Migrate Maps to including a uniqueness proof
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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Lattice.agda
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Lattice.agda
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@ -101,19 +101,22 @@ module IsEquivalenceInstances where
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in (v'' , (≈₂-trans v≈v' v'≈v'' , k,v''∈m₃))
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≈-refl : {m : Map} → m ≈ m
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≈-refl {m} = (⊆-refl , ⊆-refl)
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≈-refl {m} = (⊆-refl {m}, ⊆-refl {m})
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≈-sym : {m₁ m₂ : Map} → m₁ ≈ m₂ → m₂ ≈ m₁
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≈-sym (m₁⊆m₂ , m₂⊆m₁) = (m₂⊆m₁ , m₁⊆m₂)
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≈-trans : {m₁ m₂ m₃ : Map} → m₁ ≈ m₂ → m₂ ≈ m₃ → m₁ ≈ m₃
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≈-trans (m₁⊆m₂ , m₂⊆m₁) (m₂⊆m₃ , m₃⊆m₂) = (⊆-trans m₁⊆m₂ m₂⊆m₃ , ⊆-trans m₃⊆m₂ m₂⊆m₁)
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≈-trans {m₁} {m₂} {m₃} (m₁⊆m₂ , m₂⊆m₁) (m₂⊆m₃ , m₃⊆m₂) =
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( ⊆-trans {m₁} {m₂} {m₃} m₁⊆m₂ m₂⊆m₃
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, ⊆-trans {m₃} {m₂} {m₁} m₃⊆m₂ m₂⊆m₁
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)
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LiftEquivalence : IsEquivalence Map _≈_
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LiftEquivalence = record
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{ ≈-refl = ≈-refl
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; ≈-sym = ≈-sym
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; ≈-trans = ≈-trans
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{ ≈-refl = λ {m₁} → ≈-refl {m₁}
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; ≈-sym = λ {m₁} {m₂} → ≈-sym {m₁} {m₂}
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; ≈-trans = λ {m₁} {m₂} {m₃} → ≈-trans {m₁} {m₂} {m₃}
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}
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module IsSemilatticeInstances where
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56
Map.agda
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Map.agda
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@ -18,9 +18,6 @@ open import Data.List.Relation.Unary.Any using (Any; here; there) -- TODO: re-ex
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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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Map : Set (a ⊔ b)
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Map = List (A × B)
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keys : List (A × B) → List A
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keys [] = []
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keys ((k , v) ∷ xs) = k ∷ keys xs
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@ -32,6 +29,9 @@ data Unique {c} {C : Set c} : List C → Set c where
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→ Unique xs
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→ Unique (x ∷ xs)
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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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Unique-append : ∀ {c} {C : Set c} {x : C} {xs : List C} → ¬ 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') = push (help x'≢) (Unique-append (λ x∈xs' → x∉xs (there x∈xs')) uxs')
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@ -46,15 +46,6 @@ Unique-append {c} {C} {x} {x' ∷ xs'} x∉xs (push x'≢ uxs') = push (help x'
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_∈_ : (A × B) → List (A × B) → Set (a ⊔ b)
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_∈_ p m = MemProp._∈_ p m
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subset : ∀ (_≈_ : B → B → Set b) → List (A × B) → List (A × B) → Set (a ⊔ b)
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subset _≈_ m₁ m₂ = ∀ (k : A) (v : B) → (k , v) ∈ m₁ → Σ B (λ v' → v ≈ v' × ((k , v') ∈ m₂))
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lift : ∀ (_≈_ : B → B → Set b) → List (A × B) → List (A × B) → Set (a ⊔ b)
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lift _≈_ m₁ m₂ = (m₁ ⊆ m₂) × (m₂ ⊆ m₁)
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where
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_⊆_ : List (A × B) → List (A × B) → Set (a ⊔ b)
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_⊆_ = subset _≈_
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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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@ -62,7 +53,11 @@ foldr f b ((k , v) ∷ xs) = f k v (foldr f b xs)
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absurd : ∀ {a} {A : Set a} → ⊥ → A
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absurd ()
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private module Impl (f : B → B → B) where
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private module ImplRelation (_≈_ : B → B → Set b) where
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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 → 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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_∈k_ : A → List (A × B) → Set a
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_∈k_ k m = MemProp._∈_ k (keys m)
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@ -110,13 +105,32 @@ private module Impl (f : B → B → B) where
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merge-preserves-unique [] l₂ u₂ = u₂
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merge-preserves-unique ((k₁ , v₁) ∷ xs₁) l₂ u₂ = insert-preserves-unique k₁ v₁ (merge xs₁ l₂) (merge-preserves-unique xs₁ l₂ u₂)
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Map-functional : ∀ (k : A) (v v' : B) (xs : List (A × B)) → Unique (keys ((k , v) ∷ xs)) → MemProp._∈_ (k , v') ((k , v) ∷ xs) → v ≡ v'
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Map-functional k v v' _ _ (here k,v'≡k,v) = sym (cong proj₂ k,v'≡k,v)
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Map-functional k v v' xs (push k≢ _) (there k,v'∈xs) = absurd (unique-not-in xs v' (k≢ , k,v'∈xs))
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where
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unique-not-in : ∀ (xs : List (A × B)) (v' : B) → ¬ (All (λ k' → ¬ k ≡ k') (keys xs) × (k , v') ∈ xs)
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unique-not-in ((k' , _) ∷ xs) v' (k≢k' ∷ _ , here k',≡x) = k≢k' (cong proj₁ k',≡x)
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unique-not-in (_ ∷ xs) v' (_ ∷ rest , there k,v'∈xs) = unique-not-in xs v' (rest , k,v'∈xs)
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-- Map-functional : ∀ (k : A) (v v' : B) (xs : List (A × B)) → Unique (keys ((k , v) ∷ xs)) → MemProp._∈_ (k , v') ((k , v) ∷ xs) → v ≡ v'
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-- Map-functional k v v' _ _ (here k,v'≡k,v) = sym (cong proj₂ k,v'≡k,v)
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-- Map-functional k v v' xs (push k≢ _) (there k,v'∈xs) = absurd (unique-not-in xs v' (k≢ , k,v'∈xs))
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-- where
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-- unique-not-in : ∀ (xs : List (A × B)) (v' : B) → ¬ (All (λ k' → ¬ k ≡ k') (keys xs) × (k , v') ∈ xs)
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-- unique-not-in ((k' , _) ∷ xs) v' (k≢k' ∷ _ , here k',≡x) = k≢k' (cong proj₁ k',≡x)
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-- unique-not-in (_ ∷ xs) v' (_ ∷ rest , there k,v'∈xs) = unique-not-in xs v' (rest , k,v'∈xs)
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module _ (f : B → B → B) where
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open Impl f public using (insert; merge)
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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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module _ (_≈_ : B → B → Set b) where
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open ImplRelation _≈_ renaming (subset to subset-impl)
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subset : Map → Map → Set (a ⊔ b)
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subset (kvs₁ , _) (kvs₂ , _) = subset-impl kvs₁ kvs₂
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lift : Map → Map → Set (a ⊔ b)
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lift m₁ m₂ = subset m₁ m₂ × subset m₂ m₁
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