Prove exercise 4.26 from the textbook
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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Lattice/Map.agda
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Lattice/Map.agda
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@ -9,7 +9,7 @@ module Lattice.Map {a b : Level} (A : Set a) (B : Set b)
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(≡-dec-A : Decidable (_≡_ {a} {A}))
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(≡-dec-A : Decidable (_≡_ {a} {A}))
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(lB : IsLattice B _≈₂_ _⊔₂_ _⊓₂_) where
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(lB : IsLattice B _≈₂_ _⊔₂_ _⊓₂_) where
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import Data.List.Membership.Propositional as MemProp
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open import Data.List.Membership.Propositional as MemProp using () renaming (_∈_ to _∈ˡ_)
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open import Relation.Nullary using (¬_; Dec; yes; no)
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open import Relation.Nullary using (¬_; Dec; yes; no)
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open import Data.Nat using (ℕ)
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open import Data.Nat using (ℕ)
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@ -27,6 +27,7 @@ open IsLattice lB using () renaming
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; ⊔-idemp to ⊔₂-idemp; ⊔-comm to ⊔₂-comm; ⊔-assoc to ⊔₂-assoc
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; ⊔-idemp to ⊔₂-idemp; ⊔-comm to ⊔₂-comm; ⊔-assoc to ⊔₂-assoc
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; ⊓-idemp to ⊓₂-idemp; ⊓-comm to ⊓₂-comm; ⊓-assoc to ⊓₂-assoc
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; ⊓-idemp to ⊓₂-idemp; ⊓-comm to ⊓₂-comm; ⊓-assoc to ⊓₂-assoc
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; absorb-⊔-⊓ to absorb-⊔₂-⊓₂; absorb-⊓-⊔ to absorb-⊓₂-⊔₂
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; absorb-⊔-⊓ to absorb-⊔₂-⊓₂; absorb-⊓-⊔ to absorb-⊓₂-⊔₂
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; _≼_ to _≼₂_
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)
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)
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private module ImplKeys where
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private module ImplKeys where
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@ -55,6 +56,15 @@ private module _ where
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ListAB-functional {l = _ ∷ xs } (push _ uxs) (there k,v∈xs) (there 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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ListAB-functional uxs k,v∈xs k,v'∈xs
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k∈-dec : ∀ (k : A) (l : List A) → Dec (k ∈ l)
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k∈-dec k [] = no (λ ())
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k∈-dec k (x ∷ xs)
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with (≡-dec-A k x)
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... | yes refl = yes (here refl)
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... | no k≢x with (k∈-dec k xs)
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... | yes k∈xs = yes (there k∈xs)
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... | no k∉xs = no (λ { (here k≡x) → k≢x k≡x; (there k∈xs) → k∉xs k∈xs })
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∈k-dec : ∀ (k : A) (l : List (A × B)) → Dec (k ∈ keys l)
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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 [] = no (λ ())
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∈k-dec k ((k' , v) ∷ xs)
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∈k-dec k ((k' , v) ∷ xs)
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@ -463,7 +473,6 @@ private module ImplInsert (f : B → B → B) where
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intersect-combines u₁ u₂ k,v₁∈l₁ k,v₂∈l₂ =
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intersect-combines u₁ u₂ k,v₁∈l₁ k,v₂∈l₂ =
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restrict-preserves-∈₂ (∈-cong proj₁ k,v₁∈l₁) (updates-combine u₁ u₂ k,v₁∈l₁ k,v₂∈l₂)
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restrict-preserves-∈₂ (∈-cong proj₁ k,v₁∈l₁) (updates-combine u₁ u₂ k,v₁∈l₁ k,v₂∈l₂)
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Map : Set (a ⊔ℓ b)
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Map : Set (a ⊔ℓ b)
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Map = Σ (List (A × B)) (λ l → Unique (ImplKeys.keys l))
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Map = Σ (List (A × B)) (λ l → Unique (ImplKeys.keys l))
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@ -532,6 +541,7 @@ open ImplInsert _⊔₂_ using
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; union-preserves-∈₁
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; union-preserves-∈₁
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; union-preserves-∈₂
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; union-preserves-∈₂
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; union-preserves-∉
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; union-preserves-∉
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; union-preserves-∈k₁
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)
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)
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⊔-combines : ∀ {k : A} {v₁ v₂ : B} {m₁ m₂ : Map} → (k , v₁) ∈ m₁ → (k , v₂) ∈ m₂ → (k , v₁ ⊔₂ v₂) ∈ (m₁ ⊔ m₂)
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⊔-combines : ∀ {k : A} {v₁ v₂ : B} {m₁ m₂ : Map} → (k , v₁) ∈ m₁ → (k , v₂) ∈ m₂ → (k , v₁ ⊔₂ v₂) ∈ (m₁ ⊔ m₂)
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@ -582,6 +592,11 @@ Expr-Provenance k (e₁ ∩ e₂) k∈ke₁e₂
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... | no k∉ke₁ | yes k∈ke₂ = ⊥-elim (intersect-preserves-∉₁ {l₂ = proj₁ ⟦ e₂ ⟧} k∉ke₁ k∈ke₁e₂)
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... | no k∉ke₁ | yes k∈ke₂ = ⊥-elim (intersect-preserves-∉₁ {l₂ = proj₁ ⟦ e₂ ⟧} k∉ke₁ k∈ke₁e₂)
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... | no k∉ke₁ | no k∉ke₂ = ⊥-elim (intersect-preserves-∉₂ {l₁ = proj₁ ⟦ e₁ ⟧} k∉ke₂ k∈ke₁e₂)
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... | no k∉ke₁ | no k∉ke₂ = ⊥-elim (intersect-preserves-∉₂ {l₁ = proj₁ ⟦ e₁ ⟧} k∉ke₂ k∈ke₁e₂)
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Expr-Provenance-≡ : ∀ (k : A) (v : B) (e : Expr) → (k , v) ∈ ⟦ e ⟧ → Provenance k v e
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Expr-Provenance-≡ k v e k,v∈e
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with (v' , (p , k,v'∈e)) ← Expr-Provenance k e (forget {m = ⟦ e ⟧} k,v∈e)
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rewrite Map-functional {m = ⟦ e ⟧} k,v∈e k,v'∈e = p
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module _ (≈₂-dec : IsDecidable _≈₂_) where
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module _ (≈₂-dec : IsDecidable _≈₂_) where
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private module _ where
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private module _ where
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data SubsetInfo (m₁ m₂ : Map) : Set (a ⊔ℓ b) where
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data SubsetInfo (m₁ m₂ : Map) : Set (a ⊔ℓ b) where
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@ -904,3 +919,184 @@ lattice = record
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⊓-equal-keys : ∀ {m₁ m₂ : Map} → keys m₁ ≡ keys m₂ → keys m₁ ≡ keys (m₁ ⊓ m₂)
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⊓-equal-keys : ∀ {m₁ m₂ : Map} → keys m₁ ≡ keys m₂ → keys m₁ ≡ keys (m₁ ⊓ m₂)
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⊓-equal-keys km₁≡km₂ = intersect-equal-keys km₁≡km₂
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⊓-equal-keys km₁≡km₂ = intersect-equal-keys km₁≡km₂
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private module _ where
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open MemProp using (_∈_)
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transform : List (A × B) → List A → (A → B) → List (A × B)
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transform [] _ _ = []
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transform ((k , v) ∷ xs) ks f
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with k∈-dec k ks
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... | yes _ = (k , f k) ∷ transform xs ks f
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... | no _ = (k , v) ∷ transform xs ks f
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transform-keys-≡ : ∀ (l : List (A × B)) (ks : List A) (f : A → B) →
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ImplKeys.keys l ≡ ImplKeys.keys (transform l ks f)
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transform-keys-≡ [] ks f = refl
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transform-keys-≡ ((k , v) ∷ xs) ks f
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with k∈-dec k ks
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... | yes _ rewrite transform-keys-≡ xs ks f = refl
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... | no _ rewrite transform-keys-≡ xs ks f = refl
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transform-∉k-forward : ∀ {l : List (A × B)} (ks : List A) (f : A → B) {k : A} →
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¬ k ∈ˡ ImplKeys.keys l → ¬ k ∈ˡ ImplKeys.keys (transform l ks f)
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transform-∉k-forward {l} ks f k∉kl rewrite transform-keys-≡ l ks f = k∉kl
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transform-∈k-forward : ∀ {l : List (A × B)} (ks : List A) (f : A → B) {k : A} →
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k ∈ˡ ImplKeys.keys l → k ∈ˡ ImplKeys.keys (transform l ks f)
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transform-∈k-forward {l} ks f k∈kl rewrite transform-keys-≡ l ks f = k∈kl
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transform-∈k-backward : ∀ {l : List (A × B)} (ks : List A) (f : A → B) {k : A} →
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k ∈ˡ ImplKeys.keys (transform l ks f) → k ∈ˡ ImplKeys.keys l
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transform-∈k-backward {l} ks f k∈kt rewrite transform-keys-≡ l ks f = k∈kt
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transform-k∈ks : ∀ (l : List (A × B)) {ks : List A} (f : A → B) {k : A} →
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k ∈ˡ ks → k ∈ˡ ImplKeys.keys (transform l ks f) → (k , f k) ∈ˡ transform l ks f
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transform-k∈ks [] _ _ ()
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transform-k∈ks ((k' , v) ∷ xs) {ks} f {k} k∈ks k∈kl
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with k∈-dec k' ks | k∈kl
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... | yes _ | here refl = here refl
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... | no k∉ks | here refl = ⊥-elim (k∉ks k∈ks)
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... | yes _ | there k∈kxs = there (transform-k∈ks xs f k∈ks k∈kxs)
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... | no _ | there k∈kxs = there (transform-k∈ks xs f k∈ks k∈kxs)
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transform-k∉ks-forward : ∀ {l : List (A × B)} {ks : List A} (f : A → B) {k : A} {v : B} →
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¬ k ∈ˡ ks → (k , v) ∈ˡ l → (k , v) ∈ˡ transform l ks f
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transform-k∉ks-forward {[]} _ _ ()
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transform-k∉ks-forward {(k' , v') ∷ xs} {ks} f {k} {v} k∉ks k,v∈l
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with k∈-dec k' ks | k,v∈l
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... | yes k∈ks | here refl = ⊥-elim (k∉ks k∈ks)
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... | no k∉ks | here refl = here refl
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... | yes _ | there k,v∈xs = there (transform-k∉ks-forward f k∉ks k,v∈xs)
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... | no _ | there k,v∈xs = there (transform-k∉ks-forward f k∉ks k,v∈xs)
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transform-k∉ks-backward : ∀ {l : List (A × B)} {ks : List A} (f : A → B) {k : A} {v : B} →
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¬ k ∈ˡ ks → (k , v) ∈ˡ transform l ks f → (k , v) ∈ˡ l
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transform-k∉ks-backward {[]} _ _ ()
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transform-k∉ks-backward {(k' , v') ∷ xs} {ks} f {k} {v} k∉ks k,v∈tl
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with k∈-dec k' ks | k,v∈tl
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... | yes k∈ks | here refl = ⊥-elim (k∉ks k∈ks)
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... | no k∉ks | here refl = here refl
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... | yes _ | there k,v∈txs = there (transform-k∉ks-backward f k∉ks k,v∈txs)
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... | no _ | there k,v∈txs = there (transform-k∉ks-backward f k∉ks k,v∈txs)
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_updating_via_ : Map → List A → (A → B) → Map
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_updating_via_ (kvs , uks) ks f =
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( transform kvs ks f
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, subst Unique (transform-keys-≡ kvs ks f) uks
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)
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updating-via-∉k-forward : ∀ (m : Map) (ks : List A) (f : A → B) {k : A} →
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¬ k ∈k m → ¬ k ∈k (m updating ks via f)
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updating-via-∉k-forward m = transform-∉k-forward
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updating-via-∈k-forward : ∀ (m : Map) (ks : List A) (f : A → B) {k : A} →
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k ∈k m → k ∈k (m updating ks via f)
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updating-via-∈k-forward m = transform-∈k-forward
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updating-via-∈k-backward : ∀ (m : Map) (ks : List A) (f : A → B) {k : A} →
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k ∈k (m updating ks via f) → k ∈k m
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updating-via-∈k-backward m = transform-∈k-backward
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updating-via-k∈ks : ∀ (m : Map) {ks : List A} (f : A → B) {k : A} →
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k ∈ˡ ks → k ∈k (m updating ks via f) → (k , f k) ∈ (m updating ks via f)
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updating-via-k∈ks m = transform-k∈ks (proj₁ m)
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updating-via-k∈ks-forward : ∀ (m : Map) {ks : List A} (f : A → B) {k : A} →
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k ∈ˡ ks → k ∈k m → (k , f k) ∈ (m updating ks via f)
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updating-via-k∈ks-forward m {ks} f k∈ks k∈km rewrite transform-keys-≡ (proj₁ m) ks f = transform-k∈ks (proj₁ m) f k∈ks k∈km
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updating-via-k∈ks-≡ : ∀ (m : Map) {ks : List A} (f : A → B) {k : A} {v : B} →
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k ∈ˡ ks → (k , v) ∈ (m updating ks via f)→ v ≡ f k
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updating-via-k∈ks-≡ m {ks} f k∈ks k,v∈um
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with updating-via-k∈ks m f k∈ks (forget {m = (m updating ks via f)} k,v∈um)
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... | k,fk∈um = Map-functional {m = (m updating ks via f)} k,v∈um k,fk∈um
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updating-via-k∉ks-forward : ∀ (m : Map) {ks : List A} (f : A → B) {k : A} {v : B} →
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¬ k ∈ˡ ks → (k , v) ∈ m → (k , v) ∈ (m updating ks via f)
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updating-via-k∉ks-forward m = transform-k∉ks-forward
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updating-via-k∉ks-backward : ∀ (m : Map) {ks : List A} (f : A → B) {k : A} {v : B} →
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¬ k ∈ˡ ks → (k , v) ∈ (m updating ks via f) → (k , v) ∈ m
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updating-via-k∉ks-backward m = transform-k∉ks-backward
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module _ {l} {L : Set l}
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{_≈ˡ_ : L → L → Set l} {_⊔ˡ_ : L → L → L} {_⊓ˡ_ : L → L → L}
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(lL : IsLattice L _≈ˡ_ _⊔ˡ_ _⊓ˡ_) where
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open IsLattice isLattice using (_≼_)
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open IsLattice lL using () renaming (_≼_ to _≼ˡ_)
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module _ (f : L → Map) (f-Monotonic : Monotonic _≼ˡ_ _≼_ f)
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(g : A → L → B) (g-Monotonicʳ : ∀ k → Monotonic _≼ˡ_ _≼₂_ (g k))
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(ks : List A) where
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updater : L → A → B
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updater l k = g k l
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f' : L → Map
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f' l = (f l) updating ks via (updater l)
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f'-Monotonic : Monotonic _≼ˡ_ _≼_ f'
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f'-Monotonic {l₁} {l₂} l₁≼l₂ = (f'l₁f'l₂⊆f'l₂ , f'l₂⊆f'l₁f'l₂)
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where
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fl₁fl₂⊆fl₂ = proj₁ (f-Monotonic l₁≼l₂)
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fl₂⊆fl₁fl₂ = proj₂ (f-Monotonic l₁≼l₂)
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f'l₁f'l₂⊆f'l₂ : ((f' l₁) ⊔ (f' l₂)) ⊆ f' l₂
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f'l₁f'l₂⊆f'l₂ k v k,v∈f'l₁f'l₂
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with Expr-Provenance-≡ k v ((` (f' l₁)) ∪ (` (f' l₂))) k,v∈f'l₁f'l₂
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... | in₁ (single k,v∈f'l₁) k∉kf'l₂ =
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let
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k∈kfl₁ = updating-via-∈k-backward (f l₁) ks (updater l₁) (forget {m = f' l₁} k,v∈f'l₁)
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k∈kfl₁fl₂ = union-preserves-∈k₁ {l₁ = proj₁ (f l₁)} {l₂ = proj₁ (f l₂)} k∈kfl₁
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(v' , k,v'∈fl₁l₂) = locate {m = (f l₁ ⊔ f l₂)} k∈kfl₁fl₂
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(v'' , (v'≈v'' , k,v''∈fl₂)) = fl₁fl₂⊆fl₂ k v' k,v'∈fl₁l₂
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k∈kf'l₂ = updating-via-∈k-forward (f l₂) ks (updater l₂) (forget {m = f l₂} k,v''∈fl₂)
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in
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⊥-elim (k∉kf'l₂ k∈kf'l₂)
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... | in₂ k∉kf'l₁ (single k,v'∈f'l₂) =
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(v , (IsLattice.≈-refl lB , k,v'∈f'l₂))
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... | bothᵘ (single {v₁} k,v₁∈f'l₁) (single {v₂} k,v₂∈f'l₂)
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with k∈-dec k ks
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... | yes k∈ks
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with refl ← updating-via-k∈ks-≡ (f l₁) (updater l₁) k∈ks k,v₁∈f'l₁
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with refl ← updating-via-k∈ks-≡ (f l₂) (updater l₂) k∈ks k,v₂∈f'l₂ =
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(updater l₂ k , (g-Monotonicʳ k l₁≼l₂ , k,v₂∈f'l₂))
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... | no k∉ks =
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let
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k,v₁∈fl₁ = updating-via-k∉ks-backward (f l₁) (updater l₁) k∉ks k,v₁∈f'l₁
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k,v₂∈fl₂ = updating-via-k∉ks-backward (f l₂) (updater l₂) k∉ks k,v₂∈f'l₂
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k,v₁v₂∈fl₁fl₂ = ⊔-combines {m₁ = f l₁} {m₂ = f l₂} k,v₁∈fl₁ k,v₂∈fl₂
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(v' , (v'≈v₁v₂ , k,v'∈fl₂)) = fl₁fl₂⊆fl₂ k _ k,v₁v₂∈fl₁fl₂
|
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k,v'∈f'l₂ = updating-via-k∉ks-forward (f l₂) (updater l₂) k∉ks k,v'∈fl₂
|
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in
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(v' , (v'≈v₁v₂ , k,v'∈f'l₂))
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f'l₂⊆f'l₁f'l₂ : f' l₂ ⊆ ((f' l₁) ⊔ (f' l₂))
|
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f'l₂⊆f'l₁f'l₂ k v k,v∈f'l₂
|
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with k∈kfl₂ ← updating-via-∈k-backward (f l₂) ks (updater l₂) (forget {m = f' l₂} k,v∈f'l₂)
|
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with (v' , k,v'∈fl₂) ← locate {m = f l₂} k∈kfl₂
|
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with (v'' , (v'≈v'' , k,v''∈fl₁fl₂)) ← fl₂⊆fl₁fl₂ k v' k,v'∈fl₂
|
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with Expr-Provenance-≡ k v'' ((` (f l₁)) ∪ (` (f l₂))) k,v''∈fl₁fl₂
|
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... | in₁ (single k,v''∈fl₁) k∉kfl₂ = ⊥-elim (k∉kfl₂ k∈kfl₂)
|
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|
... | in₂ k∉kfl₁ (single k,v''∈fl₂) =
|
||||||
|
let
|
||||||
|
k∉kf'l₁ = updating-via-∉k-forward (f l₁) ks (updater l₁) k∉kfl₁
|
||||||
|
in
|
||||||
|
(v , (IsLattice.≈-refl lB , union-preserves-∈₂ k∉kf'l₁ k,v∈f'l₂))
|
||||||
|
... | bothᵘ (single {v₁} k,v₁∈fl₁) (single {v₂} k,v₂∈fl₂)
|
||||||
|
with k∈-dec k ks
|
||||||
|
... | yes k∈ks with refl ← updating-via-k∈ks-≡ (f l₂) (updater l₂) k∈ks k,v∈f'l₂ =
|
||||||
|
let
|
||||||
|
k,uv₁∈f'l₁ = updating-via-k∈ks-forward (f l₁) (updater l₁) k∈ks (forget {m = f l₁} k,v₁∈fl₁)
|
||||||
|
k,uv₂∈f'l₂ = updating-via-k∈ks-forward (f l₂) (updater l₂) k∈ks (forget {m = f l₂} k,v₂∈fl₂)
|
||||||
|
k,uv₁uv₂∈f'l₁f'l₂ = ⊔-combines {m₁ = f' l₁} {m₂ = f' l₂} k,uv₁∈f'l₁ k,uv₂∈f'l₂
|
||||||
|
in
|
||||||
|
(updater l₁ k ⊔₂ updater l₂ k , (IsLattice.≈-sym lB (g-Monotonicʳ k l₁≼l₂) , k,uv₁uv₂∈f'l₁f'l₂))
|
||||||
|
... | no k∉ks
|
||||||
|
with k,v₁∈f'l₁ ← updating-via-k∉ks-forward (f l₁) (updater l₁) k∉ks k,v₁∈fl₁
|
||||||
|
with k,v₂∈f'l₂ ← updating-via-k∉ks-forward (f l₂) (updater l₂) k∉ks k,v₂∈fl₂
|
||||||
|
with k,v₁v₂∈f'l₁f'l₂ ← ⊔-combines {m₁ = f' l₁} {m₂ = f' l₂} k,v₁∈f'l₁ k,v₂∈f'l₂
|
||||||
|
with refl ← Map-functional {m = f' l₂} k,v∈f'l₂ k,v₂∈f'l₂
|
||||||
|
with refl ← Map-functional {m = f l₂} k,v'∈fl₂ k,v₂∈fl₂ =
|
||||||
|
(v₁ ⊔₂ v , (v'≈v'' , k,v₁v₂∈f'l₁f'l₂))
|
||||||
|
|
|
||||||
Loading…
Reference in New Issue
Block a user