Add more properties of uniqueness

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

Utils.agda

@@ -2,10 +2,11 @@ module Utils where
 
 open import Agda.Primitive using () renaming (_⊔_ to _⊔ℓ_)
 open import Data.Nat using (ℕ; suc)
-open import Data.List using (List; []; _∷_; _++_)
+open import Data.List using (List; []; _∷_; _++_) renaming (map to mapˡ)
 open import Data.List.Membership.Propositional 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)
 open import Relation.Binary.PropositionalEquality using (_≡_; sym; refl)
 open import Relation.Nullary using (¬_)
 
@@ -29,6 +30,18 @@ Unique-append {c} {C} {x} {x' ∷ xs'} x∉xs (push x'≢ uxs') =
         help {[]} _ = x'≢x ∷ []
         help {e ∷ es} (x'≢e ∷ x'≢es) = x'≢e ∷ help x'≢es
 
+All-≢-map : ∀ {c d} {C : Set c} {D : Set d} (x : C) {xs : List C} (f : C → D) →
+            Injective (_≡_ {_} {C}) (_≡_ {_} {D}) f →
+            All (λ x' → ¬ x ≡ x') xs → All (λ y' → ¬ (f x) ≡ y') (mapˡ f xs)
+All-≢-map x f f-Injecitve [] = []
+All-≢-map x {x' ∷ xs'} f f-Injecitve (x≢x' ∷ x≢xs') = (λ fx≡fx' → x≢x' (f-Injecitve fx≡fx')) ∷ All-≢-map x f f-Injecitve x≢xs'
+
+Unique-map : ∀ {c d} {C : Set c} {D : Set d} {l : List C} (f : C → D) →
+             Injective (_≡_ {_} {C}) (_≡_ {_} {D}) f →
+             Unique l → Unique (mapˡ f l)
+Unique-map {l = []} _ _ _ = empty
+Unique-map {l = x ∷ xs} f f-Injecitve (push x≢xs uxs) = push (All-≢-map x f f-Injecitve x≢xs) (Unique-map f f-Injecitve uxs)
+
 All¬-¬Any : ∀ {p c} {C : Set c} {P : C → Set p} {l : List C} → All (λ x → ¬ P x) l → ¬ Any P l
 All¬-¬Any {l = x ∷ xs} (¬Px ∷ _) (here Px) = ¬Px Px
 All¬-¬Any {l = x ∷ xs} (_ ∷ ¬Pxs) (there Pxs) = All¬-¬Any ¬Pxs Pxs