Add proofs of uniqueness preservation for map insert

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

Map.agda

@@ -10,10 +10,9 @@ module Map {a b : Level} (A : Set a) (B : Set b)
 
 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 using (List; []; _∷_)
 open import Data.List.Membership.Propositional using ()
-open import Data.List.Relation.Unary.All using (All; _∷_)
+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 (⊤)
@@ -33,6 +32,18 @@ data Unique {c} {C : Set c} : List C → Set c where
         → Unique xs
         → Unique (x ∷ xs)
 
+Unique-append : ∀ {c} {C : Set c} {x : C} {xs : List C} → ¬ Data.List.Membership.Propositional._∈_ x xs → Unique xs →  Unique (xs ++ (x ∷ []))
+Unique-append {c} {C} {x} {[]} _ _ = push [] empty
+Unique-append {c} {C} {x} {x' ∷ xs'} x∉xs (push x'≢ uxs') = push (help x'≢) (Unique-append (λ x∈xs' → x∉xs (there x∈xs')) uxs')
+    where
+        x'≢x : ¬ x' ≡ x
+        x'≢x x'≡x = x∉xs (here (sym x'≡x))
+
+        help : {l : List C} → All (λ x'' → ¬ x' ≡ x'') l → All (λ x'' → ¬ x' ≡ x'') (l ++ (x ∷ []))
+        help {[]} _ = x'≢x ∷ []
+        help {e ∷ es} (x'≢e ∷ x'≢es) = x'≢e ∷ help x'≢es
+
+
 _∈_ : (A × B) → List (A × B) → Set (a ⊔ b)
 _∈_ p m = Data.List.Membership.Propositional._∈_ p m
 
@@ -72,6 +83,29 @@ insert-keys-∈ f k v ((k' , _) ∷ xs) (there k∈kxs) with (≡-dec-A k k')
 ...                                                | yes _ = refl
 ...                                                | no _ = cong (λ xs' → k' ∷ xs') (insert-keys-∈ f k v xs k∈kxs)
 
+insert-keys-∉ : ∀ (f : B → B → B) (k : A) (v : B) (l : List (A × B)) → ¬ (k ∈k l) → (keys l ++ (k ∷ [])) ≡ keys (insert f k v l)
+insert-keys-∉ f k v [] _ = refl
+insert-keys-∉ f k v ((k' , v') ∷ xs) k∉kl with (≡-dec-A k k')
+...                                         | yes k≡k' = absurd (k∉kl (here k≡k'))
+...                                         | no _ = cong (λ xs' → k' ∷ xs') (insert-keys-∉ f k v xs (λ k∈kxs → k∉kl (there k∈kxs)))
+
+∈k-dec : ∀ (k : A) (l : List (A × B)) → Dec (k ∈k l)
+∈k-dec k [] = no (λ ())
+∈k-dec k ((k' , v) ∷ xs) with (≡-dec-A k k')
+...                        | yes k≡k' = yes (here k≡k')
+...                        | no k≢k' with (∈k-dec k xs)
+...                                    | yes k∈kxs = yes (there k∈kxs)
+...                                    | no k∉kxs = no witness
+                                           where
+                                               witness : ¬ k ∈k ((k' , v) ∷ xs)
+                                               witness (here k≡k') = k≢k' k≡k'
+                                               witness (there k∈kxs) = k∉kxs k∈kxs
+
+insert-preserves-unique : ∀ (f : B → B → B) (k : A) (v : B) (l : List (A × B)) → Unique (keys l) → Unique (keys (insert f k v l))
+insert-preserves-unique f k v l u with (∈k-dec k l)
+...                               | yes k∈kl rewrite insert-keys-∈ f k v l k∈kl = u
+...                               | no k∉kl rewrite sym (insert-keys-∉ f k v l k∉kl) = Unique-append k∉kl u
+
 Map-functional : ∀ (k : A) (v v' : B) (xs : List (A × B)) → Unique (keys ((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 (push k≢ _) (there k,v'∈xs) = absurd (unique-not-in xs v' (k≢ , k,v'∈xs))