Add a proof about filter's distributivity

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

Utils.agda

@@ -3,15 +3,16 @@ 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; []; _∷_; _++_; foldr) renaming (map to mapˡ)
+open import Data.List using (List; cartesianProduct; []; _∷_; _++_; foldr; filter) 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 Data.Sum using (_⊎_)
 open import Function.Definitions using (Injective)
-open import Relation.Binary.PropositionalEquality using (_≡_; sym; refl)
-open import Relation.Nullary using (¬_)
+open import Relation.Binary.PropositionalEquality using (_≡_; sym; refl; cong)
+open import Relation.Nullary using (¬_; yes; no)
+open import Relation.Unary using (Decidable)
 
 data Unique {c} {C : Set c} : List C → Set c where
     empty : Unique []
@@ -83,6 +84,14 @@ concat-∈ : ∀ {a} {A : Set a} {x : A} {l : List A} {ls : List (List A)} →
 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')
 
+filter-++ : ∀ {a p} {A : Set a} (l₁ l₂ : List A) {P : A → Set p} (P? : Decidable P) →
+            filter P? (l₁ ++ l₂) ≡ filter P? l₁ ++ filter P? l₂
+filter-++ [] l₂ P? = refl
+filter-++ (x ∷ xs) l₂ P?
+    with P? x
+...   | yes _ = cong (x ∷_) (filter-++ xs l₂ P?)
+...   | no _ = (filter-++ xs l₂ P?)
+
 _⇒_ : ∀ {a p₁ p₂} {A : Set a} (P : A → Set p₁) (Q : A → Set p₂) →
       Set (a ⊔ℓ p₁ ⊔ℓ p₂)
 _⇒_ P Q = ∀ a → P a → Q a