Add a proof that edges lead to 'incoming' inclusion

Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>

Language.agda

@@ -10,6 +10,7 @@ open import Data.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 (∈-filter⁺)
 open import Data.List.Relation.Unary.All using (All; []; _∷_)
 open import Data.List.Relation.Unary.Any as RelAny using ()
 open import Data.Nat using (ℕ; suc)
@@ -100,3 +101,9 @@ record Program : Set where
 
     incoming : State → List State
     incoming idx = List.filter (λ idx' → (idx' , idx) ∈? (Graph.edges graph)) states
+
+    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