diff --git a/prototyping/Luau/StrictMode.agda b/prototyping/Luau/StrictMode.agda
index d3c0f153..7c49a5ac 100644
--- a/prototyping/Luau/StrictMode.agda
+++ b/prototyping/Luau/StrictMode.agda
@@ -14,6 +14,7 @@ open import Luau.TypeCheck using (_⊢ᴮ_∈_; _⊢ᴱ_∈_; ⊢ᴴ_; ⊢ᴼ_;
 open import Properties.Contradiction using (¬)
 open import Properties.TypeCheck using (typeCheckᴮ)
 open import Properties.Product using (_,_)
+open import Properties.ResolveOverloads using (resolve)
 
 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
diff --git a/prototyping/Luau/TypeCheck.agda b/prototyping/Luau/TypeCheck.agda
index fac25296..05c90b29 100644
--- a/prototyping/Luau/TypeCheck.agda
+++ b/prototyping/Luau/TypeCheck.agda
@@ -14,8 +14,9 @@ open import Luau.Type using (Type; nil; unknown; number; boolean; string; _⇒_)
 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)
-open import Properties.DecSubtyping using (dec-subtyping; resolve)
+open import Properties.DecSubtyping using (dec-subtyping)
 open import Properties.Product using (_×_; _,_)
+open import Properties.ResolveOverloads using (resolve)
 
 orUnknown : Maybe Type → Type
 orUnknown nothing = unknown
diff --git a/prototyping/Properties/ResolveOverloads.agda b/prototyping/Properties/ResolveOverloads.agda
new file mode 100644
index 00000000..8c9bf204
--- /dev/null
+++ b/prototyping/Properties/ResolveOverloads.agda
@@ -0,0 +1,78 @@
+{-# OPTIONS --rewriting #-}
+
+module Properties.ResolveOverloads where
+
+open import FFI.Data.Either using (Left; Right)
+open import Luau.Subtyping using (_<:_; _≮:_)
+open import Luau.Type using (Type ; _⇒_; unknown; never)
+open import Luau.TypeSaturation using (saturate)
+open import Properties.Contradiction using (CONTRADICTION)
+open import Properties.DecSubtyping using (dec-subtypingⁿ)
+open import Properties.Functions using (_∘_)
+open import Properties.Subtyping using (<:-refl; <:-trans; <:-∩-left; <:-∩-right; <:-∩-glb; <:-impl-¬≮:; <:-unknown; function-≮:-never)
+open import Properties.TypeNormalization using (Normal; FunType; normal; _⇒_; _∩_; _∪_; never; unknown; <:-normalize)
+open import Properties.TypeSaturation using (Overloads; Saturated; _⊆ᵒ_; normal-saturate; saturated; defn; here; left; right)
+
+data ResolvedTo F G R : Set where
+
+  yes : ∀ Sʳ Tʳ →
+
+    Overloads F (Sʳ ⇒ Tʳ) →
+    (R <: Sʳ) → 
+    (∀ {S T} → Overloads G (S ⇒ T) → (R <: S) → (Tʳ <: T)) →
+    --------------------------------------------
+    ResolvedTo F G R
+
+  no :
+
+    (∀ {S T} → Overloads G (S ⇒ T) → (R ≮: S)) →
+    --------------------------------------------
+    ResolvedTo F G R
+
+  never :
+
+    (F <: never) →
+    --------------------------------------------
+    ResolvedTo F G R
+
+Resolved : Type → Type → Set
+Resolved F R = ResolvedTo F F R
+
+resolveˢ : ∀ {F G R} → FunType G → Saturated F → Normal R → (G ⊆ᵒ F) → ResolvedTo F G R
+resolveˢ (Sⁿ ⇒ Tⁿ) (defn sat-∩ sat-∪) Rⁿ G⊆F with dec-subtypingⁿ Rⁿ Sⁿ
+resolveˢ (Sⁿ ⇒ Tⁿ) (defn sat-∩ sat-∪) Rⁿ G⊆F | Left R≮:S = no (λ { here → R≮:S })
+resolveˢ (Sⁿ ⇒ Tⁿ) (defn sat-∩ sat-∪) Rⁿ G⊆F | Right R<:S = yes _ _ (G⊆F here) R<:S (λ { here _ → <:-refl })
+resolveˢ (Gᶠ ∩ Hᶠ) (defn sat-∩ sat-∪) Rⁿ G⊆F with resolveˢ Gᶠ (defn sat-∩ sat-∪) Rⁿ (G⊆F ∘ left) | resolveˢ Hᶠ (defn sat-∩ sat-∪) Rⁿ (G⊆F ∘ right)
+resolveˢ (Gᶠ ∩ Hᶠ) (defn sat-∩ sat-∪) Rⁿ G⊆F | yes S₁ T₁ o₁ R<:S₁ tgt₁ | yes S₂ T₂ o₂ R<:S₂ tgt₂ with sat-∩ o₁ o₂
+resolveˢ (Gᶠ ∩ Hᶠ) (defn sat-∩ sat-∪) Rⁿ G⊆F | yes S₁ T₁ o₁ R<:S₁ tgt₁ | yes S₂ T₂ o₂ R<:S₂ tgt₂ | defn o p₁ p₂ =
+  yes _ _ o (<:-trans (<:-∩-glb R<:S₁ R<:S₂) p₁) (λ { (left o) p → <:-trans p₂ (<:-trans <:-∩-left (tgt₁ o p)) ; (right o) p → <:-trans p₂ (<:-trans <:-∩-right (tgt₂ o p)) })
+resolveˢ (Gᶠ ∩ Hᶠ) (defn sat-∩ sat-∪) Rⁿ G⊆F | yes S₁ T₁ o₁ R<:S₁ tgt₁ | no src₂ =
+  yes _ _ o₁ R<:S₁ (λ { (left o) p → tgt₁ o p ; (right o) p → CONTRADICTION (<:-impl-¬≮: p (src₂ o)) })
+resolveˢ (Gᶠ ∩ Hᶠ) (defn sat-∩ sat-∪) Rⁿ G⊆F | no src₁ | yes S₂ T₂ o₂ R<:S₂ tgt₂ =
+  yes _ _ o₂ R<:S₂ (λ { (left o) p → CONTRADICTION (<:-impl-¬≮: p (src₁ o)) ; (right o) p → tgt₂ o p })
+resolveˢ (Gᶠ ∩ Hᶠ) (defn sat-∩ sat-∪) Rⁿ G⊆F | no src₁ | no src₂ =
+  no (λ { (left o) → src₁ o ; (right o) → src₂ o })
+resolveˢ (Gᶠ ∩ Hᶠ) (defn sat-∩ sat-∪) Rⁿ G⊆F | _ | never q = never q
+resolveˢ (Gᶠ ∩ Hᶠ) (defn sat-∩ sat-∪) Rⁿ G⊆F | never p | _ = never p
+
+resolveᶠ : ∀ {F R} → FunType F → Normal R → Resolved (saturate F) R
+resolveᶠ Fᶠ Rⁿ = resolveˢ (normal-saturate Fᶠ) (saturated Fᶠ) Rⁿ (λ o → o)
+
+resolveⁿ : ∀ {F R} → Normal F → Normal R → Resolved (saturate F) R
+resolveⁿ (Sⁿ ⇒ Tⁿ) Rⁿ = resolveᶠ (Sⁿ ⇒ Tⁿ) Rⁿ
+resolveⁿ (Fᶠ ∩ Gᶠ) Rⁿ = resolveᶠ (Fᶠ ∩ Gᶠ) Rⁿ
+resolveⁿ (Sⁿ ∪ Tˢ) Rⁿ = no (λ ())
+resolveⁿ never Rⁿ = never <:-refl
+resolveⁿ unknown Rⁿ = no (λ ())
+
+resolve : Type → Type → Type
+resolve F R with resolveⁿ (normal F) (normal R)
+resolve F R | yes S T o R<:S p = T
+resolve F R | no p = unknown
+resolve F R | never p = never
+
+<:-resolve : ∀ {R S T} → T <: resolve (S ⇒ T) R
+<:-resolve {R} {S} {T} with resolveⁿ (normal (S ⇒ T)) (normal R)
+<:-resolve {R} {S} {T} | yes Sⁿ Tⁿ here Rⁿ<:Sʳ p = <:-normalize T
+<:-resolve {R} {S} {T} | no p = <:-unknown
+<:-resolve {R} {S} {T} | never p = CONTRADICTION (<:-impl-¬≮: p function-≮:-never)
diff --git a/prototyping/Properties/StrictMode.agda b/prototyping/Properties/StrictMode.agda
index 32e8b169..53cbbaa7 100644
--- a/prototyping/Properties/StrictMode.agda
+++ b/prototyping/Properties/StrictMode.agda
@@ -23,10 +23,11 @@ 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.Functions using (_∘_)
-open import Properties.DecSubtyping using (dec-subtyping; resolve)
+open import Properties.DecSubtyping using (dec-subtyping)
 open import Properties.Subtyping using (unknown-≮:; ≡-trans-≮:; ≮:-trans-≡; ≮:-trans; ≮:-refl; scalar-≢-impl-≮:; function-≮:-scalar; scalar-≮:-function; function-≮:-never; unknown-≮:-scalar; scalar-≮:-never; unknown-≮:-never)
 open import Properties.FunctionTypes using (src-unknown-≮:; unknown-src-≮:)
-open import Properties.Subtyping using (unknown-≮:; ≡-trans-≮:; ≮:-trans-≡; ≮:-trans; ≮:-refl; scalar-≢-impl-≮:; function-≮:-scalar; scalar-≮:-function; function-≮:-never; unknown-≮:-scalar; scalar-≮:-never; unknown-≮:-never)
+open import Properties.ResolveOverloads using (resolve; <:-resolve)
+open import Properties.Subtyping using (unknown-≮:; ≡-trans-≮:; ≮:-trans-≡; ≮:-trans; <:-trans-≮:; ≮:-refl; scalar-≢-impl-≮:; function-≮:-scalar; scalar-≮:-function; function-≮:-never; unknown-≮:-scalar; scalar-≮:-never; unknown-≮:-never)
 open import Properties.TypeCheck 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; +; -; *; /; <; >; <=; >=; ··)
@@ -148,7 +149,7 @@ reflect-subtypingᴱ H (M $ N) (app₁ s) p | Right W = Right (app₁ W)
 reflect-subtypingᴱ H (M $ N) (app₂ v s) p with reflect-subtypingᴱ H N s (⇒-≮:-resolve⁻¹ (heap-weakeningᴱ ∅ H M (rednᴱ⊑ s) (resolve-≮:-⇒ p)))
 reflect-subtypingᴱ H (M $ N) (app₂ v s) p | Left q = Left (⇒-≮:-resolve (resolve⁻¹-≮:-⇒ q))
 reflect-subtypingᴱ H (M $ N) (app₂ v s) p | Right W = Right (app₂ W)
-reflect-subtypingᴱ H (M $ N) {T = T} (beta (function f ⟨ var y ∈ S ⟩∈ U is B end) v refl q) p = Left (≡-trans-≮: (cong (λ F → resolve F (typeOfᴱ H ∅ N)) (cong orUnknown (cong typeOfᴹᴼ q))) p)
+reflect-subtypingᴱ H (M $ N) {T = T} (beta (function f ⟨ var y ∈ S ⟩∈ U is B end) v refl q) p = Left (≡-trans-≮: (cong (λ F → resolve F (typeOfᴱ H ∅ N)) (cong orUnknown (cong typeOfᴹᴼ q))) (<:-trans-≮: (<:-resolve {typeOfᴱ H ∅ N}) p))
 reflect-subtypingᴱ H (function f ⟨ var x ∈ T ⟩∈ U is B end) (function a defn) p = Left p
 reflect-subtypingᴱ H (block var b ∈ T is B end) (block s) p = Left p
 reflect-subtypingᴱ H (block var b ∈ T is return (val v) ∙ B end) (return v) p = mapR BlockMismatch (swapLR (≮:-trans p))
diff --git a/prototyping/Properties/TypeCheck.agda b/prototyping/Properties/TypeCheck.agda
index c304e8bf..12385a5a 100644
--- a/prototyping/Properties/TypeCheck.agda
+++ b/prototyping/Properties/TypeCheck.agda
@@ -17,10 +17,10 @@ open import Luau.Var using (Var; _≡ⱽ_)
 open import Luau.Heap using (Heap; Object; function_is_end) renaming (_[_] to _[_]ᴴ)
 open import Properties.Contradiction using (CONTRADICTION)
 open import Properties.Dec using (yes; no)
-open import Properties.DecSubtyping using (resolve)
 open import Properties.Equality using (_≢_; sym; trans; cong)
 open import Properties.Product using (_×_; _,_)
 open import Properties.Remember using (Remember; remember; _,_)
+open import Properties.ResolveOverloads using (resolve)
 
 typeOfᴼ : Object yes → Type
 typeOfᴼ (function f ⟨ var x ∈ S ⟩∈ T is B end) = (S ⇒ T)