diff --git a/Equivalence.agda b/Equivalence.agda
index edac885..82abed5 100644
--- a/Equivalence.agda
+++ b/Equivalence.agda
@@ -35,44 +35,3 @@ module IsEquivalenceInstances where
                 , IsEquivalence.≈-trans eB b₁≈b₂ b₂≈b₃
                 )
             }
-
-    module ForMap {a b} (A : Set a) (B : Set b)
-        (≡-dec-A : Decidable (_≡_ {a} {A}))
-        (_≈₂_ : B → B → Set b)
-        (eB : IsEquivalence B _≈₂_) where
-
-        open import Lattice.Map A B ≡-dec-A using (Map; lift; subset)
-        open import Data.List using (_∷_; []) -- TODO: re-export these with nicer names from map
-
-        open IsEquivalence eB renaming
-            ( ≈-refl to ≈₂-refl
-            ; ≈-sym to ≈₂-sym
-            ; ≈-trans to ≈₂-trans
-            )
-
-        _≈_ : Map → Map → Set (Agda.Primitive._⊔_ a b)
-        _≈_ = lift _≈₂_
-
-        _⊆_ : Map → Map → Set (Agda.Primitive._⊔_ a b)
-        _⊆_ = subset _≈₂_
-
-        private
-            ⊆-refl : (m : Map) →  m ⊆ m
-            ⊆-refl _ k v k,v∈m = (v , (≈₂-refl , k,v∈m))
-
-            ⊆-trans : (m₁ m₂ m₃ : Map) →  m₁ ⊆ m₂ → m₂ ⊆ m₃ → m₁ ⊆ m₃
-            ⊆-trans _ _ _ m₁⊆m₂ m₂⊆m₃ k v k,v∈m₁ =
-                let
-                    (v' , (v≈v' , k,v'∈m₂)) = m₁⊆m₂ k v k,v∈m₁
-                    (v'' , (v'≈v'' , k,v''∈m₃)) = m₂⊆m₃ k v' k,v'∈m₂
-                in (v'' , (≈₂-trans v≈v' v'≈v'' , k,v''∈m₃))
-
-        LiftEquivalence : IsEquivalence Map _≈_
-        LiftEquivalence = record
-            { ≈-refl = λ {m} → (⊆-refl m , ⊆-refl m)
-            ; ≈-sym = λ {m₁} {m₂} (m₁⊆m₂ , m₂⊆m₁) → (m₂⊆m₁ , m₁⊆m₂)
-            ; ≈-trans = λ {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
-                )
-            }
diff --git a/Lattice.agda b/Lattice.agda
index 03b61b6..731e80c 100644
--- a/Lattice.agda
+++ b/Lattice.agda
@@ -215,48 +215,6 @@ module IsSemilatticeInstances where
                 )
             }
 
-    module ForMap {a} {A B : Set a}
-        (≡-dec-A : Decidable (_≡_ {a} {A}))
-        (_≈₂_ : B → B → Set a)
-        (_⊔₂_ : B → B → B)
-        (sB : IsSemilattice B _≈₂_ _⊔₂_) where
-
-        open import Lattice.Map A B ≡-dec-A
-        open IsSemilattice sB renaming
-            ( ≈-refl to ≈₂-refl; ≈-sym to ≈₂-sym; ≈-⊔-cong to ≈₂-⊔₂-cong
-            ; ⊔-assoc to ⊔₂-assoc; ⊔-comm to ⊔₂-comm; ⊔-idemp to ⊔₂-idemp
-            )
-
-        module MapEquiv = IsEquivalenceInstances.ForMap A B ≡-dec-A _≈₂_ (IsSemilattice.≈-equiv sB)
-        open MapEquiv using (_≈_) public
-
-        infixl 20 _⊔_
-        infixl 20 _⊓_
-
-        _⊔_ : Map → Map → Map
-        m₁ ⊔ m₂ = union _⊔₂_ m₁ m₂
-
-        _⊓_ : Map → Map → Map
-        m₁ ⊓ m₂ = intersect _⊔₂_ m₁ m₂
-
-        MapIsUnionSemilattice : IsSemilattice Map _≈_ _⊔_
-        MapIsUnionSemilattice = record
-            { ≈-equiv = MapEquiv.LiftEquivalence
-            ; ≈-⊔-cong = λ {m₁} {m₂} {m₃} {m₄} → union-cong _≈₂_ _⊔₂_ ≈₂-⊔₂-cong {m₁} {m₂} {m₃} {m₄}
-            ; ⊔-assoc = union-assoc _≈₂_ ≈₂-refl ≈₂-sym _⊔₂_ ⊔₂-assoc
-            ; ⊔-comm = union-comm _≈₂_ ≈₂-refl ≈₂-sym _⊔₂_ ⊔₂-comm
-            ; ⊔-idemp = union-idemp _≈₂_ ≈₂-refl ≈₂-sym _⊔₂_ ⊔₂-idemp
-            }
-
-        MapIsIntersectSemilattice : IsSemilattice Map _≈_ _⊓_
-        MapIsIntersectSemilattice = record
-            { ≈-equiv = MapEquiv.LiftEquivalence
-            ; ≈-⊔-cong = λ {m₁} {m₂} {m₃} {m₄} → intersect-cong _≈₂_ _⊔₂_ ≈₂-⊔₂-cong {m₁} {m₂} {m₃} {m₄}
-            ; ⊔-assoc = intersect-assoc _≈₂_ ≈₂-refl ≈₂-sym _⊔₂_ ⊔₂-assoc
-            ; ⊔-comm = intersect-comm _≈₂_ ≈₂-refl ≈₂-sym _⊔₂_ ⊔₂-comm
-            ; ⊔-idemp = intersect-idemp _≈₂_ ≈₂-refl ≈₂-sym _⊔₂_ ⊔₂-idemp
-            }
-
 module IsLatticeInstances where
     module ForNat where
         open Nat
@@ -329,33 +287,6 @@ module IsLatticeInstances where
                 )
             }
 
-    module ForMap {a} {A B : Set a}
-        (≡-dec-A : Decidable (_≡_ {a} {A}))
-        (_≈₂_ : B → B → Set a)
-        (_⊔₂_ : B → B → B)
-        (_⊓₂_ : B → B → B)
-        (lB : IsLattice B _≈₂_ _⊔₂_ _⊓₂_) where
-
-        open import Lattice.Map A B ≡-dec-A
-        open IsLattice lB renaming
-            ( ≈-refl to ≈₂-refl; ≈-sym to ≈₂-sym
-            ; ⊔-idemp to ⊔₂-idemp; ⊓-idemp to ⊓₂-idemp
-            ; absorb-⊔-⊓ to absorb-⊔₂-⊓₂; absorb-⊓-⊔ to absorb-⊓₂-⊔₂
-            )
-
-        module MapJoin = IsSemilatticeInstances.ForMap ≡-dec-A _≈₂_ _⊔₂_ (IsLattice.joinSemilattice lB)
-        open MapJoin using (_⊔_; _≈_) public
-
-        module MapMeet = IsSemilatticeInstances.ForMap ≡-dec-A _≈₂_ _⊓₂_ (IsLattice.meetSemilattice lB)
-        open MapMeet using (_⊓_) public
-
-        MapIsLattice : IsLattice Map _≈_ _⊔_ _⊓_
-        MapIsLattice = record
-            { joinSemilattice = MapJoin.MapIsUnionSemilattice
-            ; meetSemilattice = MapMeet.MapIsIntersectSemilattice
-            ; absorb-⊔-⊓ = union-intersect-absorb _≈₂_ ≈₂-refl ≈₂-sym _⊔₂_ _⊓₂_ ⊔₂-idemp ⊓₂-idemp absorb-⊔₂-⊓₂ absorb-⊓₂-⊔₂
-            ; absorb-⊓-⊔ = intersect-union-absorb _≈₂_ ≈₂-refl ≈₂-sym _⊔₂_ _⊓₂_ ⊔₂-idemp ⊓₂-idemp absorb-⊔₂-⊓₂ absorb-⊓₂-⊔₂
-            }
 
 module IsFiniteHeightLatticeInstances where
     module ForProd {a} {A B : Set a}
diff --git a/Lattice/Map.agda b/Lattice/Map.agda
index d7b79ee..d89dd97 100644
--- a/Lattice/Map.agda
+++ b/Lattice/Map.agda
@@ -1,3 +1,4 @@
+open import Lattice
 open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; sym; trans; cong; subst)
 open import Relation.Binary.Definitions using (Decidable)
 open import Relation.Binary.Core using (Rel)
@@ -5,8 +6,10 @@ open import Relation.Nullary using (Dec; yes; no; Reflects; ofʸ; ofⁿ)
 open import Agda.Primitive using (Level) renaming (_⊔_ to _⊔ℓ_)
 
 module Lattice.Map {a b : Level} (A : Set a) (B : Set b)
+    (_≈₂_ : B → B → Set b)
+    (_⊔₂_ : B → B → B) (_⊓₂_ : B → B → B)
     (≡-dec-A : Decidable (_≡_ {a} {A}))
-    where
+    (lB : IsLattice B _≈₂_ _⊔₂_ _⊓₂_) where
 
 import Data.List.Membership.Propositional as MemProp
 
@@ -16,7 +19,16 @@ open import Data.List using (List; map; []; _∷_; _++_)
 open import Data.List.Relation.Unary.All using (All; []; _∷_)
 open import Data.List.Relation.Unary.Any using (Any; here; there) -- TODO: re-export these with nicer names from map
 open import Data.Product using (_×_; _,_; Σ; proj₁ ; proj₂)
-open import Data.Empty using (⊥)
+open import Data.Empty using (⊥; ⊥-elim)
+open import Equivalence
+
+open IsLattice lB using () renaming
+    ( ≈-refl to ≈₂-refl; ≈-sym to ≈₂-sym; ≈-trans to ≈₂-trans
+    ; ≈-⊔-cong to ≈₂-⊔₂-cong; ≈-⊓-cong to ≈₂-⊓₂-cong
+    ; ⊔-idemp to ⊔₂-idemp; ⊔-comm to ⊔₂-comm; ⊔-assoc to ⊔₂-assoc
+    ; ⊓-idemp to ⊓₂-idemp; ⊓-comm to ⊓₂-comm; ⊓-assoc to ⊓₂-assoc
+    ; absorb-⊔-⊓ to absorb-⊔₂-⊓₂; absorb-⊓-⊔ to absorb-⊓₂-⊔₂
+    )
 
 keys : List (A × B) → List A
 keys = map proj₁
@@ -45,9 +57,6 @@ All¬-¬Any : ∀ {p c} {C : Set c} {P : C → Set p} {l : List C} → All (λ x
 All¬-¬Any {l = x ∷ xs} (¬Px ∷ _) (here Px) = ¬Px Px
 All¬-¬Any {l = x ∷ xs} (_ ∷ ¬Pxs) (there Pxs) = All¬-¬Any ¬Pxs Pxs
 
-absurd : ∀ {a} {A : Set a} →  ⊥ → A
-absurd ()
-
 private module _ where
     open MemProp using (_∈_)
 
@@ -63,9 +72,9 @@ private module _ where
     ListAB-functional _ (here k,v≡x) (here k,v'≡x) =
         cong proj₂ (trans k,v≡x (sym k,v'≡x))
     ListAB-functional (push k≢xs _) (here k,v≡x) (there k,v'∈xs)
-        rewrite sym k,v≡x = absurd (unique-not-in (k≢xs , k,v'∈xs))
+        rewrite sym k,v≡x = ⊥-elim (unique-not-in (k≢xs , k,v'∈xs))
     ListAB-functional (push k≢xs _) (there k,v∈xs) (here k,v'≡x)
-        rewrite sym k,v'≡x = absurd (unique-not-in (k≢xs , k,v∈xs))
+        rewrite sym k,v'≡x = ⊥-elim (unique-not-in (k≢xs , k,v∈xs))
     ListAB-functional {l = _ ∷ xs } (push _ uxs) (there k,v∈xs) (there k,v'∈xs) =
         ListAB-functional uxs k,v∈xs k,v'∈xs
 
@@ -91,12 +100,12 @@ private module _ where
     locate {k} {(k' , v) ∷ xs} (here k≡k') rewrite k≡k' = (v , here refl)
     locate {k} {(k' , v) ∷ xs} (there k∈kxs) = let (v , k,v∈xs) = locate k∈kxs in (v , there k,v∈xs)
 
-private module ImplRelation (_≈_ : B → B → Set b) where
+private module ImplRelation where
     open MemProp using (_∈_)
 
     subset : List (A × B) → List (A × B) → Set (a ⊔ℓ b)
     subset m₁ m₂ = ∀ (k : A) (v : B) → (k , v) ∈ m₁ →
-                   Σ B (λ v' → v ≈ v' × ((k , v') ∈ m₂))
+                   Σ B (λ v' → v ≈₂ v' × ((k , v') ∈ m₂))
 
 private module ImplInsert (f : B → B → B) where
     open import Data.List using (map)
@@ -124,18 +133,18 @@ private module ImplInsert (f : B → B → B) where
     insert-keys-∈ {k} {v} {(k' , v') ∷ xs} (here k≡k')
         with (≡-dec-A k k')
     ...   | yes _ = refl
-    ...   | no k≢k' = absurd (k≢k' k≡k')
+    ...   | no k≢k' = ⊥-elim (k≢k' k≡k')
     insert-keys-∈ {k} {v} {(k' , _) ∷ xs} (there k∈kxs)
         with (≡-dec-A k k')
     ...   | yes _ = refl
     ...   | no _ = cong (λ xs' → k' ∷ xs') (insert-keys-∈ k∈kxs)
 
-    insert-keys-∉ : ∀ {k : A} {v : B} {l : List (A × B)} → 
+    insert-keys-∉ : ∀ {k : A} {v : B} {l : List (A × B)} →
                     ¬ (k ∈k l) → (keys l ++ (k ∷ [])) ≡ keys (insert k v l)
     insert-keys-∉ {k} {v} {[]} _ = refl
     insert-keys-∉ {k} {v} {(k' , v') ∷ xs} k∉kl
         with (≡-dec-A k k')
-    ...   | yes k≡k' = absurd (k∉kl (here k≡k'))
+    ...   | yes k≡k' = ⊥-elim (k∉kl (here k≡k'))
     ...   | no _ = cong (λ xs' → k' ∷ xs')
                         (insert-keys-∉ (λ k∈kxs → k∉kl (there k∈kxs)))
 
@@ -157,7 +166,7 @@ private module ImplInsert (f : B → B → B) where
     insert-fresh {l = []} k∉kl = here refl
     insert-fresh {k} {l = (k' , v') ∷ xs} k∉kl
         with ≡-dec-A k k'
-    ...   | yes k≡k' = absurd (k∉kl (here k≡k'))
+    ...   | yes k≡k' = ⊥-elim (k∉kl (here k≡k'))
     ...   | no _ = there (insert-fresh (λ k∈kxs → k∉kl (there k∈kxs)))
 
     insert-preserves-∉k : ∀ {k k' : A} {v' : B} {l : List (A × B)} →
@@ -179,7 +188,7 @@ private module ImplInsert (f : B → B → B) where
     union-preserves-∉ {l₁ = []} _ k∉kl₂ = k∉kl₂
     union-preserves-∉ {k} {(k' , v') ∷ xs₁} k∉kl₁ k∉kl₂
         with ≡-dec-A k k'
-    ...   | yes k≡k' = absurd (k∉kl₁ (here k≡k'))
+    ...   | yes k≡k' = ⊥-elim (k∉kl₁ (here k≡k'))
     ...   | no k≢k' = insert-preserves-∉k k≢k' (union-preserves-∉ (λ k∈kxs₁ → k∉kl₁ (there k∈kxs₁)) k∉kl₂)
 
     insert-preserves-∈k : ∀ {k k' : A} {v' : B} {l : List (A × B)} →
@@ -212,9 +221,9 @@ private module ImplInsert (f : B → B → B) where
                        ¬ k ∈k union l₁ l₂ → ¬ k ∈k l₁ × ¬ k ∈k l₂
     ∉-union-∉-either {k} {l₁} {l₂} k∉l₁l₂
         with ∈k-dec k l₁
-    ...   | yes k∈kl₁ = absurd (k∉l₁l₂ (union-preserves-∈k₁ k∈kl₁))
+    ...   | yes k∈kl₁ = ⊥-elim (k∉l₁l₂ (union-preserves-∈k₁ k∈kl₁))
     ...   | no k∉kl₁ with ∈k-dec k l₂
-    ...         | yes k∈kl₂ = absurd (k∉l₁l₂ (union-preserves-∈k₂ {l₁ = l₁} k∈kl₂))
+    ...         | yes k∈kl₂ = ⊥-elim (k∉l₁l₂ (union-preserves-∈k₂ {l₁ = l₁} k∈kl₂))
     ...         | no k∉kl₂ = (k∉kl₁ , k∉kl₂)
 
 
@@ -222,7 +231,7 @@ private module ImplInsert (f : B → B → B) where
                          ¬ k ≡ k' → (k , v) ∈ l → (k , v) ∈ insert k' v' l
     insert-preserves-∈ {k} {k'} {l = x ∷ xs} k≢k' (here k,v=x)
         rewrite sym k,v=x with ≡-dec-A k' k
-    ...   | yes k'≡k = absurd (k≢k' (sym k'≡k))
+    ...   | yes k'≡k = ⊥-elim (k≢k' (sym k'≡k))
     ...   | no _ = here refl
     insert-preserves-∈ {k} {k'} {l = (k'' , _) ∷ xs} k≢k' (there k,v∈xs)
         with ≡-dec-A k' k''
@@ -245,7 +254,7 @@ private module ImplInsert (f : B → B → B) where
 
             k≢k' : ¬ k ≡ k'
             k≢k' with ≡-dec-A k k'
-            ...    | yes k≡k' rewrite k≡k' = absurd (All¬-¬Any k'≢xs₁ (∈-cong proj₁ k,v∈xs₁))
+            ...    | yes k≡k' rewrite k≡k' = ⊥-elim (All¬-¬Any k'≢xs₁ (∈-cong proj₁ k,v∈xs₁))
             ...    | no  k≢k' = k≢k'
     union-preserves-∈₁ {l₁ = (k' , v') ∷ xs₁} (push k'≢xs₁ uxs₁) (here k,v≡k',v') k∉kl₂
         rewrite cong proj₁ k,v≡k',v' rewrite cong proj₂ k,v≡k',v' =
@@ -257,10 +266,10 @@ private module ImplInsert (f : B → B → B) where
         rewrite cong proj₁ k,v'≡k',v'' rewrite cong proj₂ k,v'≡k',v''
         with ≡-dec-A k' k'
     ...   | yes _ = here refl
-    ...   | no k≢k' = absurd (k≢k' refl)
+    ...   | no k≢k' = ⊥-elim (k≢k' refl)
     insert-combines {k} {l = (k' , v'') ∷ xs} (push k'≢xs uxs) (there k,v'∈xs)
         with ≡-dec-A k k'
-    ...   | yes k≡k' rewrite k≡k' = absurd (All¬-¬Any k'≢xs (∈-cong proj₁ k,v'∈xs))
+    ...   | yes k≡k' rewrite k≡k' = ⊥-elim (All¬-¬Any k'≢xs (∈-cong proj₁ k,v'∈xs))
     ...   | no k≢k' = there (insert-combines uxs k,v'∈xs)
 
     union-combines : ∀ {k : A} {v₁ v₂ : B} {l₁ l₂ : List (A × B)} →
@@ -274,7 +283,7 @@ private module ImplInsert (f : B → B → B) where
         where
             k≢k' : ¬ k ≡ k'
             k≢k' with ≡-dec-A k k'
-            ...    | yes k≡k' rewrite k≡k' = absurd (All¬-¬Any k'≢xs₁ (∈-cong proj₁ k,v₁∈xs₁))
+            ...    | yes k≡k' rewrite k≡k' = ⊥-elim (All¬-¬Any k'≢xs₁ (∈-cong proj₁ k,v₁∈xs₁))
             ...    | no  k≢k' = k≢k'
 
     update : A → B → List (A × B) → List (A × B)
@@ -383,7 +392,7 @@ private module ImplInsert (f : B → B → B) where
         rewrite cong proj₁ k,v≡k',v' rewrite cong proj₂ k,v≡k',v'
         with ∈k-dec k' l₁
     ...   | yes _ = here refl
-    ...   | no k'∉kl₁ = absurd (k'∉kl₁ k∈kl₁)
+    ...   | no k'∉kl₁ = ⊥-elim (k'∉kl₁ k∈kl₁)
     restrict-preserves-∈₂ {l₁ = l₁} {l₂ = (k' , v') ∷ xs} k∈kl₁ (there k,v∈xs)
         with ∈k-dec k' l₁
     ...   | yes _ = there (restrict-preserves-∈₂ k∈kl₁ k,v∈xs)
@@ -394,7 +403,7 @@ private module ImplInsert (f : B → B → B) where
     update-preserves-∈ {k} {k'} {v} {v'} {(k'' , v'') ∷ xs} k≢k' (here k,v≡k'',v'')
         rewrite cong proj₁ k,v≡k'',v'' rewrite cong proj₂ k,v≡k'',v''
         with ≡-dec-A k' k''
-    ...   | yes k'≡k'' = absurd (k≢k' (sym k'≡k''))
+    ...   | yes k'≡k'' = ⊥-elim (k≢k' (sym k'≡k''))
     ...   | no _ = here refl
     update-preserves-∈ {k} {k'} {v} {v'} {(k'' , v'') ∷ xs} k≢k' (there k,v∈xs)
         with ≡-dec-A k' k''
@@ -413,10 +422,10 @@ private module ImplInsert (f : B → B → B) where
         rewrite cong proj₁ k,v=k',v'' rewrite cong proj₂ k,v=k',v''
         with ≡-dec-A k' k'
     ...   | yes _ = here refl
-    ...   | no k'≢k' = absurd (k'≢k' refl)
+    ...   | no k'≢k' = ⊥-elim (k'≢k' refl)
     update-combines {k} {v} {v'} {(k' , v'') ∷ xs} (push k'≢xs uxs) (there k,v∈xs)
         with ≡-dec-A k k'
-    ...   | yes k≡k' rewrite k≡k' = absurd (All¬-¬Any k'≢xs (∈-cong proj₁ k,v∈xs))
+    ...   | yes k≡k' rewrite k≡k' = ⊥-elim (All¬-¬Any k'≢xs (∈-cong proj₁ k,v∈xs))
     ...   | no _ = there (update-combines uxs k,v∈xs)
 
     updates-combine : ∀ {k : A} {v₁ v₂ : B} {l₁ l₂ : List (A × B)} →
@@ -430,7 +439,7 @@ private module ImplInsert (f : B → B → B) where
         where
             k≢k' : ¬ k ≡ k'
             k≢k' with ≡-dec-A k k'
-            ...    | yes k≡k' rewrite k≡k' = absurd (All¬-¬Any k'≢xs (∈-cong proj₁ k,v₁∈xs))
+            ...    | yes k≡k' rewrite k≡k' = ⊥-elim (All¬-¬Any k'≢xs (∈-cong proj₁ k,v₁∈xs))
             ...    | no  k≢k' = k≢k'
 
     intersect-combines : ∀ {k : A} {v₁ v₂ : B} {l₁ l₂ : List (A × B)} →
@@ -452,400 +461,404 @@ _∈k_ k (kvs , _) = MemProp._∈_ k (keys kvs)
 Map-functional : ∀ {k : A} {v v' : B} {m : Map} → (k , v) ∈ m → (k , v') ∈ m → v ≡ v'
 Map-functional {m = (l , ul)} k,v∈m k,v'∈m = ListAB-functional ul k,v∈m k,v'∈m
 
+open ImplRelation renaming (subset to subset-impl)
+
+_⊆_ : Map → Map → Set (a ⊔ℓ b)
+_⊆_ (kvs₁ , _) (kvs₂ , _) = subset-impl kvs₁ kvs₂
+
+⊆-refl : (m : Map) →  m ⊆ m
+⊆-refl _ k v k,v∈m = (v , (≈₂-refl , k,v∈m))
+
+⊆-trans : (m₁ m₂ m₃ : Map) →  m₁ ⊆ m₂ → m₂ ⊆ m₃ → m₁ ⊆ m₃
+⊆-trans _ _ _ m₁⊆m₂ m₂⊆m₃ k v k,v∈m₁ =
+    let
+        (v' , (v≈v' , k,v'∈m₂)) = m₁⊆m₂ k v k,v∈m₁
+        (v'' , (v'≈v'' , k,v''∈m₃)) = m₂⊆m₃ k v' k,v'∈m₂
+    in (v'' , (≈₂-trans v≈v' v'≈v'' , k,v''∈m₃))
+
+_≈_ : Map → Map → Set (a ⊔ℓ b)
+_≈_ m₁ m₂ = m₁ ⊆ m₂ × m₂ ⊆ m₁
+
+≈-equiv : IsEquivalence Map _≈_
+≈-equiv = record
+    { ≈-refl = λ {m} → (⊆-refl m , ⊆-refl m)
+    ; ≈-sym = λ {m₁} {m₂} (m₁⊆m₂ , m₂⊆m₁) → (m₂⊆m₁ , m₁⊆m₂)
+    ; ≈-trans = λ {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
+        )
+    }
+
 data Expr : Set (a ⊔ℓ b) where
     `_ : Map → Expr
     _∪_ : Expr → Expr → Expr
     _∩_ : Expr → Expr → Expr
 
-module _ (f : B → B → B) where
-    open ImplInsert f renaming
-        ( insert to insert-impl
-        ; union to union-impl
-        ; intersect to intersect-impl
-        )
+open ImplInsert _⊔₂_ using (union-preserves-Unique) renaming (insert to insert-impl; union to union-impl)
+open ImplInsert _⊓₂_ using (intersect-preserves-Unique) renaming (intersect to intersect-impl)
 
-    union : Map → Map → Map
-    union (kvs₁ , _) (kvs₂ , uks₂) = (union-impl kvs₁ kvs₂ , union-preserves-Unique kvs₁ kvs₂ uks₂)
+_⊔_ : Map → Map → Map
+_⊔_ (kvs₁ , _) (kvs₂ , uks₂) = (union-impl kvs₁ kvs₂ , union-preserves-Unique kvs₁ kvs₂ uks₂)
 
-    intersect : Map → Map → Map
-    intersect (kvs₁ , _) (kvs₂ , uks₂) = (intersect-impl kvs₁ kvs₂ , intersect-preserves-Unique {kvs₁} {kvs₂} uks₂)
+_⊓_ : Map → Map → Map
+_⊓_ (kvs₁ , _) (kvs₂ , uks₂) = (intersect-impl kvs₁ kvs₂ , intersect-preserves-Unique {kvs₁} {kvs₂} uks₂)
 
-module _ (fUnion : B → B → B) (fIntersect : B → B → B) where
-    open ImplInsert fUnion using
-        ( union-combines
-        ; union-preserves-∈₁
-        ; union-preserves-∈₂
-        ; union-preserves-∉
-        )
+open ImplInsert _⊔₂_ using
+    ( union-combines
+    ; union-preserves-∈₁
+    ; union-preserves-∈₂
+    ; union-preserves-∉
+    )
 
-    open ImplInsert fIntersect using
-        ( restrict-needs-both
-        ; updates
-        ; intersect-preserves-∉₁
-        ; intersect-preserves-∉₂
-        ; intersect-combines
-        )
+open ImplInsert _⊓₂_ using
+    ( restrict-needs-both
+    ; updates
+    ; intersect-preserves-∉₁
+    ; intersect-preserves-∉₂
+    ; intersect-combines
+    )
 
-    ⟦_⟧ : Expr -> Map
-    ⟦ ` m ⟧ = m
-    ⟦ e₁ ∪ e₂ ⟧ = union fUnion ⟦ e₁ ⟧ ⟦ e₂ ⟧
-    ⟦ e₁ ∩ e₂ ⟧ = intersect fIntersect ⟦ e₁ ⟧ ⟦ e₂ ⟧
+⟦_⟧ : Expr -> Map
+⟦ ` m ⟧ = m
+⟦ e₁ ∪ e₂ ⟧ = ⟦ e₁ ⟧ ⊔ ⟦ e₂ ⟧
+⟦ e₁ ∩ e₂ ⟧ = ⟦ e₁ ⟧ ⊓ ⟦ e₂ ⟧
 
-    data Provenance (k : A) : B → Expr → Set (a ⊔ℓ b) where
-        single : ∀ {v : B} {m : Map} → (k , v) ∈ m → Provenance k v (` m)
-        in₁ : ∀ {v : B} {e₁ e₂ : Expr} → Provenance k v e₁ → ¬ k ∈k ⟦ e₂ ⟧ → Provenance k v (e₁ ∪ e₂)
-        in₂ : ∀ {v : B} {e₁ e₂ : Expr} → ¬ k ∈k ⟦ e₁ ⟧ → Provenance k v e₂ → Provenance k v (e₁ ∪ e₂)
-        bothᵘ : ∀ {v₁ v₂ : B} {e₁ e₂ : Expr} → Provenance k v₁ e₁ → Provenance k v₂ e₂ → Provenance k (fUnion v₁ v₂) (e₁ ∪ e₂)
-        bothⁱ : ∀ {v₁ v₂ : B} {e₁ e₂ : Expr} → Provenance k v₁ e₁ → Provenance k v₂ e₂ → Provenance k (fIntersect v₁ v₂) (e₁ ∩ e₂)
+data Provenance (k : A) : B → Expr → Set (a ⊔ℓ b) where
+    single : ∀ {v : B} {m : Map} → (k , v) ∈ m → Provenance k v (` m)
+    in₁ : ∀ {v : B} {e₁ e₂ : Expr} → Provenance k v e₁ → ¬ k ∈k ⟦ e₂ ⟧ → Provenance k v (e₁ ∪ e₂)
+    in₂ : ∀ {v : B} {e₁ e₂ : Expr} → ¬ k ∈k ⟦ e₁ ⟧ → Provenance k v e₂ → Provenance k v (e₁ ∪ e₂)
+    bothᵘ : ∀ {v₁ v₂ : B} {e₁ e₂ : Expr} → Provenance k v₁ e₁ → Provenance k v₂ e₂ → Provenance k (v₁ ⊔₂ v₂) (e₁ ∪ e₂)
+    bothⁱ : ∀ {v₁ v₂ : B} {e₁ e₂ : Expr} → Provenance k v₁ e₁ → Provenance k v₂ e₂ → Provenance k (v₁ ⊓₂ v₂) (e₁ ∩ e₂)
 
-    Expr-Provenance : ∀ (k : A) (e : Expr) → k ∈k ⟦ e ⟧ → Σ B (λ v → (Provenance k v e × (k , v) ∈ ⟦ e ⟧))
-    Expr-Provenance k (` m) k∈km = let (v , k,v∈m) = locate k∈km in (v , (single k,v∈m , k,v∈m))
-    Expr-Provenance k (e₁ ∪ e₂) k∈ke₁e₂
-        with ∈k-dec k (proj₁ ⟦ e₁ ⟧) | ∈k-dec k (proj₁ ⟦ e₂ ⟧)
-    ...   | yes k∈ke₁ | yes k∈ke₂ =
-            let (v₁ , (g₁ , k,v₁∈e₁)) = Expr-Provenance k e₁ k∈ke₁
-                (v₂ , (g₂ , k,v₂∈e₂)) = Expr-Provenance k e₂ k∈ke₂
-            in  (fUnion v₁ v₂ , (bothᵘ g₁ g₂ , union-combines (proj₂ ⟦ e₁ ⟧) (proj₂ ⟦ e₂ ⟧) k,v₁∈e₁ k,v₂∈e₂))
-    ...   | yes k∈ke₁ | no k∉ke₂ =
-            let (v₁ , (g₁ , k,v₁∈e₁)) = Expr-Provenance k e₁ k∈ke₁
-            in (v₁ , (in₁ g₁ k∉ke₂ , union-preserves-∈₁ (proj₂ ⟦ e₁ ⟧) k,v₁∈e₁ k∉ke₂))
-    ...   | no k∉ke₁ | yes k∈ke₂ =
-            let (v₂ , (g₂ , k,v₂∈e₂)) = Expr-Provenance k e₂ k∈ke₂
-            in (v₂ , (in₂ k∉ke₁ g₂ , union-preserves-∈₂ k∉ke₁ k,v₂∈e₂))
-    ...   | no k∉ke₁ | no k∉ke₂  = absurd (union-preserves-∉ k∉ke₁ k∉ke₂ k∈ke₁e₂)
-    Expr-Provenance k (e₁ ∩ e₂) k∈ke₁e₂
-        with ∈k-dec k (proj₁ ⟦ e₁ ⟧) | ∈k-dec k (proj₁ ⟦ e₂ ⟧)
-    ...   | yes k∈ke₁ | yes k∈ke₂ =
-            let (v₁ , (g₁ , k,v₁∈e₁)) = Expr-Provenance k e₁ k∈ke₁
-                (v₂ , (g₂ , k,v₂∈e₂)) = Expr-Provenance k e₂ k∈ke₂
-            in (fIntersect v₁ v₂ , (bothⁱ g₁ g₂ , intersect-combines (proj₂ ⟦ e₁ ⟧) (proj₂ ⟦ e₂ ⟧) k,v₁∈e₁ k,v₂∈e₂))
-    ...   | yes k∈ke₁ | no k∉ke₂ = absurd (intersect-preserves-∉₂ {l₁ = proj₁ ⟦ e₁ ⟧} k∉ke₂ k∈ke₁e₂)
-    ...   | no k∉ke₁ | yes k∈ke₂ = absurd (intersect-preserves-∉₁ {l₂ = proj₁ ⟦ e₂ ⟧} k∉ke₁ k∈ke₁e₂)
-    ...   | no k∉ke₁ | no k∉ke₂ = absurd (intersect-preserves-∉₂ {l₁ = proj₁ ⟦ e₁ ⟧} k∉ke₂ k∈ke₁e₂)
+Expr-Provenance : ∀ (k : A) (e : Expr) → k ∈k ⟦ e ⟧ → Σ B (λ v → (Provenance k v e × (k , v) ∈ ⟦ e ⟧))
+Expr-Provenance k (` m) k∈km = let (v , k,v∈m) = locate k∈km in (v , (single k,v∈m , k,v∈m))
+Expr-Provenance k (e₁ ∪ e₂) k∈ke₁e₂
+    with ∈k-dec k (proj₁ ⟦ e₁ ⟧) | ∈k-dec k (proj₁ ⟦ e₂ ⟧)
+...   | yes k∈ke₁ | yes k∈ke₂ =
+        let (v₁ , (g₁ , k,v₁∈e₁)) = Expr-Provenance k e₁ k∈ke₁
+            (v₂ , (g₂ , k,v₂∈e₂)) = Expr-Provenance k e₂ k∈ke₂
+        in  (v₁ ⊔₂ v₂ , (bothᵘ g₁ g₂ , union-combines (proj₂ ⟦ e₁ ⟧) (proj₂ ⟦ e₂ ⟧) k,v₁∈e₁ k,v₂∈e₂))
+...   | yes k∈ke₁ | no k∉ke₂ =
+        let (v₁ , (g₁ , k,v₁∈e₁)) = Expr-Provenance k e₁ k∈ke₁
+        in (v₁ , (in₁ g₁ k∉ke₂ , union-preserves-∈₁ (proj₂ ⟦ e₁ ⟧) k,v₁∈e₁ k∉ke₂))
+...   | no k∉ke₁ | yes k∈ke₂ =
+        let (v₂ , (g₂ , k,v₂∈e₂)) = Expr-Provenance k e₂ k∈ke₂
+        in (v₂ , (in₂ k∉ke₁ g₂ , union-preserves-∈₂ k∉ke₁ k,v₂∈e₂))
+...   | no k∉ke₁ | no k∉ke₂  = ⊥-elim (union-preserves-∉ k∉ke₁ k∉ke₂ k∈ke₁e₂)
+Expr-Provenance k (e₁ ∩ e₂) k∈ke₁e₂
+    with ∈k-dec k (proj₁ ⟦ e₁ ⟧) | ∈k-dec k (proj₁ ⟦ e₂ ⟧)
+...   | yes k∈ke₁ | yes k∈ke₂ =
+        let (v₁ , (g₁ , k,v₁∈e₁)) = Expr-Provenance k e₁ k∈ke₁
+            (v₂ , (g₂ , k,v₂∈e₂)) = Expr-Provenance k e₂ k∈ke₂
+        in (v₁ ⊓₂ v₂ , (bothⁱ g₁ g₂ , intersect-combines (proj₂ ⟦ e₁ ⟧) (proj₂ ⟦ e₂ ⟧) k,v₁∈e₁ k,v₂∈e₂))
+...   | yes k∈ke₁ | no k∉ke₂ = ⊥-elim (intersect-preserves-∉₂ {l₁ = proj₁ ⟦ e₁ ⟧} k∉ke₂ k∈ke₁e₂)
+...   | no k∉ke₁ | yes k∈ke₂ = ⊥-elim (intersect-preserves-∉₁ {l₂ = proj₁ ⟦ e₂ ⟧} k∉ke₁ k∈ke₁e₂)
+...   | no k∉ke₁ | no k∉ke₂ = ⊥-elim (intersect-preserves-∉₂ {l₁ = proj₁ ⟦ e₁ ⟧} k∉ke₂ k∈ke₁e₂)
 
+data SubsetInfo (m₁ m₂ : Map) : Set (a ⊔ℓ b) where
+    extra : (k : A) → k ∈k m₁ → ¬ k ∈k m₂ → SubsetInfo m₁ m₂
+    mismatch : (k : A) (v₁ v₂ : B) → (k , v₁) ∈ m₁ → (k , v₂) ∈ m₂ → ¬ v₁ ≈₂ v₂ → SubsetInfo m₁ m₂
+    fine : m₁ ⊆ m₂ → SubsetInfo m₁ m₂
 
-module _ (_≈_ : B → B → Set b) where
-    open ImplRelation _≈_ renaming (subset to subset-impl)
+SubsetInfo-to-dec : ∀ {m₁ m₂ : Map} → SubsetInfo m₁ m₂ → Dec (m₁ ⊆ m₂)
+SubsetInfo-to-dec (extra k k∈km₁ k∉km₂) =
+    let (v , k,v∈m₁) = locate k∈km₁
+    in no (λ m₁⊆m₂ →
+           let (v' , (_ , k,v'∈m₂)) = m₁⊆m₂ k v k,v∈m₁
+           in k∉km₂ (∈-cong proj₁ k,v'∈m₂))
+SubsetInfo-to-dec {m₁} {m₂} (mismatch k v₁ v₂ k,v₁∈m₁ k,v₂∈m₂ v₁̷≈v₂) =
+    no (λ m₁⊆m₂ →
+        let (v' , (v₁≈v' , k,v'∈m₂)) = m₁⊆m₂ k v₁ k,v₁∈m₁
+        in v₁̷≈v₂ (subst (λ v'' → v₁ ≈₂ v'') (Map-functional {k} {v'} {v₂} {m₂} k,v'∈m₂ k,v₂∈m₂) v₁≈v')) -- for some reason, can't just use subst...
+SubsetInfo-to-dec (fine m₁⊆m₂) = yes m₁⊆m₂
 
-    subset : Map → Map → Set (a ⊔ℓ b)
-    subset (kvs₁ , _) (kvs₂ , _) = subset-impl kvs₁ kvs₂
+module _ (≈₂-dec : ∀ (b₁ b₂ : B) → Dec (b₁ ≈₂ b₂)) where
+    compute-SubsetInfo : ∀ m₁ m₂ → SubsetInfo m₁ m₂
+    compute-SubsetInfo ([] , _) m₂ = fine (λ k v ())
+    compute-SubsetInfo m₁@((k , v) ∷ xs₁ , push k≢xs₁ uxs₁) m₂@(l₂ , u₂)
+        with compute-SubsetInfo (xs₁ , uxs₁) m₂
+    ...   | extra k' k'∈kxs₁ k'∉km₂ = extra k' (there k'∈kxs₁) k'∉km₂
+    ...   | mismatch k' v₁ v₂ k',v₁∈xs₁ k',v₂∈m₂ v₁̷≈v₂ =
+            mismatch k' v₁ v₂ (there k',v₁∈xs₁) k',v₂∈m₂ v₁̷≈v₂
+    ...   | fine xs₁⊆m₂ with ∈k-dec k l₂
+    ...       | no k∉km₂ = extra k (here refl) k∉km₂
+    ...       | yes k∈km₂ with locate k∈km₂
+    ...           | (v' , k,v'∈m₂) with ≈₂-dec v v'
+    ...               | no v̷≈v' = mismatch k v v' (here refl) (k,v'∈m₂) v̷≈v'
+    ...               | yes v≈v' = fine m₁⊆m₂
+                            where
+                                m₁⊆m₂ : m₁ ⊆ m₂
+                                m₁⊆m₂ k' v'' (here k,v≡k',v'')
+                                    rewrite cong proj₁ k,v≡k',v''
+                                    rewrite cong proj₂ k,v≡k',v'' =
+                                    (v' , (v≈v' , k,v'∈m₂))
+                                m₁⊆m₂ k' v'' (there k,v≡k',v'') =
+                                    xs₁⊆m₂ k' v'' k,v≡k',v''
 
-    lift : Map → Map → Set (a ⊔ℓ b)
-    lift m₁ m₂ = subset m₁ m₂ × subset m₂ m₁
+    ⊆-dec : ∀ m₁ m₂ → Dec (m₁ ⊆ m₂)
+    ⊆-dec m₁ m₂ = SubsetInfo-to-dec (compute-SubsetInfo m₁ m₂)
 
-    private
-        data SubsetInfo (m₁ m₂ : Map) : Set (a ⊔ℓ b) where
-            extra : (k : A) → k ∈k m₁ → ¬ k ∈k m₂ → SubsetInfo m₁ m₂
-            mismatch : (k : A) (v₁ v₂ : B) → (k , v₁) ∈ m₁ → (k , v₂) ∈ m₂ → ¬ v₁ ≈ v₂ → SubsetInfo m₁ m₂
-            fine : subset m₁ m₂ → SubsetInfo m₁ m₂
+    ≈-dec : ∀ m₁ m₂ → Dec (m₁ ≈ m₂)
+    ≈-dec m₁ m₂
+        with ⊆-dec m₁ m₂ | ⊆-dec m₂ m₁
+    ...   | yes m₁⊆m₂ | yes m₂⊆m₁ = yes (m₁⊆m₂ , m₂⊆m₁)
+    ...   | _         | no  m₂̷⊆m₁ = no (λ (_ , m₂⊆m₁) → m₂̷⊆m₁ m₂⊆m₁)
+    ...   | no  m₁̷⊆m₂ | _         = no (λ (m₁⊆m₂ , _) → m₁̷⊆m₂ m₁⊆m₂)
 
-        SubsetInfo-to-dec : ∀ {m₁ m₂ : Map} → SubsetInfo m₁ m₂ → Dec (subset m₁ m₂)
-        SubsetInfo-to-dec (extra k k∈km₁ k∉km₂) =
-            let (v , k,v∈m₁) = locate k∈km₁
-            in no (λ m₁⊆m₂ →
-                   let (v' , (_ , k,v'∈m₂)) = m₁⊆m₂ k v k,v∈m₁
-                   in k∉km₂ (∈-cong proj₁ k,v'∈m₂))
-        SubsetInfo-to-dec {m₁} {m₂} (mismatch k v₁ v₂ k,v₁∈m₁ k,v₂∈m₂ v₁̷≈v₂) =
-            no (λ m₁⊆m₂ →
-                let (v' , (v₁≈v' , k,v'∈m₂)) = m₁⊆m₂ k v₁ k,v₁∈m₁
-                in v₁̷≈v₂ (subst (λ v'' → v₁ ≈ v'') (Map-functional {k} {v'} {v₂} {m₂} k,v'∈m₂ k,v₂∈m₂) v₁≈v')) -- for some reason, can't just use subst...
-        SubsetInfo-to-dec (fine m₁⊆m₂) = yes m₁⊆m₂
+private module I⊔ = ImplInsert _⊔₂_
+private module I⊓ = ImplInsert _⊓₂_
 
+≈-⊔-cong : ∀ {m₁ m₂ m₃ m₄ : Map} → m₁ ≈ m₂ → m₃ ≈ m₄ → (m₁ ⊔ m₃) ≈ (m₂ ⊔ m₄)
+≈-⊔-cong {m₁} {m₂} {m₃} {m₄} (m₁⊆m₂ , m₂⊆m₁) (m₃⊆m₄ , m₄⊆m₃) =
+    ( ⊔-⊆ m₁ m₂ m₃ m₄ (m₁⊆m₂ , m₂⊆m₁) (m₃⊆m₄ , m₄⊆m₃)
+    , ⊔-⊆ m₂ m₁ m₄ m₃ (m₂⊆m₁ , m₁⊆m₂) (m₄⊆m₃ , m₃⊆m₄)
+    )
+    where
+        ≈-∉-cong : ∀ {m₁ m₂ : Map} {k : A} → m₁ ≈ m₂ → ¬ k ∈k m₁ → ¬ k ∈k m₂
+        ≈-∉-cong (m₁⊆m₂ , m₂⊆m₁) k∉km₁ k∈km₂ =
+            let (v₂ , k,v₂∈m₂) = locate k∈km₂
+                (_ , (_ , k,v₁∈m₁)) = m₂⊆m₁ _ v₂ k,v₂∈m₂
+            in k∉km₁ (∈-cong proj₁ k,v₁∈m₁)
 
-    module _ (≈-dec : ∀ (b₁ b₂ : B) → Dec (b₁ ≈ b₂)) where
-        private
-            compute-SubsetInfo : ∀ m₁ m₂ → SubsetInfo m₁ m₂
-            compute-SubsetInfo ([] , _) m₂ = fine (λ k v ())
-            compute-SubsetInfo m₁@((k , v) ∷ xs₁ , push k≢xs₁ uxs₁) m₂@(l₂ , u₂)
-                with compute-SubsetInfo (xs₁ , uxs₁) m₂
-            ...   | extra k' k'∈kxs₁ k'∉km₂ = extra k' (there k'∈kxs₁) k'∉km₂
-            ...   | mismatch k' v₁ v₂ k',v₁∈xs₁ k',v₂∈m₂ v₁̷≈v₂ =
-                    mismatch k' v₁ v₂ (there k',v₁∈xs₁) k',v₂∈m₂ v₁̷≈v₂
-            ...   | fine xs₁⊆m₂ with ∈k-dec k l₂
-            ...       | no k∉km₂ = extra k (here refl) k∉km₂
-            ...       | yes k∈km₂ with locate k∈km₂
-            ...           | (v' , k,v'∈m₂) with ≈-dec v v'
-            ...               | no v̷≈v' = mismatch k v v' (here refl) (k,v'∈m₂) v̷≈v'
-            ...               | yes v≈v' = fine m₁⊆m₂
-                                    where
-                                        m₁⊆m₂ : subset m₁ m₂
-                                        m₁⊆m₂ k' v'' (here k,v≡k',v'')
-                                            rewrite cong proj₁ k,v≡k',v''
-                                            rewrite cong proj₂ k,v≡k',v'' =
-                                            (v' , (v≈v' , k,v'∈m₂))
-                                        m₁⊆m₂ k' v'' (there k,v≡k',v'') =
-                                            xs₁⊆m₂ k' v'' k,v≡k',v''
+        ⊔-⊆ : ∀ (m₁ m₂ m₃ m₄ : Map) → m₁ ≈ m₂ → m₃ ≈ m₄ → (m₁ ⊔ m₃) ⊆ (m₂ ⊔ m₄)
+        ⊔-⊆ m₁ m₂ m₃ m₄ m₁≈m₂ m₃≈m₄ k v k,v∈m₁m₃
+            with Expr-Provenance k ((` m₁) ∪ (` m₃)) (∈-cong proj₁ k,v∈m₁m₃)
+        ...   | (_ , (bothᵘ (single {v₁} v₁∈m₁) (single {v₃} v₃∈m₃) , v₁v₃∈m₁m₃))
+                rewrite Map-functional {m = m₁ ⊔ m₃} k,v∈m₁m₃ v₁v₃∈m₁m₃ =
+                let (v₂ , (v₁≈v₂ , k,v₂∈m₂)) = proj₁ m₁≈m₂ k v₁ v₁∈m₁
+                    (v₄ , (v₃≈v₄ , k,v₄∈m₄)) = proj₁ m₃≈m₄ k v₃ v₃∈m₃
+                in (v₂ ⊔₂ v₄ , (≈₂-⊔₂-cong v₁≈v₂ v₃≈v₄ , I⊔.union-combines (proj₂ m₂) (proj₂ m₄) k,v₂∈m₂ k,v₄∈m₄))
+        ...   | (_ , (in₁ (single {v₁} v₁∈m₁) k∉km₃ , v₁∈m₁m₃))
+                rewrite Map-functional {m = m₁ ⊔ m₃} k,v∈m₁m₃ v₁∈m₁m₃ =
+                let (v₂ , (v₁≈v₂ , k,v₂∈m₂)) = proj₁ m₁≈m₂ k v₁ v₁∈m₁
+                    k∉km₄ = ≈-∉-cong {m₃} {m₄} m₃≈m₄ k∉km₃
+                in (v₂ , (v₁≈v₂ , I⊔.union-preserves-∈₁ (proj₂ m₂) k,v₂∈m₂ k∉km₄))
+        ...   | (_ , (in₂ k∉km₁ (single {v₃} v₃∈m₃) , v₃∈m₁m₃))
+                rewrite Map-functional {m = m₁ ⊔ m₃} k,v∈m₁m₃ v₃∈m₁m₃ =
+                let (v₄ , (v₃≈v₄ , k,v₄∈m₄)) = proj₁ m₃≈m₄ k v₃ v₃∈m₃
+                    k∉km₂ = ≈-∉-cong {m₁} {m₂} m₁≈m₂ k∉km₁
+                in (v₄ , (v₃≈v₄ , I⊔.union-preserves-∈₂ k∉km₂ k,v₄∈m₄))
 
-        subset-dec : ∀ m₁ m₂ → Dec (subset m₁ m₂)
-        subset-dec m₁ m₂ = SubsetInfo-to-dec (compute-SubsetInfo m₁ m₂)
+≈-⊓-cong : ∀ {m₁ m₂ m₃ m₄ : Map} → m₁ ≈ m₂ → m₃ ≈ m₄ → (m₁ ⊓ m₃) ≈ (m₂ ⊓ m₄)
+≈-⊓-cong {m₁} {m₂} {m₃} {m₄} (m₁⊆m₂ , m₂⊆m₁) (m₃⊆m₄ , m₄⊆m₃) =
+    ( ⊓-⊆ m₁ m₂ m₃ m₄ (m₁⊆m₂ , m₂⊆m₁) (m₃⊆m₄ , m₄⊆m₃)
+    , ⊓-⊆ m₂ m₁ m₄ m₃ (m₂⊆m₁ , m₁⊆m₂) (m₄⊆m₃ , m₃⊆m₄)
+    )
+    where
+        ⊓-⊆ : ∀ (m₁ m₂ m₃ m₄ : Map) → m₁ ≈ m₂ → m₃ ≈ m₄ → (m₁ ⊓ m₃) ⊆ (m₂ ⊓ m₄)
+        ⊓-⊆ m₁ m₂ m₃ m₄ m₁≈m₂ m₃≈m₄ k v k,v∈m₁m₃
+            with Expr-Provenance k ((` m₁) ∩ (` m₃)) (∈-cong proj₁ k,v∈m₁m₃)
+        ...   | (_ , (bothⁱ (single {v₁} v₁∈m₁) (single {v₃} v₃∈m₃) , v₁v₃∈m₁m₃))
+                rewrite Map-functional {m = m₁ ⊓ m₃} k,v∈m₁m₃ v₁v₃∈m₁m₃ =
+                let (v₂ , (v₁≈v₂ , k,v₂∈m₂)) = proj₁ m₁≈m₂ k v₁ v₁∈m₁
+                    (v₄ , (v₃≈v₄ , k,v₄∈m₄)) = proj₁ m₃≈m₄ k v₃ v₃∈m₃
+                in (v₂ ⊓₂ v₄ , (≈₂-⊓₂-cong v₁≈v₂ v₃≈v₄ , I⊓.intersect-combines (proj₂ m₂) (proj₂ m₄) k,v₂∈m₂ k,v₄∈m₄))
 
-        lift-dec : ∀ m₁ m₂ → Dec (lift m₁ m₂)
-        lift-dec m₁ m₂
-            with subset-dec m₁ m₂ | subset-dec m₂ m₁
-        ...   | yes m₁⊆m₂ | yes m₂⊆m₁ = yes (m₁⊆m₂ , m₂⊆m₁)
-        ...   | _         | no  m₂̷⊆m₁ = no (λ (_ , m₂⊆m₁) → m₂̷⊆m₁ m₂⊆m₁)
-        ...   | no  m₁̷⊆m₂ | _         = no (λ (m₁⊆m₂ , _) → m₁̷⊆m₂ m₁⊆m₂)
+⊔-idemp : ∀ (m : Map) → (m ⊔ m) ≈ m
+⊔-idemp m@(l , u) = (mm-m-⊆ , m-mm-⊆)
+    where
+        mm-m-⊆ : (m ⊔ m) ⊆ m
+        mm-m-⊆ k v k,v∈mm
+            with Expr-Provenance k ((` m) ∪ (` m)) (∈-cong proj₁ k,v∈mm)
+        ...   | (_ , (bothᵘ (single {v'} v'∈m) (single {v''} v''∈m) , v'v''∈mm))
+                rewrite Map-functional {m = m} v'∈m v''∈m
+                rewrite Map-functional {m = m ⊔ m} k,v∈mm v'v''∈mm =
+                (v'' , (⊔₂-idemp v'' , v''∈m))
+        ...   | (_ , (in₁ (single {v'} v'∈m) k∉km , _)) = ⊥-elim (k∉km (∈-cong proj₁ v'∈m))
+        ...   | (_ , (in₂ k∉km (single {v''} v''∈m) , _)) = ⊥-elim (k∉km (∈-cong proj₁ v''∈m))
 
-    -- The Provenance type requires both union and intersection functions,
-    -- but sometimes here we're working with one operation only. Just use the
-    -- union/intersection function for both -- it doesn't matter, since we don't
-    -- use the dual operations in these proofs.
+        m-mm-⊆ : m ⊆ (m ⊔ m)
+        m-mm-⊆ k v k,v∈m = (v ⊔₂ v , (≈₂-sym (⊔₂-idemp v) , I⊔.union-combines u u k,v∈m k,v∈m))
 
-    module _ (f : B → B → B)
-             (≈-f-cong : ∀ {b₁ b₂ b₃ b₄} → b₁ ≈ b₂ → b₃ ≈ b₄ → f b₁ b₃ ≈ f b₂ b₄) where
-        private module I = ImplInsert f
+⊔-comm : ∀ (m₁ m₂ : Map) → (m₁ ⊔ m₂) ≈ (m₂ ⊔ m₁)
+⊔-comm m₁ m₂ = (⊔-comm-⊆ m₁ m₂ , ⊔-comm-⊆ m₂ m₁)
+    where
+        ⊔-comm-⊆ : ∀ (m₁ m₂ : Map) → (m₁ ⊔ m₂) ⊆ (m₂ ⊔ m₁)
+        ⊔-comm-⊆ m₁@(l₁ , u₁) m₂@(l₂ , u₂) k v k,v∈m₁m₂
+            with Expr-Provenance k ((` m₁) ∪ (` m₂)) (∈-cong proj₁ k,v∈m₁m₂)
+        ...   | (_ , (bothᵘ {v₁} {v₂} (single v₁∈m₁) (single v₂∈m₂) , v₁v₂∈m₁m₂))
+                rewrite Map-functional {m = m₁ ⊔ m₂} k,v∈m₁m₂ v₁v₂∈m₁m₂ =
+                (v₂ ⊔₂ v₁ , (⊔₂-comm v₁ v₂ , I⊔.union-combines u₂ u₁ v₂∈m₂ v₁∈m₁))
+        ...   | (_ , (in₁ {v₁} (single v₁∈m₁) k∉km₂ , v₁∈m₁m₂))
+                rewrite Map-functional {m = m₁ ⊔ m₂} k,v∈m₁m₂ v₁∈m₁m₂ =
+                (v₁ , (≈₂-refl , I⊔.union-preserves-∈₂ k∉km₂ v₁∈m₁))
+        ...   | (_ , (in₂ {v₂} k∉km₁ (single v₂∈m₂) , v₂∈m₁m₂))
+                rewrite Map-functional {m = m₁ ⊔ m₂} k,v∈m₁m₂ v₂∈m₁m₂ =
+                (v₂ , (≈₂-refl , I⊔.union-preserves-∈₁ u₂ v₂∈m₂ k∉km₁))
 
-        union-cong : ∀ {m₁ m₂ m₃ m₄ : Map} → lift m₁ m₂ → lift m₃ m₄ → lift (union f m₁ m₃) (union f m₂ m₄)
-        union-cong {m₁} {m₂} {m₃} {m₄} (m₁⊆m₂ , m₂⊆m₁) (m₃⊆m₄ , m₄⊆m₃) =
-            ( union-subset m₁ m₂ m₃ m₄ (m₁⊆m₂ , m₂⊆m₁) (m₃⊆m₄ , m₄⊆m₃)
-            , union-subset m₂ m₁ m₄ m₃ (m₂⊆m₁ , m₁⊆m₂) (m₄⊆m₃ , m₃⊆m₄)
-            )
-            where
-                ≈-∉-cong : ∀ {m₁ m₂ : Map} {k : A} → lift m₁ m₂ → ¬ k ∈k m₁ → ¬ k ∈k m₂
-                ≈-∉-cong (m₁⊆m₂ , m₂⊆m₁) k∉km₁ k∈km₂ =
-                    let (v₂ , k,v₂∈m₂) = locate k∈km₂
-                        (_ , (_ , k,v₁∈m₁)) = m₂⊆m₁ _ v₂ k,v₂∈m₂
-                    in k∉km₁ (∈-cong proj₁ k,v₁∈m₁)
+⊔-assoc : ∀ (m₁ m₂ m₃ : Map) → ((m₁ ⊔ m₂) ⊔ m₃) ≈ (m₁ ⊔ (m₂ ⊔ m₃))
+⊔-assoc m₁@(l₁ , u₁) m₂@(l₂ , u₂) m₃@(l₃ , u₃) = (⊔-assoc₁ , ⊔-assoc₂)
+    where
+        ⊔-assoc₁ : ((m₁ ⊔ m₂) ⊔ m₃) ⊆ (m₁ ⊔ (m₂ ⊔ m₃))
+        ⊔-assoc₁ k v k,v∈m₁₂m₃
+            with Expr-Provenance k (((` m₁) ∪ (` m₂)) ∪ (` m₃)) (∈-cong proj₁ k,v∈m₁₂m₃)
+        ...   | (_ , (in₂ k∉ke₁₂ (single {v₃} v₃∈e₃) , v₃∈m₁₂m₃))
+                rewrite Map-functional {m = (m₁ ⊔ m₂) ⊔ m₃} k,v∈m₁₂m₃ v₃∈m₁₂m₃ =
+                let (k∉ke₁ , k∉ke₂) = I⊔.∉-union-∉-either {l₁ = l₁} {l₂ = l₂} k∉ke₁₂
+                in (v₃ , (≈₂-refl , I⊔.union-preserves-∈₂ k∉ke₁ (I⊔.union-preserves-∈₂ k∉ke₂ v₃∈e₃)))
+        ...   | (_ , (in₁ (in₂ k∉ke₁ (single {v₂} v₂∈e₂)) k∉ke₃ , v₂∈m₁₂m₃))
+                rewrite Map-functional {m = (m₁ ⊔ m₂) ⊔ m₃} k,v∈m₁₂m₃ v₂∈m₁₂m₃ =
+                (v₂ , (≈₂-refl , I⊔.union-preserves-∈₂ k∉ke₁ (I⊔.union-preserves-∈₁ u₂ v₂∈e₂ k∉ke₃)))
+        ...   | (_ , (bothᵘ (in₂ k∉ke₁ (single {v₂} v₂∈e₂)) (single {v₃} v₃∈e₃) , v₂v₃∈m₁₂m₃))
+                rewrite Map-functional {m = (m₁ ⊔ m₂) ⊔ m₃} k,v∈m₁₂m₃ v₂v₃∈m₁₂m₃ =
+                (v₂ ⊔₂ v₃ , (≈₂-refl , I⊔.union-preserves-∈₂ k∉ke₁  (I⊔.union-combines u₂ u₃ v₂∈e₂ v₃∈e₃)))
+        ...   | (_ , (in₁ (in₁ (single {v₁} v₁∈e₁) k∉ke₂) k∉ke₃ , v₁∈m₁₂m₃))
+                rewrite Map-functional {m = (m₁ ⊔ m₂) ⊔ m₃} k,v∈m₁₂m₃ v₁∈m₁₂m₃ =
+                (v₁ , (≈₂-refl , I⊔.union-preserves-∈₁ u₁ v₁∈e₁ (I⊔.union-preserves-∉ k∉ke₂ k∉ke₃)))
+        ...   | (_ , (bothᵘ (in₁ (single {v₁} v₁∈e₁) k∉ke₂) (single {v₃} v₃∈e₃) , v₁v₃∈m₁₂m₃))
+                rewrite Map-functional {m = (m₁ ⊔ m₂) ⊔ m₃} k,v∈m₁₂m₃ v₁v₃∈m₁₂m₃ =
+                (v₁ ⊔₂ v₃ , (≈₂-refl , I⊔.union-combines u₁ (I⊔.union-preserves-Unique l₂ l₃ u₃) v₁∈e₁ (I⊔.union-preserves-∈₂ k∉ke₂ v₃∈e₃)))
+        ...   | (_ , (in₁ (bothᵘ (single {v₁} v₁∈e₁) (single {v₂} v₂∈e₂)) k∉ke₃), v₁v₂∈m₁₂m₃)
+                rewrite Map-functional {m = (m₁ ⊔ m₂) ⊔ m₃} k,v∈m₁₂m₃ v₁v₂∈m₁₂m₃ =
+                (v₁ ⊔₂ v₂ , (≈₂-refl , I⊔.union-combines u₁ (I⊔.union-preserves-Unique l₂ l₃ u₃) v₁∈e₁ (I⊔.union-preserves-∈₁ u₂ v₂∈e₂ k∉ke₃)))
+        ...   | (_ , (bothᵘ (bothᵘ (single {v₁} v₁∈e₁) (single {v₂} v₂∈e₂)) (single {v₃} v₃∈e₃) , v₁v₂v₃∈m₁₂m₃))
+                rewrite Map-functional {m = (m₁ ⊔ m₂) ⊔ m₃} k,v∈m₁₂m₃ v₁v₂v₃∈m₁₂m₃ =
+                (v₁ ⊔₂ (v₂ ⊔₂ v₃) , (⊔₂-assoc v₁ v₂ v₃ , I⊔.union-combines u₁ (I⊔.union-preserves-Unique l₂ l₃ u₃) v₁∈e₁ (I⊔.union-combines u₂ u₃ v₂∈e₂ v₃∈e₃)))
 
-                union-subset : ∀ (m₁ m₂ m₃ m₄ : Map) → lift m₁ m₂ → lift m₃ m₄ → subset (union f m₁ m₃) (union f m₂ m₄)
-                union-subset m₁ m₂ m₃ m₄ m₁≈m₂ m₃≈m₄ k v k,v∈m₁m₃
-                    with Expr-Provenance f f k ((` m₁) ∪ (` m₃)) (∈-cong proj₁ k,v∈m₁m₃)
-                ...   | (_ , (bothᵘ (single {v₁} v₁∈m₁) (single {v₃} v₃∈m₃) , v₁v₃∈m₁m₃))
-                        rewrite Map-functional {m = union f m₁ m₃} k,v∈m₁m₃ v₁v₃∈m₁m₃ =
-                        let (v₂ , (v₁≈v₂ , k,v₂∈m₂)) = proj₁ m₁≈m₂ k v₁ v₁∈m₁
-                            (v₄ , (v₃≈v₄ , k,v₄∈m₄)) = proj₁ m₃≈m₄ k v₃ v₃∈m₃
-                        in (f v₂ v₄ , (≈-f-cong v₁≈v₂ v₃≈v₄ , I.union-combines (proj₂ m₂) (proj₂ m₄) k,v₂∈m₂ k,v₄∈m₄))
-                ...   | (_ , (in₁ (single {v₁} v₁∈m₁) k∉km₃ , v₁∈m₁m₃))
-                        rewrite Map-functional {m = union f m₁ m₃} k,v∈m₁m₃ v₁∈m₁m₃ =
-                        let (v₂ , (v₁≈v₂ , k,v₂∈m₂)) = proj₁ m₁≈m₂ k v₁ v₁∈m₁
-                            k∉km₄ = ≈-∉-cong {m₃} {m₄} m₃≈m₄ k∉km₃
-                        in (v₂ , (v₁≈v₂ , I.union-preserves-∈₁ (proj₂ m₂) k,v₂∈m₂ k∉km₄))
-                ...   | (_ , (in₂ k∉km₁ (single {v₃} v₃∈m₃) , v₃∈m₁m₃))
-                        rewrite Map-functional {m = union f m₁ m₃} k,v∈m₁m₃ v₃∈m₁m₃ =
-                        let (v₄ , (v₃≈v₄ , k,v₄∈m₄)) = proj₁ m₃≈m₄ k v₃ v₃∈m₃
-                            k∉km₂ = ≈-∉-cong {m₁} {m₂} m₁≈m₂ k∉km₁
-                        in (v₄ , (v₃≈v₄ , I.union-preserves-∈₂ k∉km₂ k,v₄∈m₄))
+        ⊔-assoc₂ : (m₁ ⊔ (m₂ ⊔ m₃)) ⊆ ((m₁ ⊔ m₂) ⊔ m₃)
+        ⊔-assoc₂ k v k,v∈m₁m₂₃
+            with Expr-Provenance k ((` m₁) ∪ ((` m₂) ∪ (` m₃))) (∈-cong proj₁ k,v∈m₁m₂₃)
+        ...   | (_ , (in₂ k∉ke₁ (in₂ k∉ke₂ (single {v₃} v₃∈e₃)) , v₃∈m₁m₂₃))
+                rewrite Map-functional {m = m₁ ⊔ (m₂ ⊔ m₃)} k,v∈m₁m₂₃ v₃∈m₁m₂₃ =
+                (v₃ , (≈₂-refl , I⊔.union-preserves-∈₂ (I⊔.union-preserves-∉ k∉ke₁ k∉ke₂) v₃∈e₃))
+        ...   | (_ , (in₂ k∉ke₁ (in₁ (single {v₂} v₂∈e₂) k∉ke₃) , v₂∈m₁m₂₃))
+                rewrite Map-functional {m = m₁ ⊔ (m₂ ⊔ m₃)} k,v∈m₁m₂₃ v₂∈m₁m₂₃ =
+                (v₂ , (≈₂-refl , I⊔.union-preserves-∈₁ (I⊔.union-preserves-Unique l₁ l₂ u₂) (I⊔.union-preserves-∈₂ k∉ke₁ v₂∈e₂) k∉ke₃))
+        ...   | (_ , (in₂ k∉ke₁ (bothᵘ (single {v₂} v₂∈e₂) (single {v₃} v₃∈e₃)) , v₂v₃∈m₁m₂₃))
+                rewrite Map-functional {m = m₁ ⊔ (m₂ ⊔ m₃)} k,v∈m₁m₂₃ v₂v₃∈m₁m₂₃ =
+                (v₂ ⊔₂ v₃ , (≈₂-refl , I⊔.union-combines (I⊔.union-preserves-Unique l₁ l₂ u₂) u₃ (I⊔.union-preserves-∈₂ k∉ke₁ v₂∈e₂) v₃∈e₃))
+        ...   | (_ , (in₁ (single {v₁} v₁∈e₁) k∉ke₂₃ , v₁∈m₁m₂₃))
+                rewrite Map-functional {m = m₁ ⊔ (m₂ ⊔ m₃)} k,v∈m₁m₂₃ v₁∈m₁m₂₃ =
+                let (k∉ke₂ , k∉ke₃) = I⊔.∉-union-∉-either {l₁ = l₂} {l₂ = l₃} k∉ke₂₃
+                in (v₁ , (≈₂-refl , I⊔.union-preserves-∈₁ (I⊔.union-preserves-Unique l₁ l₂ u₂) (I⊔.union-preserves-∈₁ u₁ v₁∈e₁ k∉ke₂) k∉ke₃))
+        ...   | (_ , (bothᵘ (single {v₁} v₁∈e₁) (in₂ k∉ke₂ (single {v₃} v₃∈e₃)) , v₁v₃∈m₁m₂₃))
+                rewrite Map-functional {m = m₁ ⊔ (m₂ ⊔ m₃)} k,v∈m₁m₂₃ v₁v₃∈m₁m₂₃ =
+                (v₁ ⊔₂ v₃ , (≈₂-refl , I⊔.union-combines (I⊔.union-preserves-Unique l₁ l₂ u₂) u₃ (I⊔.union-preserves-∈₁ u₁ v₁∈e₁ k∉ke₂) v₃∈e₃))
+        ...   | (_ , (bothᵘ (single {v₁} v₁∈e₁) (in₁ (single {v₂} v₂∈e₂) k∉ke₃) , v₁v₂∈m₁m₂₃))
+                rewrite Map-functional {m = m₁ ⊔ (m₂ ⊔ m₃)} k,v∈m₁m₂₃ v₁v₂∈m₁m₂₃ =
+                (v₁ ⊔₂ v₂ , (≈₂-refl , I⊔.union-preserves-∈₁ (I⊔.union-preserves-Unique l₁ l₂ u₂) (I⊔.union-combines u₁ u₂ v₁∈e₁ v₂∈e₂) k∉ke₃))
+        ...   | (_ , (bothᵘ (single {v₁} v₁∈e₁) (bothᵘ (single {v₂} v₂∈e₂) (single {v₃} v₃∈e₃)) , v₁v₂v₃∈m₁m₂₃))
+                rewrite Map-functional {m = m₁ ⊔ (m₂ ⊔ m₃)} k,v∈m₁m₂₃ v₁v₂v₃∈m₁m₂₃ =
+                ((v₁ ⊔₂ v₂) ⊔₂ v₃ , (≈₂-sym (⊔₂-assoc v₁ v₂ v₃) , I⊔.union-combines (I⊔.union-preserves-Unique l₁ l₂ u₂) u₃ (I⊔.union-combines u₁ u₂ v₁∈e₁ v₂∈e₂) v₃∈e₃))
 
-        intersect-cong : ∀ {m₁ m₂ m₃ m₄ : Map} → lift m₁ m₂ → lift m₃ m₄ → lift (intersect f m₁ m₃) (intersect f m₂ m₄)
-        intersect-cong {m₁} {m₂} {m₃} {m₄} (m₁⊆m₂ , m₂⊆m₁) (m₃⊆m₄ , m₄⊆m₃) =
-            ( intersect-subset m₁ m₂ m₃ m₄ (m₁⊆m₂ , m₂⊆m₁) (m₃⊆m₄ , m₄⊆m₃)
-            , intersect-subset m₂ m₁ m₄ m₃ (m₂⊆m₁ , m₁⊆m₂) (m₄⊆m₃ , m₃⊆m₄)
-            )
-            where
-                intersect-subset : ∀ (m₁ m₂ m₃ m₄ : Map) → lift m₁ m₂ → lift m₃ m₄ → subset (intersect f m₁ m₃) (intersect f m₂ m₄)
-                intersect-subset m₁ m₂ m₃ m₄ m₁≈m₂ m₃≈m₄ k v k,v∈m₁m₃
-                    with Expr-Provenance f f k ((` m₁) ∩ (` m₃)) (∈-cong proj₁ k,v∈m₁m₃)
-                ...   | (_ , (bothⁱ (single {v₁} v₁∈m₁) (single {v₃} v₃∈m₃) , v₁v₃∈m₁m₃))
-                        rewrite Map-functional {m = intersect f m₁ m₃} k,v∈m₁m₃ v₁v₃∈m₁m₃ =
-                        let (v₂ , (v₁≈v₂ , k,v₂∈m₂)) = proj₁ m₁≈m₂ k v₁ v₁∈m₁
-                            (v₄ , (v₃≈v₄ , k,v₄∈m₄)) = proj₁ m₃≈m₄ k v₃ v₃∈m₃
-                        in (f v₂ v₄ , (≈-f-cong v₁≈v₂ v₃≈v₄ , I.intersect-combines (proj₂ m₂) (proj₂ m₄) k,v₂∈m₂ k,v₄∈m₄))
+⊓-idemp : ∀ (m : Map) → (m ⊓ m) ≈ m
+⊓-idemp m@(l , u) = (mm-m-⊆ , m-mm-⊆)
+    where
+        mm-m-⊆ : (m ⊓ m) ⊆ m
+        mm-m-⊆ k v k,v∈mm
+            with Expr-Provenance k ((` m) ∩ (` m)) (∈-cong proj₁ k,v∈mm)
+        ...   | (_ , (bothⁱ (single {v'} v'∈m) (single {v''} v''∈m) , v'v''∈mm))
+                rewrite Map-functional {m = m} v'∈m v''∈m
+                rewrite Map-functional {m = m ⊓ m} k,v∈mm v'v''∈mm =
+                (v'' , (⊓₂-idemp v'' , v''∈m))
 
-    module _ (≈-refl : ∀ {b : B} → b ≈ b)
-             (≈-sym : ∀ {b₁ b₂ : B} → b₁ ≈ b₂ → b₂ ≈ b₁)
-             (f : B → B → B) where
-        private module I = ImplInsert f
+        m-mm-⊆ : m ⊆ (m ⊓ m)
+        m-mm-⊆ k v k,v∈m = (v ⊓₂ v , (≈₂-sym (⊓₂-idemp v) , I⊓.intersect-combines u u k,v∈m k,v∈m))
 
-        module _ (f-idemp : ∀ (b : B) → f b b ≈ b) where
-            union-idemp : ∀ (m : Map) → lift (union f m m) m
-            union-idemp m@(l , u) = (mm-m-subset , m-mm-subset)
-                where
-                    mm-m-subset : subset (union f m m) m
-                    mm-m-subset k v k,v∈mm
-                        with Expr-Provenance f f k ((` m) ∪ (` m)) (∈-cong proj₁ k,v∈mm)
-                    ...   | (_ , (bothᵘ (single {v'} v'∈m) (single {v''} v''∈m) , v'v''∈mm))
-                            rewrite Map-functional {m = m} v'∈m v''∈m
-                            rewrite Map-functional {m = union f m m} k,v∈mm v'v''∈mm =
-                            (v'' , (f-idemp v'' , v''∈m))
-                    ...   | (_ , (in₁ (single {v'} v'∈m) k∉km , _)) = absurd (k∉km (∈-cong proj₁ v'∈m))
-                    ...   | (_ , (in₂ k∉km (single {v''} v''∈m) , _)) = absurd (k∉km (∈-cong proj₁ v''∈m))
+⊓-comm : ∀ (m₁ m₂ : Map) → (m₁ ⊓ m₂) ≈ (m₂ ⊓ m₁)
+⊓-comm m₁ m₂ = (⊓-comm-⊆ m₁ m₂ , ⊓-comm-⊆ m₂ m₁)
+    where
+        ⊓-comm-⊆ : ∀ (m₁ m₂ : Map) → (m₁ ⊓ m₂) ⊆ (m₂ ⊓ m₁)
+        ⊓-comm-⊆ m₁@(l₁ , u₁) m₂@(l₂ , u₂) k v k,v∈m₁m₂
+            with Expr-Provenance k ((` m₁) ∩ (` m₂)) (∈-cong proj₁ k,v∈m₁m₂)
+        ...   | (_ , (bothⁱ {v₁} {v₂} (single v₁∈m₁) (single v₂∈m₂) , v₁v₂∈m₁m₂))
+                rewrite Map-functional {m = m₁ ⊓ m₂} k,v∈m₁m₂ v₁v₂∈m₁m₂ =
+                (v₂ ⊓₂ v₁ , (⊓₂-comm v₁ v₂ , I⊓.intersect-combines u₂ u₁ v₂∈m₂ v₁∈m₁))
 
-                    m-mm-subset : subset m (union f m m)
-                    m-mm-subset k v k,v∈m = (f v v , (≈-sym (f-idemp v) , I.union-combines u u k,v∈m k,v∈m))
+⊓-assoc : ∀ (m₁ m₂ m₃ : Map) → ((m₁ ⊓ m₂) ⊓ m₃) ≈ (m₁ ⊓ (m₂ ⊓ m₃))
+⊓-assoc m₁@(l₁ , u₁) m₂@(l₂ , u₂) m₃@(l₃ , u₃) = (⊓-assoc₁ , ⊓-assoc₂)
+    where
+        ⊓-assoc₁ : ((m₁ ⊓ m₂) ⊓ m₃) ⊆ (m₁ ⊓ (m₂ ⊓ m₃))
+        ⊓-assoc₁ k v k,v∈m₁₂m₃
+            with Expr-Provenance k (((` m₁) ∩ (` m₂)) ∩ (` m₃)) (∈-cong proj₁ k,v∈m₁₂m₃)
+        ...   | (_ , (bothⁱ (bothⁱ (single {v₁} v₁∈e₁) (single {v₂} v₂∈e₂)) (single {v₃} v₃∈e₃) , v₁v₂v₃∈m₁₂m₃))
+                rewrite Map-functional {m = (m₁ ⊓ m₂) ⊓ m₃} k,v∈m₁₂m₃ v₁v₂v₃∈m₁₂m₃ =
+                (v₁ ⊓₂ (v₂ ⊓₂ v₃) , (⊓₂-assoc v₁ v₂ v₃ , I⊓.intersect-combines u₁ (I⊓.intersect-preserves-Unique {l₂} {l₃} u₃) v₁∈e₁ (I⊓.intersect-combines u₂ u₃ v₂∈e₂ v₃∈e₃)))
 
-        module _ (f-comm : ∀ (b₁ b₂ : B) → f b₁ b₂ ≈ f b₂ b₁) where
-            union-comm : ∀ (m₁ m₂ : Map) → lift (union f m₁ m₂) (union f m₂ m₁)
-            union-comm m₁ m₂ = (union-comm-subset m₁ m₂ , union-comm-subset m₂ m₁)
-                where
-                    union-comm-subset : ∀ (m₁ m₂ : Map) → subset (union f m₁ m₂) (union f m₂ m₁)
-                    union-comm-subset m₁@(l₁ , u₁) m₂@(l₂ , u₂) k v k,v∈m₁m₂
-                        with Expr-Provenance f f k ((` m₁) ∪ (` m₂)) (∈-cong proj₁ k,v∈m₁m₂)
-                    ...   | (_ , (bothᵘ {v₁} {v₂} (single v₁∈m₁) (single v₂∈m₂) , v₁v₂∈m₁m₂))
-                            rewrite Map-functional {m = union f m₁ m₂} k,v∈m₁m₂ v₁v₂∈m₁m₂ =
-                            (f v₂ v₁ , (f-comm v₁ v₂ , I.union-combines u₂ u₁ v₂∈m₂ v₁∈m₁))
-                    ...   | (_ , (in₁ {v₁} (single v₁∈m₁) k∉km₂ , v₁∈m₁m₂))
-                            rewrite Map-functional {m = union f m₁ m₂} k,v∈m₁m₂ v₁∈m₁m₂ =
-                            (v₁ , (≈-refl , I.union-preserves-∈₂ k∉km₂ v₁∈m₁))
-                    ...   | (_ , (in₂ {v₂} k∉km₁ (single v₂∈m₂) , v₂∈m₁m₂))
-                            rewrite Map-functional {m = union f m₁ m₂} k,v∈m₁m₂ v₂∈m₁m₂ =
-                            (v₂ , (≈-refl , I.union-preserves-∈₁ u₂ v₂∈m₂ k∉km₁))
+        ⊓-assoc₂ : (m₁ ⊓ (m₂ ⊓ m₃)) ⊆ ((m₁ ⊓ m₂) ⊓ m₃)
+        ⊓-assoc₂ k v k,v∈m₁m₂₃
+            with Expr-Provenance k ((` m₁) ∩ ((` m₂) ∩ (` m₃))) (∈-cong proj₁ k,v∈m₁m₂₃)
+        ...   | (_ , (bothⁱ (single {v₁} v₁∈e₁) (bothⁱ (single {v₂} v₂∈e₂) (single {v₃} v₃∈e₃)) , v₁v₂v₃∈m₁m₂₃))
+                rewrite Map-functional {m = m₁ ⊓ (m₂ ⊓ m₃)} k,v∈m₁m₂₃ v₁v₂v₃∈m₁m₂₃ =
+                ((v₁ ⊓₂ v₂) ⊓₂ v₃ , (≈₂-sym (⊓₂-assoc v₁ v₂ v₃) , I⊓.intersect-combines (I⊓.intersect-preserves-Unique {l₁} {l₂} u₂) u₃ (I⊓.intersect-combines u₁ u₂ v₁∈e₁ v₂∈e₂) v₃∈e₃))
 
-        module _ (f-assoc : ∀ (b₁ b₂ b₃ : B) → f (f b₁ b₂) b₃ ≈ f b₁ (f b₂ b₃))  where
-            union-assoc : ∀ (m₁ m₂ m₃ : Map) → lift (union f (union f m₁ m₂) m₃) (union f m₁ (union f m₂ m₃))
-            union-assoc m₁@(l₁ , u₁) m₂@(l₂ , u₂) m₃@(l₃ , u₃) = (union-assoc₁ , union-assoc₂)
-                where
-                    union-assoc₁ : subset (union f (union f m₁ m₂) m₃) (union f m₁ (union f m₂ m₃))
-                    union-assoc₁ k v k,v∈m₁₂m₃
-                        with Expr-Provenance f f k (((` m₁) ∪ (` m₂)) ∪ (` m₃)) (∈-cong proj₁ k,v∈m₁₂m₃)
-                    ...   | (_ , (in₂ k∉ke₁₂ (single {v₃} v₃∈e₃) , v₃∈m₁₂m₃))
-                            rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₃∈m₁₂m₃ =
-                            let (k∉ke₁ , k∉ke₂) = I.∉-union-∉-either {l₁ = l₁} {l₂ = l₂} k∉ke₁₂
-                            in (v₃ , (≈-refl , I.union-preserves-∈₂ k∉ke₁ (I.union-preserves-∈₂ k∉ke₂ v₃∈e₃)))
-                    ...   | (_ , (in₁ (in₂ k∉ke₁ (single {v₂} v₂∈e₂)) k∉ke₃ , v₂∈m₁₂m₃))
-                            rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₂∈m₁₂m₃ =
-                            (v₂ , (≈-refl , I.union-preserves-∈₂ k∉ke₁ (I.union-preserves-∈₁ u₂ v₂∈e₂ k∉ke₃)))
-                    ...   | (_ , (bothᵘ (in₂ k∉ke₁ (single {v₂} v₂∈e₂)) (single {v₃} v₃∈e₃) , v₂v₃∈m₁₂m₃))
-                            rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₂v₃∈m₁₂m₃ =
-                            (f v₂ v₃ , (≈-refl , I.union-preserves-∈₂ k∉ke₁  (I.union-combines u₂ u₃ v₂∈e₂ v₃∈e₃)))
-                    ...   | (_ , (in₁ (in₁ (single {v₁} v₁∈e₁) k∉ke₂) k∉ke₃ , v₁∈m₁₂m₃))
-                            rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁∈m₁₂m₃ =
-                            (v₁ , (≈-refl , I.union-preserves-∈₁ u₁ v₁∈e₁ (I.union-preserves-∉ k∉ke₂ k∉ke₃)))
-                    ...   | (_ , (bothᵘ (in₁ (single {v₁} v₁∈e₁) k∉ke₂) (single {v₃} v₃∈e₃) , v₁v₃∈m₁₂m₃))
-                            rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁v₃∈m₁₂m₃ =
-                            (f v₁ v₃ , (≈-refl , I.union-combines u₁ (I.union-preserves-Unique l₂ l₃ u₃) v₁∈e₁ (I.union-preserves-∈₂ k∉ke₂ v₃∈e₃)))
-                    ...   | (_ , (in₁ (bothᵘ (single {v₁} v₁∈e₁) (single {v₂} v₂∈e₂)) k∉ke₃), v₁v₂∈m₁₂m₃)
-                            rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁v₂∈m₁₂m₃ =
-                            (f v₁ v₂ , (≈-refl , I.union-combines u₁ (I.union-preserves-Unique l₂ l₃ u₃) v₁∈e₁ (I.union-preserves-∈₁ u₂ v₂∈e₂ k∉ke₃)))
-                    ...   | (_ , (bothᵘ (bothᵘ (single {v₁} v₁∈e₁) (single {v₂} v₂∈e₂)) (single {v₃} v₃∈e₃) , v₁v₂v₃∈m₁₂m₃))
-                            rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁v₂v₃∈m₁₂m₃ =
-                            (f v₁ (f v₂ v₃) , (f-assoc v₁ v₂ v₃ , I.union-combines u₁ (I.union-preserves-Unique l₂ l₃ u₃) v₁∈e₁ (I.union-combines u₂ u₃ v₂∈e₂ v₃∈e₃)))
+absorb-⊓-⊔ : ∀ (m₁ m₂ : Map) →  (m₁ ⊓ (m₁ ⊔ m₂)) ≈ m₁
+absorb-⊓-⊔ m₁@(l₁ , u₁) m₂@(l₂ , u₂) = (absorb-⊓-⊔¹ , absorb-⊓-⊔²)
+    where
+        absorb-⊓-⊔¹ : (m₁ ⊓ (m₁ ⊔ m₂)) ⊆ m₁
+        absorb-⊓-⊔¹ k v k,v∈m₁m₁₂
+            with Expr-Provenance k ((` m₁) ∩ ((` m₁) ∪ (` m₂))) (∈-cong proj₁ k,v∈m₁m₁₂)
+        ...   | (_ , (bothⁱ (single {v₁} k,v₁∈m₁)
+                            (bothᵘ (single {v₁'} k,v₁'∈m₁)
+                                   (single {v₂} v₂∈m₂)) , v₁v₁'v₂∈m₁m₁₂))
+                rewrite Map-functional {m = m₁} k,v₁∈m₁ k,v₁'∈m₁
+                rewrite Map-functional {m = m₁ ⊓ (m₁ ⊔ m₂)} k,v∈m₁m₁₂ v₁v₁'v₂∈m₁m₁₂ =
+                (v₁' , (absorb-⊓₂-⊔₂ v₁' v₂ , k,v₁'∈m₁))
+        ...   | (_ , (bothⁱ (single {v₁} k,v₁∈m₁)
+                            (in₁ (single {v₁'} k,v₁'∈m₁) _) , v₁v₁'∈m₁m₁₂))
+                rewrite Map-functional {m = m₁} k,v₁∈m₁ k,v₁'∈m₁
+                rewrite Map-functional {m = m₁ ⊓ (m₁ ⊔ m₂)} k,v∈m₁m₁₂ v₁v₁'∈m₁m₁₂ =
+                (v₁' , (⊓₂-idemp v₁' , k,v₁'∈m₁))
+        ...   | (_ , (bothⁱ (single {v₁} k,v₁∈m₁)
+                            (in₂ k∉m₁ _ ) , _)) = ⊥-elim (k∉m₁ (∈-cong proj₁ k,v₁∈m₁))
 
-                    union-assoc₂ : subset (union f m₁ (union f m₂ m₃)) (union f (union f m₁ m₂) m₃)
-                    union-assoc₂ k v k,v∈m₁m₂₃
-                        with Expr-Provenance f f k ((` m₁) ∪ ((` m₂) ∪ (` m₃))) (∈-cong proj₁ k,v∈m₁m₂₃)
-                    ...   | (_ , (in₂ k∉ke₁ (in₂ k∉ke₂ (single {v₃} v₃∈e₃)) , v₃∈m₁m₂₃))
-                            rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₃∈m₁m₂₃ =
-                            (v₃ , (≈-refl , I.union-preserves-∈₂ (I.union-preserves-∉ k∉ke₁ k∉ke₂) v₃∈e₃))
-                    ...   | (_ , (in₂ k∉ke₁ (in₁ (single {v₂} v₂∈e₂) k∉ke₃) , v₂∈m₁m₂₃))
-                            rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₂∈m₁m₂₃ =
-                            (v₂ , (≈-refl , I.union-preserves-∈₁ (I.union-preserves-Unique l₁ l₂ u₂) (I.union-preserves-∈₂ k∉ke₁ v₂∈e₂) k∉ke₃))
-                    ...   | (_ , (in₂ k∉ke₁ (bothᵘ (single {v₂} v₂∈e₂) (single {v₃} v₃∈e₃)) , v₂v₃∈m₁m₂₃))
-                            rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₂v₃∈m₁m₂₃ =
-                            (f v₂ v₃ , (≈-refl , I.union-combines (I.union-preserves-Unique l₁ l₂ u₂) u₃ (I.union-preserves-∈₂ k∉ke₁ v₂∈e₂) v₃∈e₃))
-                    ...   | (_ , (in₁ (single {v₁} v₁∈e₁) k∉ke₂₃ , v₁∈m₁m₂₃))
-                            rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁∈m₁m₂₃ =
-                            let (k∉ke₂ , k∉ke₃) = I.∉-union-∉-either {l₁ = l₂} {l₂ = l₃} k∉ke₂₃
-                            in (v₁ , (≈-refl , I.union-preserves-∈₁ (I.union-preserves-Unique l₁ l₂ u₂) (I.union-preserves-∈₁ u₁ v₁∈e₁ k∉ke₂) k∉ke₃))
-                    ...   | (_ , (bothᵘ (single {v₁} v₁∈e₁) (in₂ k∉ke₂ (single {v₃} v₃∈e₃)) , v₁v₃∈m₁m₂₃))
-                            rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁v₃∈m₁m₂₃ =
-                            (f v₁ v₃ , (≈-refl , I.union-combines (I.union-preserves-Unique l₁ l₂ u₂) u₃ (I.union-preserves-∈₁ u₁ v₁∈e₁ k∉ke₂) v₃∈e₃))
-                    ...   | (_ , (bothᵘ (single {v₁} v₁∈e₁) (in₁ (single {v₂} v₂∈e₂) k∉ke₃) , v₁v₂∈m₁m₂₃))
-                            rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁v₂∈m₁m₂₃ =
-                            (f v₁ v₂ , (≈-refl , I.union-preserves-∈₁ (I.union-preserves-Unique l₁ l₂ u₂) (I.union-combines u₁ u₂ v₁∈e₁ v₂∈e₂) k∉ke₃))
-                    ...   | (_ , (bothᵘ (single {v₁} v₁∈e₁) (bothᵘ (single {v₂} v₂∈e₂) (single {v₃} v₃∈e₃)) , v₁v₂v₃∈m₁m₂₃))
-                            rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁v₂v₃∈m₁m₂₃ =
-                            (f (f v₁ v₂) v₃ , (≈-sym (f-assoc v₁ v₂ v₃) , I.union-combines (I.union-preserves-Unique l₁ l₂ u₂) u₃ (I.union-combines u₁ u₂ v₁∈e₁ v₂∈e₂) v₃∈e₃))
+        absorb-⊓-⊔² : m₁ ⊆ (m₁ ⊓ (m₁ ⊔ m₂))
+        absorb-⊓-⊔² k v k,v∈m₁
+            with ∈k-dec k l₂
+        ...   | yes k∈km₂ =
+                let (v₂ , k,v₂∈m₂) = locate k∈km₂
+                in (v ⊓₂ (v ⊔₂ v₂) , (≈₂-sym (absorb-⊓₂-⊔₂ v v₂) , I⊓.intersect-combines u₁ (I⊔.union-preserves-Unique l₁ l₂ u₂) k,v∈m₁ (I⊔.union-combines u₁ u₂ k,v∈m₁ k,v₂∈m₂)))
+        ...   | no k∉km₂ = (v ⊓₂ v , (≈₂-sym (⊓₂-idemp v) , I⊓.intersect-combines u₁ (I⊔.union-preserves-Unique l₁ l₂ u₂) k,v∈m₁ (I⊔.union-preserves-∈₁ u₁ k,v∈m₁ k∉km₂)))
 
-        module _ (f-idemp : ∀ (b : B) → f b b ≈ b) where
-            intersect-idemp : ∀ (m : Map) → lift (intersect f m m) m
-            intersect-idemp m@(l , u) = (mm-m-subset , m-mm-subset)
-                where
-                    mm-m-subset : subset (intersect f m m) m
-                    mm-m-subset k v k,v∈mm
-                        with Expr-Provenance f f k ((` m) ∩ (` m)) (∈-cong proj₁ k,v∈mm)
-                    ...   | (_ , (bothⁱ (single {v'} v'∈m) (single {v''} v''∈m) , v'v''∈mm))
-                            rewrite Map-functional {m = m} v'∈m v''∈m
-                            rewrite Map-functional {m = intersect f m m} k,v∈mm v'v''∈mm =
-                            (v'' , (f-idemp v'' , v''∈m))
+absorb-⊔-⊓ : ∀ (m₁ m₂ : Map) →  (m₁ ⊔ (m₁ ⊓ m₂)) ≈ m₁
+absorb-⊔-⊓ m₁@(l₁ , u₁) m₂@(l₂ , u₂) = (absorb-⊔-⊓¹ , absorb-⊔-⊓²)
+    where
+        absorb-⊔-⊓¹ : (m₁ ⊔ (m₁ ⊓ m₂)) ⊆ m₁
+        absorb-⊔-⊓¹ k v k,v∈m₁m₁₂
+            with Expr-Provenance k ((` m₁) ∪ ((` m₁) ∩ (` m₂))) (∈-cong proj₁ k,v∈m₁m₁₂)
+        ...   | (_ , (bothᵘ (single {v₁} k,v₁∈m₁)
+                            (bothⁱ (single {v₁'} k,v₁'∈m₁)
+                                   (single {v₂} k,v₂∈m₂)) , v₁v₁'v₂∈m₁m₁₂))
+                rewrite Map-functional {m = m₁} k,v₁∈m₁ k,v₁'∈m₁
+                rewrite Map-functional {m = m₁ ⊔ (m₁ ⊓ m₂)} k,v∈m₁m₁₂ v₁v₁'v₂∈m₁m₁₂ =
+                (v₁' , (absorb-⊔₂-⊓₂ v₁' v₂ , k,v₁'∈m₁))
+        ...   | (_ , (in₁ (single {v₁} k,v₁∈m₁) k∉km₁₂ , k,v₁∈m₁m₁₂))
+                rewrite Map-functional {m = m₁ ⊔ (m₁ ⊓ m₂)} k,v∈m₁m₁₂ k,v₁∈m₁m₁₂ =
+                (v₁ , (≈₂-refl , k,v₁∈m₁))
+        ...   | (_ , (in₂ k∉km₁ (bothⁱ (single {v₁'} k,v₁'∈m₁)
+                                      (single {v₂} k,v₂∈m₂)) , _)) =
+                ⊥-elim (k∉km₁ (∈-cong proj₁ k,v₁'∈m₁))
 
-                    m-mm-subset : subset m (intersect f m m)
-                    m-mm-subset k v k,v∈m = (f v v , (≈-sym (f-idemp v) , I.intersect-combines u u k,v∈m k,v∈m))
+        absorb-⊔-⊓² : m₁ ⊆ (m₁ ⊔ (m₁ ⊓ m₂))
+        absorb-⊔-⊓² k v k,v∈m₁
+            with ∈k-dec k l₂
+        ...   | yes k∈km₂ =
+                let (v₂ , k,v₂∈m₂) = locate k∈km₂
+                in (v ⊔₂ (v ⊓₂ v₂) , (≈₂-sym (absorb-⊔₂-⊓₂ v v₂) , I⊔.union-combines u₁ (I⊓.intersect-preserves-Unique {l₁} {l₂} u₂) k,v∈m₁ (I⊓.intersect-combines u₁ u₂ k,v∈m₁ k,v₂∈m₂)))
+        ...   | no k∉km₂ = (v , (≈₂-refl , I⊔.union-preserves-∈₁ u₁ k,v∈m₁ (I⊓.intersect-preserves-∉₂ {k} {l₁} {l₂} k∉km₂)))
 
-        module _ (f-comm : ∀ (b₁ b₂ : B) → f b₁ b₂ ≈ f b₂ b₁) where
-            intersect-comm : ∀ (m₁ m₂ : Map) → lift (intersect f m₁ m₂) (intersect f m₂ m₁)
-            intersect-comm m₁ m₂ = (intersect-comm-subset m₁ m₂ , intersect-comm-subset m₂ m₁)
-                where
-                    intersect-comm-subset : ∀ (m₁ m₂ : Map) → subset (intersect f m₁ m₂) (intersect f m₂ m₁)
-                    intersect-comm-subset m₁@(l₁ , u₁) m₂@(l₂ , u₂) k v k,v∈m₁m₂
-                        with Expr-Provenance f f k ((` m₁) ∩ (` m₂)) (∈-cong proj₁ k,v∈m₁m₂)
-                    ...   | (_ , (bothⁱ {v₁} {v₂} (single v₁∈m₁) (single v₂∈m₂) , v₁v₂∈m₁m₂))
-                            rewrite Map-functional {m = intersect f m₁ m₂} k,v∈m₁m₂ v₁v₂∈m₁m₂ =
-                            (f v₂ v₁ , (f-comm v₁ v₂ , I.intersect-combines u₂ u₁ v₂∈m₂ v₁∈m₁))
+isUnionSemilattice : IsSemilattice Map _≈_ _⊔_
+isUnionSemilattice = record
+    { ≈-equiv = ≈-equiv
+    ; ≈-⊔-cong = λ {m₁} {m₂} {m₃} {m₄} m₁≈m₂ m₃≈m₄ → ≈-⊔-cong {m₁} {m₂} {m₃} {m₄} m₁≈m₂ m₃≈m₄
+    ; ⊔-assoc = ⊔-assoc
+    ; ⊔-comm = ⊔-comm
+    ; ⊔-idemp = ⊔-idemp
+    }
 
-        module _ (f-assoc : ∀ (b₁ b₂ b₃ : B) → f (f b₁ b₂) b₃ ≈ f b₁ (f b₂ b₃))  where
-            intersect-assoc : ∀ (m₁ m₂ m₃ : Map) → lift (intersect f (intersect f m₁ m₂) m₃) (intersect f m₁ (intersect f m₂ m₃))
-            intersect-assoc m₁@(l₁ , u₁) m₂@(l₂ , u₂) m₃@(l₃ , u₃) = (intersect-assoc₁ , intersect-assoc₂)
-                where
-                    intersect-assoc₁ : subset (intersect f (intersect f m₁ m₂) m₃) (intersect f m₁ (intersect f m₂ m₃))
-                    intersect-assoc₁ k v k,v∈m₁₂m₃
-                        with Expr-Provenance f f k (((` m₁) ∩ (` m₂)) ∩ (` m₃)) (∈-cong proj₁ k,v∈m₁₂m₃)
-                    ...   | (_ , (bothⁱ (bothⁱ (single {v₁} v₁∈e₁) (single {v₂} v₂∈e₂)) (single {v₃} v₃∈e₃) , v₁v₂v₃∈m₁₂m₃))
-                            rewrite Map-functional {m = intersect f (intersect f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁v₂v₃∈m₁₂m₃ =
-                            (f v₁ (f v₂ v₃) , (f-assoc v₁ v₂ v₃ , I.intersect-combines u₁ (I.intersect-preserves-Unique {l₂} {l₃} u₃) v₁∈e₁ (I.intersect-combines u₂ u₃ v₂∈e₂ v₃∈e₃)))
+isIntersectSemilattice : IsSemilattice Map _≈_ _⊓_
+isIntersectSemilattice = record
+    { ≈-equiv = ≈-equiv
+    ; ≈-⊔-cong = λ {m₁} {m₂} {m₃} {m₄} m₁≈m₂ m₃≈m₄ → ≈-⊓-cong {m₁} {m₂} {m₃} {m₄} m₁≈m₂ m₃≈m₄
+    ; ⊔-assoc = ⊓-assoc
+    ; ⊔-comm = ⊓-comm
+    ; ⊔-idemp = ⊓-idemp
+    }
 
-                    intersect-assoc₂ : subset (intersect f m₁ (intersect f m₂ m₃)) (intersect f (intersect f m₁ m₂) m₃)
-                    intersect-assoc₂ k v k,v∈m₁m₂₃
-                        with Expr-Provenance f f k ((` m₁) ∩ ((` m₂) ∩ (` m₃))) (∈-cong proj₁ k,v∈m₁m₂₃)
-                    ...   | (_ , (bothⁱ (single {v₁} v₁∈e₁) (bothⁱ (single {v₂} v₂∈e₂) (single {v₃} v₃∈e₃)) , v₁v₂v₃∈m₁m₂₃))
-                            rewrite Map-functional {m = intersect f m₁ (intersect f m₂ m₃)} k,v∈m₁m₂₃ v₁v₂v₃∈m₁m₂₃ =
-                            (f (f v₁ v₂) v₃ , (≈-sym (f-assoc v₁ v₂ v₃) , I.intersect-combines (I.intersect-preserves-Unique {l₁} {l₂} u₂) u₃ (I.intersect-combines u₁ u₂ v₁∈e₁ v₂∈e₂) v₃∈e₃))
-
-    module _ (≈-refl : ∀ {b : B} → b ≈ b)
-             (≈-sym : ∀ {b₁ b₂ : B} → b₁ ≈ b₂ → b₂ ≈ b₁)
-             (_⊔₂_ : B → B → B) (_⊓₂_ : B → B → B)
-             (⊔₂-idemp :  ∀ (b : B) → (b ⊔₂ b) ≈ b)
-             (⊓₂-idemp :  ∀ (b : B) → (b ⊓₂ b) ≈ b)
-             (⊔₂-⊓₂-absorb : ∀ (b₁ b₂ : B) → (b₁ ⊔₂ (b₁ ⊓₂ b₂)) ≈ b₁)
-             (⊓₂-⊔₂-absorb : ∀ (b₁ b₂ : B) → (b₁ ⊓₂ (b₁ ⊔₂ b₂)) ≈ b₁)
-             where
-        private module I⊔ = ImplInsert _⊔₂_
-        private module I⊓ = ImplInsert _⊓₂_
-
-        private
-            _⊔_ = union _⊔₂_
-            _⊓_ = intersect _⊓₂_
-
-        intersect-union-absorb : ∀ (m₁ m₂ : Map) →  lift (m₁ ⊓ (m₁ ⊔ m₂)) m₁
-        intersect-union-absorb m₁@(l₁ , u₁) m₂@(l₂ , u₂) = (intersect-union-absorb₁ , intersect-union-absorb₂)
-            where
-                intersect-union-absorb₁ : subset (m₁ ⊓ (m₁ ⊔ m₂)) m₁
-                intersect-union-absorb₁ k v k,v∈m₁m₁₂
-                    with Expr-Provenance _ _ k ((` m₁) ∩ ((` m₁) ∪ (` m₂))) (∈-cong proj₁ k,v∈m₁m₁₂)
-                ...   | (_ , (bothⁱ (single {v₁} k,v₁∈m₁)
-                                    (bothᵘ (single {v₁'} k,v₁'∈m₁)
-                                           (single {v₂} v₂∈m₂)) , v₁v₁'v₂∈m₁m₁₂))
-                        rewrite Map-functional {m = m₁} k,v₁∈m₁ k,v₁'∈m₁
-                        rewrite Map-functional {m = m₁ ⊓ (m₁ ⊔ m₂)} k,v∈m₁m₁₂ v₁v₁'v₂∈m₁m₁₂ =
-                        (v₁' , (⊓₂-⊔₂-absorb v₁' v₂ , k,v₁'∈m₁))
-                ...   | (_ , (bothⁱ (single {v₁} k,v₁∈m₁)
-                                    (in₁ (single {v₁'} k,v₁'∈m₁) _) , v₁v₁'∈m₁m₁₂))
-                        rewrite Map-functional {m = m₁} k,v₁∈m₁ k,v₁'∈m₁
-                        rewrite Map-functional {m = m₁ ⊓ (m₁ ⊔ m₂)} k,v∈m₁m₁₂ v₁v₁'∈m₁m₁₂ =
-                        (v₁' , (⊓₂-idemp v₁' , k,v₁'∈m₁))
-                ...   | (_ , (bothⁱ (single {v₁} k,v₁∈m₁)
-                                    (in₂ k∉m₁ _ ) , _)) = absurd (k∉m₁ (∈-cong proj₁ k,v₁∈m₁))
-
-                intersect-union-absorb₂ : subset m₁ (m₁ ⊓ (m₁ ⊔ m₂))
-                intersect-union-absorb₂ k v k,v∈m₁
-                    with ∈k-dec k l₂
-                ...   | yes k∈km₂ =
-                        let (v₂ , k,v₂∈m₂) = locate k∈km₂
-                        in (v ⊓₂ (v ⊔₂ v₂) , (≈-sym (⊓₂-⊔₂-absorb v v₂) , I⊓.intersect-combines u₁ (I⊔.union-preserves-Unique l₁ l₂ u₂) k,v∈m₁ (I⊔.union-combines u₁ u₂ k,v∈m₁ k,v₂∈m₂)))
-                ...   | no k∉km₂ = (v ⊓₂ v , (≈-sym (⊓₂-idemp v) , I⊓.intersect-combines u₁ (I⊔.union-preserves-Unique l₁ l₂ u₂) k,v∈m₁ (I⊔.union-preserves-∈₁ u₁ k,v∈m₁ k∉km₂)))
-
-        union-intersect-absorb : ∀ (m₁ m₂ : Map) →  lift (m₁ ⊔ (m₁ ⊓ m₂)) m₁
-        union-intersect-absorb m₁@(l₁ , u₁) m₂@(l₂ , u₂) = (union-intersect-absorb₁ , union-intersect-absorb₂)
-            where
-                union-intersect-absorb₁ : subset (m₁ ⊔ (m₁ ⊓ m₂)) m₁
-                union-intersect-absorb₁ k v k,v∈m₁m₁₂
-                    with Expr-Provenance _ _ k ((` m₁) ∪ ((` m₁) ∩ (` m₂))) (∈-cong proj₁ k,v∈m₁m₁₂)
-                ...   | (_ , (bothᵘ (single {v₁} k,v₁∈m₁)
-                                    (bothⁱ (single {v₁'} k,v₁'∈m₁)
-                                           (single {v₂} k,v₂∈m₂)) , v₁v₁'v₂∈m₁m₁₂))
-                        rewrite Map-functional {m = m₁} k,v₁∈m₁ k,v₁'∈m₁
-                        rewrite Map-functional {m = m₁ ⊔ (m₁ ⊓ m₂)} k,v∈m₁m₁₂ v₁v₁'v₂∈m₁m₁₂ =
-                        (v₁' , (⊔₂-⊓₂-absorb v₁' v₂ , k,v₁'∈m₁))
-                ...   | (_ , (in₁ (single {v₁} k,v₁∈m₁) k∉km₁₂ , k,v₁∈m₁m₁₂))
-                        rewrite Map-functional {m = m₁ ⊔ (m₁ ⊓ m₂)} k,v∈m₁m₁₂ k,v₁∈m₁m₁₂ =
-                        (v₁ , (≈-refl , k,v₁∈m₁))
-                ...   | (_ , (in₂ k∉km₁ (bothⁱ (single {v₁'} k,v₁'∈m₁)
-                                              (single {v₂} k,v₂∈m₂)) , _)) =
-                        absurd (k∉km₁ (∈-cong proj₁ k,v₁'∈m₁))
-
-                union-intersect-absorb₂ : subset m₁ (m₁ ⊔ (m₁ ⊓ m₂))
-                union-intersect-absorb₂ k v k,v∈m₁
-                    with ∈k-dec k l₂
-                ...   | yes k∈km₂ =
-                        let (v₂ , k,v₂∈m₂) = locate k∈km₂
-                        in (v ⊔₂ (v ⊓₂ v₂) , (≈-sym (⊔₂-⊓₂-absorb v v₂) , I⊔.union-combines u₁ (I⊓.intersect-preserves-Unique {l₁} {l₂} u₂) k,v∈m₁ (I⊓.intersect-combines u₁ u₂ k,v∈m₁ k,v₂∈m₂)))
-                ...   | no k∉km₂ = (v , (≈-refl , I⊔.union-preserves-∈₁ u₁ k,v∈m₁ (I⊓.intersect-preserves-∉₂ {k} {l₁} {l₂} k∉km₂)))
+isLattice : IsLattice Map _≈_ _⊔_ _⊓_
+isLattice = record
+    { joinSemilattice = isUnionSemilattice
+    ; meetSemilattice = isIntersectSemilattice
+    ; absorb-⊔-⊓ = absorb-⊔-⊓
+    ; absorb-⊓-⊔ = absorb-⊓-⊔
+    }