diff --git a/Lattice.agda b/Lattice.agda
index 8c08f0e..3194c1d 100644
--- a/Lattice.agda
+++ b/Lattice.agda
@@ -101,19 +101,22 @@ module IsEquivalenceInstances where
                 in (v'' , (≈₂-trans v≈v' v'≈v'' , k,v''∈m₃))
 
             ≈-refl : {m : Map} → m ≈ m
-            ≈-refl {m} = (⊆-refl , ⊆-refl)
+            ≈-refl {m} = (⊆-refl {m}, ⊆-refl {m})
 
             ≈-sym : {m₁ m₂ : Map} → m₁ ≈ m₂ → m₂ ≈ m₁
             ≈-sym (m₁⊆m₂ , m₂⊆m₁) = (m₂⊆m₁ , m₁⊆m₂)
 
             ≈-trans : {m₁ m₂ m₃ : Map} → m₁ ≈ m₂ → m₂ ≈ m₃ → m₁ ≈ m₃
-            ≈-trans (m₁⊆m₂ , m₂⊆m₁) (m₂⊆m₃ , m₃⊆m₂) = (⊆-trans m₁⊆m₂ m₂⊆m₃ , ⊆-trans m₃⊆m₂ m₂⊆m₁)
+            ≈-trans {m₁} {m₂} {m₃} (m₁⊆m₂ , m₂⊆m₁) (m₂⊆m₃ , m₃⊆m₂) =
+                ( ⊆-trans {m₁} {m₂} {m₃} m₁⊆m₂ m₂⊆m₃
+                , ⊆-trans {m₃} {m₂} {m₁} m₃⊆m₂ m₂⊆m₁
+                )
 
             LiftEquivalence : IsEquivalence Map _≈_
             LiftEquivalence = record
-                { ≈-refl = ≈-refl
-                ; ≈-sym = ≈-sym
-                ; ≈-trans = ≈-trans
+                { ≈-refl = λ {m₁} → ≈-refl {m₁}
+                ; ≈-sym = λ {m₁} {m₂} → ≈-sym {m₁} {m₂}
+                ; ≈-trans = λ {m₁} {m₂} {m₃} → ≈-trans {m₁} {m₂} {m₃}
                 }
 
 module IsSemilatticeInstances where
diff --git a/Map.agda b/Map.agda
index 164d139..59dfc34 100644
--- a/Map.agda
+++ b/Map.agda
@@ -18,9 +18,6 @@ open import Data.List.Relation.Unary.Any using (Any; here; there) -- TODO: re-ex
 open import Data.Product using (_×_; _,_; Σ; proj₁ ; proj₂)
 open import Data.Empty using (⊥)
 
-Map : Set (a ⊔ b)
-Map = List (A × B)
-
 keys : List (A × B) → List A
 keys [] = []
 keys ((k , v) ∷ xs) = k ∷ keys xs
@@ -32,6 +29,9 @@ data Unique {c} {C : Set c} : List C → Set c where
         → Unique xs
         → Unique (x ∷ xs)
 
+Map : Set (a ⊔ b)
+Map = Σ (List (A × B)) (λ l → Unique (keys l))
+
 Unique-append : ∀ {c} {C : Set c} {x : C} {xs : List C} → ¬ MemProp._∈_ x xs → Unique xs →  Unique (xs ++ (x ∷ []))
 Unique-append {c} {C} {x} {[]} _ _ = push [] empty
 Unique-append {c} {C} {x} {x' ∷ xs'} x∉xs (push x'≢ uxs') = push (help x'≢) (Unique-append (λ x∈xs' → x∉xs (there x∈xs')) uxs')
@@ -46,15 +46,6 @@ Unique-append {c} {C} {x} {x' ∷ xs'} x∉xs (push x'≢ uxs') = push (help x'
 _∈_ : (A × B) → List (A × B) → Set (a ⊔ b)
 _∈_ p m = MemProp._∈_ p m
 
-subset : ∀ (_≈_ : B → B → Set b) → List (A × B) → List (A × B) → Set (a ⊔ b)
-subset _≈_ m₁ m₂ = ∀ (k : A) (v : B) → (k , v) ∈ m₁ → Σ B (λ v' → v ≈ v' × ((k , v') ∈ m₂))
-
-lift : ∀ (_≈_ : B → B → Set b) → List (A × B) → List (A × B) → Set (a ⊔ b)
-lift _≈_ m₁ m₂ = (m₁ ⊆ m₂) × (m₂ ⊆ m₁)
-    where
-        _⊆_ : List (A × B) → List (A × B) → Set (a ⊔ b)
-        _⊆_ = subset _≈_
-
 foldr : ∀ {c} {C : Set c} → (A → B → C → C) -> C -> List (A × B) -> C
 foldr f b [] = b
 foldr f b ((k , v) ∷ xs) = f k v (foldr f b xs)
@@ -62,7 +53,11 @@ foldr f b ((k , v) ∷ xs) = f k v (foldr f b xs)
 absurd : ∀ {a} {A : Set a} →  ⊥ → A
 absurd ()
 
-private module Impl (f : B → B → B) where
+private module ImplRelation (_≈_ : B → B → Set b) where
+    subset : List (A × B) → List (A × B) → Set (a ⊔ b)
+    subset m₁ m₂ = ∀ (k : A) (v : B) → (k , v) ∈ m₁ → Σ B (λ v' → v ≈ v' × ((k , v') ∈ m₂))
+
+private module ImplInsert (f : B → B → B) where
     _∈k_ : A → List (A × B) → Set a
     _∈k_ k m = MemProp._∈_ k (keys m)
 
@@ -110,13 +105,32 @@ private module Impl (f : B → B → B) where
     merge-preserves-unique [] l₂ u₂ = u₂
     merge-preserves-unique ((k₁ , v₁) ∷ xs₁) l₂ u₂ = insert-preserves-unique k₁ v₁ (merge xs₁ l₂) (merge-preserves-unique xs₁ l₂ u₂)
 
-Map-functional : ∀ (k : A) (v v' : B) (xs : List (A × B)) → Unique (keys ((k , v) ∷ xs)) → MemProp._∈_ (k , v') ((k , v) ∷ xs) → v ≡ v'
-Map-functional k v v' _ _ (here k,v'≡k,v) = sym (cong proj₂ k,v'≡k,v)
-Map-functional k v v' xs (push k≢ _) (there k,v'∈xs) = absurd (unique-not-in xs v' (k≢ , k,v'∈xs))
-    where
-        unique-not-in : ∀ (xs : List (A × B)) (v' : B) → ¬ (All (λ k' → ¬ k ≡ k') (keys xs) × (k , v') ∈ xs)
-        unique-not-in ((k' , _) ∷ xs) v' (k≢k' ∷ _ , here k',≡x) = k≢k' (cong proj₁ k',≡x)
-        unique-not-in (_ ∷ xs) v' (_ ∷ rest , there k,v'∈xs) = unique-not-in xs v' (rest , k,v'∈xs)
+-- Map-functional : ∀ (k : A) (v v' : B) (xs : List (A × B)) → Unique (keys ((k , v) ∷ xs)) → MemProp._∈_ (k , v') ((k , v) ∷ xs) → v ≡ v'
+-- Map-functional k v v' _ _ (here k,v'≡k,v) = sym (cong proj₂ k,v'≡k,v)
+-- Map-functional k v v' xs (push k≢ _) (there k,v'∈xs) = absurd (unique-not-in xs v' (k≢ , k,v'∈xs))
+--     where
+--         unique-not-in : ∀ (xs : List (A × B)) (v' : B) → ¬ (All (λ k' → ¬ k ≡ k') (keys xs) × (k , v') ∈ xs)
+--         unique-not-in ((k' , _) ∷ xs) v' (k≢k' ∷ _ , here k',≡x) = k≢k' (cong proj₁ k',≡x)
+--         unique-not-in (_ ∷ xs) v' (_ ∷ rest , there k,v'∈xs) = unique-not-in xs v' (rest , k,v'∈xs)
 
 module _ (f : B → B → B) where
-    open Impl f public using (insert; merge)
+    open ImplInsert f renaming
+        ( insert to insert-impl
+        ; merge to merge-impl
+        )
+
+    insert : A → B → Map → Map
+    insert k v (kvs , uks) = (insert-impl k v kvs , insert-preserves-unique k v kvs uks)
+
+    merge : Map → Map → Map
+    merge (kvs₁ , _) (kvs₂ , uks₂) = (merge-impl kvs₁ kvs₂ , merge-preserves-unique kvs₁ kvs₂ uks₂)
+
+
+module _ (_≈_ : B → B → Set b) where
+    open ImplRelation _≈_ renaming (subset to subset-impl)
+
+    subset : Map → Map → Set (a ⊔ b)
+    subset (kvs₁ , _) (kvs₂ , _) = subset-impl kvs₁ kvs₂
+
+    lift : Map → Map → Set (a ⊔ b)
+    lift m₁ m₂ = subset m₁ m₂ × subset m₂ m₁