agda-spa/Map.agda

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open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; sym; trans; cong; subst)
open import Relation.Binary.Definitions using (Decidable)
open import Relation.Binary.Core using (Rel)
open import Relation.Nullary using (Dec; yes; no; Reflects; ofʸ; ofⁿ)
open import Agda.Primitive using (Level) renaming (_⊔_ to _⊔_)
module Map {a b : Level} (A : Set a) (B : Set b)
(≡-dec-A : Decidable (_≡_ {a} {A}))
where
import Data.List.Membership.Propositional as MemProp
open import Relation.Nullary using (¬_)
open import Data.Nat using ()
open import Data.List using (List; map; []; _∷_; _++_)
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.Empty using ()
keys : List (A × B) List A
keys = map proj₁
data Unique {c} {C : Set c} : List C Set c where
empty : Unique []
push : {x : C} {xs : List C}
All (λ x' ¬ x x') xs
Unique xs
Unique (x xs)
Unique-append : {c} {C : Set c} {x : C} {xs : List C}
¬ MemProp._∈_ 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
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
absurd : {a} {A : Set a} A
absurd ()
private module _ where
open MemProp using (_∈_)
unique-not-in : {k : A} {v : B} {l : List (A × B)}
¬ (All (λ k' ¬ k k') (keys l) × (k , v) l)
unique-not-in {l = (k' , _) xs} (k≢k' _ , here k',≡x) =
k≢k' (cong proj₁ k',≡x)
unique-not-in {l = _ xs} (_ rest , there k,v'∈xs) =
unique-not-in (rest , k,v'∈xs)
ListAB-functional : {k : A} {v v' : B} {l : List (A × B)}
Unique (keys l) (k , v) l (k , v') l v v'
ListAB-functional _ (here k,v≡x) (here k,v'≡x) =
cong proj₂ (trans k,v≡x (sym k,v'≡x))
ListAB-functional (push k≢xs _) (here k,v≡x) (there k,v'∈xs)
rewrite sym k,v≡x = absurd (unique-not-in (k≢xs , k,v'∈xs))
ListAB-functional (push k≢xs _) (there k,v∈xs) (here k,v'≡x)
rewrite sym k,v'≡x = absurd (unique-not-in (k≢xs , k,v∈xs))
ListAB-functional {l = _ xs } (push _ uxs) (there k,v∈xs) (there k,v'∈xs) =
ListAB-functional uxs k,v∈xs k,v'∈xs
∈k-dec : (k : A) (l : List (A × B)) Dec (k keys 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 keys ((k' , v) xs)
witness (here k≡k') = k≢k' k≡k'
witness (there k∈kxs) = k∉kxs k∈kxs
∈-cong : {c d} {C : Set c} {D : Set d} {c : C} {l : List C}
(f : C D) c l f c map f l
∈-cong f (here c≡c') = here (cong f c≡c')
∈-cong f (there c∈xs) = there (∈-cong f c∈xs)
locate : {k : A} {l : List (A × B)} k keys l Σ B (λ v (k , v) l)
locate {k} {(k' , v) xs} (here k≡k') rewrite k≡k' = (v , here refl)
locate {k} {(k' , v) xs} (there k∈kxs) = let (v , k,v∈xs) = locate k∈kxs in (v , there k,v∈xs)
private module ImplRelation (_≈_ : B B Set b) where
open MemProp using (_∈_)
subset : List (A × B) List (A × B) Set (a ⊔ℓ b)
subset m₁ m₂ = (k : A) (v : B) (k , v) m₁
Σ B (λ v' v v' × ((k , v') m₂))
private module ImplInsert (f : B B B) where
open import Data.List using (map)
open MemProp using (_∈_)
private
_∈k_ : A List (A × B) Set a
_∈k_ k m = k (keys m)
foldr : {c} {C : Set c} (A B C C) -> C -> List (A × B) -> C
foldr f b [] = b
foldr f b ((k , v) xs) = f k v (foldr f b xs)
insert : A B List (A × B) List (A × B)
insert k v [] = (k , v) []
insert k v (x@(k' , v') xs) with ≡-dec-A k k'
... | yes _ = (k' , f v v') xs
... | no _ = x insert k v xs
union : List (A × B) List (A × B) List (A × B)
union m₁ m₂ = foldr insert m₂ m₁
insert-keys-∈ : {k : A} {v : B} {l : List (A × B)}
k ∈k l keys l keys (insert k v l)
insert-keys-∈ {k} {v} {(k' , v') xs} (here k≡k')
with (≡-dec-A k k')
... | yes _ = refl
... | no k≢k' = absurd (k≢k' k≡k')
insert-keys-∈ {k} {v} {(k' , _) xs} (there k∈kxs)
with (≡-dec-A k k')
... | yes _ = refl
... | no _ = cong (λ xs' k' xs') (insert-keys-∈ k∈kxs)
insert-keys-∉ : {k : A} {v : B} {l : List (A × B)}
¬ (k ∈k l) (keys l ++ (k [])) keys (insert k v l)
insert-keys-∉ {k} {v} {[]} _ = refl
insert-keys-∉ {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-∉ (λ k∈kxs k∉kl (there k∈kxs)))
insert-preserves-Unique : {k : A} {v : B} {l : List (A × B)}
Unique (keys l) Unique (keys (insert k v l))
insert-preserves-Unique {k} {v} {l} u
with (∈k-dec k l)
... | yes k∈kl rewrite insert-keys-∈ {v = v} k∈kl = u
... | no k∉kl rewrite sym (insert-keys-∉ {v = v} k∉kl) = Unique-append k∉kl u
union-preserves-Unique : (l₁ l₂ : List (A × B))
Unique (keys l₂) Unique (keys (union l₁ l₂))
union-preserves-Unique [] l₂ u₂ = u₂
union-preserves-Unique ((k₁ , v₁) xs₁) l₂ u₂ =
insert-preserves-Unique (union-preserves-Unique xs₁ l₂ u₂)
2023-07-30 13:49:38 -07:00
insert-fresh : {k : A} {v : B} {l : List (A × B)}
¬ k ∈k l (k , v) insert k v l
insert-fresh {l = []} k∉kl = here refl
insert-fresh {k} {l = (k' , v') xs} k∉kl
with ≡-dec-A k k'
... | yes k≡k' = absurd (k∉kl (here k≡k'))
... | no _ = there (insert-fresh (λ k∈kxs k∉kl (there k∈kxs)))
insert-preserves-∉k : {k k' : A} {v' : B} {l : List (A × B)}
¬ k k' ¬ k ∈k l ¬ k ∈k insert k' v' l
insert-preserves-∉k {l = []} k≢k' k∉kl (here k≡k') = k≢k' k≡k'
insert-preserves-∉k {l = []} k≢k' k∉kl (there ())
insert-preserves-∉k {k} {k'} {v'} {(k'' , v'') xs} k≢k' k∉kl k∈kil
with ≡-dec-A k k''
... | yes k≡k'' = k∉kl (here k≡k'')
... | no k≢k'' with ≡-dec-A k' k'' | k∈kil
... | yes k'≡k'' | here k≡k'' = k≢k'' k≡k''
... | yes k'≡k'' | there k∈kxs = k∉kl (there k∈kxs)
... | no k'≢k'' | here k≡k'' = k∉kl (here k≡k'')
... | no k'≢k'' | there k∈kxs = insert-preserves-∉k k≢k'
(λ k∈kxs k∉kl (there k∈kxs)) k∈kxs
union-preserves-∉ : {k : A} {l₁ l₂ : List (A × B)}
¬ k ∈k l₁ ¬ k ∈k l₂ ¬ k ∈k union l₁ l₂
union-preserves-∉ {l₁ = []} _ k∉kl₂ = k∉kl₂
union-preserves-∉ {k} {(k' , v') xs₁} k∉kl₁ k∉kl₂
with ≡-dec-A k k'
... | yes k≡k' = absurd (k∉kl₁ (here k≡k'))
... | no k≢k' = insert-preserves-∉k k≢k' (union-preserves-∉ (λ k∈kxs₁ k∉kl₁ (there k∈kxs₁)) k∉kl₂)
insert-preserves-∈k : {k k' : A} {v' : B} {l : List (A × B)}
k ∈k l k ∈k insert k' v' l
insert-preserves-∈k {k} {k'} {v'} {(k'' , v'') xs} (here k≡k'')
with (≡-dec-A k' k'')
... | yes _ = here k≡k''
... | no _ = here k≡k''
insert-preserves-∈k {k} {k'} {v'} {(k'' , v'') xs} (there k∈kxs)
with (≡-dec-A k' k'')
... | yes _ = there k∈kxs
... | no _ = there (insert-preserves-∈k k∈kxs)
union-preserves-∈k₁ : {k : A} {l₁ l₂ : List (A × B)}
k ∈k l₁ k ∈k (union l₁ l₂)
union-preserves-∈k₁ {k} {(k' , v') xs} {l₂} (here k≡k')
with ∈k-dec k (union xs l₂)
... | yes k∈kxsl₂ = insert-preserves-∈k k∈kxsl₂
... | no k∉kxsl₂ rewrite k≡k' = ∈-cong proj₁ (insert-fresh k∉kxsl₂)
union-preserves-∈k₁ {k} {(k' , v') xs} {l₂} (there k∈kxs) =
insert-preserves-∈k (union-preserves-∈k₁ k∈kxs)
union-preserves-∈k₂ : {k : A} {l₁ l₂ : List (A × B)}
k ∈k l₂ k ∈k (union l₁ l₂)
union-preserves-∈k₂ {k} {[]} {l₂} k∈kl₂ = k∈kl₂
union-preserves-∈k₂ {k} {(k' , v') xs} {l₂} k∈kl₂ =
insert-preserves-∈k (union-preserves-∈k₂ {l₁ = xs} k∈kl₂)
∉-union-∉-either : {k : A} {l₁ l₂ : List (A × B)}
¬ k ∈k union l₁ l₂ ¬ k ∈k l₁ × ¬ k ∈k l₂
∉-union-∉-either {k} {l₁} {l₂} k∉l₁l₂
with ∈k-dec k l₁
... | yes k∈kl₁ = absurd (k∉l₁l₂ (union-preserves-∈k₁ k∈kl₁))
... | no k∉kl₁ with ∈k-dec k l₂
... | yes k∈kl₂ = absurd (k∉l₁l₂ (union-preserves-∈k₂ {l₁ = l₁} k∈kl₂))
... | no k∉kl₂ = (k∉kl₁ , k∉kl₂)
2023-07-30 13:49:38 -07:00
insert-preserves-∈ : {k k' : A} {v v' : B} {l : List (A × B)}
¬ k k' (k , v) l (k , v) insert k' v' l
insert-preserves-∈ {k} {k'} {l = x xs} k≢k' (here k,v=x)
rewrite sym k,v=x with ≡-dec-A k' k
... | yes k'≡k = absurd (k≢k' (sym k'≡k))
... | no _ = here refl
insert-preserves-∈ {k} {k'} {l = (k'' , _) xs} k≢k' (there k,v∈xs)
with ≡-dec-A k' k''
... | yes _ = there k,v∈xs
... | no _ = there (insert-preserves-∈ k≢k' k,v∈xs)
union-preserves-∈₂ : {k : A} {v : B} {l₁ l₂ : List (A × B)}
¬ k ∈k l₁ (k , v) l₂ (k , v) union l₁ l₂
union-preserves-∈₂ {l₁ = []} _ k,v∈l₂ = k,v∈l₂
union-preserves-∈₂ {l₁ = (k' , v') xs₁} k∉kl₁ k,v∈l₂ =
let recursion = union-preserves-∈₂ (λ k∈xs₁ k∉kl₁ (there k∈xs₁)) k,v∈l₂
in insert-preserves-∈ (λ k≡k' k∉kl₁ (here k≡k')) recursion
union-preserves-∈₁ : {k : A} {v : B} {l₁ l₂ : List (A × B)}
Unique (keys l₁) (k , v) l₁ ¬ k ∈k l₂ (k , v) union l₁ l₂
union-preserves-∈₁ {k} {v} {(k' , v') xs₁} (push k'≢xs₁ uxs₁) (there k,v∈xs₁) k∉kl₂ =
insert-preserves-∈ k≢k' k,v∈mxs₁l
where
k,v∈mxs₁l = union-preserves-∈₁ uxs₁ k,v∈xs₁ k∉kl₂
k≢k' : ¬ k k'
k≢k' with ≡-dec-A k k'
... | yes k≡k' rewrite k≡k' = absurd (All¬-¬Any k'≢xs₁ (∈-cong proj₁ k,v∈xs₁))
... | no k≢k' = k≢k'
union-preserves-∈₁ {l₁ = (k' , v') xs₁} (push k'≢xs₁ uxs₁) (here k,v≡k',v') k∉kl₂
rewrite cong proj₁ k,v≡k',v' rewrite cong proj₂ k,v≡k',v' =
insert-fresh (union-preserves-∉ (All¬-¬Any k'≢xs₁) k∉kl₂)
insert-combines : {k : A} {v v' : B} {l : List (A × B)}
Unique (keys l) (k , v') l (k , f v v') (insert k v l)
insert-combines {l = (k' , v'') xs} _ (here k,v'≡k',v'')
rewrite cong proj₁ k,v'≡k',v'' rewrite cong proj₂ k,v'≡k',v''
with ≡-dec-A k' k'
... | yes _ = here refl
... | no k≢k' = absurd (k≢k' refl)
insert-combines {k} {l = (k' , v'') xs} (push k'≢xs uxs) (there k,v'∈xs)
with ≡-dec-A k k'
... | yes k≡k' rewrite k≡k' = absurd (All¬-¬Any k'≢xs (∈-cong proj₁ k,v'∈xs))
... | no k≢k' = there (insert-combines uxs k,v'∈xs)
union-combines : {k : A} {v₁ v₂ : B} {l₁ l₂ : List (A × B)}
Unique (keys l₁) Unique (keys l₂)
(k , v₁) l₁ (k , v₂) l₂ (k , f v₁ v₂) union l₁ l₂
union-combines {l₁ = (k' , v) xs₁} {l₂} (push k'≢xs₁ uxs₁) ul₂ (here k,v₁≡k',v) k,v₂∈l₂
rewrite cong proj₁ (sym (k,v₁≡k',v)) rewrite cong proj₂ (sym (k,v₁≡k',v)) =
insert-combines (union-preserves-Unique xs₁ l₂ ul₂) (union-preserves-∈₂ (All¬-¬Any k'≢xs₁) k,v₂∈l₂)
union-combines {k} {l₁ = (k' , v) xs₁} (push k'≢xs₁ uxs₁) ul₂ (there k,v₁∈xs₁) k,v₂∈l₂ =
insert-preserves-∈ k≢k' (union-combines uxs₁ ul₂ k,v₁∈xs₁ k,v₂∈l₂)
where
k≢k' : ¬ k k'
k≢k' with ≡-dec-A k k'
... | yes k≡k' rewrite k≡k' = absurd (All¬-¬Any k'≢xs₁ (∈-cong proj₁ k,v₁∈xs₁))
... | no k≢k' = k≢k'
update : A B List (A × B) List (A × B)
update k v [] = []
update k v ((k' , v') xs) with ≡-dec-A k k'
... | yes _ = (k' , f v v') xs
... | no _ = (k' , v') update k v xs
updates : List (A × B) List (A × B) List (A × B)
updates l₁ l₂ = foldr update l₂ l₁
restrict : List (A × B) List (A × B) List (A × B)
restrict l [] = []
restrict l ((k' , v') xs) with ∈k-dec k' l
... | yes _ = (k' , v') restrict l xs
... | no _ = restrict l xs
intersect : List (A × B) List (A × B) List (A × B)
intersect l₁ l₂ = restrict l₁ (updates l₁ l₂)
update-keys : {k : A} {v : B} {l : List (A × B)}
keys l keys (update k v l)
update-keys {l = []} = refl
update-keys {k} {v} {l = (k' , v') xs}
with ≡-dec-A k k'
... | yes _ = refl
... | no _ rewrite update-keys {k} {v} {xs} = refl
updates-keys : {l₁ l₂ : List (A × B)}
keys l₂ keys (updates l₁ l₂)
updates-keys {[]} = refl
updates-keys {(k , v) xs} {l₂}
rewrite updates-keys {xs} {l₂}
rewrite update-keys {k} {v} {updates xs l₂} = refl
update-preserves-Unique : {k : A} {v : B} {l : List (A × B)}
Unique (keys l) Unique (keys (update k v l ))
update-preserves-Unique {k} {v} {l} u rewrite update-keys {k} {v} {l} = u
updates-preserve-Unique : {l₁ l₂ : List (A × B)}
Unique (keys l₂) Unique (keys (updates l₁ l₂))
updates-preserve-Unique {[]} u = u
updates-preserve-Unique {(k , v) xs} u = update-preserves-Unique (updates-preserve-Unique {xs} u)
restrict-preserves-k≢ : {k : A} {l₁ l₂ : List (A × B)}
All (λ k' ¬ k k') (keys l₂) All (λ k' ¬ k k') (keys (restrict l₁ l₂))
restrict-preserves-k≢ {k} {l₁} {[]} k≢l₂ = k≢l₂
restrict-preserves-k≢ {k} {l₁} {(k' , v') xs} (k≢k' k≢xs)
with ∈k-dec k' l₁
... | yes _ = k≢k' restrict-preserves-k≢ k≢xs
... | no _ = restrict-preserves-k≢ k≢xs
restrict-preserves-Unique : {l₁ l₂ : List (A × B)}
Unique (keys l₂) Unique (keys (restrict l₁ l₂))
restrict-preserves-Unique {l₁} {[]} _ = empty
restrict-preserves-Unique {l₁} {(k , v) xs} (push k≢xs uxs)
with ∈k-dec k l₁
... | yes _ = push (restrict-preserves-k≢ k≢xs) (restrict-preserves-Unique uxs)
... | no _ = restrict-preserves-Unique uxs
intersect-preserves-Unique : {l₁ l₂ : List (A × B)}
Unique (keys l₂) Unique (keys (intersect l₁ l₂))
intersect-preserves-Unique {l₁} u = restrict-preserves-Unique (updates-preserve-Unique {l₁} u)
updates-preserve-∉₂ : {k : A} {l₁ l₂ : List (A × B)}
¬ k ∈k l₂ ¬ k ∈k updates l₁ l₂
updates-preserve-∉₂ {k} {l₁} {l₂} k∉kl₁ k∈kl₁l₂
rewrite updates-keys {l₁} {l₂} = k∉kl₁ k∈kl₁l₂
restrict-needs-both : {k : A} {l₁ l₂ : List (A × B)}
k ∈k restrict l₁ l₂ (k ∈k l₁ × k ∈k l₂)
restrict-needs-both {k} {l₁} {[]} ()
restrict-needs-both {k} {l₁} {(k' , _) xs} k∈l₁l₂
with ∈k-dec k' l₁ | k∈l₁l₂
... | yes k'∈kl₁ | here k≡k'
rewrite k≡k' =
(k'∈kl₁ , here refl)
... | yes _ | there k∈l₁xs =
let (k∈kl₁ , k∈kxs) = restrict-needs-both k∈l₁xs
in (k∈kl₁ , there k∈kxs)
... | no k'∉kl₁ | k∈l₁xs =
let (k∈kl₁ , k∈kxs) = restrict-needs-both k∈l₁xs
in (k∈kl₁ , there k∈kxs)
restrict-preserves-∉₁ : {k : A} {l₁ l₂ : List (A × B)}
¬ k ∈k l₁ ¬ k ∈k restrict l₁ l₂
restrict-preserves-∉₁ {k} {l₁} {l₂} k∉kl₁ k∈kl₁l₂ =
let (k∈kl₁ , _) = restrict-needs-both {l₂ = l₂} k∈kl₁l₂ in k∉kl₁ k∈kl₁
restrict-preserves-∉₂ : {k : A} {l₁ l₂ : List (A × B)}
¬ k ∈k l₂ ¬ k ∈k restrict l₁ l₂
restrict-preserves-∉₂ {k} {l₁} {l₂} k∉kl₂ k∈kl₁l₂ =
let (_ , k∈kl₂) = restrict-needs-both {l₂ = l₂} k∈kl₁l₂ in k∉kl₂ k∈kl₂
intersect-preserves-∉₁ : {k : A} {l₁ l₂ : List (A × B)}
¬ k ∈k l₁ ¬ k ∈k intersect l₁ l₂
intersect-preserves-∉₁ {k} {l₁} {l₂} = restrict-preserves-∉₁ {k} {l₁} {updates l₁ l₂}
intersect-preserves-∉₂ : {k : A} {l₁ l₂ : List (A × B)}
¬ k ∈k l₂ ¬ k ∈k intersect l₁ l₂
intersect-preserves-∉₂ {k} {l₁} {l₂} k∉l₂ = restrict-preserves-∉₂ (updates-preserve-∉₂ {l₁ = l₁} k∉l₂ )
restrict-preserves-∈₂ : {k : A} {v : B} {l₁ l₂ : List (A × B)}
k ∈k l₁ (k , v) l₂ (k , v) restrict l₁ l₂
restrict-preserves-∈₂ {k} {v} {l₁} {(k' , v') xs} k∈kl₁ (here k,v≡k',v')
rewrite cong proj₁ k,v≡k',v' rewrite cong proj₂ k,v≡k',v'
with ∈k-dec k' l₁
... | yes _ = here refl
... | no k'∉kl₁ = absurd (k'∉kl₁ k∈kl₁)
restrict-preserves-∈₂ {l₁ = l₁} {l₂ = (k' , v') xs} k∈kl₁ (there k,v∈xs)
with ∈k-dec k' l₁
... | yes _ = there (restrict-preserves-∈₂ k∈kl₁ k,v∈xs)
... | no _ = restrict-preserves-∈₂ k∈kl₁ k,v∈xs
update-preserves-∈ : {k k' : A} {v v' : B} {l : List (A × B)}
¬ k k' (k , v) l (k , v) update k' v' l
update-preserves-∈ {k} {k'} {v} {v'} {(k'' , v'') xs} k≢k' (here k,v≡k'',v'')
rewrite cong proj₁ k,v≡k'',v'' rewrite cong proj₂ k,v≡k'',v''
with ≡-dec-A k' k''
... | yes k'≡k'' = absurd (k≢k' (sym k'≡k''))
... | no _ = here refl
update-preserves-∈ {k} {k'} {v} {v'} {(k'' , v'') xs} k≢k' (there k,v∈xs)
with ≡-dec-A k' k''
... | yes _ = there k,v∈xs
... | no _ = there (update-preserves-∈ k≢k' k,v∈xs)
updates-preserve-∈₂ : {k : A} {v : B} {l₁ l₂ : List (A × B)}
¬ k ∈k l₁ (k , v) l₂ (k , v) updates l₁ l₂
updates-preserve-∈₂ {k} {v} {[]} {l₂} _ k,v∈l₂ = k,v∈l₂
updates-preserve-∈₂ {k} {v} {(k' , v') xs} {l₂} k∉kl₁ k,v∈l₂ =
update-preserves-∈ (λ k≡k' k∉kl₁ (here k≡k')) (updates-preserve-∈₂ (λ k∈kxs k∉kl₁ (there k∈kxs)) k,v∈l₂)
update-combines : {k : A} {v v' : B} {l : List (A × B)}
Unique (keys l) (k , v) l (k , f v' v) update k v' l
update-combines {k} {v} {v'} {(k' , v'') xs} _ (here k,v=k',v'')
rewrite cong proj₁ k,v=k',v'' rewrite cong proj₂ k,v=k',v''
with ≡-dec-A k' k'
... | yes _ = here refl
... | no k'≢k' = absurd (k'≢k' refl)
update-combines {k} {v} {v'} {(k' , v'') xs} (push k'≢xs uxs) (there k,v∈xs)
with ≡-dec-A k k'
... | yes k≡k' rewrite k≡k' = absurd (All¬-¬Any k'≢xs (∈-cong proj₁ k,v∈xs))
... | no _ = there (update-combines uxs k,v∈xs)
updates-combine : {k : A} {v₁ v₂ : B} {l₁ l₂ : List (A × B)}
Unique (keys l₁) Unique (keys l₂)
(k , v₁) l₁ (k , v₂) l₂ (k , f v₁ v₂) updates l₁ l₂
updates-combine {k} {v₁} {v₂} {(k' , v') xs} {l₂} (push k'≢xs uxs₁) u₂ (here k,v₁≡k',v') k,v₂∈l₂
rewrite cong proj₁ k,v₁≡k',v' rewrite cong proj₂ k,v₁≡k',v' =
update-combines (updates-preserve-Unique {l₁ = xs} u₂) (updates-preserve-∈₂ (All¬-¬Any k'≢xs) k,v₂∈l₂)
updates-combine {k} {v₁} {v₂} {(k' , v') xs} {l₂} (push k'≢xs uxs₁) u₂ (there k,v₁∈xs) k,v₂∈l₂ =
update-preserves-∈ k≢k' (updates-combine uxs₁ u₂ k,v₁∈xs k,v₂∈l₂)
where
k≢k' : ¬ k k'
k≢k' with ≡-dec-A k k'
... | yes k≡k' rewrite k≡k' = absurd (All¬-¬Any k'≢xs (∈-cong proj₁ k,v₁∈xs))
... | no k≢k' = k≢k'
intersect-combines : {k : A} {v₁ v₂ : B} {l₁ l₂ : List (A × B)}
Unique (keys l₁) Unique (keys l₂)
(k , v₁) l₁ (k , v₂) l₂ (k , f v₁ v₂) intersect l₁ l₂
intersect-combines u₁ u₂ k,v₁∈l₁ k,v₂∈l₂ =
restrict-preserves-∈₂ (∈-cong proj₁ k,v₁∈l₁) (updates-combine u₁ u₂ k,v₁∈l₁ k,v₂∈l₂)
Map : Set (a ⊔ℓ b)
Map = Σ (List (A × B)) (λ l Unique (keys l))
_∈_ : (A × B) Map Set (a ⊔ℓ b)
_∈_ p (kvs , _) = MemProp._∈_ p kvs
_∈k_ : A Map Set a
_∈k_ k (kvs , _) = MemProp._∈_ k (keys kvs)
Map-functional : {k : A} {v v' : B} {m : Map} (k , v) m (k , v') m v v'
Map-functional {m = (l , ul)} k,v∈m k,v'∈m = ListAB-functional ul k,v∈m k,v'∈m
data Expr : Set (a ⊔ℓ b) where
`_ : Map Expr
__ : Expr Expr Expr
_∩_ : Expr Expr Expr
module _ (f : B B B) where
open ImplInsert f renaming
( insert to insert-impl
; union to union-impl
; intersect to intersect-impl
)
union : Map Map Map
union (kvs₁ , _) (kvs₂ , uks₂) = (union-impl kvs₁ kvs₂ , union-preserves-Unique kvs₁ kvs₂ uks₂)
intersect : Map Map Map
intersect (kvs₁ , _) (kvs₂ , uks₂) = (intersect-impl kvs₁ kvs₂ , intersect-preserves-Unique {kvs₁} {kvs₂} uks₂)
module _ (fUnion : B B B) (fIntersect : B B B) where
open ImplInsert fUnion using
( union-combines
; union-preserves-∈₁
; union-preserves-∈₂
; union-preserves-∉
)
open ImplInsert fIntersect using
( restrict-needs-both
; updates
; intersect-preserves-∉₁
; intersect-preserves-∉₂
; intersect-combines
)
⟦_⟧ : Expr -> Map
` m = m
e₁ e₂ = union fUnion e₁ e₂
e₁ e₂ = intersect fIntersect e₁ e₂
data Provenance (k : A) : B Expr Set (a ⊔ℓ b) where
single : {v : B} {m : Map} (k , v) m Provenance k v (` m)
in₁ : {v : B} {e₁ e₂ : Expr} Provenance k v e₁ ¬ k ∈k e₂ Provenance k v (e₁ e₂)
in₂ : {v : B} {e₁ e₂ : Expr} ¬ k ∈k e₁ Provenance k v e₂ Provenance k v (e₁ e₂)
bothᵘ : {v₁ v₂ : B} {e₁ e₂ : Expr} Provenance k v₁ e₁ Provenance k v₂ e₂ Provenance k (fUnion v₁ v₂) (e₁ e₂)
bothⁱ : {v₁ v₂ : B} {e₁ e₂ : Expr} Provenance k v₁ e₁ Provenance k v₂ e₂ Provenance k (fIntersect v₁ v₂) (e₁ e₂)
Expr-Provenance : (k : A) (e : Expr) k ∈k e Σ B (λ v (Provenance k v e × (k , v) e ))
Expr-Provenance k (` m) k∈km = let (v , k,v∈m) = locate k∈km in (v , (single k,v∈m , k,v∈m))
Expr-Provenance k (e₁ e₂) k∈ke₁e₂
with ∈k-dec k (proj₁ e₁ ) | ∈k-dec k (proj₁ e₂ )
... | yes k∈ke₁ | yes k∈ke₂ =
let (v₁ , (g₁ , k,v₁∈e₁)) = Expr-Provenance k e₁ k∈ke₁
(v₂ , (g₂ , k,v₂∈e₂)) = Expr-Provenance k e₂ k∈ke₂
in (fUnion v₁ v₂ , (bothᵘ g₁ g₂ , union-combines (proj₂ e₁ ) (proj₂ e₂ ) k,v₁∈e₁ k,v₂∈e₂))
... | yes k∈ke₁ | no k∉ke₂ =
let (v₁ , (g₁ , k,v₁∈e₁)) = Expr-Provenance k e₁ k∈ke₁
in (v₁ , (in g₁ k∉ke₂ , union-preserves-∈₁ (proj₂ e₁ ) k,v₁∈e₁ k∉ke₂))
... | no k∉ke₁ | yes k∈ke₂ =
let (v₂ , (g₂ , k,v₂∈e₂)) = Expr-Provenance k e₂ k∈ke₂
in (v₂ , (in k∉ke₁ g₂ , union-preserves-∈₂ k∉ke₁ k,v₂∈e₂))
... | no k∉ke₁ | no k∉ke₂ = absurd (union-preserves-∉ k∉ke₁ k∉ke₂ k∈ke₁e₂)
Expr-Provenance k (e₁ e₂) k∈ke₁e₂
with ∈k-dec k (proj₁ e₁ ) | ∈k-dec k (proj₁ e₂ )
... | yes k∈ke₁ | yes k∈ke₂ =
let (v₁ , (g₁ , k,v₁∈e₁)) = Expr-Provenance k e₁ k∈ke₁
(v₂ , (g₂ , k,v₂∈e₂)) = Expr-Provenance k e₂ k∈ke₂
in (fIntersect v₁ v₂ , (bothⁱ g₁ g₂ , intersect-combines (proj₂ e₁ ) (proj₂ e₂ ) k,v₁∈e₁ k,v₂∈e₂))
... | yes k∈ke₁ | no k∉ke₂ = absurd (intersect-preserves-∉₂ {l₁ = proj₁ e₁ } k∉ke₂ k∈ke₁e₂)
... | no k∉ke₁ | yes k∈ke₂ = absurd (intersect-preserves-∉₁ {l₂ = proj₁ e₂ } k∉ke₁ k∈ke₁e₂)
... | no k∉ke₁ | no k∉ke₂ = absurd (intersect-preserves-∉₂ {l₁ = proj₁ e₁ } k∉ke₂ k∈ke₁e₂)
module _ (_≈_ : B B Set b) where
open ImplRelation _≈_ renaming (subset to subset-impl)
subset : Map Map Set (a ⊔ℓ b)
subset (kvs₁ , _) (kvs₂ , _) = subset-impl kvs₁ kvs₂
lift : Map Map Set (a ⊔ℓ b)
lift m₁ m₂ = subset m₁ m₂ × subset m₂ m₁
private
data SubsetInfo (m₁ m₂ : Map) : Set (a ⊔ℓ b) where
extra : (k : A) k ∈k m₁ ¬ k ∈k m₂ SubsetInfo m₁ m₂
mismatch : (k : A) (v₁ v₂ : B) (k , v₁) m₁ (k , v₂) m₂ ¬ v₁ v₂ SubsetInfo m₁ m₂
fine : subset m₁ m₂ SubsetInfo m₁ m₂
SubsetInfo-to-dec