Use different graph operations to implement construction

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

Utils.agda

@@ -1,9 +1,11 @@
 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; []; _∷_; _++_) renaming (map to mapˡ)
+open import Data.List using (List; cartesianProduct; []; _∷_; _++_) 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)
 open import Data.List.Relation.Unary.Any using (Any; here; there) -- TODO: re-export these with nicer names from map
 open import Function.Definitions using (Injective)
@@ -78,3 +80,9 @@ proj₁ (v , _) = v
 
 proj₂ : ∀ {a p q} {A : Set a} {P : A → Set p} {Q : A → Set q} {a : A} → (P ⊗ Q) a → Q a
 proj₂ (_ , v) = v
+
+∈-cartesianProduct : ∀ {a b} {A : Set a} {B : Set b}
+                     {x : A} {xs : List A} {y : B} {ys : List B} →
+                     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)