diff --git a/Language.agda b/Language.agda
index dae3134..473d218 100644
--- a/Language.agda
+++ b/Language.agda
@@ -11,11 +11,12 @@ open import Data.Vec.Relation.Binary.Equality.Cast using (cast-is-id)
 open import Data.List using ([]; _∷_; List) renaming (foldr to foldrˡ; map to mapˡ; _++_ to _++ˡ_)
 open import Data.List.Properties using () renaming (++-assoc to ++ˡ-assoc; map-++ to mapˡ-++ˡ; ++-identityʳ to ++ˡ-identityʳ)
 open import Data.List.Membership.Propositional as MemProp using () renaming (_∈_ to _∈ˡ_)
+open import Data.List.Membership.Propositional.Properties using () renaming (∈-++⁺ʳ to ∈ˡ-++⁺ʳ)
 open import Data.List.Relation.Unary.All using (All; []; _∷_)
 open import Data.List.Relation.Unary.Any as RelAny using ()
 open import Data.List.Relation.Unary.Any.Properties using (++⁺ʳ)
 open import Data.Fin using (Fin; suc; zero; fromℕ; inject₁; inject≤; _↑ʳ_; _↑ˡ_) renaming (_≟_ to _≟ᶠ_; cast to castᶠ)
-open import Data.Fin.Properties using (suc-injective)
+open import Data.Fin.Properties using (suc-injective) renaming (cast-is-id to castᶠ-is-id)
 open import Relation.Binary.PropositionalEquality as Eq using (subst; cong; _≡_; sym; trans; refl)
 open import Relation.Nullary using (¬_)
 open import Function using (_∘_)
@@ -103,9 +104,42 @@ module Graphs where
             nodes : Vec (List BasicStmt) size
             edges : List Edge
 
+    castᵉ : ∀ {n m : ℕ} .(p : n ≡ m) → (Fin n × Fin n) → (Fin m × Fin m)
+    castᵉ p (idx₁ , idx₂) = (castᶠ p idx₁ , castᶠ p idx₂)
+
     ↑ˡ-Edge : ∀ {n} → (Fin n × Fin n) → ∀ m → (Fin (n +ⁿ m) × Fin (n +ⁿ m))
     ↑ˡ-Edge (idx₁ , idx₂) m = (idx₁ ↑ˡ m , idx₂ ↑ˡ m)
 
+    ↑ˡ-assoc : ∀ {s n₁ n₂} (f : Fin s) (p : s +ⁿ (n₁ +ⁿ n₂) ≡ s +ⁿ n₁ +ⁿ n₂) →
+               f ↑ˡ n₁ ↑ˡ n₂ ≡ castᶠ p (f ↑ˡ (n₁ +ⁿ n₂))
+    ↑ˡ-assoc zero p = refl
+    ↑ˡ-assoc {suc s'} {n₁} {n₂} (suc f') p rewrite ↑ˡ-assoc f' (sym (+-assoc s' n₁ n₂)) = refl
+
+    ↑ˡ-Edge-assoc : ∀ {s n₁ n₂} (e : Fin s × Fin s) (p : s +ⁿ (n₁ +ⁿ n₂) ≡ s +ⁿ n₁ +ⁿ n₂) →
+                    ↑ˡ-Edge (↑ˡ-Edge e n₁) n₂ ≡ castᵉ p (↑ˡ-Edge e (n₁ +ⁿ n₂))
+    ↑ˡ-Edge-assoc (idx₁ , idx₂) p
+        rewrite ↑ˡ-assoc idx₁ p
+        rewrite ↑ˡ-assoc idx₂ p = refl
+
+    ↑ˡ-identityʳ : ∀ {s} (f : Fin s) (p : s +ⁿ 0 ≡ s) →
+               f ≡ castᶠ p (f ↑ˡ 0)
+    ↑ˡ-identityʳ zero p = refl
+    ↑ˡ-identityʳ {suc s'} (suc f') p rewrite sym (↑ˡ-identityʳ f' (+-comm s' 0)) = refl
+
+    ↑ˡ-Edge-identityʳ : ∀ {s} (e : Fin s × Fin s) (p : s +ⁿ 0 ≡ s) →
+                        e ≡ castᵉ p (↑ˡ-Edge e 0)
+    ↑ˡ-Edge-identityʳ (idx₁ , idx₂) p
+        rewrite sym (↑ˡ-identityʳ idx₁ p)
+        rewrite sym (↑ˡ-identityʳ idx₂ p) = refl
+
+    cast∈⇒∈subst : ∀ {n m : ℕ} (p : n ≡ m) (q : m ≡ n)
+                   (e : Fin n × Fin n) (es : List (Fin m × Fin m)) →
+                   castᵉ p e ∈ˡ es →
+                   e ∈ˡ subst (λ m → List (Fin m × Fin m)) q es
+    cast∈⇒∈subst refl refl (idx₁ , idx₂) es e∈es
+        rewrite castᶠ-is-id refl idx₁
+        rewrite castᶠ-is-id refl idx₂ = e∈es
+
     _[_] : ∀ (g : Graph) → Graph.Index g → List BasicStmt
     _[_] g idx = lookup (Graph.nodes g) idx
 
@@ -133,17 +167,12 @@ module Graphs where
             ; sg₂≡sg₁+n = +-assoc s₁ n₁ n₂
             ; newNodes = newNodes₁ ++ newNodes₂
             ; nsg₂≡nsg₁++newNodes = ++-assoc (+-assoc s₁ n₁ n₂) ns₁ newNodes₁ newNodes₂
-            ; e∈g₁⇒e∈g₂ = {!!}
+            ; e∈g₁⇒e∈g₂ = λ {e} e∈g₁ →
+                cast∈⇒∈subst (sym (+-assoc s₁ n₁ n₂)) (+-assoc s₁ n₁ n₂) _ _
+                             (subst (λ e' → e' ∈ˡ es₃)
+                                    (↑ˡ-Edge-assoc e (sym (+-assoc s₁ n₁ n₂)))
+                                    (e∈g₂⇒e∈g₃ (e∈g₁⇒e∈g₂ e∈g₁)))
             }
-        where
-            ↑ˡ-assoc : ∀ {s₁ s₂ s₃ n₁ n₂ : ℕ}
-                            (p : s₂ +ⁿ n₂ ≡ s₃) (q : s₁ +ⁿ n₁ ≡ s₂)
-                            (r : s₁ +ⁿ (n₁ +ⁿ n₂) ≡ s₃)
-                            (idx : Fin s₁) →
-                            castᶠ p ((castᶠ q (idx ↑ˡ n₁)) ↑ˡ n₂) ≡ castᶠ r (idx ↑ˡ (n₁ +ⁿ n₂))
-            ↑ˡ-assoc refl refl r zero = refl
-            ↑ˡ-assoc {(suc s₁)} {s₂} {s₃} {n₁} {n₂} refl refl r (suc idx')
-                rewrite ↑ˡ-assoc refl refl (sym (+-assoc s₁ n₁ n₂)) idx' = refl
 
     record Relaxable (T : Graph → Set) : Set where
         field relax : ∀ {g₁ g₂ : Graph} → g₁ ⊆ g₂ → T g₁ → T g₂
@@ -238,15 +267,11 @@ module Graphs where
                 ; sg₂≡sg₁+n = +-comm 0 s
                 ; newNodes = []
                 ; nsg₂≡nsg₁++newNodes = cast-sym _ (++-identityʳ (+-comm s 0) ns)
-                ; e∈g₁⇒e∈g₂ = {!!}
+                ; e∈g₁⇒e∈g₂ = λ {e} e∈es →
+                    cast∈⇒∈subst (+-comm s 0) (+-comm 0 s) _ _
+                                 (subst (λ e' → e' ∈ˡ _) (↑ˡ-Edge-identityʳ e (+-comm s 0)) (∈ˡ-++⁺ʳ es' e∈es))
                 }
             )
-            where
-                ↑ˡ-identityʳ : ∀ {s} (p : s +ⁿ 0 ≡ s) (idx : Fin s) →
-                           idx ≡ castᶠ p (idx ↑ˡ 0)
-                ↑ˡ-identityʳ p zero = refl
-                ↑ˡ-identityʳ {suc s'} p (suc f')
-                    rewrite sym (↑ˡ-identityʳ (+-comm s' 0) f') = refl
 
         pushEmptyBlock : MonotonicGraphFunction Graph.Index
         pushEmptyBlock = pushBasicBlock []