Danila Fedorin
4033a1b33d
The "no" case still needs to be dismissed, but that can probably be done by some lemma about keys in maps. Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
343 lines
19 KiB
Agda
343 lines
19 KiB
Agda
open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; sym; trans; cong)
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open import Relation.Binary.Definitions using (Decidable)
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open import Relation.Binary.Core using (Rel)
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open import Relation.Nullary using (Dec; yes; no; Reflects; ofʸ; ofⁿ)
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open import Agda.Primitive using (Level; _⊔_)
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module Map {a b : Level} (A : Set a) (B : Set b)
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(≡-dec-A : Decidable (_≡_ {a} {A}))
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where
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import Data.List.Membership.Propositional as MemProp
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open import Relation.Nullary using (¬_)
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open import Data.Nat using (ℕ)
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open import Data.List using (List; map; []; _∷_; _++_)
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open import Data.List.Relation.Unary.All using (All; []; _∷_)
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open import Data.List.Relation.Unary.Any using (Any; here; there) -- TODO: re-export these with nicer names from map
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open import Data.Product using (_×_; _,_; Σ; proj₁ ; proj₂)
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open import Data.Empty using (⊥)
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keys : List (A × B) → List A
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keys = map proj₁
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data Unique {c} {C : Set c} : List C → Set c where
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empty : Unique []
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push : forall {x : C} {xs : List C}
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→ All (λ x' → ¬ x ≡ x') xs
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→ Unique xs
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→ Unique (x ∷ xs)
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Unique-append : ∀ {c} {C : Set c} {x : C} {xs : List C} →
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¬ MemProp._∈_ x xs → Unique xs → Unique (xs ++ (x ∷ []))
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Unique-append {c} {C} {x} {[]} _ _ = push [] empty
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Unique-append {c} {C} {x} {x' ∷ xs'} x∉xs (push x'≢ uxs') =
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push (help x'≢) (Unique-append (λ x∈xs' → x∉xs (there x∈xs')) uxs')
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where
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x'≢x : ¬ x' ≡ x
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x'≢x x'≡x = x∉xs (here (sym x'≡x))
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help : {l : List C} → All (λ x'' → ¬ x' ≡ x'') l → All (λ x'' → ¬ x' ≡ x'') (l ++ (x ∷ []))
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help {[]} _ = x'≢x ∷ []
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help {e ∷ es} (x'≢e ∷ x'≢es) = x'≢e ∷ help x'≢es
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All¬-¬Any : ∀ {p c} {C : Set c} {P : C → Set p} {l : List C} → All (λ x → ¬ P x) l → ¬ Any P l
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All¬-¬Any {l = x ∷ xs} (¬Px ∷ _) (here Px) = ¬Px Px
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All¬-¬Any {l = x ∷ xs} (_ ∷ ¬Pxs) (there Pxs) = All¬-¬Any ¬Pxs Pxs
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absurd : ∀ {a} {A : Set a} → ⊥ → A
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absurd ()
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private module _ where
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open MemProp using (_∈_)
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unique-not-in : ∀ {k : A} {v : B} {l : List (A × B)} →
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¬ (All (λ k' → ¬ k ≡ k') (keys l) × (k , v) ∈ l)
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unique-not-in {l = (k' , _) ∷ xs} (k≢k' ∷ _ , here k',≡x) =
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k≢k' (cong proj₁ k',≡x)
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unique-not-in {l = _ ∷ xs} (_ ∷ rest , there k,v'∈xs) =
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unique-not-in (rest , k,v'∈xs)
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ListAB-functional : ∀ {k : A} {v v' : B} {l : List (A × B)} →
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Unique (keys l) → (k , v) ∈ l → (k , v') ∈ l → v ≡ v'
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ListAB-functional _ (here k,v≡x) (here k,v'≡x) =
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cong proj₂ (trans k,v≡x (sym k,v'≡x))
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ListAB-functional (push k≢xs _) (here k,v≡x) (there k,v'∈xs)
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rewrite sym k,v≡x = absurd (unique-not-in (k≢xs , k,v'∈xs))
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ListAB-functional (push k≢xs _) (there k,v∈xs) (here k,v'≡x)
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rewrite sym k,v'≡x = absurd (unique-not-in (k≢xs , k,v∈xs))
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ListAB-functional {l = _ ∷ xs } (push _ uxs) (there k,v∈xs) (there k,v'∈xs) =
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ListAB-functional uxs k,v∈xs k,v'∈xs
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∈k-dec : ∀ (k : A) (l : List (A × B)) → Dec (k ∈ keys l)
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∈k-dec k [] = no (λ ())
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∈k-dec k ((k' , v) ∷ xs)
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with (≡-dec-A k k')
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... | yes k≡k' = yes (here k≡k')
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... | no k≢k' with (∈k-dec k xs)
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... | yes k∈kxs = yes (there k∈kxs)
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... | no k∉kxs = no witness
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where
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witness : ¬ k ∈ keys ((k' , v) ∷ xs)
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witness (here k≡k') = k≢k' k≡k'
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witness (there k∈kxs) = k∉kxs k∈kxs
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∈-cong : ∀ {c d} {C : Set c} {D : Set d} {c : C} {l : List C} →
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(f : C → D) → c ∈ l → f c ∈ map f l
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∈-cong f (here c≡c') = here (cong f c≡c')
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∈-cong f (there c∈xs) = there (∈-cong f c∈xs)
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locate : ∀ {k : A} {l : List (A × B)} → k ∈ keys l → Σ B (λ v → (k , v) ∈ l)
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locate {k} {(k' , v) ∷ xs} (here k≡k') rewrite k≡k' = (v , here refl)
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locate {k} {(k' , v) ∷ xs} (there k∈kxs) = let (v , k,v∈xs) = locate k∈kxs in (v , there k,v∈xs)
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private module ImplRelation (_≈_ : B → B → Set b) where
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open MemProp using (_∈_)
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subset : List (A × B) → List (A × B) → Set (a ⊔ b)
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subset m₁ m₂ = ∀ (k : A) (v : B) → (k , v) ∈ m₁ →
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Σ B (λ v' → v ≈ v' × ((k , v') ∈ m₂))
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private module ImplInsert (f : B → B → B) where
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open import Data.List using (map)
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open MemProp using (_∈_)
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private
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_∈k_ : A → List (A × B) → Set a
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_∈k_ k m = k ∈ (keys m)
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foldr : ∀ {c} {C : Set c} → (A → B → C → C) -> C -> List (A × B) -> C
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foldr f b [] = b
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foldr f b ((k , v) ∷ xs) = f k v (foldr f b xs)
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insert : A → B → List (A × B) → List (A × B)
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insert k v [] = (k , v) ∷ []
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insert k v (x@(k' , v') ∷ xs) with ≡-dec-A k k'
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... | yes _ = (k' , f v v') ∷ xs
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... | no _ = x ∷ insert k v xs
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merge : List (A × B) → List (A × B) → List (A × B)
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merge m₁ m₂ = foldr insert m₂ m₁
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insert-keys-∈ : ∀ (k : A) (v : B) (l : List (A × B)) →
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k ∈k l → keys l ≡ keys (insert k v l)
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insert-keys-∈ k v ((k' , v') ∷ xs) (here k≡k')
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with (≡-dec-A k k')
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... | yes _ = refl
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... | no k≢k' = absurd (k≢k' k≡k')
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insert-keys-∈ k v ((k' , _) ∷ xs) (there k∈kxs)
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with (≡-dec-A k k')
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... | yes _ = refl
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... | no _ = cong (λ xs' → k' ∷ xs') (insert-keys-∈ k v xs k∈kxs)
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insert-keys-∉ : ∀ (k : A) (v : B) (l : List (A × B)) →
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¬ (k ∈k l) → (keys l ++ (k ∷ [])) ≡ keys (insert k v l)
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insert-keys-∉ k v [] _ = refl
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insert-keys-∉ k v ((k' , v') ∷ xs) k∉kl
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with (≡-dec-A k k')
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... | yes k≡k' = absurd (k∉kl (here k≡k'))
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... | no _ = cong (λ xs' → k' ∷ xs')
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(insert-keys-∉ k v xs (λ k∈kxs → k∉kl (there k∈kxs)))
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insert-preserves-Unique : ∀ (k : A) (v : B) (l : List (A × B))
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→ Unique (keys l) → Unique (keys (insert k v l))
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insert-preserves-Unique k v l u
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with (∈k-dec k l)
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... | yes k∈kl rewrite insert-keys-∈ k v l k∈kl = u
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... | no k∉kl rewrite sym (insert-keys-∉ k v l k∉kl) = Unique-append k∉kl u
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merge-preserves-Unique : ∀ (l₁ l₂ : List (A × B)) →
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Unique (keys l₂) → Unique (keys (merge l₁ l₂))
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merge-preserves-Unique [] l₂ u₂ = u₂
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merge-preserves-Unique ((k₁ , v₁) ∷ xs₁) l₂ u₂ =
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insert-preserves-Unique k₁ v₁ (merge xs₁ l₂)
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(merge-preserves-Unique xs₁ l₂ u₂)
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insert-preserves-∈-right : ∀ (k k' : A) (v v' : B) (l : List (A × B)) →
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¬ k ≡ k' → (k , v) ∈ l → (k , v) ∈ insert k' v' l
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insert-preserves-∈-right k k' v v' (x ∷ xs) k≢k' (here k,v=x)
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rewrite sym k,v=x with ≡-dec-A k' k
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... | yes k'≡k = absurd (k≢k' (sym k'≡k))
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... | no _ = here refl
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insert-preserves-∈-right k k' v v' ((k'' , _) ∷ xs) k≢k' (there k,v∈xs)
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with ≡-dec-A k' k''
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... | yes _ = there k,v∈xs
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... | no _ = there (insert-preserves-∈-right k k' v v' xs k≢k' k,v∈xs)
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insert-preserves-∈k-right : ∀ (k k' : A) (v' : B) (l : List (A × B)) →
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¬ k ≡ k' → k ∈k l → k ∈k insert k' v' l
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insert-preserves-∈k-right k k' v' l k≢k' k∈kl =
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let (v , k,v∈l) = locate k∈kl
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in ∈-cong proj₁ (insert-preserves-∈-right k k' v v' l k≢k' k,v∈l)
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insert-preserves-∉-right : ∀ (k k' : A) (v' : B) (l : List (A × B)) →
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¬ k ≡ k' → ¬ k ∈k l → ¬ k ∈k insert k' v' l
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insert-preserves-∉-right k k' v' [] k≢k' k∉kl (here k≡k') = k≢k' k≡k'
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insert-preserves-∉-right k k' v' [] k≢k' k∉kl (there ())
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insert-preserves-∉-right k k' v' ((k'' , v'') ∷ xs) k≢k' k∉kl k∈kil
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with ≡-dec-A k k''
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... | yes k≡k'' = k∉kl (here k≡k'')
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... | no k≢k'' with ≡-dec-A k' k'' | k∈kil
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... | yes k'≡k'' | here k≡k'' = k≢k'' k≡k''
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... | yes k'≡k'' | there k∈kxs = k∉kl (there k∈kxs)
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... | no k'≢k'' | here k≡k'' = k∉kl (here k≡k'')
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... | no k'≢k'' | there k∈kxs = insert-preserves-∉-right k k' v' xs k≢k'
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(λ k∈kxs → k∉kl (there k∈kxs)) k∈kxs
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merge-preserves-∉ : ∀ (k : A) (l₁ l₂ : List (A × B)) →
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¬ k ∈k l₁ → ¬ k ∈k l₂ → ¬ k ∈k merge l₁ l₂
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merge-preserves-∉ k [] l₂ _ k∉kl₂ = k∉kl₂
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merge-preserves-∉ k ((k' , v') ∷ xs₁) l₂ k∉kl₁ k∉kl₂
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with ≡-dec-A k k'
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... | yes k≡k' = absurd (k∉kl₁ (here k≡k'))
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... | no k≢k' = insert-preserves-∉-right k k' v' _ k≢k' (merge-preserves-∉ k xs₁ l₂ (λ k∈kxs₁ → k∉kl₁ (there k∈kxs₁)) k∉kl₂)
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merge-preserves-keys₁ : ∀ (k : A) (v : B) (l₁ l₂ : List (A × B)) →
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¬ k ∈k l₁ → (k , v) ∈ l₂ → (k , v) ∈ merge l₁ l₂
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merge-preserves-keys₁ k v [] l₂ _ k,v∈l₂ = k,v∈l₂
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merge-preserves-keys₁ k v ((k' , v') ∷ xs₁) l₂ k∉kl₁ k,v∈l₂ =
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let recursion = merge-preserves-keys₁ k v xs₁ l₂ (λ k∈xs₁ → k∉kl₁ (there k∈xs₁)) k,v∈l₂
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in insert-preserves-∈-right k k' v v' _ (λ k≡k' → k∉kl₁ (here k≡k')) recursion
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insert-fresh : ∀ (k : A) (v : B) (l : List (A × B)) →
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¬ k ∈k l → (k , v) ∈ insert k v l
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insert-fresh k v [] k∉kl = here refl
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insert-fresh k v ((k' , v') ∷ xs) k∉kl
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with ≡-dec-A k k'
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... | yes k≡k' = absurd (k∉kl (here k≡k'))
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... | no _ = there (insert-fresh k v xs (λ k∈kxs → k∉kl (there k∈kxs)))
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merge-preserves-keys₂ : ∀ (k : A) (v : B) (l₁ l₂ : List (A × B)) →
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Unique (keys l₁) → (k , v) ∈ l₁ → ¬ k ∈k l₂ → (k , v) ∈ merge l₁ l₂
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merge-preserves-keys₂ k v ((k' , v') ∷ xs₁) l₂ (push k'≢xs₁ uxs₁) (there k,v∈xs₁) k∉kl₂ =
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insert-preserves-∈-right k k' v v' (merge xs₁ l₂) k≢k' k,v∈mxs₁l
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where
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k,v∈mxs₁l = merge-preserves-keys₂ k v xs₁ l₂ uxs₁ k,v∈xs₁ k∉kl₂
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k≢k' : ¬ k ≡ k'
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k≢k' with ≡-dec-A k k'
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... | yes k≡k' rewrite k≡k' = absurd (All¬-¬Any k'≢xs₁ (∈-cong proj₁ k,v∈xs₁))
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... | no k≢k' = k≢k'
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merge-preserves-keys₂ k v ((k' , v') ∷ xs₁) l₂ (push k'≢xs₁ uxs₁) (here k,v≡k',v') k∉kl₂
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rewrite cong proj₁ k,v≡k',v' rewrite cong proj₂ k,v≡k',v' =
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insert-fresh k' v' _ (merge-preserves-∉ k' xs₁ l₂ (All¬-¬Any k'≢xs₁) k∉kl₂)
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insert-combines : ∀ (k : A) (v v' : B) (l : List (A × B)) →
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Unique (keys l) → (k , v') ∈ l → (k , f v v') ∈ (insert k v l)
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insert-combines k v v' ((k' , v'') ∷ xs) _ (here k,v'≡k',v'')
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rewrite cong proj₁ k,v'≡k',v'' rewrite cong proj₂ k,v'≡k',v''
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with ≡-dec-A k' k'
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... | yes _ = here refl
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... | no k≢k' = absurd (k≢k' refl)
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insert-combines k v v' ((k' , v'') ∷ xs) (push k'≢xs uxs) (there k,v'∈xs)
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with ≡-dec-A k k'
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... | yes k≡k' rewrite k≡k' = absurd (All¬-¬Any k'≢xs (∈-cong proj₁ k,v'∈xs))
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... | no k≢k' = there (insert-combines k v v' xs uxs k,v'∈xs)
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merge-combines : forall (k : A) (v₁ v₂ : B) (l₁ l₂ : List (A × B)) →
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Unique (keys l₁) → Unique (keys l₂) →
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(k , v₁) ∈ l₁ → (k , v₂) ∈ l₂ → (k , f v₁ v₂) ∈ merge l₁ l₂
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merge-combines k v₁ v₂ ((k' , v) ∷ xs₁) l₂ (push k'≢xs₁ uxs₁) ul₂ (here k,v₁≡k',v) k,v₂∈l₂
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rewrite cong proj₁ (sym (k,v₁≡k',v)) rewrite cong proj₂ (sym (k,v₁≡k',v)) =
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insert-combines k v₁ v₂ _ (merge-preserves-Unique xs₁ l₂ ul₂) (merge-preserves-keys₁ k v₂ xs₁ l₂ (All¬-¬Any k'≢xs₁) k,v₂∈l₂)
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merge-combines k v₁ v₂ ((k' , v) ∷ xs₁) l₂ (push k'≢xs₁ uxs₁) ul₂ (there k,v₁∈xs₁) k,v₂∈l₂ =
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insert-preserves-∈-right k k' (f v₁ v₂) v _ k≢k' (merge-combines k v₁ v₂ xs₁ l₂ uxs₁ ul₂ k,v₁∈xs₁ k,v₂∈l₂)
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where
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k≢k' : ¬ k ≡ k'
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k≢k' with ≡-dec-A k k'
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... | yes k≡k' rewrite k≡k' = absurd (All¬-¬Any k'≢xs₁ (∈-cong proj₁ k,v₁∈xs₁))
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... | no k≢k' = k≢k'
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Map : Set (a ⊔ b)
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Map = Σ (List (A × B)) (λ l → Unique (keys l))
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_∈_ : (A × B) → Map → Set (a ⊔ b)
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_∈_ p (kvs , _) = MemProp._∈_ p kvs
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_∈k_ : A → Map → Set a
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_∈k_ k (kvs , _) = MemProp._∈_ k (keys kvs)
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Map-functional : ∀ {k : A} {v v' : B} {m : Map} → (k , v) ∈ m → (k , v') ∈ m → v ≡ v'
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Map-functional {m = (l , ul)} k,v∈m k,v'∈m = ListAB-functional ul k,v∈m k,v'∈m
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data Provenance (k : A) (m₁ m₂ : Map) : Set (a ⊔ b) where
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both : (v₁ v₂ : B) → (k , v₁) ∈ m₁ → (k , v₂) ∈ m₂ → Provenance k m₁ m₂
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in₁ : (v₁ : B) → (k , v₁) ∈ m₁ → ¬ k ∈k m₂ → Provenance k m₁ m₂
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in₂ : (v₂ : B) → ¬ k ∈k m₁ → (k , v₂) ∈ m₂ → Provenance k m₁ m₂
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module _ (f : B → B → B) where
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open ImplInsert f renaming
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( insert to insert-impl
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; merge to merge-impl
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)
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insert : A → B → Map → Map
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insert k v (kvs , uks) = (insert-impl k v kvs , insert-preserves-Unique k v kvs uks)
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merge : Map → Map → Map
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merge (kvs₁ , _) (kvs₂ , uks₂) = (merge-impl kvs₁ kvs₂ , merge-preserves-Unique kvs₁ kvs₂ uks₂)
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MergeResult : {k : A} {m₁ m₂ : Map} → Provenance k m₁ m₂ → Set (a ⊔ b)
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MergeResult {k} {m₁} {m₂} (both v₁ v₂ _ _) = (k , f v₁ v₂) ∈ merge m₁ m₂
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MergeResult {k} {m₁} {m₂} (in₁ v₁ _ _) = (k , v₁) ∈ merge m₁ m₂
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MergeResult {k} {m₁} {m₂} (in₂ v₂ _ _) = (k , v₂) ∈ merge m₁ m₂
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merge-provenance : ∀ (m₁ m₂ : Map) (k : A) → k ∈k merge m₁ m₂ → Σ (Provenance k m₁ m₂) MergeResult
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merge-provenance m₁@(l₁ , u₁) m₂@(l₂ , u₂) k k∈km₁m₂
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with ∈k-dec k l₁ | ∈k-dec k l₂
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... | yes k∈kl₁ | yes k∈kl₂ =
|
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let
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(v₁ , k,v₁∈l₁) = locate k∈kl₁
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(v₂ , k,v₂∈l₂) = locate k∈kl₂
|
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in
|
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(both v₁ v₂ k,v₁∈l₁ k,v₂∈l₂ , merge-combines k v₁ v₂ l₁ l₂ u₁ u₂ k,v₁∈l₁ k,v₂∈l₂)
|
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... | yes k∈kl₁ | no k∉kl₂ =
|
||
let
|
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(v₁ , k,v₁∈l₁) = locate k∈kl₁
|
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in
|
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(in₁ v₁ k,v₁∈l₁ k∉kl₂ , merge-preserves-keys₂ k v₁ l₁ l₂ u₁ k,v₁∈l₁ k∉kl₂)
|
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... | no k∉kl₁ | yes k∈kl₂ =
|
||
let
|
||
(v₂ , k,v₂∈l₂) = locate k∈kl₂
|
||
in
|
||
(in₂ v₂ k∉kl₁ k,v₂∈l₂ , merge-preserves-keys₁ k v₂ l₁ l₂ k∉kl₁ k,v₂∈l₂)
|
||
... | no k∉kl₁ | no k∉kl₂ = absurd (merge-preserves-∉ k l₁ l₂ k∉kl₁ k∉kl₂ k∈km₁m₂)
|
||
|
||
module _ (_≈_ : B → B → Set b) where
|
||
open ImplRelation _≈_ renaming (subset to subset-impl)
|
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|
||
subset : Map → Map → Set (a ⊔ b)
|
||
subset (kvs₁ , _) (kvs₂ , _) = subset-impl kvs₁ kvs₂
|
||
|
||
lift : Map → Map → Set (a ⊔ b)
|
||
lift m₁ m₂ = subset m₁ m₂ × subset m₂ m₁
|
||
|
||
module _ (f : B → B → B) where
|
||
module _ (f-comm : ∀ (b₁ b₂ : B) → f b₁ b₂ ≡ f b₂ b₁) where
|
||
merge-comm : forall (m₁ m₂ : Map) → lift (_≡_) (merge f m₁ m₂) (merge f m₂ m₁)
|
||
merge-comm m₁ m₂ = (merge-comm-subset m₁ m₂ , merge-comm-subset m₂ m₁)
|
||
where
|
||
merge-comm-subset : ∀ (m₁ m₂ : Map) → subset (_≡_) (merge f m₁ m₂) (merge f m₂ m₁)
|
||
merge-comm-subset m₁ m₂ k v k,v∈m₁m₂
|
||
with ∈k-dec k (proj₁ (merge f m₂ m₁) )
|
||
... | no k∉km₂m₁ = {!!}
|
||
... | yes k∈km₂m₁ with merge-provenance f m₁ m₂ k (∈-cong proj₁ k,v∈m₁m₂) | merge-provenance f m₂ m₁ k k∈km₂m₁
|
||
... | (both v₁ v₂ v₁∈m₁ v₂∈m₂ , v₁v₂∈m₁m₂) | (both v₂' v₁' v₂'∈m₂ v₁'∈m₁ , v₂'v₁'∈m₂m₁)
|
||
rewrite Map-functional {m = merge f m₁ m₂} k,v∈m₁m₂ v₁v₂∈m₁m₂
|
||
rewrite Map-functional {m = m₁} v₁∈m₁ v₁'∈m₁
|
||
rewrite Map-functional {m = m₂} v₂∈m₂ v₂'∈m₂ = (f v₂' v₁' , (f-comm v₁' v₂' , v₂'v₁'∈m₂m₁))
|
||
... | (in₁ v₁ v₁∈m₁ _ , v₁∈m₁m₂) | (in₂ v₁' _ v₁'∈m₁ , v₁'∈m₂m₁)
|
||
rewrite Map-functional {m = merge f m₁ m₂} k,v∈m₁m₂ v₁∈m₁m₂ =
|
||
(v₁' , (Map-functional {m = m₁} v₁∈m₁ v₁'∈m₁ , v₁'∈m₂m₁))
|
||
... | (in₂ v₂ _ v₂∈m₂ , v₂∈m₁m₂) | (in₁ v₂' v₂'∈m₂ _ , v₂'∈m₂m₁)
|
||
rewrite Map-functional {m = merge f m₁ m₂} k,v∈m₁m₂ v₂∈m₁m₂ =
|
||
(v₂' , (Map-functional {m = m₂} v₂∈m₂ v₂'∈m₂ , v₂'∈m₂m₁))
|
||
-- The rest of the cases are to be dismissed.
|
||
... | (both v₁ v₂ v₁∈m₁ v₂∈m₂ , _) | (in₁ v₂' v₂'∈m₂ k∉km₁ , _) = absurd (k∉km₁ (∈-cong proj₁ v₁∈m₁))
|
||
... | (both v₁ v₂ v₁∈m₁ v₂∈m₂ , _) | (in₂ v₁' k∉km₂ v₁'∈m₁ , _) = absurd (k∉km₂ (∈-cong proj₁ v₂∈m₂))
|
||
... | (in₁ v₁ v₁∈m₁ k∉km₂ , _) | (both v₂' v₁' v₂'∈m₂ v₁'∈m₁ , _) = absurd (k∉km₂ (∈-cong proj₁ v₂'∈m₂))
|
||
... | (in₁ v₁ v₁∈m₁ k∉km₂ , _) | (in₁ v₂' v₂'∈m₂ k∉km₁ , _) = absurd (k∉km₂ (∈-cong proj₁ v₂'∈m₂))
|
||
... | (in₂ v₂ k∉km₁ v₂∈m₂ , v₂∈m₁m₂) | (both v₂' v₁' v₂'∈m₂ v₁'∈m₁ , _) = absurd (k∉km₁ (∈-cong proj₁ v₁'∈m₁))
|
||
... | (in₂ v₂ k∉km₁ v₂∈m₂ , v₂∈m₁m₂) | (in₂ v₁' _ v₁'∈m₁ , v₁'∈m₂m₁) = absurd (k∉km₁ (∈-cong proj₁ v₁'∈m₁))
|