diff --git a/Analysis/Sign.agda b/Analysis/Sign.agda
index f267c4c..3ebdee6 100644
--- a/Analysis/Sign.agda
+++ b/Analysis/Sign.agda
@@ -3,6 +3,7 @@ module Analysis.Sign where
 open import Data.String using (String) renaming (_≟_ to _≟ˢ_)
 open import Data.Nat using (suc)
 open import Data.Product using (_×_; proj₁; _,_)
+open import Data.Empty using (⊥; ⊥-elim)
 open import Data.List using (List; _∷_; []; foldr; cartesianProduct; cartesianProductWith)
 open import Data.List.Membership.Propositional as MemProp using () renaming (_∈_ to _∈ˡ_)
 open import Relation.Binary.PropositionalEquality using (_≡_; refl; sym; trans; subst)
@@ -61,8 +62,7 @@ open import Lattice.AboveBelow Sign _≡_ (record { ≈-refl = refl; ≈-sym = s
 -- 'sign' has no underlying lattice structure, so use the 'plain' above-below lattice.
 open AB.Plain 0ˢ using ()
     renaming
-        ( finiteHeightLattice to finiteHeightLatticeᵍ
-        ; isLattice to isLatticeᵍ
+        ( isLattice to isLatticeᵍ
         ; fixedHeight to fixedHeightᵍ
         ; _≼_ to _≼ᵍ_
         ; _⊔_ to _⊔ᵍ_
@@ -130,6 +130,8 @@ module WithProg (prog : Program) where
             ; _updating_via_ to _updatingᵛ_via_
             ; locate to locateᵛ
             ; m₁≼m₂⇒m₁[k]≼m₂[k] to m₁≼m₂⇒m₁[k]ᵛ≼m₂[k]ᵛ
+            ; all-equal-keys to all-equal-keysᵛ
+            ; ∈k-dec to ∈k-decᵛ
             )
     open IsLattice isLatticeᵛ
         using ()
@@ -206,57 +208,57 @@ module WithProg (prog : Program) where
     states-in-Map : ∀ (s : State) (sv : StateVariables) → s ∈kᵐ sv
     states-in-Map s sv@(m , ksv≡states) rewrite ksv≡states = states-complete s
 
-    eval : ∀ (e : Expr) → (∀ k → k ∈ᵉ e → k ∈ˡ vars) → VariableSigns → SignLattice
-    eval (e₁ + e₂) k∈e⇒k∈vars vs =
-        plus (eval e₁ (λ k k∈e₁ → k∈e⇒k∈vars k (in⁺₁ k∈e₁)) vs)
-             (eval e₂ (λ k k∈e₂ → k∈e⇒k∈vars k (in⁺₂ k∈e₂)) vs)
-    eval (e₁ - e₂) k∈e⇒k∈vars vs =
-        minus (eval e₁ (λ k k∈e₁ → k∈e⇒k∈vars k (in⁻₁ k∈e₁)) vs)
-              (eval e₂ (λ k k∈e₂ → k∈e⇒k∈vars k (in⁻₂ k∈e₂)) vs)
-    eval (` k) k∈e⇒k∈vars vs = proj₁ (locateᵛ {k} {vs} (vars-in-Map k vs (k∈e⇒k∈vars k here)))
-    eval (# 0) _ _ = [ 0ˢ ]ᵍ
-    eval (# (suc n')) _ _ = [ + ]ᵍ
+    eval : ∀ (e : Expr) → VariableSigns → SignLattice
+    eval (e₁ + e₂) vs = plus (eval e₁ vs) (eval e₂ vs)
+    eval (e₁ - e₂) vs = minus (eval e₁ vs) (eval e₂ vs)
+    eval (` k) vs
+        with ∈k-decᵛ k (proj₁ (proj₁ vs))
+    ...   | yes k∈vs = proj₁ (locateᵛ {k} {vs} k∈vs)
+    ...   | no _ = ⊤ᵍ
+    eval (# 0) _ = [ 0ˢ ]ᵍ
+    eval (# (suc n')) _ = [ + ]ᵍ
 
-    eval-Mono : ∀ (e : Expr) (k∈e⇒k∈vars : ∀ k → k ∈ᵉ e → k ∈ˡ vars) → Monotonic _≼ᵛ_ _≼ᵍ_ (eval e k∈e⇒k∈vars)
-    eval-Mono (e₁ + e₂) k∈e⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂ =
+    eval-Mono : ∀ (e : Expr) → Monotonic _≼ᵛ_ _≼ᵍ_ (eval e)
+    eval-Mono (e₁ + e₂) {vs₁} {vs₂} vs₁≼vs₂ =
         let
             -- TODO: can this be done with less boilerplate?
-            k∈e₁⇒k∈vars = λ k k∈e₁ → k∈e⇒k∈vars k (in⁺₁ k∈e₁)
-            k∈e₂⇒k∈vars = λ k k∈e₂ → k∈e⇒k∈vars k (in⁺₂ k∈e₂)
-            g₁vs₁ = eval e₁ k∈e₁⇒k∈vars vs₁
-            g₂vs₁ = eval e₂ k∈e₂⇒k∈vars vs₁
-            g₁vs₂ = eval e₁ k∈e₁⇒k∈vars vs₂
-            g₂vs₂ = eval e₂ k∈e₂⇒k∈vars vs₂
+            g₁vs₁ = eval e₁ vs₁
+            g₂vs₁ = eval e₂ vs₁
+            g₁vs₂ = eval e₁ vs₂
+            g₂vs₂ = eval e₂ vs₂
         in
             ≼ᵍ-trans
                 {plus g₁vs₁ g₂vs₁} {plus g₁vs₂ g₂vs₁} {plus g₁vs₂ g₂vs₂}
-                (plus-Monoˡ g₂vs₁ {g₁vs₁} {g₁vs₂} (eval-Mono e₁ k∈e₁⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂))
-                (plus-Monoʳ g₁vs₂ {g₂vs₁} {g₂vs₂} (eval-Mono e₂ k∈e₂⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂))
-    eval-Mono (e₁ - e₂) k∈e⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂ =
+                (plus-Monoˡ g₂vs₁ {g₁vs₁} {g₁vs₂} (eval-Mono e₁ {vs₁} {vs₂} vs₁≼vs₂))
+                (plus-Monoʳ g₁vs₂ {g₂vs₁} {g₂vs₂} (eval-Mono e₂ {vs₁} {vs₂} vs₁≼vs₂))
+    eval-Mono (e₁ - e₂) {vs₁} {vs₂} vs₁≼vs₂ =
         let
             -- TODO: here too -- can this be done with less boilerplate?
-            k∈e₁⇒k∈vars = λ k k∈e₁ → k∈e⇒k∈vars k (in⁻₁ k∈e₁)
-            k∈e₂⇒k∈vars = λ k k∈e₂ → k∈e⇒k∈vars k (in⁻₂ k∈e₂)
-            g₁vs₁ = eval e₁ k∈e₁⇒k∈vars vs₁
-            g₂vs₁ = eval e₂ k∈e₂⇒k∈vars vs₁
-            g₁vs₂ = eval e₁ k∈e₁⇒k∈vars vs₂
-            g₂vs₂ = eval e₂ k∈e₂⇒k∈vars vs₂
+            g₁vs₁ = eval e₁ vs₁
+            g₂vs₁ = eval e₂ vs₁
+            g₁vs₂ = eval e₁ vs₂
+            g₂vs₂ = eval e₂ vs₂
         in
             ≼ᵍ-trans
                 {minus g₁vs₁ g₂vs₁} {minus g₁vs₂ g₂vs₁} {minus g₁vs₂ g₂vs₂}
-                (minus-Monoˡ g₂vs₁ {g₁vs₁} {g₁vs₂} (eval-Mono e₁ k∈e₁⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂))
-                (minus-Monoʳ g₁vs₂ {g₂vs₁} {g₂vs₂} (eval-Mono e₂ k∈e₂⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂))
-    eval-Mono (` k) k∈e⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂ =
-        let
-            (v₁ , k,v₁∈vs₁) = locateᵛ {k} {vs₁} (vars-in-Map k vs₁ (k∈e⇒k∈vars k here))
-            (v₂ , k,v₂∈vs₂) = locateᵛ {k} {vs₂} (vars-in-Map k vs₂ (k∈e⇒k∈vars k here))
-        in
-            m₁≼m₂⇒m₁[k]ᵛ≼m₂[k]ᵛ vs₁ vs₂ vs₁≼vs₂ k,v₁∈vs₁ k,v₂∈vs₂
-    eval-Mono (# 0) _ _ = ≈ᵍ-refl
-    eval-Mono (# (suc n')) _ _ = ≈ᵍ-refl
+                (minus-Monoˡ g₂vs₁ {g₁vs₁} {g₁vs₂} (eval-Mono e₁ {vs₁} {vs₂} vs₁≼vs₂))
+                (minus-Monoʳ g₁vs₂ {g₂vs₁} {g₂vs₂} (eval-Mono e₂ {vs₁} {vs₂} vs₁≼vs₂))
+    eval-Mono (` k) {vs₁@((kvs₁ , _) , _)} {vs₂@((kvs₂ , _), _)} vs₁≼vs₂
+        with ∈k-decᵛ k kvs₁ | ∈k-decᵛ k kvs₂
+    ...   | yes k∈kvs₁ | yes k∈kvs₂ =
+            let
+                (v₁ , k,v₁∈vs₁) = locateᵛ {k} {vs₁} k∈kvs₁
+                (v₂ , k,v₂∈vs₂) = locateᵛ {k} {vs₂} k∈kvs₂
+            in
+                m₁≼m₂⇒m₁[k]ᵛ≼m₂[k]ᵛ vs₁ vs₂ vs₁≼vs₂ k,v₁∈vs₁ k,v₂∈vs₂
+    ...   | yes k∈kvs₁ | no k∉kvs₂ = ⊥-elim (k∉kvs₂ (subst (λ l → k ∈ˡ l) (all-equal-keysᵛ vs₁ vs₂) k∈kvs₁))
+    ...   | no k∉kvs₁ | yes k∈kvs₂ = ⊥-elim (k∉kvs₁ (subst (λ l → k ∈ˡ l) (all-equal-keysᵛ vs₂ vs₁) k∈kvs₂))
+    ...   | no k∉kvs₁ | no k∉kvs₂ = IsLattice.≈-refl isLatticeᵍ
+    eval-Mono (# 0) _ = ≈ᵍ-refl
+    eval-Mono (# (suc n')) _ = ≈ᵍ-refl
 
-    private module _ (k : String) (e : Expr) (k∈e⇒k∈vars : ∀ k → k ∈ᵉ e → k ∈ˡ vars) where
-        open VariableSignsFiniteMap.GeneralizedUpdate vars isLatticeᵛ (λ x → x) (λ a₁≼a₂ → a₁≼a₂) (λ _ → eval e k∈e⇒k∈vars) (λ _ {vs₁} {vs₂} vs₁≼vs₂ → eval-Mono e k∈e⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂) (k ∷ [])
+    private module _ (k : String) (e : Expr) where
+        open VariableSignsFiniteMap.GeneralizedUpdate vars isLatticeᵛ (λ x → x) (λ a₁≼a₂ → a₁≼a₂) (λ _ → eval e) (λ _ {vs₁} {vs₂} vs₁≼vs₂ → eval-Mono e {vs₁} {vs₂} vs₁≼vs₂) (k ∷ [])
             renaming
                 ( f' to updateVariablesFromExpression
                 ; f'-Monotonic to updateVariablesFromExpression-Mono
@@ -265,28 +267,23 @@ module WithProg (prog : Program) where
 
     updateVariablesForState : State → StateVariables → VariableSigns
     updateVariablesForState s sv
-        -- More weirdness here. Apparently, capturing the with-equality proof
-        -- using 'in p' makes code that reasons about this function (below)
-        -- throw ill-typed with-abstraction errors. Instead, make use of the
-        -- fact that later with-clauses are generalized over earlier ones to
-        -- construct a specialization of vars-complete for (code s).
-        with code s | (λ k → vars-complete {k} s)
-    ...   | k ← e | k∈codes⇒k∈vars =
+        with code s
+    ...   | k ← e =
             let
                 (vs , s,vs∈sv) = locateᵐ {s} {sv} (states-in-Map s sv)
             in
-                updateVariablesFromExpression k e (λ k k∈e → k∈codes⇒k∈vars k (in←₂ k∈e)) vs
+                updateVariablesFromExpression k e vs
 
     updateVariablesForState-Monoʳ : ∀ (s : State) → Monotonic _≼ᵐ_ _≼ᵛ_ (updateVariablesForState s)
     updateVariablesForState-Monoʳ s {sv₁} {sv₂} sv₁≼sv₂
-        with code s | (λ k → vars-complete {k} s)
-    ...   | k ← e | k∈codes⇒k∈vars =
+        with code s
+    ...   | k ← e =
             let
                 (vs₁ , s,vs₁∈sv₁) = locateᵐ {s} {sv₁} (states-in-Map s sv₁)
                 (vs₂ , s,vs₂∈sv₂) = locateᵐ {s} {sv₂} (states-in-Map s sv₂)
                 vs₁≼vs₂ = m₁≼m₂⇒m₁[k]ᵐ≼m₂[k]ᵐ sv₁ sv₂ sv₁≼sv₂ s,vs₁∈sv₁ s,vs₂∈sv₂
             in
-                updateVariablesFromExpression-Mono k e (λ k k∈e → k∈codes⇒k∈vars k (in←₂ k∈e)) {vs₁} {vs₂} vs₁≼vs₂
+                updateVariablesFromExpression-Mono k e {vs₁} {vs₂} vs₁≼vs₂
 
     open StateVariablesFiniteMap.GeneralizedUpdate states isLatticeᵐ (λ x → x) (λ a₁≼a₂ → a₁≼a₂) updateVariablesForState updateVariablesForState-Monoʳ states
         renaming
diff --git a/Lattice/FiniteMap.agda b/Lattice/FiniteMap.agda
index 59d2f36..ca4cb7c 100644
--- a/Lattice/FiniteMap.agda
+++ b/Lattice/FiniteMap.agda
@@ -12,7 +12,7 @@ module Lattice.FiniteMap {a b : Level} {A : Set a} {B : Set b}
 
 open IsLattice lB using () renaming (_≼_ to _≼₂_)
 open import Lattice.Map ≡-dec-A lB as Map
-    using (Map; ⊔-equal-keys; ⊓-equal-keys; ∈k-dec)
+    using (Map; ⊔-equal-keys; ⊓-equal-keys)
     renaming
         ( _≈_ to _≈ᵐ_
         ; _⊔_ to _⊔ᵐ_
@@ -37,6 +37,7 @@ open import Lattice.Map ≡-dec-A lB as Map
         ; updating-via-keys-≡ to updatingᵐ-via-keys-≡
         ; f'-Monotonic to f'-Monotonicᵐ
         ; _≼_ to _≼ᵐ_
+        ; ∈k-dec to ∈k-decᵐ
         )
 open import Data.List.Membership.Propositional using () renaming (_∈_ to _∈ˡ_)
 open import Data.Product using (_×_; _,_; Σ; proj₁ ; proj₂)
@@ -82,6 +83,8 @@ module WithKeys (ks : List A) where
     _∈k_ : A → FiniteMap → Set a
     _∈k_ k (m₁ , _) = k ∈ˡ (keysᵐ m₁)
 
+    ∈k-dec = ∈k-decᵐ
+
     locate : ∀ {k : A} {fm : FiniteMap} → k ∈k fm → Σ B (λ v → (k , v) ∈ fm)
     locate {k} {fm = (m , _)} k∈kfm = locateᵐ {k} {m} k∈kfm
 
@@ -182,7 +185,7 @@ module WithKeys (ks : List A) where
                           fm₁ ≼ fm₂ → Pairwise _≼₂_ (fm₁ [ ks' ]) (fm₂ [ ks' ])
     m₁≼m₂⇒m₁[ks]≼m₂[ks] _ _ [] _ = []
     m₁≼m₂⇒m₁[ks]≼m₂[ks] fm₁@(m₁ , km₁≡ks) fm₂@(m₂ , km₂≡ks) (k ∷ ks'') m₁≼m₂
-        with ∈k-dec k (proj₁ m₁) | ∈k-dec k (proj₁ m₂)
+        with ∈k-decᵐ k (proj₁ m₁) | ∈k-decᵐ k (proj₁ m₂)
     ...   | yes k∈km₁ | yes k∈km₂ =
             let
                 (v₁ , k,v₁∈m₁) = locateᵐ {m = m₁} k∈km₁