WIP

prototyping/FFI/Data/Aeson.agda

@@ -1,6 +1,9 @@
+{-# OPTIONS --rewriting #-}
+
 module FFI.Data.Aeson where
 
 open import Agda.Builtin.Equality using (_≡_)
+open import Agda.Builtin.Equality.Rewrite using ()
 open import Agda.Builtin.Bool using (Bool)
 open import Agda.Builtin.String using (String)
 
@@ -42,6 +45,8 @@ postulate lookup-insert : ∀ {A} k v (m : KeyMap A) → (lookup k (insert k v m
 postulate lookup-empty : ∀ {A} k → (lookup {A} k empty ≡ nothing)
 postulate singleton-insert-empty : ∀ {A} k (v : A) → (singleton k v ≡ insert k v empty)
 
+{-# REWRITE lookup-insert lookup-empty singleton-insert-empty #-}
+
 data Value : Set where
   object : KeyMap Value → Value
   array : Vector Value → Value

prototyping/FFI/Data/Vector.agda

@@ -1,6 +1,9 @@
+{-# OPTIONS --rewriting #-}
+
 module FFI.Data.Vector where
 
 open import Agda.Builtin.Equality using (_≡_)
+open import Agda.Builtin.Equality.Rewrite using ()
 open import Agda.Builtin.Int using (Int; pos; negsuc)
 open import Agda.Builtin.Nat using (Nat)
 open import FFI.Data.Bool using (Bool; false; true)
@@ -33,6 +36,8 @@ postulate length-empty : ∀ {A} → (length (empty {A}) ≡ 0)
 postulate lookup-snoc : ∀ {A} (x : A) (v : Vector A) → (lookup (snoc v x) (length v) ≡ just x)
 postulate lookup-snoc-empty : ∀ {A} (x : A) → (lookup (snoc empty x) 0 ≡ just x)
 
+{-# REWRITE length-empty lookup-snoc lookup-snoc-empty #-}
+
 head : ∀ {A} → (Vector A) → (Maybe A)
 head vec with null vec
 head vec | false = just (unsafeHead vec)

prototyping/Luau/AddrCtxt.agda

@@ -1,3 +1,5 @@
+{-# OPTIONS --rewriting #-}
+
 module Luau.AddrCtxt where
 
 open import Luau.Type using (Type)

prototyping/Luau/Heap.agda

@@ -1,3 +1,5 @@
+{-# OPTIONS --rewriting #-}
+
 module Luau.Heap where
 
 open import Agda.Builtin.Equality using (_≡_)
@@ -37,13 +39,3 @@ next = length
 
 allocated : ∀ {a} → Heap a → HeapValue a → Heap a
 allocated = snoc
-
--- next-emp : (length empty ≡ 0)
-next-emp = FFI.Data.Vector.length-empty
-
--- lookup-next : ∀ V H → (lookup (allocated H V) (next H) ≡ just V)
-lookup-next = FFI.Data.Vector.lookup-snoc
-
--- lookup-next-emp : ∀ V → (lookup (allocated emp V) 0 ≡ just V)
-lookup-next-emp = FFI.Data.Vector.lookup-snoc-empty
-

prototyping/Luau/OpSem.agda

@@ -1,3 +1,5 @@
+{-# OPTIONS --rewriting #-}
+
 module Luau.OpSem where
 
 open import Agda.Builtin.Equality using (_≡_)

prototyping/Luau/RuntimeError.agda

@@ -1,3 +1,5 @@
+{-# OPTIONS --rewriting #-}
+
 module Luau.RuntimeError where
 
 open import Agda.Builtin.Equality using (_≡_)

prototyping/Luau/StrictMode.agda

@@ -1,81 +1,108 @@
+{-# OPTIONS --rewriting #-}
+
 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; yes; nil; addr; var; var_∈_; _⟨_⟩∈_; function_is_end; _$_; block_is_end; local_←_; _∙_; done; return; name)
 open import Luau.Type using (Type; strict; bot; top; nil; _⇒_; tgt)
-open import Luau.Heap using (Heap) renaming (_[_] to _[_]ᴴ)
+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_∋_∈_⊣_; var; addr; app; block; return; local; function)
+open import Luau.TypeCheck(strict) using (_⊢ᴮ_∈_; _⊢ᴱ_∈_; var; addr; app; block; return; local; function)
 open import Properties.Equality using (_≢_)
-open import Properties.TypeCheck(strict) using (typeOfᴴ)
+open import Properties.TypeCheck(strict) using (typeOfᴴ; typeCheckᴮ)
 
 src : Type → Type
 src = Luau.Type.src strict
 
-data Warningᴱ (H : Heap yes) {Γ S} : ∀ {M T Δ} → (Γ ⊢ᴱ S ∋ M ∈ T ⊣ Δ) → Set
+data Warningᴱ (H : Heap yes) {Γ} : ∀ {M T} → (Γ ⊢ᴱ M ∈ T) → Set
-data Warningᴮ (H : Heap yes) {Γ S} : ∀ {B T Δ} → (Γ ⊢ᴮ S ∋ B ∈ T ⊣ Δ) → Set
+data Warningᴮ (H : Heap yes) {Γ} : ∀ {B T} → (Γ ⊢ᴮ B ∈ T) → Set
 
-data Warningᴱ H {Γ S} where
+data Warningᴱ H {Γ} where
 
-  bot : ∀ {M T Δ} {D : Γ ⊢ᴱ S ∋ M ∈ T ⊣ Δ} →
+  BadlyTypedFunctionAddress : ∀ a f {x S T U B} →
 
-    (T ≡ bot) →
+    (H [ a ]ᴴ ≡ just (function f ⟨ var x ∈ T ⟩∈ U is B end)) →
-    ------------
+    Warningᴮ H (typeCheckᴮ H (x ↦ T) B) →
-    Warningᴱ H D
+    --------------------------------------------------------
+    Warningᴱ H (addr a S)
 
-  addr : ∀ a T →
+  UnallocatedAddress : ∀ a {T} →
 
-    (T ≢ typeOfᴴ(H [ a ]ᴴ)) →
+    (H [ a ]ᴴ ≡ nothing) →
-    -------------------------
+    --------------------------------------------------------
     Warningᴱ H (addr a T)
 
-  app₁ : ∀ {M N T U Δ₁ Δ₂} {D₁ : Γ ⊢ᴱ (U ⇒ S) ∋ M ∈ T ⊣ Δ₁} {D₂ : Γ ⊢ᴱ (src T) ∋ N ∈ U ⊣ Δ₂} →
+  app₀ : ∀ {M N T U} {D₁ : Γ ⊢ᴱ M ∈ T} {D₂ : Γ ⊢ᴱ N ∈ U} →
+
+    (src T ≢ U) →
+    -----------------
+    Warningᴱ H (app D₁ D₂)
+  
+  app₁ : ∀ {M N T U} {D₁ : Γ ⊢ᴱ M ∈ T} {D₂ : Γ ⊢ᴱ N ∈ U} →
 
     Warningᴱ H D₁ →
     -----------------
     Warningᴱ H (app D₁ D₂)
 
-  app₂ : ∀ {M N T U Δ₁ Δ₂} {D₁ : Γ ⊢ᴱ (U ⇒ S) ∋ M ∈ T ⊣ Δ₁} {D₂ : Γ ⊢ᴱ (src T) ∋ N ∈ U ⊣ Δ₂} →
+  app₂ : ∀ {M N T U} {D₁ : Γ ⊢ᴱ M ∈ T} {D₂ : Γ ⊢ᴱ N ∈ U} →
 
     Warningᴱ H D₂ →
     -----------------
-    Warningᴱ H(app D₁ D₂)
+    Warningᴱ H (app D₁ D₂)
 
-  block : ∀ b {B T Δ} {D : Γ ⊢ᴮ S ∋ B ∈ T ⊣ Δ} →
+  function₀ : ∀ f {x B T U V} {D : (Γ ⊕ x ↦ T) ⊢ᴮ B ∈ V} →
+
+    (U ≢ V) →
+    -------------------------
+    Warningᴱ H (function f {U = U} D)
+
+  function₁ : ∀ f {x B T U V} {D : (Γ ⊕ x ↦ T) ⊢ᴮ B ∈ V} →
+
+    Warningᴮ H D →
+    -------------------------
+    Warningᴱ H (function f {U = U} D)
+
+  block : ∀ b {B T} {D : Γ ⊢ᴮ B ∈ T} →
 
     Warningᴮ H D →
     -----------------
-    Warningᴱ H(block b D)
+    Warningᴱ H (block b D)
 
-data Warningᴮ H {Γ S} where
+data Warningᴮ H {Γ} where
 
-  disagree : ∀ {B T Δ} {D : Γ ⊢ᴮ S ∋ B ∈ T ⊣ Δ} →
+  return : ∀ {M B T U} {D₁ : Γ ⊢ᴱ M ∈ T} {D₂ : Γ ⊢ᴮ B ∈ U} →
-
-    (S ≢ T) →
-    -----------
-    Warningᴮ H D
-
-  return : ∀ {M B T U Δ₁ Δ₂} {D₁ : Γ ⊢ᴱ S ∋ M ∈ T ⊣ Δ₁} {D₂ : Γ ⊢ᴮ nil ∋ B ∈ U ⊣ Δ₂} →
 
     Warningᴱ H D₁ →
     ------------------
     Warningᴮ H (return D₁ D₂)
 
-  local₁ : ∀ {x M B T U V Δ₁ Δ₂} {D₁ : Γ ⊢ᴱ T ∋ M ∈ U ⊣ Δ₁} {D₂ : (Γ ⊕ x ↦ T) ⊢ᴮ S ∋ B ∈ V ⊣ Δ₂} →
+  local₀ : ∀ {x M B T U V} {D₁ : Γ ⊢ᴱ M ∈ U} {D₂ : (Γ ⊕ x ↦ T) ⊢ᴮ B ∈ V} →
+
+    (T ≢ U) →
+    --------------------
+    Warningᴮ H (local D₁ D₂)
+
+  local₁ : ∀ {x M B T U V} {D₁ : Γ ⊢ᴱ M ∈ U} {D₂ : (Γ ⊕ x ↦ T) ⊢ᴮ B ∈ V} →
 
     Warningᴱ H D₁ →
     --------------------
     Warningᴮ H (local D₁ D₂)
 
--- data Warningᴴ {H} : ∀ {V T} → (H ▷ V ∈ T) → Set where
+  local₂ : ∀ {x M B T U V} {D₁ : Γ ⊢ᴱ M ∈ U} {D₂ : (Γ ⊕ x ↦ T) ⊢ᴮ B ∈ V} →
 
---   nothing :
+    Warningᴮ H D₂ →
+    --------------------
+    Warningᴮ H (local D₁ D₂)
 
---     -----------------
+  function₁ : ∀ f {x B C T U V W} {D₁ : (Γ ⊕ x ↦ T) ⊢ᴮ C ∈ V} {D₂ : (Γ ⊕ f ↦ (T ⇒ U)) ⊢ᴮ B ∈ W} →
---     Warningᴴ(nothing)
 
---   function : ∀ f {x B T U V W} {D : (x ↦ T) ⊢ᴮ U ∋ B ∈ V ⊣ (x ↦ W)} →
+    Warningᴮ H D₁ →
+    --------------------
+    Warningᴮ H (function f D₁ D₂)
 
---     Warningᴮ(D) →
+  function₂ : ∀ f {x B C T U V W} {D₁ : (Γ ⊕ x ↦ T) ⊢ᴮ C ∈ V} {D₂ : (Γ ⊕ f ↦ (T ⇒ U)) ⊢ᴮ B ∈ W} →
---     --------------------
+
---     Warningᴴ(function f D)
+    Warningᴮ H D₂ →
+    --------------------
+    Warningᴮ H (function f D₁ D₂)

prototyping/Luau/TypeCheck.agda

@@ -1,3 +1,5 @@
+{-# OPTIONS --rewriting #-}
+
 open import Luau.Type using (Mode)
 
 module Luau.TypeCheck (m : Mode) where
@@ -17,70 +19,138 @@ open import FFI.Data.Maybe using (Maybe; just; nothing)
 src : Type → Type
 src = Luau.Type.src m
 
-data _⊢ᴮ_∋_∈_⊣_ : VarCtxt → Type → Block yes → Type → VarCtxt → Set
+data _⊢ᴮ_∈_ : VarCtxt → Block yes → Type → Set
-data _⊢ᴱ_∋_∈_⊣_ : VarCtxt → Type → Expr yes → Type → VarCtxt → Set
+data _⊢ᴱ_∈_ : VarCtxt → Expr yes → Type → Set
 
-data _⊢ᴮ_∋_∈_⊣_ where
+data _⊢ᴮ_∈_ where
 
-  done : ∀ {S Γ} →
+  done : ∀ {Γ} →
 
-    ----------------------
+    ---------------
-    Γ ⊢ᴮ S ∋ done ∈ nil ⊣ ∅
+    Γ ⊢ᴮ done ∈ nil
 
-  return : ∀ {M B S T U Γ Δ₁ Δ₂} →
+  return : ∀ {M B T U Γ} →
 
-    Γ ⊢ᴱ S ∋ M ∈ T ⊣ Δ₁ →
+    Γ ⊢ᴱ M ∈ T →
-    Γ ⊢ᴮ nil ∋ B ∈ U ⊣ Δ₂ →
+    Γ ⊢ᴮ B ∈ U →
-    ---------------------------------
+    ---------------------
-    Γ ⊢ᴮ S ∋ return M ∙ B ∈ T ⊣ Δ₁
+    Γ ⊢ᴮ return M ∙ B ∈ T
 
-  local : ∀ {x M B S T U V Γ Δ₁ Δ₂} →
+  local : ∀ {x M B T U V Γ} →
 
-    Γ ⊢ᴱ T ∋ M ∈ U ⊣ Δ₁ →
+    Γ ⊢ᴱ M ∈ U →
-    (Γ ⊕ x ↦ T) ⊢ᴮ S ∋ B ∈ V ⊣ Δ₂ →
+    (Γ ⊕ x ↦ T) ⊢ᴮ B ∈ V →
-    ----------------------------------------------------------
+    --------------------------------
-    Γ ⊢ᴮ S ∋ local var x ∈ T ← M ∙ B ∈ V ⊣ (Δ₁ ⋒ (Δ₂ ⊝ x))
+    Γ ⊢ᴮ local var x ∈ T ← M ∙ B ∈ V
 
-  function : ∀ {f x B C S T U V W Γ Δ₁ Δ₂} →
+  function : ∀ f {x B C T U V W Γ} →
 
-    (Γ ⊕ x ↦ T) ⊢ᴮ U ∋ C ∈ V ⊣ Δ₁ →
+    (Γ ⊕ x ↦ T) ⊢ᴮ C ∈ V →
-    (Γ ⊕ f ↦ (T ⇒ U)) ⊢ᴮ S ∋ B ∈ W ⊣ Δ₂ →
+    (Γ ⊕ f ↦ (T ⇒ U)) ⊢ᴮ B ∈ W →
-    ---------------------------------------------------------------------------------
+    -------------------------------------------------
-    Γ ⊢ᴮ S ∋ function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B ∈ W ⊣ ((Δ₁ ⊝ x) ⋒ (Δ₂ ⊝ f))
+    Γ ⊢ᴮ function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B ∈ W
 
-data _⊢ᴱ_∋_∈_⊣_  where
+data _⊢ᴱ_∈_ where
 
-  nil : ∀ {S Γ} →
+  nil : ∀ {Γ} →
 
-    ----------------------
+    --------------
-    Γ ⊢ᴱ S ∋ nil ∈ nil ⊣ ∅
+    Γ ⊢ᴱ nil ∈ nil
 
-  var : ∀ x {S T Γ} →
+  var : ∀ x {T Γ} →
 
     T ≡ Γ [ x ]ⱽ →
-    ----------------------------
+    --------------
-    Γ ⊢ᴱ S ∋ var x ∈ T ⊣ (x ↦ S)
+    Γ ⊢ᴱ var x ∈ T
 
-  addr : ∀ a T {S Γ} →
+  addr : ∀ a T {Γ} →
 
-    -------------------------
+    -----------------
-    Γ ⊢ᴱ S ∋ (addr a) ∈ T ⊣ ∅
+    Γ ⊢ᴱ (addr a) ∈ T
 
-  app : ∀ {M N S T U Γ Δ₁ Δ₂} →
+  app : ∀ {M N T U Γ} →
 
-    Γ ⊢ᴱ (U ⇒ S) ∋ M ∈ T ⊣ Δ₁ →
+    Γ ⊢ᴱ M ∈ T →
-    Γ ⊢ᴱ (src T) ∋ N ∈ U ⊣ Δ₂ →
+    Γ ⊢ᴱ N ∈ U →
-    --------------------------------------
+    ----------------------
-    Γ ⊢ᴱ S ∋ (M $ N) ∈ (tgt T) ⊣ (Δ₁ ⋒ Δ₂)
+    Γ ⊢ᴱ (M $ N) ∈ (tgt T)
 
-  function : ∀ {f x B S T U V Γ Δ} →
+  function : ∀ f {x B T U V Γ} →
 
-    (Γ ⊕ x ↦ T) ⊢ᴮ U ∋ B ∈ V ⊣ Δ →
+    (Γ ⊕ x ↦ T) ⊢ᴮ B ∈ V →
-    -----------------------------------------------------------------------
+    -----------------------------------------------------
-    Γ ⊢ᴱ S ∋ (function f ⟨ var x ∈ T ⟩∈ U is B end) ∈ (T ⇒ U) ⊣ (Δ ⊝ x)
+    Γ ⊢ᴱ (function f ⟨ var x ∈ T ⟩∈ U is B end) ∈ (T ⇒ U)
 
-  block : ∀ b {B S T Γ Δ} →
+  block : ∀ b {B T Γ} →
 
-    Γ ⊢ᴮ S ∋ B ∈ T ⊣ Δ →
+    Γ ⊢ᴮ B ∈ T →
-    ----------------------------------------------------
+    ---------------------------
-    Γ ⊢ᴱ S ∋ (block b is B end) ∈ T ⊣ Δ
+    Γ ⊢ᴱ (block b is B end) ∈ T
+
+-- data _⊢ᴮ_∋_∈_⊣_ : VarCtxt → Type → Block yes → Type → VarCtxt → Set
+-- data _⊢ᴱ_∋_∈_⊣_ : VarCtxt → Type → Expr yes → Type → VarCtxt → Set
+
+-- data _⊢ᴮ_∋_∈_⊣_ where
+
+--   done : ∀ {S Γ} →
+
+--     ----------------------
+--     Γ ⊢ᴮ S ∋ done ∈ nil ⊣ ∅
+
+--   return : ∀ {M B S T U Γ Δ₁ Δ₂} →
+
+--     Γ ⊢ᴱ S ∋ M ∈ T ⊣ Δ₁ →
+--     Γ ⊢ᴮ nil ∋ B ∈ U ⊣ Δ₂ →
+--     ---------------------------------
+--     Γ ⊢ᴮ S ∋ return M ∙ B ∈ T ⊣ Δ₁
+
+--   local : ∀ {x M B S T U V Γ Δ₁ Δ₂} →
+
+--     Γ ⊢ᴱ T ∋ M ∈ U ⊣ Δ₁ →
+--     (Γ ⊕ x ↦ T) ⊢ᴮ S ∋ B ∈ V ⊣ Δ₂ →
+--     ----------------------------------------------------------
+--     Γ ⊢ᴮ S ∋ local var x ∈ T ← M ∙ B ∈ V ⊣ (Δ₁ ⋒ (Δ₂ ⊝ x))
+
+--   function : ∀ {f x B C S T U V W Γ Δ₁ Δ₂} →
+
+--     (Γ ⊕ x ↦ T) ⊢ᴮ U ∋ C ∈ V ⊣ Δ₁ →
+--     (Γ ⊕ f ↦ (T ⇒ U)) ⊢ᴮ S ∋ B ∈ W ⊣ Δ₂ →
+--     ---------------------------------------------------------------------------------
+--     Γ ⊢ᴮ S ∋ function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B ∈ W ⊣ ((Δ₁ ⊝ x) ⋒ (Δ₂ ⊝ f))
+
+-- data _⊢ᴱ_∋_∈_⊣_  where
+
+--   nil : ∀ {S Γ} →
+
+--     ----------------------
+--     Γ ⊢ᴱ S ∋ nil ∈ nil ⊣ ∅
+
+--   var : ∀ x {S T Γ} →
+
+--     T ≡ Γ [ x ]ⱽ →
+--     ----------------------------
+--     Γ ⊢ᴱ S ∋ var x ∈ T ⊣ (x ↦ S)
+
+--   addr : ∀ a T {S Γ} →
+
+--     -------------------------
+--     Γ ⊢ᴱ S ∋ (addr a) ∈ T ⊣ ∅
+
+--   app : ∀ {M N S T U Γ Δ₁ Δ₂} →
+
+--     Γ ⊢ᴱ (U ⇒ S) ∋ M ∈ T ⊣ Δ₁ →
+--     Γ ⊢ᴱ (src T) ∋ N ∈ U ⊣ Δ₂ →
+--     --------------------------------------
+--     Γ ⊢ᴱ S ∋ (M $ N) ∈ (tgt T) ⊣ (Δ₁ ⋒ Δ₂)
+
+--   function : ∀ {f x B S T U V Γ Δ} →
+
+--     (Γ ⊕ x ↦ T) ⊢ᴮ U ∋ B ∈ V ⊣ Δ →
+--     -----------------------------------------------------------------------
+--     Γ ⊢ᴱ S ∋ (function f ⟨ var x ∈ T ⟩∈ U is B end) ∈ (T ⇒ U) ⊣ (Δ ⊝ x)
+
+--   block : ∀ b {B S T Γ Δ} →
+
+--     Γ ⊢ᴮ S ∋ B ∈ T ⊣ Δ →
+--     ----------------------------------------------------
+--     Γ ⊢ᴱ S ∋ (block b is B end) ∈ T ⊣ Δ

prototyping/Luau/VarCtxt.agda

@@ -1,3 +1,5 @@
+{-# OPTIONS --rewriting #-}
+
 module Luau.VarCtxt where
 
 open import Agda.Builtin.Equality using (_≡_)
@@ -34,10 +36,3 @@ x ↦ T = singleton (fromString x) T
 
 _⊕_↦_ : VarCtxt → Var → Type → VarCtxt
 Γ ⊕ x ↦ T = insert (fromString x) T Γ
-
--- ⊕-[] : ∀ (Γ : VarCtxt) x T → (((Γ ⊕ x ↦ T) [ x ]) ≡ T)
-⊕-[] = λ (Γ : VarCtxt) x T → cong orBot (lookup-insert (fromString x) T Γ)
-
--- ∅-[] : ∀ x → ∅ [ x ] ≡ bot
-∅-[]  = λ (x : Var) → cong orBot (lookup-empty (fromString x))
-

prototyping/Properties/Step.agda

@@ -1,3 +1,5 @@
+{-# OPTIONS --rewriting #-}
+
 module Properties.Step where
 
 open import Agda.Builtin.Equality using (_≡_; refl)

prototyping/Properties/StrictMode.agda

@@ -5,102 +5,188 @@ module Properties.StrictMode where
 import Agda.Builtin.Equality.Rewrite
 open import Agda.Builtin.Equality using (_≡_; refl)
 open import FFI.Data.Maybe using (Maybe; just; nothing)
-open import Luau.Heap using (Heap; HeapValue; function_is_end; defn; alloc; ok; next; lookup-next) renaming (_≡_⊕_↦_ to _≡ᴴ_⊕_↦_; _[_] to _[_]ᴴ)
+open import Luau.Heap using (Heap; HeapValue; function_is_end; defn; alloc; ok; next) renaming (_≡_⊕_↦_ to _≡ᴴ_⊕_↦_; _[_] to _[_]ᴴ)
-open import Luau.StrictMode using (Warningᴱ; Warningᴮ; bot; disagree; addr; app₁; app₂; block; return; local₁)
+open import Luau.StrictMode using (Warningᴱ; Warningᴮ; BadlyTypedFunctionAddress; UnallocatedAddress; app₀; app₁; app₂; block; return; local₀; local₁; local₂; function₀; function₁; function₂)
-open import Luau.Substitution using (_[_/_]ᴮ; _[_/_]ᴱ)
+open import Luau.Substitution using (_[_/_]ᴮ; _[_/_]ᴱ; _[_/_]ᴮunless_; var_[_/_]ᴱwhenever_)
-open import Luau.Syntax using (Expr; yes; var_∈_; _⟨_⟩∈_; _$_; addr; nil; function_is_end; block_is_end; done; return; local_←_; _∙_; fun; arg)
+open import Luau.Syntax using (Expr; yes; var; var_∈_; _⟨_⟩∈_; _$_; addr; nil; function_is_end; block_is_end; done; return; local_←_; _∙_; fun; arg)
 open import Luau.Type using (Type; strict; nil; _⇒_; bot; tgt)
-open import Luau.TypeCheck(strict) using (_⊢ᴮ_∋_∈_⊣_; _⊢ᴱ_∋_∈_⊣_; nil; var; addr; app; function; block; done; return; local)
+open import Luau.TypeCheck(strict) using (_⊢ᴮ_∈_; _⊢ᴱ_∈_; nil; var; addr; app; function; block; done; return; local)
 open import Luau.Value using (val; nil; addr)
+open import Luau.Var using (_≡ⱽ_)
 open import Luau.Addr using (_≡ᴬ_)
 open import Luau.AddrCtxt using (AddrCtxt)
-open import Luau.VarCtxt using (VarCtxt; ∅; _⋒_; _↦_; _⊕_↦_; _⊝_; ∅-[]) renaming (_[_] to _[_]ⱽ)
+open import Luau.VarCtxt using (VarCtxt; ∅; _⋒_; _↦_; _⊕_↦_; _⊝_) renaming (_[_] to _[_]ⱽ)
 open import Luau.VarCtxt using (VarCtxt; ∅)
 open import Properties.Remember using (remember; _,_)
-open import Properties.Equality using (sym; cong; trans; subst₁)
+open import Properties.Equality using (_≢_; sym; cong; trans; subst₁)
 open import Properties.Dec using (Dec; yes; no)
 open import Properties.Contradiction using (CONTRADICTION)
-open import Properties.TypeCheck(strict) using (typeOfᴴ; typeOfᴱ; typeOfᴮ; typeCheckᴱ; typeCheckᴮ)
+open import Properties.TypeCheck(strict) using (declaredTypeᴴ; typeOfⱽ; typeOfᴴ; typeOfᴱ; typeOfᴮ; typeOfᴱⱽ; typeCheckᴱ; typeCheckᴮ)
 open import Luau.OpSem using (_⊢_⟶ᴮ_⊣_; _⊢_⟶ᴱ_⊣_; app; function; beta; return; block; done; local; subst)
 
-{-# REWRITE lookup-next #-}
-
 src = Luau.Type.src strict
 
+_≡ᵀ_ : ∀ (T U : Type) → Dec(T ≡ U)
+_≡ᵀ_ = {!!}
+
 data _⊑_ (H : Heap yes) : Heap yes → Set where
   refl : (H ⊑ H)
   snoc : ∀ {H′ H″ a V} → (H ⊑ H′) → (H″ ≡ᴴ H′ ⊕ a ↦ V) → (H ⊑ H″)
 
-warning-⊑ : ∀ {H H′ Γ Δ S T M} {D : Γ ⊢ᴱ S ∋ M ∈ T ⊣ Δ} → (H ⊑ H′) → (Warningᴱ H′ D) → Warningᴱ H D
+warning-⊑ : ∀ {H H′ Γ T M} {D : Γ ⊢ᴱ M ∈ T} → (H ⊑ H′) → (Warningᴱ H′ D) → Warningᴱ H D
 warning-⊑ = {!!}
 
-data TypeOfᴱ-⊑-Result H H′ Γ M : Set where
+data LookupResult (H : Heap yes) a V : Set where
-  ok : (typeOfᴱ H Γ M ≡ typeOfᴱ H′ Γ M) → TypeOfᴱ-⊑-Result H H′ Γ M
+  just : (H [ a ]ᴴ ≡ just V) → LookupResult H a V
-  warning : (∀ {S} → Warningᴱ H (typeCheckᴱ H Γ S M)) → TypeOfᴱ-⊑-Result H H′ Γ M
+  nothing : (H [ a ]ᴴ ≡ nothing) → LookupResult H a V
 
-data TypeOfᴮ-⊑-Result H H′ Γ B : Set where
+lookup-⊑-just : ∀ {H H′ V} a → (H ⊑ H′) → (H′ [ a ]ᴴ ≡ just V) → LookupResult H a V
-  ok : (typeOfᴮ H Γ B ≡ typeOfᴮ H′ Γ B) → TypeOfᴮ-⊑-Result H H′ Γ B
+lookup-⊑-just = {!!}
-  warning : (∀ {S} → Warningᴮ H (typeCheckᴮ H Γ S B)) → TypeOfᴮ-⊑-Result H H′ Γ B
 
-typeOfᴱ-⊑ : ∀ {H H′ Γ M} → (H ⊑ H′) → (TypeOfᴱ-⊑-Result H H′ Γ M)
+lookup-⊑-nothing : ∀ {H H′} a → (H ⊑ H′) → (H′ [ a ]ᴴ ≡ nothing) → (H [ a ]ᴴ ≡ nothing)
-typeOfᴱ-⊑ = {!!}
+lookup-⊑-nothing = {!!}
 
-typeOfᴮ-⊑ : ∀ {H H′ Γ B} → (H ⊑ H′) → (TypeOfᴮ-⊑-Result H H′ Γ B)
+data OrWarningᴱ {Γ M T} (H : Heap yes) (D : Γ ⊢ᴱ M ∈ T) A : Set where
-typeOfᴮ-⊑ = {!!}
+  ok : A → OrWarningᴱ H D A
+  warning : Warningᴱ H D → OrWarningᴱ H D A
 
-blah : ∀ {H H′ Γ S S′ M} →  (H ⊑ H′) → (S ≡ S′) → (Warningᴱ H′ (typeCheckᴱ H′ Γ S′ M)) → (Warningᴱ H (typeCheckᴱ H Γ S M))
+data OrWarningᴮ {Γ B T} (H : Heap yes) (D : Γ ⊢ᴮ B ∈ T) A : Set where
-blah = {!!}
+  ok : A → OrWarningᴮ H D A
-
+  warning : Warningᴮ H D → OrWarningᴮ H D A
-bloz : ∀ {H Γ S S′ M} → (S ≡ S′) → (Warningᴱ H (typeCheckᴱ H Γ S′ M)) → (Warningᴱ H (typeCheckᴱ H Γ S M))
-bloz = {!!}
 
 redn-⊑ : ∀ {H H′ M M′} → (H ⊢ M ⟶ᴱ M′ ⊣ H′) → (H ⊑ H′)
 redn-⊑ = {!!}
 
+⊕-overwrite : ∀ {Γ x y T U} → (x ≡ y) → ((Γ ⊕ x ↦ T) ⊕ y ↦ U) ≡ (Γ ⊕ y ↦ U)
+⊕-overwrite = {!!}
+
+⊕-swap : ∀ {Γ x y T U} → (x ≢ y) → ((Γ ⊕ x ↦ T) ⊕ y ↦ U) ≡ ((Γ ⊕ y ↦ U) ⊕ x ↦ T)
+⊕-swap = {!!}
+
 substitutivityᴱ : ∀ {Γ T H M v x} → (T ≡ typeOfᴱ H Γ (val v)) → (typeOfᴱ H (Γ ⊕ x ↦ T) M ≡ typeOfᴱ H Γ (M [ v / x ]ᴱ))
 substitutivityᴮ : ∀ {Γ T H B v x} → (T ≡ typeOfᴱ H Γ (val v)) → (typeOfᴮ H (Γ ⊕ x ↦ T) B ≡ typeOfᴮ H Γ (B [ v / x ]ᴮ))
 
 substitutivityᴱ = {!!}
 substitutivityᴮ = {!!}
 
-preservationᴱ : ∀ {H H′ M M′ Γ} → (H ⊢ M ⟶ᴱ M′ ⊣ H′) → (typeOfᴱ H Γ M ≡ typeOfᴱ H′ Γ M′)
+heap-weakeningᴱ : ∀ {H H′ M Γ} → (H ⊑ H′) → OrWarningᴱ H (typeCheckᴱ H Γ M) (typeOfᴱ H Γ M ≡ typeOfᴱ H′ Γ M)
-preservationᴮ : ∀ {H H′ B B′ Γ} → (H ⊢ B ⟶ᴮ B′ ⊣ H′) → (typeOfᴮ H Γ B ≡ typeOfᴮ H′ Γ B′)
+heap-weakeningᴮ : ∀ {H H′ B Γ} → (H ⊑ H′) → OrWarningᴮ H (typeCheckᴮ H Γ B) (typeOfᴮ H Γ B ≡ typeOfᴮ H′ Γ B)
 
-preservationᴱ (function {F = f ⟨ var x ∈ S ⟩∈ T} defn) = refl
+heap-weakeningᴱ = {!!}
-preservationᴱ (app s) = cong tgt (preservationᴱ s)
+heap-weakeningᴮ = {!!}
-preservationᴱ (beta {F = f ⟨ var x ∈ S ⟩∈ T} p) = trans (cong tgt (cong typeOfᴴ p)) {!!}
-preservationᴱ (block s) = preservationᴮ s
-preservationᴱ (return p) = refl
-preservationᴱ done = refl
 
-preservationᴮ (local {x = var x ∈ T} {B = B} s) with typeOfᴮ-⊑ {B = B} (redn-⊑ s)
+preservationᴱ : ∀ {H H′ M M′ Γ} → (H ⊢ M ⟶ᴱ M′ ⊣ H′) → OrWarningᴱ H (typeCheckᴱ H Γ M) (typeOfᴱ H Γ M ≡ typeOfᴱ H′ Γ M′)
-preservationᴮ (local {x = var x ∈ T} s) | ok p = p
+preservationᴮ : ∀ {H H′ B B′ Γ} → (H ⊢ B ⟶ᴮ B′ ⊣ H′) → OrWarningᴮ H (typeCheckᴮ H Γ B) (typeOfᴮ H Γ B ≡ typeOfᴮ H′ Γ B′)
-preservationᴮ (local {x = var x ∈ T} s) | warning W = {!!}
-preservationᴮ (subst {x = var x ∈ T} {B = B}) = substitutivityᴮ {B = B} {!!}
-preservationᴮ (function {F = f ⟨ var x ∈ S ⟩∈ T} {B = B} defn) with typeOfᴮ-⊑ {B = B} (snoc refl defn)
-preservationᴮ (function {F = f ⟨ var x ∈ S ⟩∈ T} {B = B} defn) | ok r = trans r (substitutivityᴮ {T = S ⇒ T} {B = B} refl)
-preservationᴮ (function {F = f ⟨ var x ∈ S ⟩∈ T} {B = B} defn) | warning W = {!!}
-preservationᴮ (return s) = preservationᴱ s
 
-reflectᴱ : ∀ {H H′ M M′ S} → (H ⊢ M ⟶ᴱ M′ ⊣ H′) → Warningᴱ H′ (typeCheckᴱ H′ ∅ S M′) → Warningᴱ H (typeCheckᴱ H ∅ S M)
+preservationᴱ (function {F = f ⟨ var x ∈ S ⟩∈ T} defn) = ok refl
-reflectᴮ : ∀ {H H′ B B′ S} → (H ⊢ B ⟶ᴮ B′ ⊣ H′) → Warningᴮ H′ (typeCheckᴮ H′ ∅ S B′) → Warningᴮ H (typeCheckᴮ H ∅ S B)
+preservationᴱ (app s) with preservationᴱ s
+preservationᴱ (app s) | ok p = ok (cong tgt p)
+preservationᴱ (app s) | warning W = warning (app₁ W)
+preservationᴱ (beta {F = f ⟨ var x ∈ S ⟩∈ T} p) = {!!} -- ok (trans (cong tgt (cong typeOfᴴ p)) {!!})
+preservationᴱ (block s) with preservationᴮ s
+preservationᴱ (block s) | ok p = ok p
+preservationᴱ (block {b = b} s) | warning W = warning (block b W)
+preservationᴱ (return p) = ok refl
+preservationᴱ done = ok refl
 
-reflectᴱ s W with redn-⊑ s
+preservationᴮ (local {x = var x ∈ T} s) with heap-weakeningᴮ (redn-⊑ s)
-reflectᴱ (function {F = f ⟨ var x ∈ S ⟩∈ T} defn) (addr a _ r) | p  = CONTRADICTION (r refl)
+preservationᴮ (local {x = var x ∈ T} s) | ok p = ok p
-reflectᴱ (app s) (bot x) | p  = {!x!}
+preservationᴮ (local {x = var x ∈ T} s) | warning W = warning (local₂ W)
-reflectᴱ (app s) (app₁ W) | p with typeOfᴱ-⊑ p
+preservationᴮ (subst {x = var x ∈ T} {B = B}) = ok (substitutivityᴮ {B = B} {!!})
-reflectᴱ (app s) (app₁ W) | p | ok q = app₁ (bloz (cong (λ ∙ → ∙ ⇒ _) q) (reflectᴱ s W))
+preservationᴮ (function {F = f ⟨ var x ∈ S ⟩∈ T} {B = B} defn) with heap-weakeningᴮ (snoc refl defn)
-reflectᴱ (app s) (app₁ W) | p | warning W′ = app₂ W′
+preservationᴮ (function {F = f ⟨ var x ∈ S ⟩∈ T} {B = B} defn) | ok r = ok (trans r (substitutivityᴮ {T = S ⇒ T} {B = B} refl))
-reflectᴱ (app s) (app₂ W) | p = app₂ (blah p (cong src (preservationᴱ s)) W)
+preservationᴮ (function {F = f ⟨ var x ∈ S ⟩∈ T} {B = B} defn) | warning W = warning (function₂ f W)
-reflectᴱ (beta s) (bot x₁) | p = {!!}
+preservationᴮ (return s) with preservationᴱ s
-reflectᴱ (beta {F = f ⟨ var x ∈ T ⟩∈ U} q) (block _ (disagree x₁)) | p = {!!}
+preservationᴮ (return s) | ok p = ok p
-reflectᴱ (beta {F = f ⟨ var x ∈ T ⟩∈ U} q) (block _ (local₁ W)) | p = app₂ (bloz (cong src (cong typeOfᴴ q)) W)
+preservationᴮ (return s) | warning W = warning (return W)
-reflectᴱ (block s) (bot x₁) | p = {!!}
-reflectᴱ (block s) (block b W) | p = block b (reflectᴮ s W)
-reflectᴱ (return q) W | p  = block _ (return W)
-reflectᴱ done (bot x) | p = {!!}
 
-reflectᴮ s = {!!}
+reflect-substitutionᴱ : ∀ {H Γ Γ′ T} M v x → (T ≡ typeOfⱽ H v) → (Γ′ ≡ Γ ⊕ x ↦ T) → Warningᴱ H (typeCheckᴱ H Γ (M [ v / x ]ᴱ)) → Warningᴱ H (typeCheckᴱ H Γ′ M)
+reflect-substitutionᴱ-whenever-yes : ∀ {H Γ Γ′ T} v x y (p : x ≡ y) → (typeOfᴱ H Γ (val v) ≡ T) → (Γ′ ≡ Γ ⊕ x ↦ T) → Warningᴱ H (typeCheckᴱ H Γ (var y [ v / x ]ᴱwhenever yes p)) → Warningᴱ H (typeCheckᴱ H Γ′ (var y))
+reflect-substitutionᴱ-whenever-no : ∀ {H Γ Γ′ T} v x y (p : x ≢ y) → (typeOfᴱ H Γ (val v) ≡ T) → (Γ′ ≡ Γ ⊕ x ↦ T) → Warningᴱ H (typeCheckᴱ H Γ (var y [ v / x ]ᴱwhenever no p)) → Warningᴱ H (typeCheckᴱ H Γ′ (var y))
+reflect-substitutionᴮ : ∀ {H Γ Γ′ T} B v x → (T ≡ typeOfⱽ H v) → (Γ′ ≡ Γ ⊕ x ↦ T) → Warningᴮ H (typeCheckᴮ H Γ (B [ v / x ]ᴮ)) → Warningᴮ H (typeCheckᴮ H Γ′ B)
+reflect-substitutionᴮ-unless-yes : ∀ {H Γ Γ′ T} B v x y (r : x ≡ y) → (T ≡ typeOfⱽ H v) → (Γ′ ≡ Γ) → Warningᴮ H (typeCheckᴮ H Γ (B [ v / x ]ᴮunless yes r)) → Warningᴮ H (typeCheckᴮ H Γ′ B)
+
+reflect-substitutionᴱ (var y) v x refl q W with x ≡ⱽ y
+reflect-substitutionᴱ (var y) v x refl q W | yes r = reflect-substitutionᴱ-whenever-yes v x y r (typeOfᴱⱽ v) q W
+reflect-substitutionᴱ (var y) v x refl q W | no r = reflect-substitutionᴱ-whenever-no v x y r (typeOfᴱⱽ v) q W
+reflect-substitutionᴱ (addr a) v x p q (BadlyTypedFunctionAddress a f r W) = BadlyTypedFunctionAddress a f r W
+reflect-substitutionᴱ (addr a) v x p q (UnallocatedAddress a r) = UnallocatedAddress a r
+reflect-substitutionᴱ (M $ N) v x p q (app₀ r) = {!!}
+reflect-substitutionᴱ (M $ N) v x p q (app₁ W) = app₁ (reflect-substitutionᴱ M v x p q W)
+reflect-substitutionᴱ (M $ N) v x p q (app₂ W) = app₂ (reflect-substitutionᴱ N v x p q W)
+reflect-substitutionᴱ (function f ⟨ var y ∈ T ⟩∈ U is B end) v x p q (function₀ f r) = {!!}
+reflect-substitutionᴱ (function f ⟨ var y ∈ T ⟩∈ U is B end) v x p refl (function₁ f W) with (x ≡ⱽ y)
+reflect-substitutionᴱ (function f ⟨ var y ∈ T ⟩∈ U is B end) v x p refl (function₁ f W) | yes r = function₁ f (reflect-substitutionᴮ-unless-yes B v x y r p (⊕-overwrite r) W)
+reflect-substitutionᴱ (function f ⟨ var y ∈ T ⟩∈ U is B end) v x p refl (function₁ f W) | no r = function₁ f (reflect-substitutionᴮ B v x p (⊕-swap r) W)
+reflect-substitutionᴱ (block b is B end) v x p q (block b W) = block b (reflect-substitutionᴮ B v x p q W)
+
+reflect-substitutionᴱ-whenever-no v x y r refl refl ()
+reflect-substitutionᴱ-whenever-yes (addr a) x x refl refl refl (BadlyTypedFunctionAddress a f p W) = {!!}
+reflect-substitutionᴱ-whenever-yes (addr a) x x refl refl refl (UnallocatedAddress a p) = {!!}
+
+reflect-substitutionᴮ (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) v x p q (function₁ f W) = {!!}
+reflect-substitutionᴮ (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) v x p q (function₂ f W) = {!!}
+reflect-substitutionᴮ (local var y ∈ T ← M ∙ B) v x p q (local₀ r) = {!!}
+reflect-substitutionᴮ (local var y ∈ T ← M ∙ B) v x p q (local₁ W) = local₁ (reflect-substitutionᴱ M v x p q W)
+reflect-substitutionᴮ (local var y ∈ T ← M ∙ B) v x p q (local₂ W) = {!!}
+reflect-substitutionᴮ (return M ∙ B) v x p q (return W) = return (reflect-substitutionᴱ M v x p q W)
+
+reflect-substitutionᴮ-unless-yes B v x y r p refl W = W
+
+reflect-weakeningᴱ : ∀ {H H′ Γ M} → (H ⊑ H′) → Warningᴱ H′ (typeCheckᴱ H′ Γ M) → Warningᴱ H (typeCheckᴱ H Γ M)
+reflect-weakeningᴮ : ∀ {H H′ Γ B} → (H ⊑ H′) → Warningᴮ H′ (typeCheckᴮ H′ Γ B) → Warningᴮ H (typeCheckᴮ H Γ B)
+
+reflect-weakeningᴱ h (BadlyTypedFunctionAddress a f p W) with lookup-⊑-just a h p
+reflect-weakeningᴱ h (BadlyTypedFunctionAddress a f p W) | just q = BadlyTypedFunctionAddress a f q (reflect-weakeningᴮ h W)
+reflect-weakeningᴱ h (BadlyTypedFunctionAddress a f p W) | nothing q = UnallocatedAddress a q
+reflect-weakeningᴱ h (UnallocatedAddress a p) = UnallocatedAddress a (lookup-⊑-nothing a h p)
+reflect-weakeningᴱ h (app₀ p) with heap-weakeningᴱ h |  heap-weakeningᴱ h
+reflect-weakeningᴱ h (app₀ p) | ok q₁ | ok q₂ = app₀ (λ r → p (trans (cong src (sym q₁)) (trans r q₂)))
+reflect-weakeningᴱ h (app₀ p) | warning W | _ = app₁ W
+reflect-weakeningᴱ h (app₀ p) | _ | warning W = app₂ W
+reflect-weakeningᴱ h (app₁ W) = app₁ (reflect-weakeningᴱ h W)
+reflect-weakeningᴱ h (app₂ W) = app₂ (reflect-weakeningᴱ h W)
+reflect-weakeningᴱ h (function₀ f p) with heap-weakeningᴮ h
+reflect-weakeningᴱ h (function₀ f p) | ok q = function₀ f (λ r → p (trans r q))
+reflect-weakeningᴱ h (function₀ f p) | warning W = function₁ f W
+reflect-weakeningᴱ h (function₁ f W) = function₁ f (reflect-weakeningᴮ h W)
+reflect-weakeningᴱ h (block b W) = block b (reflect-weakeningᴮ h W)
+
+reflect-weakeningᴮ h (return W) = return (reflect-weakeningᴱ h W)
+reflect-weakeningᴮ h (local₀ p) with heap-weakeningᴱ h
+reflect-weakeningᴮ h (local₀ p) | ok q = local₀ (λ r → p (trans r q))
+reflect-weakeningᴮ h (local₀ p) | warning W = local₁ W
+reflect-weakeningᴮ h (local₁ W) = local₁ (reflect-weakeningᴱ h W)
+reflect-weakeningᴮ h (local₂ W) = local₂ (reflect-weakeningᴮ h W)
+reflect-weakeningᴮ h (function₁ f W) = function₁ f (reflect-weakeningᴮ h W)
+reflect-weakeningᴮ h (function₂ f W) = function₂ f (reflect-weakeningᴮ h W)
+
+reflectᴱ : ∀ {H H′ M M′} → (H ⊢ M ⟶ᴱ M′ ⊣ H′) → Warningᴱ H′ (typeCheckᴱ H′ ∅ M′) → Warningᴱ H (typeCheckᴱ H ∅ M)
+reflectᴮ : ∀ {H H′ B B′} → (H ⊢ B ⟶ᴮ B′ ⊣ H′) → Warningᴮ H′ (typeCheckᴮ H′ ∅ B′) → Warningᴮ H (typeCheckᴮ H ∅ B)
+
+reflectᴱ (function {F = f ⟨ var x ∈ S ⟩∈ T} defn) (BadlyTypedFunctionAddress a f refl W) = function₁ f (reflect-weakeningᴮ (snoc refl defn) W)
+reflectᴱ (app s) (app₀ p) with preservationᴱ s | heap-weakeningᴱ (redn-⊑ s)
+reflectᴱ (app s) (app₀ p) | ok q | ok q′ = app₀ (λ r → p (trans (trans (cong src (sym q)) r) q′))
+reflectᴱ (app s) (app₀ p) | warning W | _ = app₁ W
+reflectᴱ (app s) (app₀ p) | _ | warning W  = app₂ W
+reflectᴱ (app s) (app₁ W) = app₁ (reflectᴱ s W)
+reflectᴱ (app s) (app₂ W) = app₂ (reflect-weakeningᴱ (redn-⊑ s) W)
+reflectᴱ (beta {a = a} {F = f ⟨ var x ∈ T ⟩∈ U} q) (block f (local₀ p)) = app₀ (λ r → p (trans (sym (cong src (cong declaredTypeᴴ q))) r))
+reflectᴱ (beta {a = a} {F = f ⟨ var x ∈ T ⟩∈ U} q) (block f (local₁ W)) = app₂ W
+reflectᴱ (beta {a = a} {F = f ⟨ var x ∈ T ⟩∈ U} q) (block f (local₂ {T = T′} W)) = app₁ (BadlyTypedFunctionAddress a f q W)
+reflectᴱ (block s) (block b W)  = block b (reflectᴮ s W)
+reflectᴱ (return q) W = block _ (return W)
+
+reflectᴮ (local s) (local₀ p) with preservationᴱ s
+reflectᴮ (local s) (local₀ p) | ok q = local₀ (λ r → p (trans r q))
+reflectᴮ (local s) (local₀ p) | warning W = local₁ W
+reflectᴮ (local s) (local₁ W) = local₁ (reflectᴱ s W)
+reflectᴮ (local s) (local₂ W) = local₂ (reflect-weakeningᴮ (redn-⊑ s) W)
+reflectᴮ (subst {H = H} {x = var x ∈ T} {v = v}) W with T ≡ᵀ (typeOfᴱ H ∅ (val v))
+reflectᴮ (subst {x = var x ∈ T} {v = v}) W | yes refl = local₂ (reflect-substitutionᴮ _ v x (typeOfᴱⱽ v) refl W)
+reflectᴮ (subst {x = var x ∈ T} {v = v}) W | no p = local₀ p
+reflectᴮ (function {F = f ⟨ var x ∈ S ⟩∈ T} defn) W = function₂ f (reflect-weakeningᴮ (snoc refl defn) (reflect-substitutionᴮ _ _ f refl refl W))
+reflectᴮ (return s) (return W) = return (reflectᴱ s W)
 
 -- reflectᴱ (function {F = f ⟨ var x ∈ S ⟩∈ T} defn) (bot ())
 -- reflectᴱ (function defn) (addr a T q) = CONTRADICTION (q refl)
@@ -168,12 +254,6 @@ reflectᴮ s = {!!}
 -- progressᴮ H h (function D₁ D₂) q with alloc H _
 -- progressᴮ H h (function D₁ D₂) q | ok a H′ r = step (function r)
 
-import FFI.Data.Aeson
-{-# REWRITE FFI.Data.Aeson.singleton-insert-empty #-}
-
-_≡ᵀ_ : (T U : Type) → Dec (T ≡ U)
-_≡ᵀ_ = {!!}
-
 -- data LookupResult {Σ V S} (D : Σ ▷ V ∈ S) : Set where
 
 --   function : ∀ f {x B T U W} →

prototyping/Properties/TypeCheck.agda

@@ -1,3 +1,5 @@
+{-# OPTIONS --rewriting #-}
+
 open import Luau.Type using (Mode)
 
 module Properties.TypeCheck (m : Mode) where
@@ -5,13 +7,14 @@ module Properties.TypeCheck (m : Mode) where
 open import Agda.Builtin.Equality using (_≡_; refl)
 open import FFI.Data.Maybe using (Maybe; just; nothing)
 open import FFI.Data.Either using (Either)
-open import Luau.TypeCheck(m) using (_⊢ᴱ_∋_∈_⊣_; _⊢ᴮ_∋_∈_⊣_; nil; var; addr; app; function; block; done; return; local)
+open import Luau.TypeCheck(m) using (_⊢ᴱ_∈_; _⊢ᴮ_∈_; nil; var; addr; app; function; block; done; return; local)
 open import Luau.Syntax using (Block; Expr; yes; nil; var; addr; _$_; function_is_end; block_is_end; _∙_; return; done; local_←_; _⟨_⟩; _⟨_⟩∈_; var_∈_; name; fun; arg)
 open import Luau.Type using (Type; nil; top; bot; _⇒_; tgt)
-open import Luau.VarCtxt using (VarCtxt; ∅; _↦_; _⊕_↦_; _⋒_; _⊝_; ⊕-[]) renaming (_[_] to _[_]ⱽ)
+open import Luau.VarCtxt using (VarCtxt; ∅; _↦_; _⊕_↦_; _⋒_; _⊝_) renaming (_[_] to _[_]ⱽ)
 open import Luau.AddrCtxt using (AddrCtxt) renaming (_[_] to _[_]ᴬ)
 open import Luau.Addr using (Addr)
 open import Luau.Var using (Var; _≡ⱽ_)
+open import Luau.Value using (Value; nil; addr; val)
 open import Luau.Heap using (Heap; HeapValue; function_is_end) renaming (_[_] to _[_]ᴴ)
 open import Properties.Dec using (yes; no)
 open import Properties.Equality using (_≢_; sym; trans; cong)
@@ -20,16 +23,20 @@ open import Properties.Remember using (remember; _,_)
 src : Type → Type
 src = Luau.Type.src m
 
-typeOfᴴ : Maybe(HeapValue yes) → Type
+declaredTypeᴴ : Maybe(HeapValue yes) → Type
-typeOfᴴ nothing = bot
+declaredTypeᴴ nothing = bot
-typeOfᴴ (just function f ⟨ var x ∈ S ⟩∈ T is B end) = (S ⇒ T)
+declaredTypeᴴ (just function f ⟨ var x ∈ S ⟩∈ T is B end) = (S ⇒ T)
+
+typeOfⱽ : Heap yes → Value → Type
+typeOfⱽ H nil = nil
+typeOfⱽ H (addr a) = declaredTypeᴴ (H [ a ]ᴴ)
 
 typeOfᴱ : Heap yes → VarCtxt → (Expr yes) → Type
 typeOfᴮ : Heap yes → VarCtxt → (Block yes) → Type
 
 typeOfᴱ H Γ nil = nil
 typeOfᴱ H Γ (var x) = Γ [ x ]ⱽ
-typeOfᴱ H Γ (addr a) = typeOfᴴ (H [ a ]ᴴ)
+typeOfᴱ H Γ (addr a) = declaredTypeᴴ (H [ a ]ᴴ)
 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 b is B end) = typeOfᴮ H Γ B
@@ -39,32 +46,25 @@ typeOfᴮ H Γ (local var x ∈ T ← M ∙ B) = typeOfᴮ H (Γ ⊕ x ↦ T) B
 typeOfᴮ H Γ (return M ∙ B) = typeOfᴱ H Γ M
 typeOfᴮ H Γ done = nil
 
-contextOfᴱ : Heap yes → VarCtxt → Type → (Expr yes) → VarCtxt
+typeOfᴴ : Heap yes → Maybe(HeapValue yes) → Type
-contextOfᴮ : Heap yes → VarCtxt → Type → (Block yes) → VarCtxt
+typeOfᴴ H nothing = bot
+typeOfᴴ H (just function f ⟨ var x ∈ S ⟩∈ T is B end) = (S ⇒ typeOfᴮ H (x ↦ S) B)
 
-contextOfᴱ H Γ S nil = ∅
+typeOfᴱⱽ : ∀ {H Γ} v → (typeOfᴱ H Γ (val v) ≡ typeOfⱽ H v)
-contextOfᴱ H Γ S (var x) = (x ↦ S)
+typeOfᴱⱽ nil = refl
-contextOfᴱ H Γ S (addr a) = ∅
+typeOfᴱⱽ (addr a) = refl
-contextOfᴱ H Γ S (M $ N) = (contextOfᴱ H Γ (U ⇒ S) M) ⋒ (contextOfᴱ H Γ (src T) N) where T = typeOfᴱ H Γ M; U = typeOfᴱ H Γ N
-contextOfᴱ H Γ S (function f ⟨ var x ∈ T ⟩∈ U is B end) = (contextOfᴮ H (Γ ⊕ x ↦ T) U B) ⊝ x
-contextOfᴱ H Γ S (block b is B end) = (contextOfᴮ H Γ S B)
 
-contextOfᴮ H Γ S (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) = ((contextOfᴮ H (Γ ⊕ x ↦ T) U C) ⊝ x) ⋒ ((contextOfᴮ H (Γ ⊕ f ↦ (T ⇒ U)) S B) ⊝ f)
+typeCheckᴱ : ∀ H Γ M → (Γ ⊢ᴱ M ∈ (typeOfᴱ H Γ M))
-contextOfᴮ H Γ S (local var x ∈ T ← M ∙ B) = (contextOfᴱ H Γ T M) ⋒ ((contextOfᴮ H (Γ ⊕ x ↦ T)S B) ⊝ x)
+typeCheckᴮ : ∀ H Γ B → (Γ ⊢ᴮ B ∈ (typeOfᴮ H Γ B))
-contextOfᴮ H Γ S (return M ∙ B) = (contextOfᴱ H Γ S M)
-contextOfᴮ H Γ S done = ∅
 
-typeCheckᴱ : ∀ H Γ S M → (Γ ⊢ᴱ S ∋ M ∈ (typeOfᴱ H Γ M) ⊣ (contextOfᴱ H Γ S M))
+typeCheckᴱ H Γ nil = nil
-typeCheckᴮ : ∀ H Γ S B → (Γ ⊢ᴮ S ∋ B ∈ (typeOfᴮ H Γ B) ⊣ (contextOfᴮ H Γ S B))
+typeCheckᴱ H Γ (var x) = var x refl
+typeCheckᴱ H Γ (addr a) = addr a (declaredTypeᴴ (H [ a ]ᴴ))
+typeCheckᴱ H Γ (M $ N) = app (typeCheckᴱ H Γ M) (typeCheckᴱ H Γ N)
+typeCheckᴱ H Γ (function f ⟨ var x ∈ T ⟩∈ U is B end) = function f (typeCheckᴮ H (Γ ⊕ x ↦ T) B)
+typeCheckᴱ H Γ (block b is B end) = block b (typeCheckᴮ H Γ B)
 
-typeCheckᴱ H Γ S nil = nil
+typeCheckᴮ H Γ (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) = function f (typeCheckᴮ H (Γ ⊕ x ↦ T) C) (typeCheckᴮ H (Γ ⊕ f ↦ (T ⇒ U)) B)
-typeCheckᴱ H Γ S (var x) = var x refl
+typeCheckᴮ H Γ (local var x ∈ T ← M ∙ B) = local (typeCheckᴱ H Γ M) (typeCheckᴮ H (Γ ⊕ x ↦ T) B)
-typeCheckᴱ H Γ S (addr a) = addr a (typeOfᴴ (H [ a ]ᴴ))
+typeCheckᴮ H Γ (return M ∙ B) = return (typeCheckᴱ H Γ M) (typeCheckᴮ H Γ B)
-typeCheckᴱ H Γ S (M $ N) = app (typeCheckᴱ H Γ (typeOfᴱ H Γ N ⇒ S) M) (typeCheckᴱ H Γ (src (typeOfᴱ H Γ M)) N)
+typeCheckᴮ H Γ done = done
-typeCheckᴱ H Γ S (function f ⟨ var x ∈ T ⟩∈ U is B end) = function(typeCheckᴮ H (Γ ⊕ x ↦ T) U B)
-typeCheckᴱ H Γ S (block b is B end) = block b (typeCheckᴮ H Γ S B)
-
-typeCheckᴮ H Γ S (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) = function(typeCheckᴮ H (Γ ⊕ x ↦ T) U C) (typeCheckᴮ H (Γ ⊕ f ↦ (T ⇒ U)) S B)
-typeCheckᴮ H Γ S (local var x ∈ T ← M ∙ B) = local (typeCheckᴱ H Γ T M) (typeCheckᴮ H (Γ ⊕ x ↦ T) S B)
-typeCheckᴮ H Γ S (return M ∙ B) = return (typeCheckᴱ H Γ S M) (typeCheckᴮ H Γ nil B)
-typeCheckᴮ H Γ S done = done