Prototyping: add semantic subtyping (#424)

Adds subtyping to strict mode.

prototyping/FFI/Data/Aeson.agda

@@ -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)
 

prototyping/FFI/Data/Either.agda

@@ -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)
+swapLR : ∀ {A B} → Either A B → Either B A
-mapLeft f (Left x) = Left (f x)
+swapLR (Left x) = Right x
-mapLeft f (Right x) = Right x
+swapLR (Right x) = Left x
 
-mapRight : ∀ {A B C} → (B → C) → (Either A B) → (Either A C)
+mapL : ∀ {A B C} → (A → B) → Either A C → Either B C
-mapRight f (Left x) = Left x
+mapL f (Left x) = Left (f x)
-mapRight f (Right x) = Right (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

prototyping/Luau/RuntimeError.agda

@@ -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)

prototyping/Luau/StrictMode.agda

@@ -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 Luau.TypeCheck(strict) using (_⊢ᴮ_∈_; _⊢ᴱ_∈_; ⊢ᴴ_; ⊢ᴼ_; _⊢ᴴᴱ_▷_∈_; _⊢ᴴᴮ_▷_∈_; var; addr; app; binexp; block; return; local; function; srcBinOp)
-open import Properties.Equality using (_≢_)
+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)
 

prototyping/Luau/StrictMode/ToString.agda

@@ -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 (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
+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) (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
+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"

prototyping/Luau/Subtyping.agda

@@ -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

prototyping/Luau/Type.agda

@@ -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

prototyping/Luau/TypeCheck.agda

@@ -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
+orAny : Maybe Type → Type
-orNone nothing = none
+orAny nothing = any
-orNone (just T) = T
+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
 

prototyping/Properties.agda

@@ -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

prototyping/Properties/Functions.agda

@@ -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)

prototyping/Properties/StrictMode.agda

@@ -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.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; orNone; tgtBinOp)
+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.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.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
+heap-weakeningᴱ : ∀ Γ H M {H′ U} → (H ⊑ H′) → (typeOfᴱ H′ Γ M ≮: U) → (typeOfᴱ H Γ M ≮: U)
-  ok : A → OrWarningᴱ H D A
+heap-weakeningᴱ Γ H (var x) h p = p
-  warning : Warningᴱ H D → OrWarningᴱ H D A
+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
+heap-weakeningᴮ : ∀ Γ H B {H′ U} → (H ⊑ H′) → (typeOfᴮ H′ Γ B ≮: U) → (typeOfᴮ H Γ B ≮: U)
-  ok : A → OrWarningᴮ H D A
+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
-  warning : Warningᴮ H D → OrWarningᴮ H D A
+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
+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)
-  ok : A → OrWarningᴴᴱ H D A
+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)
-  warning : Warningᴴᴱ H D → OrWarningᴴᴱ H D A
+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
+substitutivityᴱ H (var y) v x p = substitutivityᴱ-whenever H v x y (x ≡ⱽ y) p
-  ok : A → OrWarningᴴᴮ H D A
+substitutivityᴱ H (val w) v x p = Left p
-  warning : Warningᴴᴮ H D → OrWarningᴴᴮ H D A
+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)
+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
-heap-weakeningᴮ : ∀ H B {H′ Γ} → (H ⊑ H′) → OrWarningᴮ H (typeCheckᴮ H Γ B) (typeOfᴮ H Γ B ≡ typeOfᴮ H′ Γ B)
+substitutivityᴮ H (local var y ∈ T ← M ∙ B) v x p = substitutivityᴮ-unless H B v x y (x ≡ⱽ y) p
-
-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 (return M ∙ B) v x p = substitutivityᴱ H M v x p
-substitutivityᴮ H done v x p = refl
+substitutivityᴮ H done v x p = Left p
-substitutivityᴮ-unless-yes H B v x x refl q refl = refl
+substitutivityᴮ-unless H B v x y (yes p) q = substitutivityᴮ-unless-yes H B v x y p (⊕-over p) q
-substitutivityᴮ-unless-no H B v x y p q refl = substitutivityᴮ H B v x 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′)
+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))
-preservationᴮ : ∀ H B {H′ B′} → (H ⊢ B ⟶ᴮ B′ ⊣ H′) → OrWarningᴴᴮ H (typeCheckᴴᴮ H ∅ B) (typeOfᴮ H ∅ B ≡ typeOfᴮ H′ ∅ B′)
+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
+reflect-subtypingᴱ H (M $ N) (app₁ s) p = mapLR none-tgt-≮: app₁ (reflect-subtypingᴱ H M s (tgt-none-≮: p))
-preservationᴱ H (M $ N) (app₁ s) with preservationᴱ H M s
+reflect-subtypingᴱ H (M $ N) (app₂ v s) p = Left (none-tgt-≮: (heap-weakeningᴱ ∅ H M (rednᴱ⊑ s) (tgt-none-≮: p)))
-preservationᴱ H (M $ N) (app₁ s) | ok p = ok (cong tgt 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)
-preservationᴱ H (M $ N) (app₁ s) | warning (expr W) = warning (expr (app₁ W))
+reflect-subtypingᴱ H (function f ⟨ var x ∈ T ⟩∈ U is B end) (function a defn) p = Left p
-preservationᴱ H (M $ N) (app₁ s) | warning (heap W) = warning (heap W)
+reflect-subtypingᴱ H (block var b ∈ T is B end) (block s) p = Left p
-preservationᴱ H (M $ N) (app₂ p s) with heap-weakeningᴱ H M (rednᴱ⊑ s)
+reflect-subtypingᴱ H (block var b ∈ T is return (val v) ∙ B end) (return v) p = mapR BlockMismatch (swapLR (≮:-trans p))
-preservationᴱ H (M $ N) (app₂ p s) | ok q = ok (cong tgt q)
+reflect-subtypingᴱ H (block var b ∈ T is done end) done p = mapR BlockMismatch (swapLR (≮:-trans p))
-preservationᴱ H (M $ N) (app₂ p s) | warning W  = warning (expr (app₁ W))
+reflect-subtypingᴱ H (binexp M op N) (binOp₀ s) p = Left (≡-trans-≮: (binOpPreservation H s) p)
-preservationᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ S ⟩∈ T is B end) v refl p) with remember (typeOfⱽ H v)
+reflect-subtypingᴱ H (binexp M op N) (binOp₁ s) p = Left p
-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
+reflect-subtypingᴱ H (binexp M op N) (binOp₂ s) p = Left 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 | 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
 
-preservationᴮ H (local var x ∈ T ← M ∙ B) (local s) with heap-weakeningᴮ H B (rednᴱ⊑ s)
+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)
-preservationᴮ H (local var x ∈ T ← M ∙ B) (local s) | ok p = ok p
+reflect-subtypingᴮ H (local var x ∈ T ← M ∙ B) (local s) p = Left (heap-weakeningᴮ (x ↦ T) H B (rednᴱ⊑ s) p)
-preservationᴮ H (local var x ∈ T ← M ∙ B) (local s) | warning W = warning (block (local₂ W))
+reflect-subtypingᴮ H (local var x ∈ T ← M ∙ B) (subst v) p = mapR LocalVarMismatch (substitutivityᴮ H B v x p) 
-preservationᴮ H (local var x ∈ T ← M ∙ B) (subst v) with remember (typeOfⱽ H v)
+reflect-subtypingᴮ H (return M ∙ B) (return s) p = mapR return (reflect-subtypingᴱ H M s p)
-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-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 : ∀ {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-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 : ∀ {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-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 → 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 : ∀ {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 : ∀ {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) → (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-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) → (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-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 W = reflect-substitutionᴱ-whenever H v x y (x ≡ⱽ y) W
-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 (val (addr a)) v x (UnallocatedAddress r) = Left (UnallocatedAddress r)
-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 (M $ N) v x (FunctionCallMismatch p) with substitutivityᴱ H N v x p
-reflect-substitutionᴱ H (val (addr a)) v x p (UnallocatedAddress r) = UnallocatedAddress r
+reflect-substitutionᴱ H (M $ N) v x (FunctionCallMismatch p) | Right W = Right (Right W)
-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 (FunctionCallMismatch p) | Left q with substitutivityᴱ H M v x (src-any-≮: q)
-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 (FunctionCallMismatch p) | Left q | Left r = Left ((FunctionCallMismatch ∘ any-src-≮: q) r)
-reflect-substitutionᴱ H (M $ N) v x p (app₂ W) = app₂ (reflect-substitutionᴱ H N v x p W)
+reflect-substitutionᴱ H (M $ N) v x (FunctionCallMismatch p) | Left q | Right W = Right (Right 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 (M $ N) v x (app₁ W) = mapL app₁ (reflect-substitutionᴱ H M v x W)
-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 (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 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 (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 p (function₁ W) with (x ≡ⱽ y)
+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 (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 (block var b ∈ T is B end) v x (BlockMismatch q) =  mapLR BlockMismatch Right (substitutivityᴮ H B v x q)
-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 (block₁ W′) = mapL block₁ (reflect-substitutionᴮ H B v x 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 (binexp M op N) v x (BinOpMismatch₁ q) = mapLR BinOpMismatch₁ Right (substitutivityᴱ H M v x q)
-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) v x (BinOpMismatch₂ q) = mapLR BinOpMismatch₂ Right (substitutivityᴱ H N v x q)
-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) v x (bin₁ W) = mapL bin₁ (reflect-substitutionᴱ H M v x W)
-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) v x (bin₂ W) = mapL bin₂ (reflect-substitutionᴱ H N v x W)
-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-whenever-no H v x y p q (UnboundVariable r) = UnboundVariable (trans (sym (⊕-lookup-miss x y _ _ p)) r)
+reflect-substitutionᴱ-whenever H a x x (yes refl) (UnallocatedAddress p) = Right (Left (UnallocatedAddress p))
-reflect-substitutionᴱ-whenever-yes H (addr a) x x refl p (UnallocatedAddress q) with trans p (cong typeOfᴹᴼ q)
+reflect-substitutionᴱ-whenever H v x y (no p) (UnboundVariable q) = Left (UnboundVariable (trans (sym (⊕-lookup-miss x y _ _ p)) q))
-reflect-substitutionᴱ-whenever-yes H (addr a) x x refl p (UnallocatedAddress 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 (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 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 (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 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 (function₂ W) = mapL function₂ (reflect-substitutionᴮ-unless H B v x f (x ≡ⱽ f) W)
-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 (local var y ∈ T ← M ∙ B) v x (LocalVarMismatch q) = mapLR LocalVarMismatch Right (substitutivityᴱ H M v x q)
-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 (local var y ∈ T ← M ∙ B) v x (local₁ W) = mapL local₁ (reflect-substitutionᴱ H M v x 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 (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 (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) v x p (function₂ W) with (x ≡ⱽ f)
+reflect-substitutionᴮ H (return M ∙ B) v x (return W) = mapL return (reflect-substitutionᴱ H M v x W)
-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-unless-yes H B v x y r p refl W = 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-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 (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-weakeningᴱ : ∀ Γ H M {H′} → (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-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 Γ 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 Γ 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 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 H (M $ N) h (FunctionCallMismatch p) | ok q₁ | ok q₂ = FunctionCallMismatch (λ r → p (trans (cong src (sym q₁)) (trans r q₂)))
+reflect-weakeningᴱ Γ H (M $ N) h (app₁ W) = app₁ (reflect-weakeningᴱ Γ H M h W)
-reflect-weakeningᴱ H (M $ N) h (FunctionCallMismatch p) | warning W | _ = app₁ W
+reflect-weakeningᴱ Γ H (M $ N) h (app₂ W) = app₂ (reflect-weakeningᴱ Γ H N h W)
-reflect-weakeningᴱ H (M $ N) h (FunctionCallMismatch p) | _ | warning W = app₂ W
+reflect-weakeningᴱ Γ H (binexp M op N) h (BinOpMismatch₁ p) = BinOpMismatch₁ (heap-weakeningᴱ Γ H M h p)
-reflect-weakeningᴱ H (M $ N) h (app₁ W) = app₁ (reflect-weakeningᴱ H M h W)
+reflect-weakeningᴱ Γ H (binexp M op N) h (BinOpMismatch₂ p) = BinOpMismatch₂ (heap-weakeningᴱ Γ H N h p)
-reflect-weakeningᴱ H (M $ N) h (app₂ W) = app₂ (reflect-weakeningᴱ H N h W)
+reflect-weakeningᴱ Γ H (binexp M op N) h (bin₁ W′) = bin₁ (reflect-weakeningᴱ Γ H M h W′)
-reflect-weakeningᴱ H (binexp M op N) h (BinOpMismatch₁ p) with heap-weakeningᴱ H M h
+reflect-weakeningᴱ Γ H (binexp M op N) h (bin₂ W′) = bin₂ (reflect-weakeningᴱ Γ H N h W′)
-reflect-weakeningᴱ H (binexp M op N) h (BinOpMismatch₁ p) | ok q = BinOpMismatch₁ (subst₁ (BinOpWarning op) (sym q) p)
+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 H (binexp M op N) h (BinOpMismatch₁ p) | warning W = bin₁ W
+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 H (binexp M op N) h (BinOpMismatch₂ p) with heap-weakeningᴱ H N h
+reflect-weakeningᴱ Γ H (block var b ∈ T is B end) h (BlockMismatch p) = BlockMismatch (heap-weakeningᴮ Γ H B h p)
-reflect-weakeningᴱ H (binexp M op N) h (BinOpMismatch₂ p) | ok q = BinOpMismatch₂ (subst₁ (BinOpWarning op) (sym q) p)
+reflect-weakeningᴱ Γ H (block var b ∈ T is B end) h (block₁ W) = block₁ (reflect-weakeningᴮ Γ H B h W)
-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-weakeningᴮ H (return M ∙ B) h (return W) = return (reflect-weakeningᴱ H M h W)
+reflect-weakeningᴮ Γ H (return M ∙ B) h (return W) = return (reflect-weakeningᴱ Γ H M h W)
-reflect-weakeningᴮ H (local var y ∈ T ← M ∙ B) h (LocalVarMismatch p) with heap-weakeningᴱ H M h
+reflect-weakeningᴮ Γ H (local var y ∈ T ← M ∙ B) h (LocalVarMismatch p) = LocalVarMismatch (heap-weakeningᴱ Γ H M h p)
-reflect-weakeningᴮ H (local var y ∈ T ← M ∙ B) h (LocalVarMismatch p) | ok q = LocalVarMismatch (λ r → p (trans r q))
+reflect-weakeningᴮ Γ H (local var y ∈ T ← M ∙ B) h (local₁ W) = local₁ (reflect-weakeningᴱ Γ H M h W)
-reflect-weakeningᴮ H (local var y ∈ T ← M ∙ B) h (LocalVarMismatch p) | warning W = local₁ 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 H (local var y ∈ T ← M ∙ B) h (local₁ W) = local₁ (reflect-weakeningᴱ H M 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 H (local var y ∈ T ← M ∙ B) h (local₂ W) = local₂ (reflect-weakeningᴮ H B h W)
+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 H (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) h (FunctionDefnMismatch p) with heap-weakeningᴮ H C h
+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-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-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) = FunctionDefnMismatch (heap-weakeningᴮ (x ↦ T) H B h p)
-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 (function₁ W) = function₁ (reflect-weakeningᴮ (x ↦ T) H B h W)
-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′)
 
-reflectᴱ : ∀ H M {H′ M′} → (H ⊢ M ⟶ᴱ M′ ⊣ H′) → Warningᴱ H′ (typeCheckᴱ H′ ∅ M′) → Warningᴴᴱ H (typeCheckᴴᴱ H ∅ M)
+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′) → Warningᴴᴮ H (typeCheckᴴᴮ H ∅ B)
+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) = 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) (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) (app₁ W′) = mapL app₁ (reflectᴱ H M s W′)
-reflectᴱ H (M $ N) (app₁ s) (FunctionCallMismatch p) | warning (expr W) | _ = expr (app₁ W)
+reflectᴱ H (M $ N) (app₁ s) (app₂ W′) = Left (app₂ (reflect-weakeningᴱ ∅ H N (rednᴱ⊑ s) W′))
-reflectᴱ H (M $ N) (app₁ s) (FunctionCallMismatch p) | warning (heap W) | _ = heap 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) (app₁ s) (FunctionCallMismatch p) | _ | warning W  = expr (app₂ W)
+reflectᴱ H (M $ N) (app₂ p s) (app₁ W′) = Left (app₁ (reflect-weakeningᴱ ∅ H M (rednᴱ⊑ s) W′))
-reflectᴱ H (M $ N) (app₁ s) (app₁ W′) with reflectᴱ H M s W′
+reflectᴱ H (M $ N) (app₂ p s) (app₂ W′) = mapL app₂ (reflectᴱ H N s W′)
-reflectᴱ H (M $ N) (app₁ s) (app₁ W′) | heap W = heap 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 (M $ N) (app₁ s) (app₁ W′) | expr W = expr (app₁ W)
+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 (M $ N) (app₁ s) (app₂ W′) = expr (app₂ (reflect-weakeningᴱ H N (rednᴱ⊑ s) W′))
+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 (M $ N) (app₂ p s) (FunctionCallMismatch p′) with heap-weakeningᴱ H (val p) (rednᴱ⊑ s) | preservationᴱ H N s
+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 (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 (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 (M $ N) (app₂ p s) (FunctionCallMismatch p′) | warning W | _ = expr (app₁ 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 (M $ N) (app₂ p s) (FunctionCallMismatch p′) | _ | warning (expr 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′) | Right (Right q) = Left (FunctionCallMismatch (≮:-trans-≡ q (cong src (cong orAny (cong typeOfᴹᴼ (sym p))))))
-reflectᴱ H (M $ N) (app₂ p s) (FunctionCallMismatch p′) | _ | warning (heap W) = heap W
+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 (M $ N) (app₂ p s) (app₁ W′) = expr (app₁ (reflect-weakeningᴱ H M (rednᴱ⊑ s) W′))
+reflectᴱ H (block var b ∈ T is B end) (block s) (block₁ W′) = mapL block₁ (reflectᴮ H B s W′)
-reflectᴱ H (M $ N) (app₂ p s) (app₂ W′) with reflectᴱ H N s W′
+reflectᴱ H (block var b ∈ T is B end) (return v) W′ = Left (block₁ (return 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 (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) = Left (cond BinOpMismatch₁ bin₁ (reflect-subtypingᴱ H M s p))
-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) (BinOpMismatch₂ p) = Left (BinOpMismatch₂ (heap-weakeningᴱ ∅ H N (rednᴱ⊑ s) p))
-reflectᴱ H (binexp M op N) (binOp₁ s) (BinOpMismatch₁ p) | warning (heap W) = heap W
+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) (BinOpMismatch₁ p) | warning (expr W) = expr (bin₁ 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) (binOp₁ s) (BinOpMismatch₂ p) with heap-weakeningᴱ H N (rednᴱ⊑ s)
+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) (binOp₁ s) (BinOpMismatch₂ p) | ok q = expr (BinOpMismatch₂ ((subst₁ (BinOpWarning op) (sym q) 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) (binOp₁ s) (BinOpMismatch₂ p) | warning W = expr (bin₂ W)
+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) (binOp₁ s) (bin₁ W′) with reflectᴱ H M s W′
+reflectᴱ H (binexp M op N) (binOp₂ s) (bin₂ W′) = mapL bin₂ (reflectᴱ H N 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)
 
-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) = Left (cond LocalVarMismatch local₁ (reflect-subtypingᴱ H M s p))
-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) (local₁ W′) = mapL local₁ (reflectᴱ H M s W′)
-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) (local₂ W′) = Left (local₂ (reflect-weakeningᴮ (x ↦ T) H B (rednᴱ⊑ s) 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) (subst v) W′ = Left (cond local₂ (cond local₁ LocalVarMismatch) (reflect-substitutionᴮ H B v x W′))
-reflectᴮ H (local var x ∈ T ← M ∙ B) (local s) (local₁ W′) with reflectᴱ H M s 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 (local var x ∈ T ← M ∙ B) (local s) (local₁ W′) | heap W = heap 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 (local var x ∈ T ← M ∙ B) (local s) (local₁ W′) | expr W = block (local₁ W)
+reflectᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) (function a defn) W′ | Right (Left (UnallocatedAddress ()))
-reflectᴮ H (local var x ∈ T ← M ∙ B) (local s) (local₂ W′) = block (local₂ (reflect-weakeningᴮ H B (rednᴱ⊑ s) W′))
+reflectᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) (function a defn) W′ | Right (Right p) = CONTRADICTION (≮:-refl p)
-reflectᴮ H (local var x ∈ T ← M ∙ B) (subst v) W′ with remember (typeOfⱽ H v)
+reflectᴮ H (return M ∙ B) (return s) (return W′) = mapL return (reflectᴱ H M s W′)
-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)
 
-reflectᴴᴱ : ∀ H M {H′ M′} → (H ⊢ M ⟶ᴱ M′ ⊣ H′) → Warningᴴᴱ H′ (typeCheckᴴᴱ H′ ∅ M′) → Warningᴴᴱ H (typeCheckᴴᴱ H ∅ M)
+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′) → Warningᴴᴮ H′ (typeCheckᴴᴮ H′ ∅ B′) → Warningᴴᴮ H (typeCheckᴴᴮ H ∅ B)
+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 (M $ N) (app₁ s) W = mapL app₁ (reflectᴴᴱ 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 (M $ N) (app₂ v s) W = mapL app₂ (reflectᴴᴱ H N s W)
-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 (M $ N) (beta O v refl p) W = Right W
-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 p) (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 | warning W = expr (function₁ W)
+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) (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 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) (heap (addr b refl W′)) | no r = heap (addr b (lookup-not-allocated p r) (reflect-weakeningᴼ H _ (snoc p) 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 (M $ N) (app₁ s) (heap W′) with reflectᴴᴱ H M s (heap W′)
+reflectᴴᴱ H (block var b ∈ T is B end) (block s) W = mapL block₁ (reflectᴴᴮ H B s W)
-reflectᴴᴱ H (M $ N) (app₁ s) (heap W′) | heap W = heap W
+reflectᴴᴱ H (block var b ∈ T is return (val v) ∙ B end) (return v) W = Right W
-reflectᴴᴱ H (M $ N) (app₁ s) (heap W′) | expr W = expr (app₁ W)
+reflectᴴᴱ H (block var b ∈ T is done end) done W = Right W
-reflectᴴᴱ H (M $ N) (app₂ p s) (heap W′) with reflectᴴᴱ H N s (heap W′)
+reflectᴴᴱ H (binexp M op N) (binOp₀ s) W = Right W
-reflectᴴᴱ H (M $ N) (app₂ p s) (heap W′) | heap W = heap W
+reflectᴴᴱ H (binexp M op N) (binOp₁ s) W = mapL bin₁ (reflectᴴᴱ H M s W)
-reflectᴴᴱ H (M $ N) (app₂ p s) (heap W′) | expr W = expr (app₂ W)
+reflectᴴᴱ H (binexp M op N) (binOp₂ s) W = mapL bin₂ (reflectᴴᴱ H N s 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)
 
-reflectᴴᴮ H B s (block W′) = reflectᴮ H B s 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 (local var x ∈ T ← M ∙ B) (local s) (heap W′) with reflectᴴᴱ H M s (heap W′)
+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 (local var x ∈ T ← M ∙ B) (local s) (heap W′) | heap W = heap W
+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 (local var x ∈ T ← M ∙ B) (local s) (heap W′) | expr W = block (local₁ 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) (subst v) (heap W′) = heap W′
+reflectᴴᴮ H (local var x ∈ T ← M ∙ B) (local s) W = mapL local₁ (reflectᴴᴱ H M s 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 (local var x ∈ T ← M ∙ B) (subst v) W = Right W
-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 (return M ∙ B) (return s) W = mapL return (reflectᴴᴱ H M s W)
-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)
 
-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))
+isntNumber : ∀ H v → (valueType v ≢ number) → (typeOfᴱ H ∅ (val v) ≮: number)
-runtimeBinOpWarning H v (+ p) = + (λ q → p (mustBeNumber H ∅ v q))
+isntNumber H nil p = scalar-≢-impl-≮: nil number (λ ())
-runtimeBinOpWarning H v (- p) = - (λ q → p (mustBeNumber H ∅ v q))
+isntNumber H (addr a) p with remember (H [ a ]ᴴ)
-runtimeBinOpWarning H v (* p) = * (λ q → p (mustBeNumber H ∅ v q))
+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)
-runtimeBinOpWarning H v (/ p) = / (λ q → p (mustBeNumber H ∅ v q))
+isntNumber H (addr a) p | (nothing , q) = ≡-trans-≮: (cong orAny (cong typeOfᴹᴼ q)) (any-≮:-scalar number)
-runtimeBinOpWarning H v (< p) = < (λ q → p (mustBeNumber H ∅ v q))
+isntNumber H (number x) p = CONTRADICTION (p refl)
-runtimeBinOpWarning H v (> p) = > (λ q → p (mustBeNumber H ∅ v q))
+isntNumber H (bool x) p = scalar-≢-impl-≮: boolean number (λ ())
-runtimeBinOpWarning H v (<= p) = <= (λ q → p (mustBeNumber H ∅ v q))
+isntNumber H (string x) p = scalar-≢-impl-≮: string number (λ ())
-runtimeBinOpWarning H v (>= p) = >= (λ q → p (mustBeNumber H ∅ v q))
+
-runtimeBinOpWarning H v (·· p) = ·· (λ q → p (mustBeString H ∅ v q))
+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) = FunctionCallMismatch (any-src-≮: (isntEmpty H w) (isntFunction H v p))
-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) (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 with reflect* ∅ᴴ B t (Left (runtimeWarningᴮ H′ B′ err))
-wellTypedProgramsDontGoWrong H′ B B′ t err | heap (addr a refl ())
+wellTypedProgramsDontGoWrong H′ B B′ t err | Right (addr a refl ())
-wellTypedProgramsDontGoWrong H′ B B′ t err | block W = W
+wellTypedProgramsDontGoWrong H′ B B′ t err | Left W = W

prototyping/Properties/Subtyping.agda

@@ -0,0 +1,221 @@
+{-# 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

prototyping/Properties/TypeCheck.agda

@@ -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 (var x) = orAny(Γ [ x ]ⱽ)
-typeOfᴱ H Γ (val v) = orNone(typeOfⱽ H 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
@@ -109,4 +109,4 @@ typeCheckᴴᴱ H Γ M = (typeCheckᴴ H , typeCheckᴱ H Γ M)
 
 typeCheckᴴᴮ : ∀ H Γ M → (Γ ⊢ᴴᴮ H ▷ M ∈ typeOfᴮ H Γ M)
 typeCheckᴴᴮ H Γ M = (typeCheckᴴ H , typeCheckᴮ H Γ M)
- 
+ 

prototyping/Tests/Interpreter/concat_number_and_string/out.txt

@@ -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