diff --git a/Language.agda b/Language.agda
index 081192f..1c88ba8 100644
--- a/Language.agda
+++ b/Language.agda
@@ -86,6 +86,4 @@ record Program : Set where
 
     edge⇒incoming : ∀ {s₁ s₂ : State} → (s₁ , s₂) ListMem.∈ (Graph.edges graph) →
                     s₁ ListMem.∈ (incoming s₂)
-    edge⇒incoming {s₁} {s₂} s₁,s₂∈es =
-        ∈-filter⁺ (λ s' → (s' , s₂) ∈? (Graph.edges graph))
-                  (states-complete s₁) s₁,s₂∈es
+    edge⇒incoming = edge⇒predecessor graph
diff --git a/Language/Graphs.agda b/Language/Graphs.agda
index bb5599f..fdf826c 100644
--- a/Language/Graphs.agda
+++ b/Language/Graphs.agda
@@ -6,7 +6,7 @@ open import Data.Fin as Fin using (Fin; suc; zero)
 open import Data.Fin.Properties as FinProp using (suc-injective)
 open import Data.List as List using (List; []; _∷_)
 open import Data.List.Membership.Propositional as ListMem using ()
-open import Data.List.Membership.Propositional.Properties as ListMemProp using ()
+open import Data.List.Membership.Propositional.Properties as ListMemProp using (∈-filter⁺)
 open import Data.List.Relation.Unary.All using (All; []; _∷_)
 open import Data.List.Relation.Unary.Any as RelAny using ()
 open import Data.Nat as Nat using (ℕ; suc)
@@ -161,3 +161,9 @@ module _ (g : Graph) where
 
     predecessors : (Graph.Index g) → List (Graph.Index g)
     predecessors idx = List.filter (λ idx' → (idx' , idx) ∈? (Graph.edges g)) indices
+
+    edge⇒predecessor : ∀ {idx₁ idx₂ : Graph.Index g} → (idx₁ , idx₂) ListMem.∈ (Graph.edges g) →
+                    idx₁ ListMem.∈ (predecessors idx₂)
+    edge⇒predecessor {idx₁} {idx₂} idx₁,idx₂∈es =
+        ∈-filter⁺ (λ idx' → (idx' , idx₂) ∈? (Graph.edges g))
+                  (indices-complete idx₁) idx₁,idx₂∈es