Simplify proofs about 'loop' using concatenation lemma

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

Utils.agda

@@ -3,7 +3,7 @@ module Utils where
 open import Agda.Primitive using () renaming (_⊔_ to _⊔ℓ_)
 open import Data.Product as Prod using ()
 open import Data.Nat using (ℕ; suc)
-open import Data.List using (List; cartesianProduct; []; _∷_; _++_) renaming (map to mapˡ)
+open import Data.List using (List; cartesianProduct; []; _∷_; _++_; foldr) renaming (map to mapˡ)
 open import Data.List.Membership.Propositional using (_∈_)
 open import Data.List.Membership.Propositional.Properties as ListMemProp using ()
 open import Data.List.Relation.Unary.All using (All; []; _∷_; map)
@@ -86,3 +86,8 @@ proj₂ (_ , v) = v
                      x ∈ xs → y ∈ ys → (x Prod., y) ∈ cartesianProduct xs ys
 ∈-cartesianProduct {x = x} (here refl) y∈ys = ListMemProp.∈-++⁺ˡ (x∈xs⇒fx∈fxs (x Prod.,_) y∈ys)
 ∈-cartesianProduct {x = x} {xs = x' ∷ _} {ys = ys} (there x∈rest) y∈ys = ListMemProp.∈-++⁺ʳ (mapˡ (x' Prod.,_) ys) (∈-cartesianProduct x∈rest y∈ys)
+
+concat-∈ : ∀ {a} {A : Set a} {x : A} {l : List A} {ls : List (List A)} →
+           x ∈ l → l ∈ ls → x ∈ foldr _++_ [] ls
+concat-∈ x∈l (here refl) = ListMemProp.∈-++⁺ˡ x∈l
+concat-∈ {ls = l' ∷ ls'} x∈l (there l∈ls') = ListMemProp.∈-++⁺ʳ l' (concat-∈ x∈l l∈ls')