diff --git a/prototyping/FFI/Data/Aeson.agda b/prototyping/FFI/Data/Aeson.agda
index 684f2177..558c8a8c 100644
--- a/prototyping/FFI/Data/Aeson.agda
+++ b/prototyping/FFI/Data/Aeson.agda
@@ -1,11 +1,12 @@
 module FFI.Data.Aeson where
 
+open import Agda.Builtin.Equality using (_≡_)
 open import Agda.Builtin.Bool using (Bool)
 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)
+open import FFI.Data.Maybe using (Maybe; just)
 open import FFI.Data.Either using (Either; mapLeft)
 open import FFI.Data.Scientific using (Scientific)
 open import FFI.Data.Vector using (Vector)
@@ -37,6 +38,8 @@ postulate
 {-# COMPILE GHC unionWith = \_ -> Data.Aeson.KeyMap.unionWith #-}
 {-# COMPILE GHC lookup = \_ -> Data.Aeson.KeyMap.lookup #-}
 
+postulate lookup-insert : ∀ {A} k v (m : KeyMap A) → (lookup k (insert k v m) ≡ just v)
+
 data Value : Set where
   object : KeyMap Value → Value
   array : Vector Value → Value
diff --git a/prototyping/Luau/TypeCheck.agda b/prototyping/Luau/TypeCheck.agda
index 8420bda8..ad1c8d4e 100644
--- a/prototyping/Luau/TypeCheck.agda
+++ b/prototyping/Luau/TypeCheck.agda
@@ -2,7 +2,7 @@ module Luau.TypeCheck where
 
 open import Agda.Builtin.Equality using (_≡_)
 open import FFI.Data.Maybe using (Maybe; just)
-open import Luau.Syntax using (Expr; Stat; Block; yes; nil; addr; var; var_∈_; _⟨_⟩∈_; anon⟨_⟩∈_; function_is_end; _$_; block_is_end; local_←_; _∙_; done; return; name)
+open import Luau.Syntax using (Expr; Stat; Block; yes; nil; addr; var; var_∈_; _⟨_⟩∈_; function_is_end; _$_; block_is_end; local_←_; _∙_; done; return; name)
 open import Luau.Var using (Var)
 open import Luau.Addr using (Addr)
 open import Luau.Heap using (Heap; HeapValue; function_is_end) renaming (_[_] to _[_]ᴴ)
@@ -40,7 +40,7 @@ data _▷_⊢ᴮ_∋_∈_⊣_ where
 
     Σ ▷ (Γ ⊕ x ↦ T) ⊢ᴮ any ∋ C ∈ U ⊣ Δ₁ →
     Σ ▷ (Γ ⊕ f ↦ (T ⇒ U)) ⊢ᴮ S ∋ B ∈ V ⊣ Δ₂ →
-    ---------------------------------------------------------------------------
+    ---------------------------------------------------------------------------------
     Σ ▷ Γ ⊢ᴮ S ∋ function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B ∈ V ⊣ ((Δ₁ ⊝ x) ⋒ (Δ₂ ⊝ f))
 
 data _▷_⊢ᴱ_∋_∈_⊣_  where
@@ -64,16 +64,16 @@ data _▷_⊢ᴱ_∋_∈_⊣_  where
 
   app : ∀ {Σ M N S T U Γ Δ₁ Δ₂} →
 
-    Σ ▷ Γ ⊢ᴱ (U ⇒ S) ∋ M ∈ T ⊣ Δ₂ →
+    Σ ▷ Γ ⊢ᴱ (U ⇒ S) ∋ M ∈ T ⊣ Δ₁ →
     Σ ▷ Γ ⊢ᴱ (src T) ∋ N ∈ U ⊣ Δ₂ →
     --------------------------------------
     Σ ▷ Γ ⊢ᴱ S ∋ (M $ N) ∈ (tgt T) ⊣ (Δ₁ ⋒ Δ₂)
 
-  function : ∀ {Σ x B S T U V Γ Δ} →
+  function : ∀ {Σ f x B S T U V Γ Δ} →
 
     Σ ▷ (Γ ⊕ x ↦ T) ⊢ᴮ U ∋ B ∈ V ⊣ Δ →
-    ----------------------------------------------------------------------
-    Σ ▷ Γ ⊢ᴱ S ∋ (function anon⟨ var x ∈ T ⟩∈ U is B end) ∈ (T ⇒ U) ⊣ (Δ ⊝ x)
+    -----------------------------------------------------------------------
+    Σ ▷ Γ ⊢ᴱ S ∋ (function f ⟨ var x ∈ T ⟩∈ U is B end) ∈ (T ⇒ U) ⊣ (Δ ⊝ x)
 
   block : ∀ {Σ b B S T Γ Δ} →
 
diff --git a/prototyping/Luau/VarCtxt.agda b/prototyping/Luau/VarCtxt.agda
index ab170d3a..4db495c6 100644
--- a/prototyping/Luau/VarCtxt.agda
+++ b/prototyping/Luau/VarCtxt.agda
@@ -1,8 +1,9 @@
 module Luau.VarCtxt where
 
+open import Agda.Builtin.Equality using (_≡_)
 open import Luau.Type using (Type; _∪_; _∩_)
 open import Luau.Var using (Var)
-open import FFI.Data.Aeson using (KeyMap; Key; empty; unionWith; singleton; insert; delete; lookup; fromString)
+open import FFI.Data.Aeson using (KeyMap; Key; empty; unionWith; singleton; insert; delete; lookup; fromString; lookup-insert)
 open import FFI.Data.Maybe using (Maybe; just; nothing)
 
 VarCtxt : Set
@@ -29,3 +30,5 @@ 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 ]) ≡ just T)
+⊕-[] = λ (Γ : VarCtxt) x T → lookup-insert (fromString x) T Γ
diff --git a/prototyping/Properties/TypeCheck.agda b/prototyping/Properties/TypeCheck.agda
new file mode 100644
index 00000000..f7d09d67
--- /dev/null
+++ b/prototyping/Properties/TypeCheck.agda
@@ -0,0 +1,140 @@
+module Properties.TypeCheck 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 using (_▷_⊢ᴱ_∋_∈_⊣_; _▷_⊢ᴮ_∋_∈_⊣_; nil; var; addr; app)
+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; none; _⇒_; src; tgt)
+open import Luau.VarCtxt using (VarCtxt; ∅; _↦_; _⊕_↦_; _⋒_; ⊕-[]) renaming (_[_] to _[_]ⱽ)
+open import Luau.Addr using (Addr)
+open import Luau.Var using (Var; _≡ⱽ_)
+open import Luau.AddrCtxt using (AddrCtxt) renaming (_[_] to _[_]ᴬ)
+open import Properties.Dec using (⊥; yes; no)
+open import Properties.Remember using (remember; _,_)
+
+sym : ∀ {A : Set} {a b : A} → (a ≡ b) → (b ≡ a)
+sym refl = refl
+
+trans : ∀ {A : Set} {a b c : A} → (a ≡ b) → (b ≡ c) → (a ≡ c)
+trans refl refl = refl
+
+cong : ∀ {A B : Set} {a b : A} (f : A → B) → (a ≡ b) → (f a ≡ f b)
+cong f refl = refl
+
+_⊆_ : ∀ {A : Set} → (A → Set) → (A → Set) → Set
+P ⊆ Q = (∀ a → P a → Q a)
+
+data _⊝_ {A : Set} (P : A → Set) (a b : A) : Set where
+  _,_ : (P b) → ((a ≡ b) → ⊥) → (P ⊝ a) b
+
+data _∪_ {A : Set} (P Q : A → Set) (a : A) : Set where
+  left : (P a) → (P ∪ Q) a
+  right : (Q a) → (P ∪ Q) a
+
+∪-⊆ : ∀ {A : Set} {P Q R : A → Set} → (P ⊆ R) → (Q ⊆ R) → ((P ∪ Q) ⊆ R)
+∪-⊆ p q a (left r) = p a r
+∪-⊆ p q a (right r) = q a r
+
+⊆-left : {A : Set} {P Q R : A → Set} → ((P ∪ Q) ⊆ R) → (P ⊆ R)
+⊆-left p a q = p a (left q)
+
+⊆-right : {A : Set} {P Q R : A → Set} → ((P ∪ Q) ⊆ R) → (Q ⊆ R)
+⊆-right p a q = p a (right q)
+
+sing : ∀ {A} → A → A → Set
+sing = _≡_
+
+data emp {A : Set} : A → Set where
+
+fvᴱ : ∀ {a} → Expr a → Var → Set
+fvᴮ : ∀ {a} → Block a → Var → Set
+
+fvᴱ nil = emp
+fvᴱ (var x) = sing x
+fvᴱ (addr x) = emp
+fvᴱ (M $ N) = fvᴱ M ∪ fvᴱ N
+fvᴱ function F is B end = fvᴮ B ⊝ name (arg F)
+fvᴱ block b is B end = fvᴮ B
+
+fvᴮ (function F is C end ∙ B) = (fvᴮ C ⊝ name (arg F)) ∪ (fvᴮ B ⊝ fun F)
+fvᴮ (local x ← M ∙ B) = fvᴱ M ∪ (fvᴮ B ⊝ name x)
+fvᴮ (return M ∙ B) = fvᴱ M ∪ fvᴮ B
+fvᴮ done = emp
+
+faᴱ : ∀ {a} → Expr a → Addr → Set
+faᴮ : ∀ {a} → Block a → Addr → Set
+
+faᴱ nil = emp
+faᴱ (var x) = emp
+faᴱ (addr a) = sing a
+faᴱ (M $ N) = faᴱ M ∪ faᴱ N
+faᴱ function F is B end = faᴮ B
+faᴱ block b is B end = faᴮ B
+
+faᴮ (function F is C end ∙ B) = faᴮ C ∪ faᴮ B
+faᴮ (local x ← M ∙ B) =  faᴱ M ∪ faᴮ B
+faᴮ (return M ∙ B) = faᴱ M ∪ faᴮ B
+faᴮ done = emp
+
+data dv (Γ : VarCtxt) (x : Var) : Set where
+  just : ∀ T → (just T ≡ Γ [ x ]ⱽ) → dv Γ x
+
+data da (Σ : AddrCtxt) (a : Addr) : Set where
+  just : ∀ T → (just T ≡ Σ [ a ]ᴬ) → da Σ a
+
+⊕-⊆-⊝ : ∀ {Γ P} x T → ((P ⊝ x) ⊆ dv Γ) → (P ⊆ dv (Γ ⊕ x ↦ T))
+⊕-⊆-⊝ x T p y q with x ≡ⱽ y
+⊕-⊆-⊝ x T p .x q | yes refl = just T (sym (⊕-[] _ x T))
+⊕-⊆-⊝ x T p y q | no r = {!!}
+
+orNone : Maybe Type → Type
+orNone nothing = none
+orNone (just T) = T
+
+dv-orNone : ∀ {Γ x} → (dv Γ x) → (just (orNone (Γ [ x ]ⱽ)) ≡ Γ [ x ]ⱽ)
+dv-orNone (just T p) = trans (sym (cong just (cong orNone p))) p
+
+da-orNone : ∀ {Σ a} → (da Σ a) → (just (orNone (Σ [ a ]ᴬ)) ≡ Σ [ a ]ᴬ)
+da-orNone (just T p) = trans (sym (cong just (cong orNone p))) p
+
+
+typeOfᴱ : AddrCtxt → VarCtxt → (Expr yes) → Type
+typeOfᴮ : AddrCtxt → VarCtxt → (Block yes) → Type
+
+typeOfᴱ Σ Γ nil = nil
+typeOfᴱ Σ Γ (var x) = orNone(Γ [ x ]ⱽ)
+typeOfᴱ Σ Γ (addr a) = orNone(Σ [ a ]ᴬ)
+typeOfᴱ Σ Γ (M $ N) = tgt(typeOfᴱ Σ Γ M)
+typeOfᴱ Σ Γ (function f ⟨ var x ∈ S ⟩∈ T is B end) = S ⇒ T
+typeOfᴱ Σ Γ (block b is B end) = typeOfᴮ Σ Γ B
+
+typeOfᴮ Σ Γ (function f ⟨ var x ∈ S ⟩∈ T is C end ∙ B) = typeOfᴮ Σ (Γ ⊕ f ↦ (S ⇒ T)) B
+typeOfᴮ Σ Γ (local var x ∈ T ← M ∙ B) = typeOfᴮ Σ (Γ ⊕ x ↦ T) B
+typeOfᴮ Σ Γ (return M ∙ B) = typeOfᴱ Σ Γ M
+typeOfᴮ Σ Γ done = nil
+
+data TypeCheckResultᴱ (Σ : AddrCtxt) (Γ : VarCtxt) (S : Type) (M : Expr yes) : Set
+data TypeCheckResultᴮ (Σ : AddrCtxt) (Γ : VarCtxt) (S : Type) (B : Block yes) : Set
+
+data TypeCheckResultᴱ Σ Γ S M where
+
+  ok : ∀ Δ → (Σ ▷ Γ ⊢ᴱ S ∋ M ∈ (typeOfᴱ Σ Γ M) ⊣ Δ) → TypeCheckResultᴱ Σ Γ S M
+  
+data TypeCheckResultᴮ Σ Γ S B where
+
+  ok : ∀ Δ → (Σ ▷ Γ ⊢ᴮ S ∋ B ∈ (typeOfᴮ Σ Γ B) ⊣ Δ) → TypeCheckResultᴮ Σ Γ S B
+  
+typeCheckᴱ : ∀ Σ Γ S M → (faᴱ M ⊆ da Σ) → (fvᴱ M ⊆ dv Γ) → (TypeCheckResultᴱ Σ Γ S M)
+typeCheckᴮ : ∀ Σ Γ S B → (faᴮ B ⊆ da Σ) → (fvᴮ B ⊆ dv Γ) → (TypeCheckResultᴮ Σ Γ S B)
+
+typeCheckᴱ Σ Γ S nil p q = ok ∅ nil
+typeCheckᴱ Σ Γ S (var x) p q = ok (x ↦ S) (var x (dv-orNone (q x refl)))
+typeCheckᴱ Σ Γ S (addr a) p q = ok ∅ (addr a (da-orNone (p a refl)))
+typeCheckᴱ Σ Γ S (M $ N) p q with typeCheckᴱ Σ Γ (typeOfᴱ Σ Γ N ⇒ S) M (⊆-left p) (⊆-left q) | typeCheckᴱ Σ Γ (src (typeOfᴱ Σ Γ M)) N (⊆-right p) (⊆-right q)
+typeCheckᴱ Σ Γ S (M $ N) p q | ok Δ₁ r | ok Δ₂ s = ok (Δ₁ ⋒ Δ₂) (app r s)
+typeCheckᴱ Σ Γ S (function f ⟨ var x ∈ T ⟩∈ U is B end) p q with typeCheckᴮ Σ (Γ ⊕ x ↦ T) U B p {!q!}
+typeCheckᴱ Σ Γ S (function f ⟨ var x ∈ T ⟩∈ U is B end) p q | R = {!!}
+typeCheckᴱ Σ Γ S (block b is B end) p q = {!!}
+
+typeCheckᴮ Σ Γ S B = {!!}