diff --git a/Lattice/Builder.agda b/Lattice/Builder.agda
index d3d97f3..36db51e 100644
--- a/Lattice/Builder.agda
+++ b/Lattice/Builder.agda
@@ -40,16 +40,12 @@ record Graph : Set where
         edges : List Edge
 
     data Path : Node → Node → Set where
-        last : ∀ {n₁ n₂ : Node} →  (n₁ , n₂) ∈ˡ edges →  Path n₁ n₂
+        done : ∀ {n : Node} → Path n n
         step : ∀ {n₁ n₂ n₃ : Node} →  (n₁ , n₂) ∈ˡ edges → Path n₂ n₃ → Path n₁ n₃
 
-    interior : ∀ {n₁ n₂} → Path n₁ n₂ → List Node
-    interior (last _) = []
-    interior (step {n₂ = n₂} _ p) = n₂ ∷ interior p
-
     _++_ : ∀ {n₁ n₂ n₃} → Path n₁ n₂ → Path n₂ n₃ → Path n₁ n₃
-    last e ++ p = step e p
-    step e p ++ p' = step e (p ++ p')
+    done ++ p = p
+    (step e p₁) ++ p₂ = step e (p₁ ++ p₂)
 
     Adjacency : Set
     Adjacency = ∀ (n₁ n₂ : Node) → List (Path n₁ n₂)
@@ -70,9 +66,9 @@ record Graph : Set where
     through-monotonic adj n p∈adjn₁n₂ = ∈ˡ-++⁺ʳ _ p∈adjn₁n₂
 
     seedWithEdges : ∀ (es : List Edge) →  (∀ {e} → e ∈ˡ es → e ∈ˡ edges) → Adjacency
-    seedWithEdges es e∈es⇒e∈edges = foldr (λ ((n₁ , n₂) , n₁n₂∈edges) → Adjacency-update n₁ n₂ ((last n₁n₂∈edges) ∷_)) (λ n₁ n₂ → []) (mapWith∈ˡ es (λ {e} e∈es →  (e , e∈es⇒e∈edges e∈es)))
+    seedWithEdges es e∈es⇒e∈edges = foldr (λ ((n₁ , n₂) , n₁n₂∈edges) → Adjacency-update n₁ n₂ ((step n₁n₂∈edges done) ∷_)) (λ n₁ n₂ → []) (mapWith∈ˡ es (λ {e} e∈es →  (e , e∈es⇒e∈edges e∈es)))
 
-    e∈seedWithEdges : ∀ {n₁ n₂ es} → (e∈es⇒e∈edges : ∀ {e} → e ∈ˡ es → e ∈ˡ edges) → ∀ (n₁n₂∈es : (n₁ , n₂) ∈ˡ es) →  (last (e∈es⇒e∈edges n₁n₂∈es)) ∈ˡ seedWithEdges es e∈es⇒e∈edges n₁ n₂
+    e∈seedWithEdges : ∀ {n₁ n₂ es} → (e∈es⇒e∈edges : ∀ {e} → e ∈ˡ es → e ∈ˡ edges) → ∀ (n₁n₂∈es : (n₁ , n₂) ∈ˡ es) →  (step (e∈es⇒e∈edges n₁n₂∈es) done) ∈ˡ seedWithEdges es e∈es⇒e∈edges n₁ n₂
     e∈seedWithEdges {es = []} e∈es⇒e∈edges ()
     e∈seedWithEdges {es = (n₁' , n₂') ∷ es} e∈es⇒e∈edges (here refl)
         with n₁' ≟ n₁' | n₂' ≟ n₂'
@@ -89,7 +85,7 @@ record Graph : Set where
     adj¹ : Adjacency
     adj¹ = seedWithEdges edges (λ x → x)
 
-    edge∈adj¹ : ∀ {n₁ n₂} (n₁n₂∈edges : (n₁ , n₂) ∈ˡ edges) →  (last n₁n₂∈edges) ∈ˡ adj¹ n₁ n₂
+    edge∈adj¹ : ∀ {n₁ n₂} (n₁n₂∈edges : (n₁ , n₂) ∈ˡ edges) →  (step n₁n₂∈edges done) ∈ˡ adj¹ n₁ n₂
     edge∈adj¹ = e∈seedWithEdges (λ x → x)
 
     throughAll : List Node → Adjacency
@@ -106,4 +102,4 @@ record Graph : Set where
     adj = throughAll (proj₁ nodes)
 
     NoCycles : Set
-    NoCycles = ∀ (n : Node) → adj n n ≡ []
+    NoCycles = ∀ (n : Node) →  All (_≡ done) (adj n n)