diff --git a/Analysis/Forward/Lattices.agda b/Analysis/Forward/Lattices.agda
index f365c83..4bbf80b 100644
--- a/Analysis/Forward/Lattices.agda
+++ b/Analysis/Forward/Lattices.agda
@@ -10,7 +10,6 @@ module Analysis.Forward.Lattices
 
 open import Data.String using (String) renaming (_≟_ to _≟ˢ_)
 open import Data.Product using (proj₁; proj₂; _,_)
-open import Data.Unit using (tt)
 open import Data.Sum using (inj₁; inj₂)
 open import Data.List using (List; _∷_; []; foldr)
 open import Data.List.Membership.Propositional using () renaming (_∈_ to _∈ˡ_)
@@ -38,7 +37,7 @@ instance
 -- with keys strings. Use a bundle to avoid explicitly specifying operators.
 -- It's helpful to export these via 'public' since consumers tend to
 -- use various variable lattice operations.
-module VariableValuesFiniteMap = Lattice.FiniteMap.WithKeys String L tt vars
+module VariableValuesFiniteMap = Lattice.FiniteMap String L vars
 open VariableValuesFiniteMap
     using ()
     renaming
@@ -65,13 +64,13 @@ open IsLattice isLatticeᵛ
         ; ⊔-idemp to ⊔ᵛ-idemp
         )
     public
-open Lattice.FiniteMap.IterProdIsomorphism String L _
+open Lattice.FiniteMap.IterProdIsomorphism String L vars
     using ()
     renaming
         ( Provenance-union to Provenance-unionᵐ
         )
     public
-open Lattice.FiniteMap.IterProdIsomorphism.WithUniqueKeysAndFixedHeight String L _ vars-Unique
+open Lattice.FiniteMap.IterProdIsomorphism.WithUniqueKeysAndFixedHeight String L vars vars-Unique
     using ()
     renaming
         ( isFiniteHeightLattice to isFiniteHeightLatticeᵛ
@@ -83,7 +82,7 @@ open Lattice.FiniteMap.IterProdIsomorphism.WithUniqueKeysAndFixedHeight String L
 ⊥ᵛ = Chain.Height.⊥ fixedHeightᵛ
 
 -- Finally, the map we care about is (state -> (variables -> value)). Bring that in.
-module StateVariablesFiniteMap = Lattice.FiniteMap.WithKeys State VariableValues tt states
+module StateVariablesFiniteMap = Lattice.FiniteMap State VariableValues states
 open StateVariablesFiniteMap
     using (_[_]; []-∈; m₁≼m₂⇒m₁[ks]≼m₂[ks]; m₁≈m₂⇒k∈m₁⇒k∈km₂⇒v₁≈v₂)
     renaming
@@ -98,7 +97,7 @@ open StateVariablesFiniteMap
         ; m₁≼m₂⇒m₁[k]≼m₂[k] to m₁≼m₂⇒m₁[k]ᵐ≼m₂[k]ᵐ
         )
     public
-open Lattice.FiniteMap.IterProdIsomorphism.WithUniqueKeysAndFixedHeight State VariableValues _ states-Unique
+open Lattice.FiniteMap.IterProdIsomorphism.WithUniqueKeysAndFixedHeight State VariableValues states states-Unique
     using ()
     renaming
         ( isFiniteHeightLattice to isFiniteHeightLatticeᵐ
diff --git a/Lattice/FiniteMap.agda b/Lattice/FiniteMap.agda
index a90e2c2..66bb5f6 100644
--- a/Lattice/FiniteMap.agda
+++ b/Lattice/FiniteMap.agda
@@ -9,10 +9,10 @@ module Lattice.FiniteMap (A : Set) (B : Set)
     {_≈₂_ : B → B → Set}
     {_⊔₂_ : B → B → B} {_⊓₂_ : B → B → B}
     {{≡-Decidable-A : IsDecidable {_} {A} _≡_}}
-    {{lB : IsLattice B _≈₂_ _⊔₂_ _⊓₂_}} (dummy : ⊤) where
+    {{lB : IsLattice B _≈₂_ _⊔₂_ _⊓₂_}} (ks : List A) where
 
 open IsLattice lB using () renaming (_≼_ to _≼₂_)
-open import Lattice.Map A B dummy as Map
+open import Lattice.Map A B _ as Map
     using
         ( Map
         ; ⊔-equal-keys
@@ -74,7 +74,7 @@ open import Showable using (Showable; show)
 open import Isomorphism using (IsInverseˡ; IsInverseʳ)
 open import Chain using (Height)
 
-module WithKeys (ks : List A) where
+private module WithKeys (ks : List A) where
     FiniteMap : Set
     FiniteMap = Σ Map (λ m → Map.keys m ≡ ks)
 
@@ -131,7 +131,7 @@ module WithKeys (ks : List A) where
 
     []-∈ : ∀ {k : A} {v : B} {ks' : List A} (fm : FiniteMap) →
            k ∈ˡ ks' → (k , v) ∈ fm → v ∈ˡ (fm [ ks' ])
-    []-∈ {k} {v} {ks'} (m , _) k∈ks' k,v∈fm = []ᵐ-∈ m k,v∈fm k∈ks' 
+    []-∈ {k} {v} {ks'} (m , _) k∈ks' k,v∈fm = []ᵐ-∈ m k,v∈fm k∈ks'
 
     ≈-equiv : IsEquivalence FiniteMap _≈_
     ≈-equiv = record
@@ -253,9 +253,8 @@ module WithKeys (ks : List A) where
     ...   | yes k∈km₁ | no k∉km₂ = ⊥-elim (∈k-exclusive fm₁ fm₂ (k∈km₁ , k∉km₂))
     ...   | no k∉km₁ | yes k∈km₂ = ⊥-elim (∈k-exclusive fm₂ fm₁ (k∈km₂ , k∉km₁))
 
-open WithKeys public
-
 module IterProdIsomorphism where
+    open WithKeys
     open import Data.Unit using (tt)
     open import Lattice.Unit using ()
         renaming
@@ -267,7 +266,7 @@ module IterProdIsomorphism where
             ; ≈-equiv to ≈ᵘ-equiv
             ; fixedHeight to fixedHeightᵘ
             )
-    open import Lattice.IterProd B ⊤ dummy
+    open import Lattice.IterProd B ⊤ _
         as IP
         using (IterProd)
     open IsLattice lB using ()
@@ -631,3 +630,5 @@ module IterProdIsomorphism where
         ⊥-contains-bottoms {k} {v} k,v∈⊥
             rewrite IP.⊥-built {length ks} {{fhB = fixedHeightᵘ}} =
             to-build uks k v k,v∈⊥
+
+open WithKeys ks public