Prove that maps are functional assuming uniqueness

Lattice.agda

@@ -76,7 +76,6 @@ module IsEquivalenceInstances where
 
         open import Map A B ≡-dec-A using (Map; lift; subset; insert)
         open import Data.List using (_∷_; []) -- TODO: re-export these with nicer names from map
-        open import Data.List.Relation.Unary.Any using (Any; here; there) -- TODO: re-export these with nicer names from map
 
         open IsEquivalence eB renaming
             ( ≈-refl to ≈₂-refl

Map.agda

@@ -1,4 +1,4 @@
-open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; sym; cong)
 open import Relation.Binary.Definitions using (Decidable)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Nullary using (Dec; yes; no)
@@ -8,26 +8,25 @@ module Map {a b : Level} (A : Set a) (B : Set b)
     (≡-dec-A : Decidable (_≡_ {a} {A}))
     where
 
+open import Relation.Nullary using (¬_)
 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.List.Relation.Unary.All using (All; _∷_)
+open import Data.List.Relation.Unary.Any using (Any; here; there) -- TODO: re-export these with nicer names from map
+open import Data.Product using (_×_; _,_; Σ; proj₁ ; proj₂)
 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
+record ⊤' : Set (a ⊔ b) where
-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
+Unique : List (A × B) → Set (a ⊔ b)
-foldr f b [] = b
+Unique [] = ⊤'
-foldr f b ((k , v) ∷ xs) = f k v (foldr f b xs)
+Unique ((k , _) ∷ xs) = All (λ (k' , _) → ¬ k ≡ k') xs × Unique xs
 
 _∈_ : (A × B) → Map → Set (a ⊔ b)
 _∈_ p m = Data.List.Membership.Propositional._∈_ p m
@@ -40,3 +39,27 @@ lift _≈_ m₁ m₂ = (m₁ ⊆ m₂) × (m₂ ⊆ m₁)
     where
         _⊆_ : Map → Map → Set (a ⊔ b)
         _⊆_ = subset _≈_
+
+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)
+
+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
+
+merge : (B → B → B) → Map → Map → Map
+merge f m₁ m₂ = foldr (insert f) m₂ m₁
+
+Map-functional : ∀ (k : A) (v v' : B) (xs : List (A × B)) → Unique ((k , v) ∷ xs) → Data.List.Membership.Propositional._∈_ (k , v') ((k , v) ∷ xs) → v ≡ v'
+Map-functional k v v' _ _ (here k,v'≡k,v) = sym (cong proj₂ k,v'≡k,v)
+Map-functional k v v' xs (k≢ , _) (there k,v'∈xs) = absurd (unique-not-in xs v' (k≢ , k,v'∈xs))
+    where
+        absurd : ∀ {a} {A : Set a} →  ⊥ → A
+        absurd ()
+
+        unique-not-in : ∀ (xs : List (A × B)) (v' : B) → ¬ (All (λ (k' , _) → ¬ k ≡ k') xs × (k , v') ∈ xs)
+        unique-not-in ((k' , _) ∷ xs) v' (k≢k' ∷ _ , here k',≡x) = k≢k' (cong proj₁ k',≡x)
+        unique-not-in (_ ∷ xs) v' (_ ∷ rest , there k,v'∈xs) = unique-not-in xs v' (rest , k,v'∈xs)