T ++ " but argument has type " ++ typeToString U ++ subtypeWarningToString p
warningToStringᴱ (M $ N) (app₁ W) = warningToStringᴱ M W
warningToStringᴱ (M $ N) (app₂ W) = warningToStringᴱ N W
warningToStringᴱ (function f ⟨ var x ∈ T ⟩∈ U is B end) (FunctionDefnMismatch {V = V} p) = "Function expresion " ++ varToString f ++ " has return type " ++ typeToString U ++ " but body returns " ++ typeToString V
warningToStringᴱ (function f ⟨ var x ∈ T ⟩∈ U is B end) (FunctionDefnMismatch {V = V} p) = "Function expresion " ++ varToString f ++ " has return type " ++ typeToString U ++ " but body returns " ++ typeToString V ++ subtypeWarningToString p
warningToStringᴱ (function f ⟨ var x ∈ T ⟩∈ U is B end) (function₁ W) = warningToStringᴮ B W ++ "\n in function expression " ++ varToString f
warningToStringᴱ block var b ∈ T is B end (BlockMismatch {U = U} p) = "Block " ++ varToString b ++ " has type " ++ typeToString T ++ " but body returns " ++ typeToString U
warningToStringᴱ block var b ∈ T is B end (BlockMismatch {U = U} p) = "Block " ++ varToString b ++ " has type " ++ typeToString T ++ " but body returns " ++ typeToString U ++ subtypeWarningToString p
warningToStringᴱ block var b ∈ T is B end (block₁ W) = warningToStringᴮ B W ++ "\n in block " ++ varToString b
warningToStringᴱ (binexp M op N) (BinOpMismatch₁ {T = T} p) = "Binary operator " ++ binOpToString op ++ " lhs has type " ++ typeToString T
warningToStringᴱ (binexp M op N) (BinOpMismatch₂ {U = U} p) = "Binary operator " ++ binOpToString op ++ " rhs has type " ++ typeToString U
warningToStringᴱ (binexp M op N) (BinOpMismatch₁ {T = T} p) = "Binary operator " ++ binOpToString op ++ " lhs has type " ++ typeToString T ++ subtypeWarningToString p
warningToStringᴱ (binexp M op N) (BinOpMismatch₂ {U = U} p) = "Binary operator " ++ binOpToString op ++ " rhs has type " ++ typeToString U ++ subtypeWarningToString p
warningToStringᴱ (binexp M op N) (bin₁ W) = warningToStringᴱ M W
warningToStringᴱ (binexp M op N) (bin₂ W) = warningToStringᴱ N W
warningToStringᴮ (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) (FunctionDefnMismatch {V = V} p) = "Function declaration " ++ varToString f ++ " has return type " ++ typeToString U ++ " but body returns " ++ typeToString V
warningToStringᴮ (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) (FunctionDefnMismatch {V = V} p) = "Function declaration " ++ varToString f ++ " has return type " ++ typeToString U ++ " but body returns " ++ typeToString V ++ subtypeWarningToString p
warningToStringᴮ (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) (function₁ W) = warningToStringᴮ C W ++ "\n in function declaration " ++ varToString f
warningToStringᴮ (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) (function₂ W) = warningToStringᴮ B W
warningToStringᴮ (local var x ∈ T ← M ∙ B) (LocalVarMismatch {U = U} p) = "Local variable " ++ varToString x ++ " has type " ++ typeToString T ++ " but expression has type " ++ typeToString U
warningToStringᴮ (local var x ∈ T ← M ∙ B) (LocalVarMismatch {U = U} p) = "Local variable " ++ varToString x ++ " has type " ++ typeToString T ++ " but expression has type " ++ typeToString U ++ subtypeWarningToString p
warningToStringᴮ (local var x ∈ T ← M ∙ B) (local₁ W) = warningToStringᴱ M W ++ "\n in local variable declaration " ++ varToString x
warningToStringᴮ (local var x ∈ T ← M ∙ B) (local₂ W) = warningToStringᴮ B W
warningToStringᴮ (return M ∙ B) (return W) = warningToStringᴱ M W ++ "\n in return statement"

62
prototyping/Luau/Subtyping.agda Normal file
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@ -0,0 +1,62 @@
{-# OPTIONS --rewriting #-}
open import Luau.Type using (Type; Scalar; nil; number; string; boolean; none; any; _⇒_; _∪_; _∩_)
open import Properties.Equality using (_≢_)
module Luau.Subtyping where
-- An implementation of semantic subtyping
-- We think of types as languages of trees
data Tree : Set where
scalar : ∀ {T} → Scalar T → Tree
function : Tree
function-ok : Tree → Tree
function-err : Tree → Tree
data Language : Type → Tree → Set
data ¬Language : Type → Tree → Set
data Language where
scalar : ∀ {T} → (s : Scalar T) → Language T (scalar s)
function : ∀ {T U} → Language (T ⇒ U) function
function-ok : ∀ {T U u} → (Language U u) → Language (T ⇒ U) (function-ok u)
function-err : ∀ {T U t} → (¬Language T t) → Language (T ⇒ U) (function-err t)
scalar-function-err : ∀ {S t} → (Scalar S) → Language S (function-err t)
left : ∀ {T U t} → Language T t → Language (T ∪ U) t
right : ∀ {T U u} → Language U u → Language (T ∪ U) u
_,_ : ∀ {T U t} → Language T t → Language U t → Language (T ∩ U) t
any : ∀ {t} → Language any t
data ¬Language where
scalar-scalar : ∀ {S T} → (s : Scalar S) → (Scalar T) → (S ≢ T) → ¬Language T (scalar s)
scalar-function : ∀ {S} → (Scalar S) → ¬Language S function
scalar-function-ok : ∀ {S u} → (Scalar S) → ¬Language S (function-ok u)
function-scalar : ∀ {S T U} (s : Scalar S) → ¬Language (T ⇒ U) (scalar s)
function-ok : ∀ {T U u} → (¬Language U u) → ¬Language (T ⇒ U) (function-ok u)
function-err : ∀ {T U t} → (Language T t) → ¬Language (T ⇒ U) (function-err t)
_,_ : ∀ {T U t} → ¬Language T t → ¬Language U t → ¬Language (T ∪ U) t
left : ∀ {T U t} → ¬Language T t → ¬Language (T ∩ U) t
right : ∀ {T U u} → ¬Language U u → ¬Language (T ∩ U) u
none : ∀ {t} → ¬Language none t
-- Subtyping as language inclusion
_<:_ : Type → Type → Set
(T <: U) = ∀ t → (Language T t) → (Language U t)
-- For warnings, we are interested in failures of subtyping,
-- which is whrn there is a tree in T's language that isn't in U's.
data _≮:_ (T U : Type) : Set where
witness : ∀ t →
Language T t →
¬Language U t →
-----------------
T ≮: U

7
prototyping/Luau/Type.agda
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@ -17,6 +17,13 @@ data Type : Set where
_∪_ : Type → Type → Type
_∩_ : Type → Type → Type
data Scalar : Type → Set where
number : Scalar number
boolean : Scalar boolean
string : Scalar string
nil : Scalar nil
lhs : Type → Type
lhs (T ⇒ _) = T
lhs (T ∪ _) = T

23
prototyping/Luau/TypeCheck.agda
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@ -10,7 +10,7 @@ open import Luau.Syntax using (Expr; Stat; Block; BinaryOperator; yes; nil; addr
open import Luau.Var using (Var)
open import Luau.Addr using (Addr)
open import Luau.Heap using (Heap; Object; function_is_end) renaming (_[_] to _[_]ᴴ)
open import Luau.Type using (Type; Mode; nil; none; number; boolean; string; _⇒_; tgt)
open import Luau.Type using (Type; Mode; nil; any; number; boolean; string; _⇒_; tgt)
open import Luau.VarCtxt using (VarCtxt; ∅; _⋒_; _↦_; _⊕_↦_; _⊝_) renaming (_[_] to _[_]ⱽ)
open import FFI.Data.Vector using (Vector)
open import FFI.Data.Maybe using (Maybe; just; nothing)
@ -19,9 +19,22 @@ open import Properties.Product using (_×_; _,_)
src : Type → Type
src = Luau.Type.src m
orNone : Maybe Type → Type
orNone nothing = none
orNone (just T) = T
orAny : Maybe Type → Type
orAny nothing = any
orAny (just T) = T
srcBinOp : BinaryOperator → Type
srcBinOp + = number
srcBinOp - = number
srcBinOp * = number
srcBinOp / = number
srcBinOp < = number
srcBinOp > = number
srcBinOp == = any
srcBinOp ~= = any
srcBinOp <= = number
srcBinOp >= = number
srcBinOp ·· = string
tgtBinOp : BinaryOperator → Type
tgtBinOp + = number
@ -76,7 +89,7 @@ data _⊢ᴱ_∈_ where
var : ∀ {x T Γ} →
T ≡ orNone(Γ [ x ]ⱽ) →
T ≡ orAny(Γ [ x ]ⱽ) →
----------------
Γ ⊢ᴱ (var x) ∈ T

2
prototyping/Properties.agda
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@ -5,7 +5,9 @@ module Properties where
import Properties.Contradiction
import Properties.Dec
import Properties.Equality
import Properties.Functions
import Properties.Remember
import Properties.Step
import Properties.StrictMode
import Properties.Subtyping
import Properties.TypeCheck

6
prototyping/Properties/Functions.agda Normal file
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@ -0,0 +1,6 @@
module Properties.Functions where
infixr 5 _∘_
_∘_ : ∀ {A B C : Set} → (B → C) → (A → B) → (A → C)
(f ∘ g) x = f (g x)

594
prototyping/Properties/StrictMode.agda
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@ -4,13 +4,15 @@ module Properties.StrictMode where
import Agda.Builtin.Equality.Rewrite
open import Agda.Builtin.Equality using (_≡_; refl)
open import FFI.Data.Either using (Either; Left; Right; mapL; mapR; mapLR; swapLR; cond)
open import FFI.Data.Maybe using (Maybe; just; nothing)
open import Luau.Heap using (Heap; Object; function_is_end; defn; alloc; ok; next; lookup-not-allocated) renaming (_≡_⊕_↦_ to _≡ᴴ_⊕_↦_; _[_] to _[_]ᴴ; ∅ to ∅ᴴ)
open import Luau.StrictMode using (Warningᴱ; Warningᴮ; Warningᴼ; Warningᴴᴱ; Warningᴴᴮ; UnallocatedAddress; UnboundVariable; FunctionCallMismatch; app₁; app₂; BinOpWarning; BinOpMismatch₁; BinOpMismatch₂; bin₁; bin₂; BlockMismatch; block₁; return; LocalVarMismatch; local₁; local₂; FunctionDefnMismatch; function₁; function₂; heap; expr; block; addr; +; -; *; /; <; >; <=; >=; ··)
open import Luau.StrictMode using (Warningᴱ; Warningᴮ; Warningᴼ; Warningᴴ; UnallocatedAddress; UnboundVariable; FunctionCallMismatch; app₁; app₂; BinOpMismatch₁; BinOpMismatch₂; bin₁; bin₂; BlockMismatch; block₁; return; LocalVarMismatch; local₁; local₂; FunctionDefnMismatch; function₁; function₂; heap; expr; block; addr)
open import Luau.Substitution using (_[_/_]ᴮ; _[_/_]ᴱ; _[_/_]ᴮunless_; var_[_/_]ᴱwhenever_)
open import Luau.Subtyping using (_≮:_; witness; any; none; scalar; function; scalar-function; scalar-function-ok; scalar-function-err; scalar-scalar; function-scalar; function-ok; function-err; left; right; _,_; Tree; Language; ¬Language)
open import Luau.Syntax using (Expr; yes; var; val; var_∈_; _⟨_⟩∈_; _$_; addr; number; bool; string; binexp; nil; function_is_end; block_is_end; done; return; local_←_; _∙_; fun; arg; name; ==; ~=)
open import Luau.Type using (Type; strict; nil; _⇒_; none; tgt; _≡ᵀ_; _≡ᴹᵀ_)
open import Luau.TypeCheck(strict) using (_⊢ᴮ_∈_; _⊢ᴱ_∈_; _⊢ᴴᴮ_▷_∈_; _⊢ᴴᴱ_▷_∈_; nil; var; addr; app; function; block; done; return; local; orNone; tgtBinOp)
open import Luau.Type using (Type; strict; nil; number; boolean; string; _⇒_; none; any; _∩_; _∪_; tgt; _≡ᵀ_; _≡ᴹᵀ_)
open import Luau.TypeCheck(strict) using (_⊢ᴮ_∈_; _⊢ᴱ_∈_; _⊢ᴴᴮ_▷_∈_; _⊢ᴴᴱ_▷_∈_; nil; var; addr; app; function; block; done; return; local; orAny; srcBinOp; tgtBinOp)
open import Luau.Var using (_≡ⱽ_)
open import Luau.Addr using (_≡ᴬ_)
open import Luau.VarCtxt using (VarCtxt; ∅; _⋒_; _↦_; _⊕_↦_; _⊝_; ⊕-lookup-miss; ⊕-swap; ⊕-over) renaming (_[_] to _[_]ⱽ)
@ -18,17 +20,19 @@ open import Luau.VarCtxt using (VarCtxt; ∅)
open import Properties.Remember using (remember; _,_)
open import Properties.Equality using (_≢_; sym; cong; trans; subst₁)
open import Properties.Dec using (Dec; yes; no)
open import Properties.Contradiction using (CONTRADICTION)
open import Properties.TypeCheck(strict) using (typeOfᴼ; typeOfᴹᴼ; typeOfⱽ; typeOfᴱ; typeOfᴮ; typeCheckᴱ; typeCheckᴮ; typeCheckᴼ; typeCheckᴴᴱ; typeCheckᴴᴮ; mustBeFunction; mustBeNumber; mustBeString)
open import Properties.Contradiction using (CONTRADICTION; ¬)
open import Properties.Functions using (_∘_)
open import Properties.Subtyping using (any-≮:; ≡-trans-≮:; ≮:-trans-≡; none-tgt-≮:; tgt-none-≮:; src-any-≮:; any-src-≮:; ≮:-trans; ≮:-refl; scalar-≢-impl-≮:; function-≮:-scalar; scalar-≮:-function; function-≮:-none; any-≮:-scalar; scalar-≮:-none; any-≮:-none)
open import Properties.TypeCheck(strict) using (typeOfᴼ; typeOfᴹᴼ; typeOfⱽ; typeOfᴱ; typeOfᴮ; typeCheckᴱ; typeCheckᴮ; typeCheckᴼ; typeCheckᴴ)
open import Luau.OpSem using (_⟦_⟧_⟶_; _⊢_⟶*_⊣_; _⊢_⟶ᴮ_⊣_; _⊢_⟶ᴱ_⊣_; app₁; app₂; function; beta; return; block; done; local; subst; binOp₀; binOp₁; binOp₂; refl; step; +; -; *; /; <; >; ==; ~=; <=; >=; ··)
open import Luau.RuntimeError using (BinOpError; RuntimeErrorᴱ; RuntimeErrorᴮ; FunctionMismatch; BinOpMismatch₁; BinOpMismatch₂; UnboundVariable; SEGV; app₁; app₂; bin₁; bin₂; block; local; return; +; -; *; /; <; >; <=; >=; ··)
open import Luau.RuntimeType using (valueType)
open import Luau.RuntimeType using (RuntimeType; valueType; number; string; boolean; nil; function)
src = Luau.Type.src strict
data _⊑_ (H : Heap yes) : Heap yes → Set where
refl : (H ⊑ H)
snoc : ∀ {H′ a V} → (H′ ≡ᴴ H ⊕ a ↦ V) → (H ⊑ H′)
snoc : ∀ {H′ a O} → (H′ ≡ᴴ H ⊕ a ↦ O) → (H ⊑ H′)
rednᴱ⊑ : ∀ {H H′ M M′} → (H ⊢ M ⟶ᴱ M′ ⊣ H′) → (H ⊑ H′)
rednᴮ⊑ : ∀ {H H′ B B′} → (H ⊢ B ⟶ᴮ B′ ⊣ H′) → (H ⊑ H′)
@ -59,90 +63,51 @@ lookup-⊑-nothing {H} a (snoc defn) p with a ≡ᴬ next H
lookup-⊑-nothing {H} a (snoc defn) p | yes refl = refl
lookup-⊑-nothing {H} a (snoc o) p | no q = trans (lookup-not-allocated o q) p
data OrWarningᴱ {Γ M T} (H : Heap yes) (D : Γ ⊢ᴱ M ∈ T) A : Set where
ok : A → OrWarningᴱ H D A
warning : Warningᴱ H D → OrWarningᴱ H D A
heap-weakeningᴱ : ∀ Γ H M {H′ U} → (H ⊑ H′) → (typeOfᴱ H′ Γ M ≮: U) → (typeOfᴱ H Γ M ≮: U)
heap-weakeningᴱ Γ H (var x) h p = p
heap-weakeningᴱ Γ H (val nil) h p = p
heap-weakeningᴱ Γ H (val (addr a)) refl p = p
heap-weakeningᴱ Γ H (val (addr a)) (snoc {a = b} q) p with a ≡ᴬ b
heap-weakeningᴱ Γ H (val (addr a)) (snoc {a = a} defn) p | yes refl = any-≮: p
heap-weakeningᴱ Γ H (val (addr a)) (snoc {a = b} q) p | no r = ≡-trans-≮: (cong orAny (cong typeOfᴹᴼ (lookup-not-allocated q r))) p
heap-weakeningᴱ Γ H (val (number x)) h p = p
heap-weakeningᴱ Γ H (val (bool x)) h p = p
heap-weakeningᴱ Γ H (val (string x)) h p = p
heap-weakeningᴱ Γ H (M $ N) h p = none-tgt-≮: (heap-weakeningᴱ Γ H M h (tgt-none-≮: p))
heap-weakeningᴱ Γ H (function f ⟨ var x ∈ T ⟩∈ U is B end) h p = p
heap-weakeningᴱ Γ H (block var b ∈ T is B end) h p = p
heap-weakeningᴱ Γ H (binexp M op N) h p = p
data OrWarningᴮ {Γ B T} (H : Heap yes) (D : Γ ⊢ᴮ B ∈ T) A : Set where
ok : A → OrWarningᴮ H D A
warning : Warningᴮ H D → OrWarningᴮ H D A
heap-weakeningᴮ : ∀ Γ H B {H′ U} → (H ⊑ H′) → (typeOfᴮ H′ Γ B ≮: U) → (typeOfᴮ H Γ B ≮: U)
heap-weakeningᴮ Γ H (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) h p = heap-weakeningᴮ (Γ ⊕ f ↦ (T ⇒ U)) H B h p
heap-weakeningᴮ Γ H (local var x ∈ T ← M ∙ B) h p = heap-weakeningᴮ (Γ ⊕ x ↦ T) H B h p
heap-weakeningᴮ Γ H (return M ∙ B) h p = heap-weakeningᴱ Γ H M h p
heap-weakeningᴮ Γ H done h p = p
data OrWarningᴴᴱ {Γ M T} H (D : Γ ⊢ᴴᴱ H ▷ M ∈ T) A : Set where
ok : A → OrWarningᴴᴱ H D A
warning : Warningᴴᴱ H D → OrWarningᴴᴱ H D A
substitutivityᴱ : ∀ {Γ T U} H M v x → (typeOfᴱ H Γ (M [ v / x ]ᴱ) ≮: U) → Either (typeOfᴱ H (Γ ⊕ x ↦ T) M ≮: U) (typeOfᴱ H ∅ (val v) ≮: T)
substitutivityᴱ-whenever : ∀ {Γ T U} H v x y (r : Dec(x ≡ y)) → (typeOfᴱ H Γ (var y [ v / x ]ᴱwhenever r) ≮: U) → Either (typeOfᴱ H (Γ ⊕ x ↦ T) (var y) ≮: U) (typeOfᴱ H ∅ (val v) ≮: T)
substitutivityᴮ : ∀ {Γ T U} H B v x → (typeOfᴮ H Γ (B [ v / x ]ᴮ) ≮: U) → Either (typeOfᴮ H (Γ ⊕ x ↦ T) B ≮: U) (typeOfᴱ H ∅ (val v) ≮: T)
substitutivityᴮ-unless : ∀ {Γ T U V} H B v x y (r : Dec(x ≡ y)) → (typeOfᴮ H (Γ ⊕ y ↦ U) (B [ v / x ]ᴮunless r) ≮: V) → Either (typeOfᴮ H ((Γ ⊕ x ↦ T) ⊕ y ↦ U) B ≮: V) (typeOfᴱ H ∅ (val v) ≮: T)
substitutivityᴮ-unless-yes : ∀ {Γ Γ′ T V} H B v x y (r : x ≡ y) → (Γ′ ≡ Γ) → (typeOfᴮ H Γ (B [ v / x ]ᴮunless yes r) ≮: V) → Either (typeOfᴮ H Γ′ B ≮: V) (typeOfᴱ H ∅ (val v) ≮: T)
substitutivityᴮ-unless-no : ∀ {Γ Γ′ T V} H B v x y (r : x ≢ y) → (Γ′ ≡ Γ ⊕ x ↦ T) → (typeOfᴮ H Γ (B [ v / x ]ᴮunless no r) ≮: V) → Either (typeOfᴮ H Γ′ B ≮: V) (typeOfᴱ H ∅ (val v) ≮: T)
data OrWarningᴴᴮ {Γ B T} H (D : Γ ⊢ᴴᴮ H ▷ B ∈ T) A : Set where
ok : A → OrWarningᴴᴮ H D A
warning : Warningᴴᴮ H D → OrWarningᴴᴮ H D A
substitutivityᴱ H (var y) v x p = substitutivityᴱ-whenever H v x y (x ≡ⱽ y) p
substitutivityᴱ H (val w) v x p = Left p
substitutivityᴱ H (binexp M op N) v x p = Left p
substitutivityᴱ H (M $ N) v x p = mapL none-tgt-≮: (substitutivityᴱ H M v x (tgt-none-≮: p))
substitutivityᴱ H (function f ⟨ var y ∈ T ⟩∈ U is B end) v x p = Left p
substitutivityᴱ H (block var b ∈ T is B end) v x p = Left p
substitutivityᴱ-whenever H v x x (yes refl) q = swapLR (≮:-trans q)
substitutivityᴱ-whenever H v x y (no p) q = Left (≡-trans-≮: (cong orAny (sym (⊕-lookup-miss x y _ _ p))) q)
heap-weakeningᴱ : ∀ H M {H′ Γ} → (H ⊑ H′) → OrWarningᴱ H (typeCheckᴱ H Γ M) (typeOfᴱ H Γ M ≡ typeOfᴱ H′ Γ M)
heap-weakeningᴮ : ∀ H B {H′ Γ} → (H ⊑ H′) → OrWarningᴮ H (typeCheckᴮ H Γ B) (typeOfᴮ H Γ B ≡ typeOfᴮ H′ Γ B)
heap-weakeningᴱ H (var x) h = ok refl
heap-weakeningᴱ H (val nil) h = ok refl
heap-weakeningᴱ H (val (addr a)) refl = ok refl
heap-weakeningᴱ H (val (addr a)) (snoc {a = b} defn) with a ≡ᴬ b
heap-weakeningᴱ H (val (addr a)) (snoc {a = a} defn) | yes refl = warning (UnallocatedAddress refl)
heap-weakeningᴱ H (val (addr a)) (snoc {a = b} p) | no q = ok (cong orNone (cong typeOfᴹᴼ (lookup-not-allocated p q)))
heap-weakeningᴱ H (val (number n)) h = ok refl
heap-weakeningᴱ H (val (bool b)) h = ok refl
heap-weakeningᴱ H (val (string x)) h = ok refl
heap-weakeningᴱ H (binexp M op N) h = ok refl
heap-weakeningᴱ H (M $ N) h with heap-weakeningᴱ H M h
heap-weakeningᴱ H (M $ N) h | ok p = ok (cong tgt p)
heap-weakeningᴱ H (M $ N) h | warning W = warning (app₁ W)
heap-weakeningᴱ H (function f ⟨ var x ∈ T ⟩∈ U is B end) h = ok refl
heap-weakeningᴱ H (block var b ∈ T is B end) h = ok refl
heap-weakeningᴮ H (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) h with heap-weakeningᴮ H B h
heap-weakeningᴮ H (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) h | ok p = ok p
heap-weakeningᴮ H (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) h | warning W = warning (function₂ W)
heap-weakeningᴮ H (local var x ∈ T ← M ∙ B) h with heap-weakeningᴮ H B h
heap-weakeningᴮ H (local var x ∈ T ← M ∙ B) h | ok p = ok p
heap-weakeningᴮ H (local var x ∈ T ← M ∙ B) h | warning W = warning (local₂ W)
heap-weakeningᴮ H (return M ∙ B) h with heap-weakeningᴱ H M h
heap-weakeningᴮ H (return M ∙ B) h | ok p = ok p
heap-weakeningᴮ H (return M ∙ B) h | warning W = warning (return W)
heap-weakeningᴮ H (done) h = ok refl
none-not-obj : ∀ O → none ≢ typeOfᴼ O
none-not-obj (function f ⟨ var x ∈ T ⟩∈ U is B end) ()
typeOf-val-not-none : ∀ {H Γ} v → OrWarningᴱ H (typeCheckᴱ H Γ (val v)) (none ≢ typeOfᴱ H Γ (val v))
typeOf-val-not-none nil = ok (λ ())
typeOf-val-not-none (number n) = ok (λ ())
typeOf-val-not-none (bool b) = ok (λ ())
typeOf-val-not-none (string x) = ok (λ ())
typeOf-val-not-none {H = H} (addr a) with remember (H [ a ]ᴴ)
typeOf-val-not-none {H = H} (addr a) | (just O , p) = ok (λ q → none-not-obj O (trans q (cong orNone (cong typeOfᴹᴼ p))))
typeOf-val-not-none {H = H} (addr a) | (nothing , p) = warning (UnallocatedAddress p)
substitutivityᴱ : ∀ {Γ T} H M v x → (just T ≡ typeOfⱽ H v) → (typeOfᴱ H (Γ ⊕ x ↦ T) M ≡ typeOfᴱ H Γ (M [ v / x ]ᴱ))
substitutivityᴱ-whenever-yes : ∀ {Γ T} H v x y (p : x ≡ y) → (just T ≡ typeOfⱽ H v) → (typeOfᴱ H (Γ ⊕ x ↦ T) (var y) ≡ typeOfᴱ H Γ (var y [ v / x ]ᴱwhenever (yes p)))
substitutivityᴱ-whenever-no : ∀ {Γ T} H v x y (p : x ≢ y) → (just T ≡ typeOfⱽ H v) → (typeOfᴱ H (Γ ⊕ x ↦ T) (var y) ≡ typeOfᴱ H Γ (var y [ v / x ]ᴱwhenever (no p)))
substitutivityᴮ : ∀ {Γ T} H B v x → (just T ≡ typeOfⱽ H v) → (typeOfᴮ H (Γ ⊕ x ↦ T) B ≡ typeOfᴮ H Γ (B [ v / x ]ᴮ))
substitutivityᴮ-unless-yes : ∀ {Γ Γ′ T} H B v x y (p : x ≡ y) → (just T ≡ typeOfⱽ H v) → (Γ′ ≡ Γ) → (typeOfᴮ H Γ′ B ≡ typeOfᴮ H Γ (B [ v / x ]ᴮunless (yes p)))
substitutivityᴮ-unless-no : ∀ {Γ Γ′ T} H B v x y (p : x ≢ y) → (just T ≡ typeOfⱽ H v) → (Γ′ ≡ Γ ⊕ x ↦ T) → (typeOfᴮ H Γ′ B ≡ typeOfᴮ H Γ (B [ v / x ]ᴮunless (no p)))
substitutivityᴱ H (var y) v x p with x ≡ⱽ y
substitutivityᴱ H (var y) v x p | yes q = substitutivityᴱ-whenever-yes H v x y q p
substitutivityᴱ H (var y) v x p | no q = substitutivityᴱ-whenever-no H v x y q p
substitutivityᴱ H (val w) v x p = refl
substitutivityᴱ H (binexp M op N) v x p = refl
substitutivityᴱ H (M $ N) v x p = cong tgt (substitutivityᴱ H M v x p)
substitutivityᴱ H (function f ⟨ var y ∈ T ⟩∈ U is B end) v x p = refl
substitutivityᴱ H (block var b ∈ T is B end) v x p = refl
substitutivityᴱ-whenever-yes H v x x refl q = cong orNone q
substitutivityᴱ-whenever-no H v x y p q = cong orNone ( sym (⊕-lookup-miss x y _ _ p))
substitutivityᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) v x p with x ≡ⱽ f
substitutivityᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) v x p | yes q = substitutivityᴮ-unless-yes H B v x f q p (⊕-over q)
substitutivityᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) v x p | no q = substitutivityᴮ-unless-no H B v x f q p (⊕-swap q)
substitutivityᴮ H (local var y ∈ T ← M ∙ B) v x p with x ≡ⱽ y
substitutivityᴮ H (local var y ∈ T ← M ∙ B) v x p | yes q = substitutivityᴮ-unless-yes H B v x y q p (⊕-over q)
substitutivityᴮ H (local var y ∈ T ← M ∙ B) v x p | no q = substitutivityᴮ-unless-no H B v x y q p (⊕-swap q)
substitutivityᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) v x p = substitutivityᴮ-unless H B v x f (x ≡ⱽ f) p
substitutivityᴮ H (local var y ∈ T ← M ∙ B) v x p = substitutivityᴮ-unless H B v x y (x ≡ⱽ y) p
substitutivityᴮ H (return M ∙ B) v x p = substitutivityᴱ H M v x p
substitutivityᴮ H done v x p = refl
substitutivityᴮ-unless-yes H B v x x refl q refl = refl
substitutivityᴮ-unless-no H B v x y p q refl = substitutivityᴮ H B v x q
substitutivityᴮ H done v x p = Left p
substitutivityᴮ-unless H B v x y (yes p) q = substitutivityᴮ-unless-yes H B v x y p (⊕-over p) q
substitutivityᴮ-unless H B v x y (no p) q = substitutivityᴮ-unless-no H B v x y p (⊕-swap p) q
substitutivityᴮ-unless-yes H B v x y refl refl p = Left p
substitutivityᴮ-unless-no H B v x y p refl q = substitutivityᴮ H B v x q
binOpPreservation : ∀ H {op v w x} → (v ⟦ op ⟧ w ⟶ x) → (tgtBinOp op ≡ typeOfᴱ H ∅ (val x))
binOpPreservation H (+ m n) = refl
@ -157,306 +122,215 @@ binOpPreservation H (== v w) = refl
binOpPreservation H (~= v w) = refl
binOpPreservation H (·· v w) = refl
preservationᴱ : ∀ H M {H′ M′} → (H ⊢ M ⟶ᴱ M′ ⊣ H′) → OrWarningᴴᴱ H (typeCheckᴴᴱ H ∅ M) (typeOfᴱ H ∅ M ≡ typeOfᴱ H′ ∅ M′)
preservationᴮ : ∀ H B {H′ B′} → (H ⊢ B ⟶ᴮ B′ ⊣ H′) → OrWarningᴴᴮ H (typeCheckᴴᴮ H ∅ B) (typeOfᴮ H ∅ B ≡ typeOfᴮ H′ ∅ B′)
reflect-subtypingᴱ : ∀ H M {H′ M′ T} → (H ⊢ M ⟶ᴱ M′ ⊣ H′) → (typeOfᴱ H′ ∅ M′ ≮: T) → Either (typeOfᴱ H ∅ M ≮: T) (Warningᴱ H (typeCheckᴱ H ∅ M))
reflect-subtypingᴮ : ∀ H B {H′ B′ T} → (H ⊢ B ⟶ᴮ B′ ⊣ H′) → (typeOfᴮ H′ ∅ B′ ≮: T) → Either (typeOfᴮ H ∅ B ≮: T) (Warningᴮ H (typeCheckᴮ H ∅ B))
preservationᴱ H (function f ⟨ var x ∈ T ⟩∈ U is B end) (function a defn) = ok refl
preservationᴱ H (M $ N) (app₁ s) with preservationᴱ H M s
preservationᴱ H (M $ N) (app₁ s) | ok p = ok (cong tgt p)
preservationᴱ H (M $ N) (app₁ s) | warning (expr W) = warning (expr (app₁ W))
preservationᴱ H (M $ N) (app₁ s) | warning (heap W) = warning (heap W)
preservationᴱ H (M $ N) (app₂ p s) with heap-weakeningᴱ H M (rednᴱ⊑ s)
preservationᴱ H (M $ N) (app₂ p s) | ok q = ok (cong tgt q)
preservationᴱ H (M $ N) (app₂ p s) | warning W = warning (expr (app₁ W))
preservationᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ S ⟩∈ T is B end) v refl p) with remember (typeOfⱽ H v)
preservationᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ S ⟩∈ T is B end) v refl p) | (just U , q) with S ≡ᵀ U | T ≡ᵀ typeOfᴮ H (x ↦ S) B
preservationᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ S ⟩∈ T is B end) v refl p) | (just U , q) | yes refl | yes refl = ok (cong tgt (cong orNone (cong typeOfᴹᴼ p)))
preservationᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ S ⟩∈ T is B end) v refl p) | (just U , q) | yes refl | no r = warning (heap (addr a p (FunctionDefnMismatch r)))
preservationᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ S ⟩∈ T is B end) v refl p) | (just U , q) | no r | _ = warning (expr (FunctionCallMismatch (λ s → r (trans (trans (sym (cong src (cong orNone (cong typeOfᴹᴼ p)))) s) (cong orNone q)))))
preservationᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ S ⟩∈ T is B end) v refl p) | (nothing , q) with typeOf-val-not-none v
preservationᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ S ⟩∈ T is B end) v refl p) | (nothing , q) | ok r = CONTRADICTION (r (sym (cong orNone q)))
preservationᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ S ⟩∈ T is B end) v refl p) | (nothing , q) | warning W = warning (expr (app₂ W))
preservationᴱ H (block var b ∈ T is B end) (block s) = ok refl
preservationᴱ H (block var b ∈ T is return M ∙ B end) (return v) with T ≡ᵀ typeOfᴱ H ∅ (val v)
preservationᴱ H (block var b ∈ T is return M ∙ B end) (return v) | yes p = ok p
preservationᴱ H (block var b ∈ T is return M ∙ B end) (return v) | no p = warning (expr (BlockMismatch p))
preservationᴱ H (block var b ∈ T is done end) (done) with T ≡ᵀ nil
preservationᴱ H (block var b ∈ T is done end) (done) | yes p = ok p
preservationᴱ H (block var b ∈ T is done end) (done) | no p = warning (expr (BlockMismatch p))
preservationᴱ H (binexp M op N) (binOp₀ s) = ok (binOpPreservation H s)
preservationᴱ H (binexp M op N) (binOp₁ s) = ok refl
preservationᴱ H (binexp M op N) (binOp₂ s) = ok refl
reflect-subtypingᴱ H (M $ N) (app₁ s) p = mapLR none-tgt-≮: app₁ (reflect-subtypingᴱ H M s (tgt-none-≮: p))
reflect-subtypingᴱ H (M $ N) (app₂ v s) p = Left (none-tgt-≮: (heap-weakeningᴱ ∅ H M (rednᴱ⊑ s) (tgt-none-≮: p)))
reflect-subtypingᴱ H (M $ N) (beta (function f ⟨ var y ∈ T ⟩∈ U is B end) v refl q) p = Left (≡-trans-≮: (cong tgt (cong orAny (cong typeOfᴹᴼ q))) p)
reflect-subtypingᴱ H (function f ⟨ var x ∈ T ⟩∈ U is B end) (function a defn) p = Left p
reflect-subtypingᴱ H (block var b ∈ T is B end) (block s) p = Left p
reflect-subtypingᴱ H (block var b ∈ T is return (val v) ∙ B end) (return v) p = mapR BlockMismatch (swapLR (≮:-trans p))
reflect-subtypingᴱ H (block var b ∈ T is done end) done p = mapR BlockMismatch (swapLR (≮:-trans p))
reflect-subtypingᴱ H (binexp M op N) (binOp₀ s) p = Left (≡-trans-≮: (binOpPreservation H s) p)
reflect-subtypingᴱ H (binexp M op N) (binOp₁ s) p = Left p
reflect-subtypingᴱ H (binexp M op N) (binOp₂ s) p = Left p
preservationᴮ H (local var x ∈ T ← M ∙ B) (local s) with heap-weakeningᴮ H B (rednᴱ⊑ s)
preservationᴮ H (local var x ∈ T ← M ∙ B) (local s) | ok p = ok p
preservationᴮ H (local var x ∈ T ← M ∙ B) (local s) | warning W = warning (block (local₂ W))
preservationᴮ H (local var x ∈ T ← M ∙ B) (subst v) with remember (typeOfⱽ H v)
preservationᴮ H (local var x ∈ T ← M ∙ B) (subst v) | (just U , p) with T ≡ᵀ U
preservationᴮ H (local var x ∈ T ← M ∙ B) (subst v) | (just T , p) | yes refl = ok (substitutivityᴮ H B v x (sym p))
preservationᴮ H (local var x ∈ T ← M ∙ B) (subst v) | (just U , p) | no q = warning (block (LocalVarMismatch (λ r → q (trans r (cong orNone p)))))
preservationᴮ H (local var x ∈ T ← M ∙ B) (subst v) | (nothing , p) with typeOf-val-not-none v
preservationᴮ H (local var x ∈ T ← M ∙ B) (subst v) | (nothing , p) | ok q = CONTRADICTION (q (sym (cong orNone p)))
preservationᴮ H (local var x ∈ T ← M ∙ B) (subst v) | (nothing , p) | warning W = warning (block (local₁ W))
preservationᴮ H (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) (function a defn) with heap-weakeningᴮ H B (snoc defn)
preservationᴮ H (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) (function a defn) | ok r = ok (trans r (substitutivityᴮ _ B (addr a) f refl))
preservationᴮ H (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) (function a defn) | warning W = warning (block (function₂ W))
preservationᴮ H (return M ∙ B) (return s) with preservationᴱ H M s
preservationᴮ H (return M ∙ B) (return s) | ok p = ok p
preservationᴮ H (return M ∙ B) (return s) | warning (expr W) = warning (block (return W))
preservationᴮ H (return M ∙ B) (return s) | warning (heap W) = warning (heap W)
reflect-subtypingᴮ H (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) (function a defn) p = mapLR (heap-weakeningᴮ _ _ B (snoc defn)) (CONTRADICTION ∘ ≮:-refl) (substitutivityᴮ _ B (addr a) f p)
reflect-subtypingᴮ H (local var x ∈ T ← M ∙ B) (local s) p = Left (heap-weakeningᴮ (x ↦ T) H B (rednᴱ⊑ s) p)
reflect-subtypingᴮ H (local var x ∈ T ← M ∙ B) (subst v) p = mapR LocalVarMismatch (substitutivityᴮ H B v x p)
reflect-subtypingᴮ H (return M ∙ B) (return s) p = mapR return (reflect-subtypingᴱ H M s p)
reflect-substitutionᴱ : ∀ {Γ T} H M v x → (just T ≡ typeOfⱽ H v) → Warningᴱ H (typeCheckᴱ H Γ (M [ v / x ]ᴱ)) → Warningᴱ H (typeCheckᴱ H (Γ ⊕ x ↦ T) M)
reflect-substitutionᴱ-whenever-yes : ∀ {Γ T} H v x y (p : x ≡ y) → (just T ≡ typeOfⱽ H v) → Warningᴱ H (typeCheckᴱ H Γ (var y [ v / x ]ᴱwhenever yes p)) → Warningᴱ H (typeCheckᴱ H (Γ ⊕ x ↦ T) (var y))
reflect-substitutionᴱ-whenever-no : ∀ {Γ T} H v x y (p : x ≢ y) → (just T ≡ typeOfⱽ H v) → Warningᴱ H (typeCheckᴱ H Γ (var y [ v / x ]ᴱwhenever no p)) → Warningᴱ H (typeCheckᴱ H (Γ ⊕ x ↦ T) (var y))
reflect-substitutionᴮ : ∀ {Γ T} H B v x → (just T ≡ typeOfⱽ H v) → Warningᴮ H (typeCheckᴮ H Γ (B [ v / x ]ᴮ)) → Warningᴮ H (typeCheckᴮ H (Γ ⊕ x ↦ T) B)
reflect-substitutionᴮ-unless-yes : ∀ {Γ Γ′ T} H B v x y (r : x ≡ y) → (just T ≡ typeOfⱽ H v) → (Γ′ ≡ Γ) → Warningᴮ H (typeCheckᴮ H Γ (B [ v / x ]ᴮunless yes r)) → Warningᴮ H (typeCheckᴮ H Γ′ B)
reflect-substitutionᴮ-unless-no : ∀ {Γ Γ′ T} H B v x y (r : x ≢ y) → (just T ≡ typeOfⱽ H v) → (Γ′ ≡ Γ ⊕ x ↦ T) → Warningᴮ H (typeCheckᴮ H Γ (B [ v / x ]ᴮunless no r)) → Warningᴮ H (typeCheckᴮ H Γ′ B)
reflect-substitutionᴱ : ∀ {Γ T} H M v x → Warningᴱ H (typeCheckᴱ H Γ (M [ v / x ]ᴱ)) → Either (Warningᴱ H (typeCheckᴱ H (Γ ⊕ x ↦ T) M)) (Either (Warningᴱ H (typeCheckᴱ H ∅ (val v))) (typeOfᴱ H ∅ (val v) ≮: T))
reflect-substitutionᴱ-whenever : ∀ {Γ T} H v x y (p : Dec(x ≡ y)) → Warningᴱ H (typeCheckᴱ H Γ (var y [ v / x ]ᴱwhenever p)) → Either (Warningᴱ H (typeCheckᴱ H (Γ ⊕ x ↦ T) (var y))) (Either (Warningᴱ H (typeCheckᴱ H ∅ (val v))) (typeOfᴱ H ∅ (val v) ≮: T))
reflect-substitutionᴮ : ∀ {Γ T} H B v x → Warningᴮ H (typeCheckᴮ H Γ (B [ v / x ]ᴮ)) → Either (Warningᴮ H (typeCheckᴮ H (Γ ⊕ x ↦ T) B)) (Either (Warningᴱ H (typeCheckᴱ H ∅ (val v))) (typeOfᴱ H ∅ (val v) ≮: T))
reflect-substitutionᴮ-unless : ∀ {Γ T U} H B v x y (r : Dec(x ≡ y)) → Warningᴮ H (typeCheckᴮ H (Γ ⊕ y ↦ U) (B [ v / x ]ᴮunless r)) → Either (Warningᴮ H (typeCheckᴮ H ((Γ ⊕ x ↦ T) ⊕ y ↦ U) B)) (Either (Warningᴱ H (typeCheckᴱ H ∅ (val v))) (typeOfᴱ H ∅ (val v) ≮: T))
reflect-substitutionᴮ-unless-yes : ∀ {Γ Γ′ T} H B v x y (r : x ≡ y) → (Γ′ ≡ Γ) → Warningᴮ H (typeCheckᴮ H Γ (B [ v / x ]ᴮunless yes r)) → Either (Warningᴮ H (typeCheckᴮ H Γ′ B)) (Either (Warningᴱ H (typeCheckᴱ H ∅ (val v))) (typeOfᴱ H ∅ (val v) ≮: T))
reflect-substitutionᴮ-unless-no : ∀ {Γ Γ′ T} H B v x y (r : x ≢ y) → (Γ′ ≡ Γ ⊕ x ↦ T) → Warningᴮ H (typeCheckᴮ H Γ (B [ v / x ]ᴮunless no r)) → Either (Warningᴮ H (typeCheckᴮ H Γ′ B)) (Either (Warningᴱ H (typeCheckᴱ H ∅ (val v))) (typeOfᴱ H ∅ (val v) ≮: T))
reflect-substitutionᴱ H (var y) v x p W with x ≡ⱽ y
reflect-substitutionᴱ H (var y) v x p W | yes r = reflect-substitutionᴱ-whenever-yes H v x y r p W
reflect-substitutionᴱ H (var y) v x p W | no r = reflect-substitutionᴱ-whenever-no H v x y r p W
reflect-substitutionᴱ H (val (addr a)) v x p (UnallocatedAddress r) = UnallocatedAddress r
reflect-substitutionᴱ H (M $ N) v x p (FunctionCallMismatch q) = FunctionCallMismatch (λ s → q (trans (cong src (sym (substitutivityᴱ H M v x p))) (trans s (substitutivityᴱ H N v x p))))
reflect-substitutionᴱ H (M $ N) v x p (app₁ W) = app₁ (reflect-substitutionᴱ H M v x p W)
reflect-substitutionᴱ H (M $ N) v x p (app₂ W) = app₂ (reflect-substitutionᴱ H N v x p W)
reflect-substitutionᴱ H (function f ⟨ var y ∈ T ⟩∈ U is B end) v x p (FunctionDefnMismatch q) with (x ≡ⱽ y)
reflect-substitutionᴱ H (function f ⟨ var y ∈ T ⟩∈ U is B end) v x p (FunctionDefnMismatch q) | yes r = FunctionDefnMismatch (λ s → q (trans s (substitutivityᴮ-unless-yes H B v x y r p (⊕-over r))))
reflect-substitutionᴱ H (function f ⟨ var y ∈ T ⟩∈ U is B end) v x p (FunctionDefnMismatch q) | no r = FunctionDefnMismatch (λ s → q (trans s (substitutivityᴮ-unless-no H B v x y r p (⊕-swap r))))
reflect-substitutionᴱ H (function f ⟨ var y ∈ T ⟩∈ U is B end) v x p (function₁ W) with (x ≡ⱽ y)
reflect-substitutionᴱ H (function f ⟨ var y ∈ T ⟩∈ U is B end) v x p (function₁ W) | yes r = function₁ (reflect-substitutionᴮ-unless-yes H B v x y r p (⊕-over r) W)
reflect-substitutionᴱ H (function f ⟨ var y ∈ T ⟩∈ U is B end) v x p (function₁ W) | no r = function₁ (reflect-substitutionᴮ-unless-no H B v x y r p (⊕-swap r) W)
reflect-substitutionᴱ H (block var b ∈ T is B end) v x p (BlockMismatch q) = BlockMismatch (λ r → q (trans r (substitutivityᴮ H B v x p)))
reflect-substitutionᴱ H (block var b ∈ T is B end) v x p (block₁ W) = block₁ (reflect-substitutionᴮ H B v x p W)
reflect-substitutionᴱ H (binexp M op N) x v p (BinOpMismatch₁ q) = BinOpMismatch₁ (subst₁ (BinOpWarning op) (sym (substitutivityᴱ H M x v p)) q)
reflect-substitutionᴱ H (binexp M op N) x v p (BinOpMismatch₂ q) = BinOpMismatch₂ (subst₁ (BinOpWarning op) (sym (substitutivityᴱ H N x v p)) q)
reflect-substitutionᴱ H (binexp M op N) x v p (bin₁ W) = bin₁ (reflect-substitutionᴱ H M x v p W)
reflect-substitutionᴱ H (binexp M op N) x v p (bin₂ W) = bin₂ (reflect-substitutionᴱ H N x v p W)
reflect-substitutionᴱ H (var y) v x W = reflect-substitutionᴱ-whenever H v x y (x ≡ⱽ y) W
reflect-substitutionᴱ H (val (addr a)) v x (UnallocatedAddress r) = Left (UnallocatedAddress r)
reflect-substitutionᴱ H (M $ N) v x (FunctionCallMismatch p) with substitutivityᴱ H N v x p
reflect-substitutionᴱ H (M $ N) v x (FunctionCallMismatch p) | Right W = Right (Right W)
reflect-substitutionᴱ H (M $ N) v x (FunctionCallMismatch p) | Left q with substitutivityᴱ H M v x (src-any-≮: q)
reflect-substitutionᴱ H (M $ N) v x (FunctionCallMismatch p) | Left q | Left r = Left ((FunctionCallMismatch ∘ any-src-≮: q) r)
reflect-substitutionᴱ H (M $ N) v x (FunctionCallMismatch p) | Left q | Right W = Right (Right W)
reflect-substitutionᴱ H (M $ N) v x (app₁ W) = mapL app₁ (reflect-substitutionᴱ H M v x W)
reflect-substitutionᴱ H (M $ N) v x (app₂ W) = mapL app₂ (reflect-substitutionᴱ H N v x W)
reflect-substitutionᴱ H (function f ⟨ var y ∈ T ⟩∈ U is B end) v x (FunctionDefnMismatch q) = mapLR FunctionDefnMismatch Right (substitutivityᴮ-unless H B v x y (x ≡ⱽ y) q)
reflect-substitutionᴱ H (function f ⟨ var y ∈ T ⟩∈ U is B end) v x (function₁ W) = mapL function₁ (reflect-substitutionᴮ-unless H B v x y (x ≡ⱽ y) W)
reflect-substitutionᴱ H (block var b ∈ T is B end) v x (BlockMismatch q) = mapLR BlockMismatch Right (substitutivityᴮ H B v x q)
reflect-substitutionᴱ H (block var b ∈ T is B end) v x (block₁ W′) = mapL block₁ (reflect-substitutionᴮ H B v x W′)
reflect-substitutionᴱ H (binexp M op N) v x (BinOpMismatch₁ q) = mapLR BinOpMismatch₁ Right (substitutivityᴱ H M v x q)
reflect-substitutionᴱ H (binexp M op N) v x (BinOpMismatch₂ q) = mapLR BinOpMismatch₂ Right (substitutivityᴱ H N v x q)
reflect-substitutionᴱ H (binexp M op N) v x (bin₁ W) = mapL bin₁ (reflect-substitutionᴱ H M v x W)
reflect-substitutionᴱ H (binexp M op N) v x (bin₂ W) = mapL bin₂ (reflect-substitutionᴱ H N v x W)
reflect-substitutionᴱ-whenever-no H v x y p q (UnboundVariable r) = UnboundVariable (trans (sym (⊕-lookup-miss x y _ _ p)) r)
reflect-substitutionᴱ-whenever-yes H (addr a) x x refl p (UnallocatedAddress q) with trans p (cong typeOfᴹᴼ q)
reflect-substitutionᴱ-whenever-yes H (addr a) x x refl p (UnallocatedAddress q) | ()
reflect-substitutionᴱ-whenever H a x x (yes refl) (UnallocatedAddress p) = Right (Left (UnallocatedAddress p))
reflect-substitutionᴱ-whenever H v x y (no p) (UnboundVariable q) = Left (UnboundVariable (trans (sym (⊕-lookup-miss x y _ _ p)) q))
reflect-substitutionᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) v x p (FunctionDefnMismatch q) with (x ≡ⱽ y)
reflect-substitutionᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) v x p (FunctionDefnMismatch q) | yes r = FunctionDefnMismatch (λ s → q (trans s (substitutivityᴮ-unless-yes H C v x y r p (⊕-over r))))
reflect-substitutionᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) v x p (FunctionDefnMismatch q) | no r = FunctionDefnMismatch (λ s → q (trans s (substitutivityᴮ-unless-no H C v x y r p (⊕-swap r))))
reflect-substitutionᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) v x p (function₁ W) with (x ≡ⱽ y)
reflect-substitutionᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) v x p (function₁ W) | yes r = function₁ (reflect-substitutionᴮ-unless-yes H C v x y r p (⊕-over r) W)
reflect-substitutionᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) v x p (function₁ W) | no r = function₁ (reflect-substitutionᴮ-unless-no H C v x y r p (⊕-swap r) W)
reflect-substitutionᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) v x p (function₂ W) with (x ≡ⱽ f)
reflect-substitutionᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) v x p (function₂ W)| yes r = function₂ (reflect-substitutionᴮ-unless-yes H B v x f r p (⊕-over r) W)
reflect-substitutionᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) v x p (function₂ W)| no r = function₂ (reflect-substitutionᴮ-unless-no H B v x f r p (⊕-swap r) W)
reflect-substitutionᴮ H (local var y ∈ T ← M ∙ B) v x p (LocalVarMismatch q) = LocalVarMismatch (λ r → q (trans r (substitutivityᴱ H M v x p)))
reflect-substitutionᴮ H (local var y ∈ T ← M ∙ B) v x p (local₁ W) = local₁ (reflect-substitutionᴱ H M v x p W)
reflect-substitutionᴮ H (local var y ∈ T ← M ∙ B) v x p (local₂ W) with (x ≡ⱽ y)
reflect-substitutionᴮ H (local var y ∈ T ← M ∙ B) v x p (local₂ W) | yes r = local₂ (reflect-substitutionᴮ-unless-yes H B v x y r p (⊕-over r) W)
reflect-substitutionᴮ H (local var y ∈ T ← M ∙ B) v x p (local₂ W) | no r = local₂ (reflect-substitutionᴮ-unless-no H B v x y r p (⊕-swap r) W)
reflect-substitutionᴮ H (return M ∙ B) v x p (return W) = return (reflect-substitutionᴱ H M v x p W)
reflect-substitutionᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) v x (FunctionDefnMismatch q) = mapLR FunctionDefnMismatch Right (substitutivityᴮ-unless H C v x y (x ≡ⱽ y) q)
reflect-substitutionᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) v x (function₁ W) = mapL function₁ (reflect-substitutionᴮ-unless H C v x y (x ≡ⱽ y) W)
reflect-substitutionᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) v x (function₂ W) = mapL function₂ (reflect-substitutionᴮ-unless H B v x f (x ≡ⱽ f) W)
reflect-substitutionᴮ H (local var y ∈ T ← M ∙ B) v x (LocalVarMismatch q) = mapLR LocalVarMismatch Right (substitutivityᴱ H M v x q)
reflect-substitutionᴮ H (local var y ∈ T ← M ∙ B) v x (local₁ W) = mapL local₁ (reflect-substitutionᴱ H M v x W)
reflect-substitutionᴮ H (local var y ∈ T ← M ∙ B) v x (local₂ W) = mapL local₂ (reflect-substitutionᴮ-unless H B v x y (x ≡ⱽ y) W)
reflect-substitutionᴮ H (return M ∙ B) v x (return W) = mapL return (reflect-substitutionᴱ H M v x W)
reflect-substitutionᴮ-unless-yes H B v x y r p refl W = W
reflect-substitutionᴮ-unless-no H B v x y r p refl W = reflect-substitutionᴮ H B v x p W
reflect-substitutionᴮ-unless H B v x y (yes p) W = reflect-substitutionᴮ-unless-yes H B v x y p (⊕-over p) W
reflect-substitutionᴮ-unless H B v x y (no p) W = reflect-substitutionᴮ-unless-no H B v x y p (⊕-swap p) W
reflect-substitutionᴮ-unless-yes H B v x x refl refl W = Left W
reflect-substitutionᴮ-unless-no H B v x y p refl W = reflect-substitutionᴮ H B v x W
reflect-weakeningᴱ : ∀ H M {H′ Γ} → (H ⊑ H′) → Warningᴱ H′ (typeCheckᴱ H′ Γ M) → Warningᴱ H (typeCheckᴱ H Γ M)
reflect-weakeningᴮ : ∀ H B {H′ Γ} → (H ⊑ H′) → Warningᴮ H′ (typeCheckᴮ H′ Γ B) → Warningᴮ H (typeCheckᴮ H Γ B)
reflect-weakeningᴱ : ∀ Γ H M {H′} → (H ⊑ H′) → Warningᴱ H′ (typeCheckᴱ H′ Γ M) → Warningᴱ H (typeCheckᴱ H Γ M)
reflect-weakeningᴮ : ∀ Γ H B {H′} → (H ⊑ H′) → Warningᴮ H′ (typeCheckᴮ H′ Γ B) → Warningᴮ H (typeCheckᴮ H Γ B)
reflect-weakeningᴱ H (var x) h (UnboundVariable p) = (UnboundVariable p)
reflect-weakeningᴱ H (val (addr a)) h (UnallocatedAddress p) = UnallocatedAddress (lookup-⊑-nothing a h p)
reflect-weakeningᴱ H (M $ N) h (FunctionCallMismatch p) with heap-weakeningᴱ H M h | heap-weakeningᴱ H N h
reflect-weakeningᴱ H (M $ N) h (FunctionCallMismatch p) | ok q₁ | ok q₂ = FunctionCallMismatch (λ r → p (trans (cong src (sym q₁)) (trans r q₂)))
reflect-weakeningᴱ H (M $ N) h (FunctionCallMismatch p) | warning W | _ = app₁ W
reflect-weakeningᴱ H (M $ N) h (FunctionCallMismatch p) | _ | warning W = app₂ W
reflect-weakeningᴱ H (M $ N) h (app₁ W) = app₁ (reflect-weakeningᴱ H M h W)
reflect-weakeningᴱ H (M $ N) h (app₂ W) = app₂ (reflect-weakeningᴱ H N h W)
reflect-weakeningᴱ H (binexp M op N) h (BinOpMismatch₁ p) with heap-weakeningᴱ H M h
reflect-weakeningᴱ H (binexp M op N) h (BinOpMismatch₁ p) | ok q = BinOpMismatch₁ (subst₁ (BinOpWarning op) (sym q) p)
reflect-weakeningᴱ H (binexp M op N) h (BinOpMismatch₁ p) | warning W = bin₁ W
reflect-weakeningᴱ H (binexp M op N) h (BinOpMismatch₂ p) with heap-weakeningᴱ H N h
reflect-weakeningᴱ H (binexp M op N) h (BinOpMismatch₂ p) | ok q = BinOpMismatch₂ (subst₁ (BinOpWarning op) (sym q) p)
reflect-weakeningᴱ H (binexp M op N) h (BinOpMismatch₂ p) | warning W = bin₂ W
reflect-weakeningᴱ H (binexp M op N) h (bin₁ W′) = bin₁ (reflect-weakeningᴱ H M h W′)
reflect-weakeningᴱ H (binexp M op N) h (bin₂ W′) = bin₂ (reflect-weakeningᴱ H N h W′)
reflect-weakeningᴱ H (function f ⟨ var y ∈ T ⟩∈ U is B end) h (FunctionDefnMismatch p) with heap-weakeningᴮ H B h
reflect-weakeningᴱ H (function f ⟨ var y ∈ T ⟩∈ U is B end) h (FunctionDefnMismatch p) | ok q = FunctionDefnMismatch (λ r → p (trans r q))
reflect-weakeningᴱ H (function f ⟨ var y ∈ T ⟩∈ U is B end) h (FunctionDefnMismatch p) | warning W = function₁ W
reflect-weakeningᴱ H (function f ⟨ var y ∈ T ⟩∈ U is B end) h (function₁ W) = function₁ (reflect-weakeningᴮ H B h W)
reflect-weakeningᴱ H (block var b ∈ T is B end) h (BlockMismatch p) with heap-weakeningᴮ H B h
reflect-weakeningᴱ H (block var b ∈ T is B end) h (BlockMismatch p) | ok q = BlockMismatch (λ r → p (trans r q))
reflect-weakeningᴱ H (block var b ∈ T is B end) h (BlockMismatch p) | warning W = block₁ W
reflect-weakeningᴱ H (block var b ∈ T is B end) h (block₁ W) = block₁ (reflect-weakeningᴮ H B h W)
reflect-weakeningᴱ Γ H (var x) h (UnboundVariable p) = (UnboundVariable p)
reflect-weakeningᴱ Γ H (val (addr a)) h (UnallocatedAddress p) = UnallocatedAddress (lookup-⊑-nothing a h p)
reflect-weakeningᴱ Γ H (M $ N) h (FunctionCallMismatch p) = FunctionCallMismatch (heap-weakeningᴱ Γ H N h (any-src-≮: p (heap-weakeningᴱ Γ H M h (src-any-≮: p))))
reflect-weakeningᴱ Γ H (M $ N) h (app₁ W) = app₁ (reflect-weakeningᴱ Γ H M h W)
reflect-weakeningᴱ Γ H (M $ N) h (app₂ W) = app₂ (reflect-weakeningᴱ Γ H N h W)
reflect-weakeningᴱ Γ H (binexp M op N) h (BinOpMismatch₁ p) = BinOpMismatch₁ (heap-weakeningᴱ Γ H M h p)
reflect-weakeningᴱ Γ H (binexp M op N) h (BinOpMismatch₂ p) = BinOpMismatch₂ (heap-weakeningᴱ Γ H N h p)
reflect-weakeningᴱ Γ H (binexp M op N) h (bin₁ W′) = bin₁ (reflect-weakeningᴱ Γ H M h W′)
reflect-weakeningᴱ Γ H (binexp M op N) h (bin₂ W′) = bin₂ (reflect-weakeningᴱ Γ H N h W′)
reflect-weakeningᴱ Γ H (function f ⟨ var y ∈ T ⟩∈ U is B end) h (FunctionDefnMismatch p) = FunctionDefnMismatch (heap-weakeningᴮ (Γ ⊕ y ↦ T) H B h p)
reflect-weakeningᴱ Γ H (function f ⟨ var y ∈ T ⟩∈ U is B end) h (function₁ W) = function₁ (reflect-weakeningᴮ (Γ ⊕ y ↦ T) H B h W)
reflect-weakeningᴱ Γ H (block var b ∈ T is B end) h (BlockMismatch p) = BlockMismatch (heap-weakeningᴮ Γ H B h p)
reflect-weakeningᴱ Γ H (block var b ∈ T is B end) h (block₁ W) = block₁ (reflect-weakeningᴮ Γ H B h W)
reflect-weakeningᴮ H (return M ∙ B) h (return W) = return (reflect-weakeningᴱ H M h W)
reflect-weakeningᴮ H (local var y ∈ T ← M ∙ B) h (LocalVarMismatch p) with heap-weakeningᴱ H M h
reflect-weakeningᴮ H (local var y ∈ T ← M ∙ B) h (LocalVarMismatch p) | ok q = LocalVarMismatch (λ r → p (trans r q))
reflect-weakeningᴮ H (local var y ∈ T ← M ∙ B) h (LocalVarMismatch p) | warning W = local₁ W
reflect-weakeningᴮ H (local var y ∈ T ← M ∙ B) h (local₁ W) = local₁ (reflect-weakeningᴱ H M h W)
reflect-weakeningᴮ H (local var y ∈ T ← M ∙ B) h (local₂ W) = local₂ (reflect-weakeningᴮ H B h W)
reflect-weakeningᴮ H (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) h (FunctionDefnMismatch p) with heap-weakeningᴮ H C h
reflect-weakeningᴮ H (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) h (FunctionDefnMismatch p) | ok q = FunctionDefnMismatch (λ r → p (trans r q))
reflect-weakeningᴮ H (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) h (FunctionDefnMismatch p) | warning W = function₁ W
reflect-weakeningᴮ H (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) h (function₁ W) = function₁ (reflect-weakeningᴮ H C h W)
reflect-weakeningᴮ H (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) h (function₂ W) = function₂ (reflect-weakeningᴮ H B h W)
reflect-weakeningᴮ Γ H (return M ∙ B) h (return W) = return (reflect-weakeningᴱ Γ H M h W)
reflect-weakeningᴮ Γ H (local var y ∈ T ← M ∙ B) h (LocalVarMismatch p) = LocalVarMismatch (heap-weakeningᴱ Γ H M h p)
reflect-weakeningᴮ Γ H (local var y ∈ T ← M ∙ B) h (local₁ W) = local₁ (reflect-weakeningᴱ Γ H M h W)
reflect-weakeningᴮ Γ H (local var y ∈ T ← M ∙ B) h (local₂ W) = local₂ (reflect-weakeningᴮ (Γ ⊕ y ↦ T) H B h W)
reflect-weakeningᴮ Γ H (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) h (FunctionDefnMismatch p) = FunctionDefnMismatch (heap-weakeningᴮ (Γ ⊕ x ↦ T) H C h p)
reflect-weakeningᴮ Γ H (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) h (function₁ W) = function₁ (reflect-weakeningᴮ (Γ ⊕ x ↦ T) H C h W)
reflect-weakeningᴮ Γ H (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) h (function₂ W) = function₂ (reflect-weakeningᴮ (Γ ⊕ f ↦ (T ⇒ U)) H B h W)
reflect-weakeningᴼ : ∀ H O {H′} → (H ⊑ H′) → Warningᴼ H′ (typeCheckᴼ H′ O) → Warningᴼ H (typeCheckᴼ H O)
reflect-weakeningᴼ H (just (function f ⟨ var x ∈ T ⟩∈ U is B end)) h (FunctionDefnMismatch p) with heap-weakeningᴮ H B h
reflect-weakeningᴼ H (just (function f ⟨ var x ∈ T ⟩∈ U is B end)) h (FunctionDefnMismatch p) | ok q = FunctionDefnMismatch (λ r → p (trans r q))
reflect-weakeningᴼ H (just (function f ⟨ var x ∈ T ⟩∈ U is B end)) h (FunctionDefnMismatch p) | warning W = function₁ W
reflect-weakeningᴼ H (just (function f ⟨ var x ∈ T ⟩∈ U is B end)) h (function₁ W′) = function₁ (reflect-weakeningᴮ H B h W′)
reflect-weakeningᴼ H (just function f ⟨ var x ∈ T ⟩∈ U is B end) h (FunctionDefnMismatch p) = FunctionDefnMismatch (heap-weakeningᴮ (x ↦ T) H B h p)
reflect-weakeningᴼ H (just function f ⟨ var x ∈ T ⟩∈ U is B end) h (function₁ W) = function₁ (reflect-weakeningᴮ (x ↦ T) H B h W)
reflectᴱ : ∀ H M {H′ M′} → (H ⊢ M ⟶ᴱ M′ ⊣ H′) → Warningᴱ H′ (typeCheckᴱ H′ ∅ M′) → Warningᴴᴱ H (typeCheckᴴᴱ H ∅ M)
reflectᴮ : ∀ H B {H′ B′} → (H ⊢ B ⟶ᴮ B′ ⊣ H′) → Warningᴮ H′ (typeCheckᴮ H′ ∅ B′) → Warningᴴᴮ H (typeCheckᴴᴮ H ∅ B)
reflectᴱ : ∀ H M {H′ M′} → (H ⊢ M ⟶ᴱ M′ ⊣ H′) → Warningᴱ H′ (typeCheckᴱ H′ ∅ M′) → Either (Warningᴱ H (typeCheckᴱ H ∅ M)) (Warningᴴ H (typeCheckᴴ H))
reflectᴮ : ∀ H B {H′ B′} → (H ⊢ B ⟶ᴮ B′ ⊣ H′) → Warningᴮ H′ (typeCheckᴮ H′ ∅ B′) → Either (Warningᴮ H (typeCheckᴮ H ∅ B)) (Warningᴴ H (typeCheckᴴ H))
reflectᴱ H (M $ N) (app₁ s) (FunctionCallMismatch p) with preservationᴱ H M s | heap-weakeningᴱ H N (rednᴱ⊑ s)
reflectᴱ H (M $ N) (app₁ s) (FunctionCallMismatch p) | ok q | ok q′ = expr (FunctionCallMismatch (λ r → p (trans (trans (cong src (sym q)) r) q′)))
reflectᴱ H (M $ N) (app₁ s) (FunctionCallMismatch p) | warning (expr W) | _ = expr (app₁ W)
reflectᴱ H (M $ N) (app₁ s) (FunctionCallMismatch p) | warning (heap W) | _ = heap W
reflectᴱ H (M $ N) (app₁ s) (FunctionCallMismatch p) | _ | warning W = expr (app₂ W)
reflectᴱ H (M $ N) (app₁ s) (app₁ W′) with reflectᴱ H M s W′
reflectᴱ H (M $ N) (app₁ s) (app₁ W′) | heap W = heap W
reflectᴱ H (M $ N) (app₁ s) (app₁ W′) | expr W = expr (app₁ W)
reflectᴱ H (M $ N) (app₁ s) (app₂ W′) = expr (app₂ (reflect-weakeningᴱ H N (rednᴱ⊑ s) W′))
reflectᴱ H (M $ N) (app₂ p s) (FunctionCallMismatch p′) with heap-weakeningᴱ H (val p) (rednᴱ⊑ s) | preservationᴱ H N s
reflectᴱ H (M $ N) (app₂ p s) (FunctionCallMismatch p′) | ok q | ok q′ = expr (FunctionCallMismatch (λ r → p′ (trans (trans (cong src (sym q)) r) q′)))
reflectᴱ H (M $ N) (app₂ p s) (FunctionCallMismatch p′) | warning W | _ = expr (app₁ W)
reflectᴱ H (M $ N) (app₂ p s) (FunctionCallMismatch p′) | _ | warning (expr W) = expr (app₂ W)
reflectᴱ H (M $ N) (app₂ p s) (FunctionCallMismatch p′) | _ | warning (heap W) = heap W
reflectᴱ H (M $ N) (app₂ p s) (app₁ W′) = expr (app₁ (reflect-weakeningᴱ H M (rednᴱ⊑ s) W′))
reflectᴱ H (M $ N) (app₂ p s) (app₂ W′) with reflectᴱ H N s W′
reflectᴱ H (M $ N) (app₂ p s) (app₂ W′) | heap W = heap W
reflectᴱ H (M $ N) (app₂ p s) (app₂ W′) | expr W = expr (app₂ W)
reflectᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ T ⟩∈ U is B end) v refl p) (BlockMismatch q) with remember (typeOfⱽ H v)
reflectᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ T ⟩∈ U is B end) v refl p) (BlockMismatch q) | (just S , r) with S ≡ᵀ T
reflectᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ T ⟩∈ U is B end) v refl p) (BlockMismatch q) | (just T , r) | yes refl = heap (addr a p (FunctionDefnMismatch (λ s → q (trans s (substitutivityᴮ H B v x (sym r))))))
reflectᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ T ⟩∈ U is B end) v refl p) (BlockMismatch q) | (just S , r) | no s = expr (FunctionCallMismatch (λ t → s (trans (cong orNone (sym r)) (trans (sym t) (cong src (cong orNone (cong typeOfᴹᴼ p)))))))
reflectᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ T ⟩∈ U is B end) v refl p) (BlockMismatch q) | (nothing , r) with typeOf-val-not-none v
reflectᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ T ⟩∈ U is B end) v refl p) (BlockMismatch q) | (nothing , r) | ok s = CONTRADICTION (s (cong orNone (sym r)))
reflectᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ T ⟩∈ U is B end) v refl p) (BlockMismatch q) | (nothing , r) | warning W = expr (app₂ W)
reflectᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ T ⟩∈ U is B end) v refl p) (block₁ W′) with remember (typeOfⱽ H v)
reflectᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ T ⟩∈ U is B end) v refl p) (block₁ W′) | (just S , q) with S ≡ᵀ T
reflectᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ T ⟩∈ U is B end) v refl p) (block₁ W′) | (just T , q) | yes refl = heap (addr a p (function₁ (reflect-substitutionᴮ H B v x (sym q) W′)))
reflectᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ T ⟩∈ U is B end) v refl p) (block₁ W′) | (just S , q) | no r = expr (FunctionCallMismatch (λ s → r (trans (cong orNone (sym q)) (trans (sym s) (cong src (cong orNone (cong typeOfᴹᴼ p)))))))
reflectᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ T ⟩∈ U is B end) v refl p) (block₁ W′) | (nothing , q) with typeOf-val-not-none v
reflectᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ T ⟩∈ U is B end) v refl p) (block₁ W′) | (nothing , q) | ok r = CONTRADICTION (r (cong orNone (sym q)))
reflectᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ T ⟩∈ U is B end) v refl p) (block₁ W′) | (nothing , q) | warning W = expr (app₂ W)
reflectᴱ H (block var b ∈ T is B end) (block s) (BlockMismatch p) with preservationᴮ H B s
reflectᴱ H (block var b ∈ T is B end) (block s) (BlockMismatch p) | ok q = expr (BlockMismatch (λ r → p (trans r q)))
reflectᴱ H (block var b ∈ T is B end) (block s) (BlockMismatch p) | warning (heap W) = heap W
reflectᴱ H (block var b ∈ T is B end) (block s) (BlockMismatch p) | warning (block W) = expr (block₁ W)
reflectᴱ H (block var b ∈ T is B end) (block s) (block₁ W′) with reflectᴮ H B s W′
reflectᴱ H (block var b ∈ T is B end) (block s) (block₁ W′) | heap W = heap W
reflectᴱ H (block var b ∈ T is B end) (block s) (block₁ W′) | block W = expr (block₁ W)
reflectᴱ H (block var b ∈ T is B end) (return v) W′ = expr (block₁ (return W′))
reflectᴱ H (M $ N) (app₁ s) (FunctionCallMismatch p) = cond (Left ∘ FunctionCallMismatch ∘ heap-weakeningᴱ ∅ H N (rednᴱ⊑ s) ∘ any-src-≮: p) (Left ∘ app₁) (reflect-subtypingᴱ H M s (src-any-≮: p))
reflectᴱ H (M $ N) (app₁ s) (app₁ W′) = mapL app₁ (reflectᴱ H M s W′)
reflectᴱ H (M $ N) (app₁ s) (app₂ W′) = Left (app₂ (reflect-weakeningᴱ ∅ H N (rednᴱ⊑ s) W′))
reflectᴱ H (M $ N) (app₂ p s) (FunctionCallMismatch q) = cond (λ r → Left (FunctionCallMismatch (any-src-≮: r (heap-weakeningᴱ ∅ H M (rednᴱ⊑ s) (src-any-≮: r))))) (Left ∘ app₂) (reflect-subtypingᴱ H N s q)
reflectᴱ H (M $ N) (app₂ p s) (app₁ W′) = Left (app₁ (reflect-weakeningᴱ ∅ H M (rednᴱ⊑ s) W′))
reflectᴱ H (M $ N) (app₂ p s) (app₂ W′) = mapL app₂ (reflectᴱ H N s W′)
reflectᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ T ⟩∈ U is B end) v refl p) (BlockMismatch q) with substitutivityᴮ H B v x q
reflectᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ T ⟩∈ U is B end) v refl p) (BlockMismatch q) | Left r = Right (addr a p (FunctionDefnMismatch r))
reflectᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ T ⟩∈ U is B end) v refl p) (BlockMismatch q) | Right r = Left (FunctionCallMismatch (≮:-trans-≡ r ((cong src (cong orAny (cong typeOfᴹᴼ (sym p)))))))
reflectᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ T ⟩∈ U is B end) v refl p) (block₁ W′) with reflect-substitutionᴮ _ B v x W′
reflectᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ T ⟩∈ U is B end) v refl p) (block₁ W′) | Left W = Right (addr a p (function₁ W))
reflectᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ T ⟩∈ U is B end) v refl p) (block₁ W′) | Right (Left W) = Left (app₂ W)
reflectᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ T ⟩∈ U is B end) v refl p) (block₁ W′) | Right (Right q) = Left (FunctionCallMismatch (≮:-trans-≡ q (cong src (cong orAny (cong typeOfᴹᴼ (sym p))))))
reflectᴱ H (block var b ∈ T is B end) (block s) (BlockMismatch p) = Left (cond BlockMismatch block₁ (reflect-subtypingᴮ H B s p))
reflectᴱ H (block var b ∈ T is B end) (block s) (block₁ W′) = mapL block₁ (reflectᴮ H B s W′)
reflectᴱ H (block var b ∈ T is B end) (return v) W′ = Left (block₁ (return W′))
reflectᴱ H (function f ⟨ var x ∈ T ⟩∈ U is B end) (function a defn) (UnallocatedAddress ())
reflectᴱ H (binexp M op N) (binOp₀ ()) (UnallocatedAddress p)
reflectᴱ H (binexp M op N) (binOp₁ s) (BinOpMismatch₁ p) with preservationᴱ H M s
reflectᴱ H (binexp M op N) (binOp₁ s) (BinOpMismatch₁ p) | ok q = expr (BinOpMismatch₁ (subst₁ (BinOpWarning op) (sym q) p))
reflectᴱ H (binexp M op N) (binOp₁ s) (BinOpMismatch₁ p) | warning (heap W) = heap W
reflectᴱ H (binexp M op N) (binOp₁ s) (BinOpMismatch₁ p) | warning (expr W) = expr (bin₁ W)
reflectᴱ H (binexp M op N) (binOp₁ s) (BinOpMismatch₂ p) with heap-weakeningᴱ H N (rednᴱ⊑ s)
reflectᴱ H (binexp M op N) (binOp₁ s) (BinOpMismatch₂ p) | ok q = expr (BinOpMismatch₂ ((subst₁ (BinOpWarning op) (sym q) p)))
reflectᴱ H (binexp M op N) (binOp₁ s) (BinOpMismatch₂ p) | warning W = expr (bin₂ W)
reflectᴱ H (binexp M op N) (binOp₁ s) (bin₁ W′) with reflectᴱ H M s W′
reflectᴱ H (binexp M op N) (binOp₁ s) (bin₁ W′) | heap W = heap W
reflectᴱ H (binexp M op N) (binOp₁ s) (bin₁ W′) | expr W = expr (bin₁ W)
reflectᴱ H (binexp M op N) (binOp₁ s) (bin₂ W′) = expr (bin₂ (reflect-weakeningᴱ H N (rednᴱ⊑ s) W′))
reflectᴱ H (binexp M op N) (binOp₂ s) (BinOpMismatch₁ p) with heap-weakeningᴱ H M (rednᴱ⊑ s)
reflectᴱ H (binexp M op N) (binOp₂ s) (BinOpMismatch₁ p) | ok q = expr (BinOpMismatch₁ (subst₁ (BinOpWarning op) (sym q) p))
reflectᴱ H (binexp M op N) (binOp₂ s) (BinOpMismatch₁ p) | warning W = expr (bin₁ W)
reflectᴱ H (binexp M op N) (binOp₂ s) (BinOpMismatch₂ p) with preservationᴱ H N s
reflectᴱ H (binexp M op N) (binOp₂ s) (BinOpMismatch₂ p) | ok q = expr (BinOpMismatch₂ (subst₁ (BinOpWarning op) (sym q) p))
reflectᴱ H (binexp M op N) (binOp₂ s) (BinOpMismatch₂ p) | warning (heap W) = heap W
reflectᴱ H (binexp M op N) (binOp₂ s) (BinOpMismatch₂ p) | warning (expr W) = expr (bin₂ W)
reflectᴱ H (binexp M op N) (binOp₂ s) (bin₁ W′) = expr (bin₁ (reflect-weakeningᴱ H M (rednᴱ⊑ s) W′))
reflectᴱ H (binexp M op N) (binOp₂ s) (bin₂ W′) with reflectᴱ H N s W′
reflectᴱ H (binexp M op N) (binOp₂ s) (bin₂ W′) | heap W = heap W
reflectᴱ H (binexp M op N) (binOp₂ s) (bin₂ W′) | expr W = expr (bin₂ W)
reflectᴱ H (binexp M op N) (binOp₁ s) (BinOpMismatch₁ p) = Left (cond BinOpMismatch₁ bin₁ (reflect-subtypingᴱ H M s p))
reflectᴱ H (binexp M op N) (binOp₁ s) (BinOpMismatch₂ p) = Left (BinOpMismatch₂ (heap-weakeningᴱ ∅ H N (rednᴱ⊑ s) p))
reflectᴱ H (binexp M op N) (binOp₁ s) (bin₁ W′) = mapL bin₁ (reflectᴱ H M s W′)
reflectᴱ H (binexp M op N) (binOp₁ s) (bin₂ W′) = Left (bin₂ (reflect-weakeningᴱ ∅ H N (rednᴱ⊑ s) W′))
reflectᴱ H (binexp M op N) (binOp₂ s) (BinOpMismatch₁ p) = Left (BinOpMismatch₁ (heap-weakeningᴱ ∅ H M (rednᴱ⊑ s) p))
reflectᴱ H (binexp M op N) (binOp₂ s) (BinOpMismatch₂ p) = Left (cond BinOpMismatch₂ bin₂ (reflect-subtypingᴱ H N s p))
reflectᴱ H (binexp M op N) (binOp₂ s) (bin₁ W′) = Left (bin₁ (reflect-weakeningᴱ ∅ H M (rednᴱ⊑ s) W′))
reflectᴱ H (binexp M op N) (binOp₂ s) (bin₂ W′) = mapL bin₂ (reflectᴱ H N s W′)
reflectᴮ H (local var x ∈ T ← M ∙ B) (local s) (LocalVarMismatch p) with preservationᴱ H M s
reflectᴮ H (local var x ∈ T ← M ∙ B) (local s) (LocalVarMismatch p) | ok q = block (LocalVarMismatch (λ r → p (trans r q)))
reflectᴮ H (local var x ∈ T ← M ∙ B) (local s) (LocalVarMismatch p) | warning (expr W) = block (local₁ W)
reflectᴮ H (local var x ∈ T ← M ∙ B) (local s) (LocalVarMismatch p) | warning (heap W) = heap W
reflectᴮ H (local var x ∈ T ← M ∙ B) (local s) (local₁ W′) with reflectᴱ H M s W′
reflectᴮ H (local var x ∈ T ← M ∙ B) (local s) (local₁ W′) | heap W = heap W
reflectᴮ H (local var x ∈ T ← M ∙ B) (local s) (local₁ W′) | expr W = block (local₁ W)
reflectᴮ H (local var x ∈ T ← M ∙ B) (local s) (local₂ W′) = block (local₂ (reflect-weakeningᴮ H B (rednᴱ⊑ s) W′))
reflectᴮ H (local var x ∈ T ← M ∙ B) (subst v) W′ with remember (typeOfⱽ H v)
reflectᴮ H (local var x ∈ T ← M ∙ B) (subst v) W′ | (just S , p) with S ≡ᵀ T
reflectᴮ H (local var x ∈ T ← M ∙ B) (subst v) W′ | (just T , p) | yes refl = block (local₂ (reflect-substitutionᴮ H B v x (sym p) W′))
reflectᴮ H (local var x ∈ T ← M ∙ B) (subst v) W′ | (just S , p) | no q = block (LocalVarMismatch (λ r → q (trans (cong orNone (sym p)) (sym r))))
reflectᴮ H (local var x ∈ T ← M ∙ B) (subst v) W′ | (nothing , p) with typeOf-val-not-none v
reflectᴮ H (local var x ∈ T ← M ∙ B) (subst v) W′ | (nothing , p) | ok r = CONTRADICTION (r (cong orNone (sym p)))
reflectᴮ H (local var x ∈ T ← M ∙ B) (subst v) W′ | (nothing , p) | warning W = block (local₁ W)
reflectᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) (function a defn) W′ = block (function₂ (reflect-weakeningᴮ H B (snoc defn) (reflect-substitutionᴮ _ B (addr a) f refl W′)))
reflectᴮ H (return M ∙ B) (return s) (return W′) with reflectᴱ H M s W′
reflectᴮ H (return M ∙ B) (return s) (return W′) | heap W = heap W
reflectᴮ H (return M ∙ B) (return s) (return W′) | expr W = block (return W)
reflectᴮ H (local var x ∈ T ← M ∙ B) (local s) (LocalVarMismatch p) = Left (cond LocalVarMismatch local₁ (reflect-subtypingᴱ H M s p))
reflectᴮ H (local var x ∈ T ← M ∙ B) (local s) (local₁ W′) = mapL local₁ (reflectᴱ H M s W′)
reflectᴮ H (local var x ∈ T ← M ∙ B) (local s) (local₂ W′) = Left (local₂ (reflect-weakeningᴮ (x ↦ T) H B (rednᴱ⊑ s) W′))
reflectᴮ H (local var x ∈ T ← M ∙ B) (subst v) W′ = Left (cond local₂ (cond local₁ LocalVarMismatch) (reflect-substitutionᴮ H B v x W′))
reflectᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) (function a defn) W′ with reflect-substitutionᴮ _ B (addr a) f W′
reflectᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) (function a defn) W′ | Left W = Left (function₂ (reflect-weakeningᴮ (f ↦ (T ⇒ U)) H B (snoc defn) W))
reflectᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) (function a defn) W′ | Right (Left (UnallocatedAddress ()))
reflectᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) (function a defn) W′ | Right (Right p) = CONTRADICTION (≮:-refl p)
reflectᴮ H (return M ∙ B) (return s) (return W′) = mapL return (reflectᴱ H M s W′)
reflectᴴᴱ : ∀ H M {H′ M′} → (H ⊢ M ⟶ᴱ M′ ⊣ H′) → Warningᴴᴱ H′ (typeCheckᴴᴱ H′ ∅ M′) → Warningᴴᴱ H (typeCheckᴴᴱ H ∅ M)
reflectᴴᴮ : ∀ H B {H′ B′} → (H ⊢ B ⟶ᴮ B′ ⊣ H′) → Warningᴴᴮ H′ (typeCheckᴴᴮ H′ ∅ B′) → Warningᴴᴮ H (typeCheckᴴᴮ H ∅ B)
reflectᴴᴱ : ∀ H M {H′ M′} → (H ⊢ M ⟶ᴱ M′ ⊣ H′) → Warningᴴ H′ (typeCheckᴴ H′) → Either (Warningᴱ H (typeCheckᴱ H ∅ M)) (Warningᴴ H (typeCheckᴴ H))
reflectᴴᴮ : ∀ H B {H′ B′} → (H ⊢ B ⟶ᴮ B′ ⊣ H′) → Warningᴴ H′ (typeCheckᴴ H′) → Either (Warningᴮ H (typeCheckᴮ H ∅ B)) (Warningᴴ H (typeCheckᴴ H))
reflectᴴᴱ H M s (expr W′) = reflectᴱ H M s W′
reflectᴴᴱ H (function f ⟨ var x ∈ T ⟩∈ U is B end) (function a p) (heap (addr b refl W′)) with b ≡ᴬ a
reflectᴴᴱ H (function f ⟨ var x ∈ T ⟩∈ U is B end) (function a defn) (heap (addr a refl (FunctionDefnMismatch p))) | yes refl with heap-weakeningᴮ H B (snoc defn)
reflectᴴᴱ H (function f ⟨ var x ∈ T ⟩∈ U is B end) (function a defn) (heap (addr a refl (FunctionDefnMismatch p))) | yes refl | ok r = expr (FunctionDefnMismatch λ q → p (trans q r))
reflectᴴᴱ H (function f ⟨ var x ∈ T ⟩∈ U is B end) (function a defn) (heap (addr a refl (FunctionDefnMismatch p))) | yes refl | warning W = expr (function₁ W)
reflectᴴᴱ H (function f ⟨ var x ∈ T ⟩∈ U is B end) (function a defn) (heap (addr a refl (function₁ W′))) | yes refl = expr (function₁ (reflect-weakeningᴮ H B (snoc defn) W′))
reflectᴴᴱ H (function f ⟨ var x ∈ T ⟩∈ U is B end) (function a p) (heap (addr b refl W′)) | no r = heap (addr b (lookup-not-allocated p r) (reflect-weakeningᴼ H _ (snoc p) W′))
reflectᴴᴱ H (M $ N) (app₁ s) (heap W′) with reflectᴴᴱ H M s (heap W′)
reflectᴴᴱ H (M $ N) (app₁ s) (heap W′) | heap W = heap W
reflectᴴᴱ H (M $ N) (app₁ s) (heap W′) | expr W = expr (app₁ W)
reflectᴴᴱ H (M $ N) (app₂ p s) (heap W′) with reflectᴴᴱ H N s (heap W′)
reflectᴴᴱ H (M $ N) (app₂ p s) (heap W′) | heap W = heap W
reflectᴴᴱ H (M $ N) (app₂ p s) (heap W′) | expr W = expr (app₂ W)
reflectᴴᴱ H (M $ N) (beta O v p q) (heap W′) = heap W′
reflectᴴᴱ H (block var b ∈ T is B end) (block s) (heap W′) with reflectᴴᴮ H B s (heap W′)
reflectᴴᴱ H (block var b ∈ T is B end) (block s) (heap W′) | heap W = heap W
reflectᴴᴱ H (block var b ∈ T is B end) (block s) (heap W′) | block W = expr (block₁ W)
reflectᴴᴱ H (block var b ∈ T is return N ∙ B end) (return v) (heap W′) = heap W′
reflectᴴᴱ H (block var b ∈ T is done end) done (heap W′) = heap W′
reflectᴴᴱ H (binexp M op N) (binOp₀ s) (heap W′) = heap W′
reflectᴴᴱ H (binexp M op N) (binOp₁ s) (heap W′) with reflectᴴᴱ H M s (heap W′)
reflectᴴᴱ H (binexp M op N) (binOp₁ s) (heap W′) | heap W = heap W
reflectᴴᴱ H (binexp M op N) (binOp₁ s) (heap W′) | expr W = expr (bin₁ W)
reflectᴴᴱ H (binexp M op N) (binOp₂ s) (heap W′) with reflectᴴᴱ H N s (heap W′)
reflectᴴᴱ H (binexp M op N) (binOp₂ s) (heap W′) | heap W = heap W
reflectᴴᴱ H (binexp M op N) (binOp₂ s) (heap W′) | expr W = expr (bin₂ W)
reflectᴴᴱ H (M $ N) (app₁ s) W = mapL app₁ (reflectᴴᴱ H M s W)
reflectᴴᴱ H (M $ N) (app₂ v s) W = mapL app₂ (reflectᴴᴱ H N s W)
reflectᴴᴱ H (M $ N) (beta O v refl p) W = Right W
reflectᴴᴱ H (function f ⟨ var x ∈ T ⟩∈ U is B end) (function a p) (addr b refl W) with b ≡ᴬ a
reflectᴴᴱ H (function f ⟨ var x ∈ T ⟩∈ U is B end) (function a defn) (addr b refl (FunctionDefnMismatch p)) | yes refl = Left (FunctionDefnMismatch (heap-weakeningᴮ (x ↦ T) H B (snoc defn) p))
reflectᴴᴱ H (function f ⟨ var x ∈ T ⟩∈ U is B end) (function a defn) (addr b refl (function₁ W)) | yes refl = Left (function₁ (reflect-weakeningᴮ (x ↦ T) H B (snoc defn) W))
reflectᴴᴱ H (function f ⟨ var x ∈ T ⟩∈ U is B end) (function a p) (addr b refl W) | no q = Right (addr b (lookup-not-allocated p q) (reflect-weakeningᴼ H _ (snoc p) W))
reflectᴴᴱ H (block var b ∈ T is B end) (block s) W = mapL block₁ (reflectᴴᴮ H B s W)
reflectᴴᴱ H (block var b ∈ T is return (val v) ∙ B end) (return v) W = Right W
reflectᴴᴱ H (block var b ∈ T is done end) done W = Right W
reflectᴴᴱ H (binexp M op N) (binOp₀ s) W = Right W
reflectᴴᴱ H (binexp M op N) (binOp₁ s) W = mapL bin₁ (reflectᴴᴱ H M s W)
reflectᴴᴱ H (binexp M op N) (binOp₂ s) W = mapL bin₂ (reflectᴴᴱ H N s W)
reflectᴴᴮ H B s (block W′) = reflectᴮ H B s W′
reflectᴴᴮ H (local var x ∈ T ← M ∙ B) (local s) (heap W′) with reflectᴴᴱ H M s (heap W′)
reflectᴴᴮ H (local var x ∈ T ← M ∙ B) (local s) (heap W′) | heap W = heap W
reflectᴴᴮ H (local var x ∈ T ← M ∙ B) (local s) (heap W′) | expr W = block (local₁ W)
reflectᴴᴮ H (local var x ∈ T ← M ∙ B) (subst v) (heap W′) = heap W′
reflectᴴᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) (function a p) (heap (addr b refl W′)) with b ≡ᴬ a
reflectᴴᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) (function a defn) (heap (addr a refl (FunctionDefnMismatch p))) | yes refl with heap-weakeningᴮ H C (snoc defn)
reflectᴴᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) (function a defn) (heap (addr a refl (FunctionDefnMismatch p))) | yes refl | ok r = block (FunctionDefnMismatch (λ q → p (trans q r)))
reflectᴴᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) (function a defn) (heap (addr a refl (FunctionDefnMismatch p))) | yes refl | warning W = block (function₁ W)
reflectᴴᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) (function a defn) (heap (addr a refl (function₁ W′))) | yes refl = block (function₁ (reflect-weakeningᴮ H C (snoc defn) W′))
reflectᴴᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) (function a p) (heap (addr b refl W′)) | no r = heap (addr b (lookup-not-allocated p r) (reflect-weakeningᴼ H _ (snoc p) W′))
reflectᴴᴮ H (return M ∙ B) (return s) (heap W′) with reflectᴴᴱ H M s (heap W′)
reflectᴴᴮ H (return M ∙ B) (return s) (heap W′) | heap W = heap W
reflectᴴᴮ H (return M ∙ B) (return s) (heap W′) | expr W = block (return W)
reflectᴴᴮ H (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) (function a p) (addr b refl W) with b ≡ᴬ a
reflectᴴᴮ H (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) (function a defn) (addr b refl (FunctionDefnMismatch p)) | yes refl = Left (FunctionDefnMismatch (heap-weakeningᴮ (x ↦ T) H C (snoc defn) p))
reflectᴴᴮ H (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) (function a defn) (addr b refl (function₁ W)) | yes refl = Left (function₁ (reflect-weakeningᴮ (x ↦ T) H C (snoc defn) W))
reflectᴴᴮ H (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) (function a p) (addr b refl W) | no q = Right (addr b (lookup-not-allocated p q) (reflect-weakeningᴼ H _ (snoc p) W))
reflectᴴᴮ H (local var x ∈ T ← M ∙ B) (local s) W = mapL local₁ (reflectᴴᴱ H M s W)
reflectᴴᴮ H (local var x ∈ T ← M ∙ B) (subst v) W = Right W
reflectᴴᴮ H (return M ∙ B) (return s) W = mapL return (reflectᴴᴱ H M s W)
reflect* : ∀ H B {H′ B′} → (H ⊢ B ⟶* B′ ⊣ H′) → Warningᴴᴮ H′ (typeCheckᴴᴮ H′ ∅ B′) → Warningᴴᴮ H (typeCheckᴴᴮ H ∅ B)
reflect* : ∀ H B {H′ B′} → (H ⊢ B ⟶* B′ ⊣ H′) → Either (Warningᴮ H′ (typeCheckᴮ H′ ∅ B′)) (Warningᴴ H′ (typeCheckᴴ H′)) → Either (Warningᴮ H (typeCheckᴮ H ∅ B)) (Warningᴴ H (typeCheckᴴ H))
reflect* H B refl W = W
reflect* H B (step s t) W = reflectᴴᴮ H B s (reflect* _ _ t W)
reflect* H B (step s t) W = cond (reflectᴮ H B s) (reflectᴴᴮ H B s) (reflect* _ _ t W)
runtimeBinOpWarning : ∀ H {op} v → BinOpError op (valueType v) → BinOpWarning op (orNone (typeOfⱽ H v))
runtimeBinOpWarning H v (+ p) = + (λ q → p (mustBeNumber H ∅ v q))
runtimeBinOpWarning H v (- p) = - (λ q → p (mustBeNumber H ∅ v q))
runtimeBinOpWarning H v (* p) = * (λ q → p (mustBeNumber H ∅ v q))
runtimeBinOpWarning H v (/ p) = / (λ q → p (mustBeNumber H ∅ v q))
runtimeBinOpWarning H v (< p) = < (λ q → p (mustBeNumber H ∅ v q))
runtimeBinOpWarning H v (> p) = > (λ q → p (mustBeNumber H ∅ v q))
runtimeBinOpWarning H v (<= p) = <= (λ q → p (mustBeNumber H ∅ v q))
runtimeBinOpWarning H v (>= p) = >= (λ q → p (mustBeNumber H ∅ v q))
runtimeBinOpWarning H v (·· p) = ·· (λ q → p (mustBeString H ∅ v q))
isntNumber : ∀ H v → (valueType v ≢ number) → (typeOfᴱ H ∅ (val v) ≮: number)
isntNumber H nil p = scalar-≢-impl-≮: nil number (λ ())
isntNumber H (addr a) p with remember (H [ a ]ᴴ)
isntNumber H (addr a) p | (just (function f ⟨ var x ∈ T ⟩∈ U is B end) , q) = ≡-trans-≮: (cong orAny (cong typeOfᴹᴼ q)) (function-≮:-scalar number)
isntNumber H (addr a) p | (nothing , q) = ≡-trans-≮: (cong orAny (cong typeOfᴹᴼ q)) (any-≮:-scalar number)
isntNumber H (number x) p = CONTRADICTION (p refl)
isntNumber H (bool x) p = scalar-≢-impl-≮: boolean number (λ ())
isntNumber H (string x) p = scalar-≢-impl-≮: string number (λ ())
isntString : ∀ H v → (valueType v ≢ string) → (typeOfᴱ H ∅ (val v) ≮: string)
isntString H nil p = scalar-≢-impl-≮: nil string (λ ())
isntString H (addr a) p with remember (H [ a ]ᴴ)
isntString H (addr a) p | (just (function f ⟨ var x ∈ T ⟩∈ U is B end) , q) = ≡-trans-≮: (cong orAny (cong typeOfᴹᴼ q)) (function-≮:-scalar string)
isntString H (addr a) p | (nothing , q) = ≡-trans-≮: (cong orAny (cong typeOfᴹᴼ q)) (any-≮:-scalar string)
isntString H (number x) p = scalar-≢-impl-≮: number string (λ ())
isntString H (bool x) p = scalar-≢-impl-≮: boolean string (λ ())
isntString H (string x) p = CONTRADICTION (p refl)
isntFunction : ∀ H v {T U} → (valueType v ≢ function) → (typeOfᴱ H ∅ (val v) ≮: (T ⇒ U))
isntFunction H nil p = scalar-≮:-function nil
isntFunction H (addr a) p = CONTRADICTION (p refl)
isntFunction H (number x) p = scalar-≮:-function number
isntFunction H (bool x) p = scalar-≮:-function boolean
isntFunction H (string x) p = scalar-≮:-function string
isntEmpty : ∀ H v → (typeOfᴱ H ∅ (val v) ≮: none)
isntEmpty H nil = scalar-≮:-none nil
isntEmpty H (addr a) with remember (H [ a ]ᴴ)
isntEmpty H (addr a) | (just (function f ⟨ var x ∈ T ⟩∈ U is B end) , p) = ≡-trans-≮: (cong orAny (cong typeOfᴹᴼ p)) function-≮:-none
isntEmpty H (addr a) | (nothing , p) = ≡-trans-≮: (cong orAny (cong typeOfᴹᴼ p)) any-≮:-none
isntEmpty H (number x) = scalar-≮:-none number
isntEmpty H (bool x) = scalar-≮:-none boolean
isntEmpty H (string x) = scalar-≮:-none string
runtimeBinOpWarning : ∀ H {op} v → BinOpError op (valueType v) → (typeOfᴱ H ∅ (val v) ≮: srcBinOp op)
runtimeBinOpWarning H v (+ p) = isntNumber H v p
runtimeBinOpWarning H v (- p) = isntNumber H v p
runtimeBinOpWarning H v (* p) = isntNumber H v p
runtimeBinOpWarning H v (/ p) = isntNumber H v p
runtimeBinOpWarning H v (< p) = isntNumber H v p
runtimeBinOpWarning H v (> p) = isntNumber H v p
runtimeBinOpWarning H v (<= p) = isntNumber H v p
runtimeBinOpWarning H v (>= p) = isntNumber H v p
runtimeBinOpWarning H v (·· p) = isntString H v p
runtimeWarningᴱ : ∀ H M → RuntimeErrorᴱ H M → Warningᴱ H (typeCheckᴱ H ∅ M)
runtimeWarningᴮ : ∀ H B → RuntimeErrorᴮ H B → Warningᴮ H (typeCheckᴮ H ∅ B)
runtimeWarningᴱ H (var x) UnboundVariable = UnboundVariable refl
runtimeWarningᴱ H (val (addr a)) (SEGV p) = UnallocatedAddress p
runtimeWarningᴱ H (M $ N) (FunctionMismatch v w p) with typeOf-val-not-none w
runtimeWarningᴱ H (M $ N) (FunctionMismatch v w p) | ok q = FunctionCallMismatch (λ r → p (mustBeFunction H ∅ v (λ r′ → q (trans r′ r))))
runtimeWarningᴱ H (M $ N) (FunctionMismatch v w p) | warning W = app₂ W
runtimeWarningᴱ H (M $ N) (FunctionMismatch v w p) = FunctionCallMismatch (any-src-≮: (isntEmpty H w) (isntFunction H v p))
runtimeWarningᴱ H (M $ N) (app₁ err) = app₁ (runtimeWarningᴱ H M err)
runtimeWarningᴱ H (M $ N) (app₂ err) = app₂ (runtimeWarningᴱ H N err)
runtimeWarningᴱ H (block var b ∈ T is B end) (block err) = block₁ (runtimeWarningᴮ H B err)
@ -469,6 +343,6 @@ runtimeWarningᴮ H (local var x ∈ T ← M ∙ B) (local err) = local₁ (runt
runtimeWarningᴮ H (return M ∙ B) (return err) = return (runtimeWarningᴱ H M err)
wellTypedProgramsDontGoWrong : ∀ H′ B B′ → (∅ᴴ ⊢ B ⟶* B′ ⊣ H′) → (RuntimeErrorᴮ H′ B′) → Warningᴮ ∅ᴴ (typeCheckᴮ ∅ᴴ ∅ B)
wellTypedProgramsDontGoWrong H′ B B′ t err with reflect* ∅ᴴ B t (block (runtimeWarningᴮ H′ B′ err))
wellTypedProgramsDontGoWrong H′ B B′ t err | heap (addr a refl ())
wellTypedProgramsDontGoWrong H′ B B′ t err | block W = W
wellTypedProgramsDontGoWrong H′ B B′ t err with reflect* ∅ᴴ B t (Left (runtimeWarningᴮ H′ B′ err))
wellTypedProgramsDontGoWrong H′ B B′ t err | Right (addr a refl ())
wellTypedProgramsDontGoWrong H′ B B′ t err | Left W = W

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{-# 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; any; none; 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; none; any; _⇒_; _∪_; _∩_; tgt)
open import Properties.Contradiction using (CONTRADICTION; ¬)
open import Properties.Equality using (_≢_)
open import Properties.Functions 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 none t = Left none
dec-language any t = Right any
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) none (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)
any-≮:-scalar : ∀ {U} → (Scalar U) → (any ≮: U)
any-≮:-scalar s = witness (function-ok (scalar s)) any (scalar-function-ok s)
scalar-≮:-none : ∀ {U} → (Scalar U) → (U ≮: none)
scalar-≮:-none s = witness (scalar s) (scalar s) none
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 = none} (scalar ())
tgt-function-ok {T = any} p = any
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 any = any
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-none-≮: : ∀ {T U} → (tgt T ≮: U) → (T ≮: (skalar ∪ (none ⇒ U)))
tgt-none-≮: (witness t p q) = witness (function-ok t) (tgt-function-ok p) (skalar-function-ok , function-ok q)
none-tgt-≮: : ∀ {T U} → (T ≮: (skalar ∪ (none ⇒ U))) → (tgt T ≮: U)
none-tgt-≮: (witness (scalar s) p (q₁ , q₂)) = CONTRADICTION (≮:-refl (witness (scalar s) (skalar-scalar s) q₁))
none-tgt-≮: (witness function p (q₁ , scalar-function ()))
none-tgt-≮: (witness (function-ok t) p (q₁ , function-ok q₂)) = witness t (function-ok-tgt p) q₂
none-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} none = scalar-function-err nil
function-err-src {T = T₁ ⇒ T₂} p = function-err p
function-err-src {T = none} (scalar-scalar number () p)
function-err-src {T = none} (scalar-function-ok ())
function-err-src {T = any} none = any
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 = none} any = none
¬function-err-src {T = any} (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 = none
src-¬function-err {T = T₁ ⇒ T₂} (function-err p) = p
src-¬function-err {T = none} (scalar-function-err ())
src-¬function-err {T = any} p = none
src-¬function-err {T = boolean} p = none
src-¬function-err {T = number} p = none
src-¬function-err {T = string} p = none
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) = none
src-¬scalar boolean (scalar boolean) = none
src-¬scalar string (scalar string) = none
src-¬scalar nil (scalar nil) = none
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 any = none
src-any-≮: : ∀ {T U} → (T ≮: src U) → (U ≮: (T ⇒ any))
src-any-≮: (witness t p q) = witness (function-err t) (function-err-src q) (¬function-err-src p)
any-src-≮: : ∀ {S T U} → (U ≮: S) → (T ≮: (U ⇒ any)) → (U ≮: src T)
any-src-≮: (witness t x x₁) (witness (scalar s) p (function-scalar s)) = witness t x (src-¬scalar s p)
any-src-≮: r (witness (function-ok (scalar s)) p (function-ok (scalar-scalar s () q)))
any-src-≮: r (witness (function-ok (function-ok _)) p (function-ok (scalar-function-ok ())))
any-src-≮: r (witness (function-err t) p (function-err q)) = witness t q (src-¬function-err p)
-- Properties of any and none
any-≮: : ∀ {T U} → (T ≮: U) → (any ≮: U)
any-≮: (witness t p q) = witness t any q
none-≮: : ∀ {T U} → (T ≮: U) → (T ≮: none)
none-≮: (witness t p q) = witness t p none
any-≮:-none : (any ≮: none)
any-≮:-none = witness (scalar nil) any none
function-≮:-none : ∀ {T U} → ((T ⇒ U) ≮: none)
function-≮:-none = witness function function none

16
prototyping/Properties/TypeCheck.agda
View File

@ -8,7 +8,7 @@ open import Agda.Builtin.Equality using (_≡_; refl)
open import Agda.Builtin.Bool using (Bool; true; false)
open import FFI.Data.Maybe using (Maybe; just; nothing)
open import FFI.Data.Either using (Either)
open import Luau.TypeCheck(m) using (_⊢ᴱ_∈_; _⊢ᴮ_∈_; ⊢ᴼ_; ⊢ᴴ_; _⊢ᴴᴱ_▷_∈_; _⊢ᴴᴮ_▷_∈_; nil; var; addr; number; bool; string; app; function; block; binexp; done; return; local; nothing; orNone; tgtBinOp)
open import Luau.TypeCheck(m) using (_⊢ᴱ_∈_; _⊢ᴮ_∈_; ⊢ᴼ_; ⊢ᴴ_; _⊢ᴴᴱ_▷_∈_; _⊢ᴴᴮ_▷_∈_; nil; var; addr; number; bool; string; app; function; block; binexp; done; return; local; nothing; orAny; tgtBinOp)
open import Luau.Syntax using (Block; Expr; Value; BinaryOperator; yes; nil; addr; number; bool; string; val; var; binexp; _$_; function_is_end; block_is_end; _∙_; return; done; local_←_; _⟨_⟩; _⟨_⟩∈_; var_∈_; name; fun; arg; +; -; *; /; <; >; ==; ~=; <=; >=)
open import Luau.Type using (Type; nil; any; none; number; boolean; string; _⇒_; tgt)
open import Luau.RuntimeType using (RuntimeType; nil; number; function; string; valueType)
@ -42,8 +42,8 @@ typeOfⱽ H (string x) = just string
typeOfᴱ : Heap yes → VarCtxt → (Expr yes) → Type
typeOfᴮ : Heap yes → VarCtxt → (Block yes) → Type
typeOfᴱ H Γ (var x) = orNone(Γ [ x ]ⱽ)
typeOfᴱ H Γ (val v) = orNone(typeOfⱽ H v)
typeOfᴱ H Γ (var x) = orAny(Γ [ x ]ⱽ)
typeOfᴱ H Γ (val v) = orAny(typeOfⱽ H v)
typeOfᴱ H Γ (M $ N) = tgt(typeOfᴱ H Γ M)
typeOfᴱ H Γ (function f ⟨ var x ∈ S ⟩∈ T is B end) = S ⇒ T
typeOfᴱ H Γ (block var b ∈ T is B end) = T
@ -64,17 +64,17 @@ mustBeFunction H Γ (string x) p = CONTRADICTION (p refl)
mustBeNumber : ∀ H Γ v → (typeOfᴱ H Γ (val v) ≡ number) → (valueType(v) ≡ number)
mustBeNumber H Γ (addr a) p with remember (H [ a ]ᴴ)
mustBeNumber H Γ (addr a) p | (just O , q) with trans (cong orNone (cong typeOfᴹᴼ (sym q))) p
mustBeNumber H Γ (addr a) p | (just O , q) with trans (cong orAny (cong typeOfᴹᴼ (sym q))) p
mustBeNumber H Γ (addr a) p | (just function f ⟨ var x ∈ T ⟩∈ U is B end , q) | ()
mustBeNumber H Γ (addr a) p | (nothing , q) with trans (cong orNone (cong typeOfᴹᴼ (sym q))) p
mustBeNumber H Γ (addr a) p | (nothing , q) with trans (cong orAny (cong typeOfᴹᴼ (sym q))) p
mustBeNumber H Γ (addr a) p | nothing , q | ()
mustBeNumber H Γ (number n) p = refl
mustBeString : ∀ H Γ v → (typeOfᴱ H Γ (val v) ≡ string) → (valueType(v) ≡ string)
mustBeString H Γ (addr a) p with remember (H [ a ]ᴴ)
mustBeString H Γ (addr a) p | (just O , q) with trans (cong orNone (cong typeOfᴹᴼ (sym q))) p
mustBeString H Γ (addr a) p | (just O , q) with trans (cong orAny (cong typeOfᴹᴼ (sym q))) p
mustBeString H Γ (addr a) p | (just function f ⟨ var x ∈ T ⟩∈ U is B end , q) | ()
mustBeString H Γ (addr a) p | (nothing , q) with trans (cong orNone (cong typeOfᴹᴼ (sym q))) p
mustBeString H Γ (addr a) p | (nothing , q) with trans (cong orAny (cong typeOfᴹᴼ (sym q))) p
mustBeString H Γ (addr a) p | (nothing , q) | ()
mustBeString H Γ (string x) p = refl
@ -83,7 +83,7 @@ typeCheckᴮ : ∀ H Γ B → (Γ ⊢ᴮ B ∈ (typeOfᴮ H Γ B))
typeCheckᴱ H Γ (var x) = var refl
typeCheckᴱ H Γ (val nil) = nil
typeCheckᴱ H Γ (val (addr a)) = addr (orNone (typeOfᴹᴼ (H [ a ]ᴴ)))
typeCheckᴱ H Γ (val (addr a)) = addr (orAny (typeOfᴹᴼ (H [ a ]ᴴ)))
typeCheckᴱ H Γ (val (number n)) = number
typeCheckᴱ H Γ (val (bool b)) = bool
typeCheckᴱ H Γ (val (string x)) = string

1
prototyping/Tests/Interpreter/concat_number_and_string/out.txt
View File

@ -9,3 +9,4 @@ value 37.0 is not a string
TYPE ERROR:
Local variable y has type string but expression has type number
because provided type contains v, where v is a number