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Add proofs of uniqueness preservation for map insert

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Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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Danila Fedorin
2023-07-24 22:51:27 -07:00
parent c50195942d
commit 87a476ab9e
1 changed files with 37 additions and 3 deletions
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Map.agda
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@@ -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))