diff --git a/Language/Semantics.agda b/Language/Semantics.agda
index b41ca21..bfd813b 100644
--- a/Language/Semantics.agda
+++ b/Language/Semantics.agda
@@ -12,7 +12,7 @@ open import Relation.Nullary using (¬_)
 open import Relation.Binary.PropositionalEquality using (_≡_)
 
 open import Lattice
-open import Utils using (_⇒_)
+open import Utils using (_⇒_; _∧_; _∨_)
 
 data Value : Set where
     ↑ᶻ : ℤ → Value
@@ -69,3 +69,5 @@ record LatticeInterpretation {l} {L : Set l} {_≈_ : L → L → Set l}
     field
         ⟦_⟧ : L → Value → Set
         ⟦⟧-respects-≈ : ∀ {l₁ l₂ : L} → l₁ ≈ l₂ → ⟦ l₁ ⟧ ⇒ ⟦ l₂ ⟧
+        ⟦⟧-⊔-∨ : ∀ {l₁ l₂ : L} → (⟦ l₁ ⟧ ∨ ⟦ l₂ ⟧) ⇒ ⟦ l₁ ⊔ l₂ ⟧
+        ⟦⟧-⊓-∧ : ∀ {l₁ l₂ : L} → (⟦ l₁ ⟧ ∧ ⟦ l₂ ⟧) ⇒ ⟦ l₁ ⊓ l₂ ⟧
diff --git a/Utils.agda b/Utils.agda
index c6689d0..0e76eb7 100644
--- a/Utils.agda
+++ b/Utils.agda
@@ -1,13 +1,14 @@
 module Utils where
 
 open import Agda.Primitive using () renaming (_⊔_ to _⊔ℓ_)
-open import Data.Product as Prod using ()
+open import Data.Product as Prod using (_×_)
 open import Data.Nat using (ℕ; suc)
 open import Data.List using (List; cartesianProduct; []; _∷_; _++_; foldr) renaming (map to mapˡ)
 open import Data.List.Membership.Propositional using (_∈_)
 open import Data.List.Membership.Propositional.Properties as ListMemProp using ()
 open import Data.List.Relation.Unary.All using (All; []; _∷_; map)
 open import Data.List.Relation.Unary.Any using (Any; here; there) -- TODO: re-export these with nicer names from map
+open import Data.Sum using (_⊎_)
 open import Function.Definitions using (Injective)
 open import Relation.Binary.PropositionalEquality using (_≡_; sym; refl)
 open import Relation.Nullary using (¬_)
@@ -85,3 +86,11 @@ concat-∈ {ls = l' ∷ ls'} x∈l (there l∈ls') = ListMemProp.∈-++⁺ʳ l'
 _⇒_ : ∀ {a p₁ p₂} {A : Set a} (P : A → Set p₁) (Q : A → Set p₂) →
       Set (a ⊔ℓ p₁ ⊔ℓ p₂)
 _⇒_ P Q = ∀ a → P a → Q a
+
+_∨_ : ∀ {a p₁ p₂} {A : Set a} (P : A → Set p₁) (Q : A → Set p₂) →
+      A → Set (p₁ ⊔ℓ p₂)
+_∨_ P Q a = P a ⊎ Q a
+
+_∧_ : ∀ {a p₁ p₂} {A : Set a} (P : A → Set p₁) (Q : A → Set p₂) →
+      A → Set (p₁ ⊔ℓ p₂)
+_∧_ P Q a = P a × Q a