diff --git a/Language.agda b/Language.agda
index d2af0dc..2b078d9 100644
--- a/Language.agda
+++ b/Language.agda
@@ -14,7 +14,7 @@ open import Data.List.Membership.Propositional as MemProp using () renaming (_
 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_)
+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 Relation.Binary.PropositionalEquality as Eq using (subst; cong; _≡_; sym; trans; refl)
 open import Relation.Nullary using (¬_)
@@ -121,6 +121,13 @@ module Graphs where
                         e ∈ˡ (Graph.edges g₁) →
                         (↑ˡ-Edge e n) ∈ˡ (subst (λ m → List (Fin m × Fin m)) sg₂≡sg₁+n (Graph.edges g₂))
 
+    flatten-casts : ∀ {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))
+    flatten-casts refl refl r zero = refl
+    flatten-casts {(suc s₁)} {s₂} {s₃} {n₁} {n₂} refl refl r (suc idx')
+        rewrite flatten-casts refl refl (sym (+-assoc s₁ n₁ n₂)) idx' = refl
 
     ⊆-trans : ∀ {g₁ g₂ g₃ : Graph} → g₁ ⊆ g₂ → g₂ ⊆ g₃ → g₁ ⊆ g₃
     ⊆-trans {MkGraph s₁ ns₁ es₁} {MkGraph s₂ ns₂ es₂} {MkGraph s₃ ns₃ es₃}
@@ -139,7 +146,9 @@ module Graphs where
 
         --     lookup (cast p₂ ns₃) ((Fin.cast (sym p₁) (idx ↑ˡ n₁) ↑ˡ  n₂)) ≡ lookup ns₃ (Fin.cast (sym p₂) ((Fin.cast (sym p₁) (idx ↑ˡ n₁) ↑ˡ  n₂))) -- by lookup-cast₂
 
-        --     -- ??
+        --     lookup ns₃ (Fin.cast (sym p₂) ((Fin.cast (sym p₁) (idx ↑ˡ n₁) ↑ˡ  n₂))) ≡ lookup ns₃ (Fin.cast (sym (+-assoc s₁ n₁ n₂)) (idx ↑ˡ (n₁ + n₂))) -- by flatten-casts
+        --
+        --     lookup ns₃ (Fin.cast (sym (+-assoc s₁ n₁ n₂)) (idx ↑ˡ (n₁ + n₂))) ≡ lookup (cast (+-assoc s₁ n₁ n₂) ns₃) (idx ↑ˡ (n₁ + n₂)) ∎
         }
 
     record Relaxable (T : Graph → Set) : Set where