Add a generic Map module and prove its induced equivalence relation

Map.agda

@@ -0,0 +1,42 @@
+open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl)
+open import Relation.Binary.Definitions using (Decidable)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Nullary using (Dec; yes; no)
+open import Agda.Primitive using (Level; _⊔_)
+
+module Map {a b : Level} (A : Set a) (B : Set b)
+    (≡-dec-A : Decidable (_≡_ {a} {A}))
+    where
+
+open import Data.Nat using (ℕ)
+open import Data.String using (String; _++_)
+open import Data.List using (List; []; _∷_)
+open import Data.List.Membership.Propositional using ()
+open import Data.Product using (_×_; _,_; Σ)
+open import Data.Unit using (⊤)
+open import Data.Empty using (⊥)
+
+Map : Set (a ⊔ b)
+Map = List (A × B)
+
+insert : (B → B → B) → A → B → Map → Map
+insert f k v [] = (k , v) ∷ []
+insert f k v (x@(k' , v') ∷ xs) with ≡-dec-A k k'
+...                             | yes _ = (k , f v v') ∷ xs
+...                             | no _ = x ∷ insert f k v xs
+
+foldr : ∀ {c} {C : Set c} → (A → B → C → C) -> C -> Map -> C
+foldr f b [] = b
+foldr f b ((k , v) ∷ xs) = f k v (foldr f b xs)
+
+_∈_ : (A × B) → Map → Set (a ⊔ b)
+_∈_ p m = Data.List.Membership.Propositional._∈_ p m
+
+subset : ∀ (_≈_ : B → B → Set b) → Map → Map → Set (a ⊔ b)
+subset _≈_ m₁ m₂ = ∀ (k : A) (v : B) → (k , v) ∈ m₁ → Σ B (λ v' → v ≈ v' × ((k , v') ∈ m₂))
+
+lift : ∀ (_≈_ : B → B → Set b) → Map → Map → Set (a ⊔ b)
+lift _≈_ m₁ m₂ = (m₁ ⊆ m₂) × (m₂ ⊆ m₁)
+    where
+        _⊆_ : Map → Map → Set (a ⊔ b)
+        _⊆_ = subset _≈_