Prove multi-key access monotonicity in finite maps
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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@ -2,7 +2,7 @@ open import Lattice
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open import Relation.Binary.PropositionalEquality as Eq
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using (_≡_;refl; sym; trans; cong; subst)
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open import Agda.Primitive using (Level) renaming (_⊔_ to _⊔ℓ_)
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open import Data.List using (List)
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open import Data.List using (List; _∷_; [])
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module Lattice.FiniteMap {a b : Level} (A : Set a) (B : Set b)
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(_≈₂_ : B → B → Set b)
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@ -10,8 +10,9 @@ module Lattice.FiniteMap {a b : Level} (A : Set a) (B : Set b)
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(≡-dec-A : IsDecidable (_≡_ {a} {A}))
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(lB : IsLattice B _≈₂_ _⊔₂_ _⊓₂_) where
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open IsLattice lB using () renaming (_≼_ to _≼₂_)
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open import Lattice.Map A B _≈₂_ _⊔₂_ _⊓₂_ ≡-dec-A lB as Map
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using (Map; ⊔-equal-keys; ⊓-equal-keys)
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using (Map; ⊔-equal-keys; ⊓-equal-keys; ∈k-dec; m₁≼m₂⇒m₁[k]≼m₂[k])
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renaming
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( _≈_ to _≈ᵐ_
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; _⊔_ to _⊔ᵐ_
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@ -28,9 +29,21 @@ open import Lattice.Map A B _≈₂_ _⊔₂_ _⊓₂_ ≡-dec-A lB as Map
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; absorb-⊔-⊓ to absorb-⊔ᵐ-⊓ᵐ
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; absorb-⊓-⊔ to absorb-⊓ᵐ-⊔ᵐ
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; ≈-dec to ≈ᵐ-dec
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; _[_] to _[_]ᵐ
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; locate to locateᵐ
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; keys to keysᵐ
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; _updating_via_ to _updatingᵐ_via_
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; updating-via-keys-≡ to updatingᵐ-via-keys-≡
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; f'-Monotonic to f'-Monotonicᵐ
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; _≼_ to _≼ᵐ_
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)
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open import Data.List.Membership.Propositional using () renaming (_∈_ to _∈ˡ_)
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open import Data.Product using (_×_; _,_; Σ; proj₁ ; proj₂)
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open import Equivalence
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open import Function using (_∘_)
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open import Relation.Nullary using (¬_; Dec; yes; no)
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open import Utils using (Pairwise; _∷_; [])
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open import Data.Empty using (⊥-elim)
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module _ (ks : List A) where
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FiniteMap : Set (a ⊔ℓ b)
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@ -56,6 +69,18 @@ module _ (ks : List A) where
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km₁≡ks
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)
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_∈k_ : A → FiniteMap → Set a
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_∈k_ k (m₁ , _) = k ∈ˡ (keysᵐ m₁)
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_updating_via_ : FiniteMap → List A → (A → B) → FiniteMap
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_updating_via_ (m₁ , ksm₁≡ks) ks f =
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( m₁ updatingᵐ ks via f
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, trans (sym (updatingᵐ-via-keys-≡ m₁ ks f)) ksm₁≡ks
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)
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_[_] : FiniteMap → List A → List B
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_[_] (m₁ , _) ks = m₁ [ ks ]ᵐ
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≈-equiv : IsEquivalence FiniteMap _≈_
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≈-equiv = record
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{ ≈-refl =
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@ -97,6 +122,8 @@ module _ (ks : List A) where
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; absorb-⊓-⊔ = λ (m₁ , _) (m₂ , _) → absorb-⊓ᵐ-⊔ᵐ m₁ m₂
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}
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open IsLattice isLattice using (_≼_) public
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lattice : Lattice FiniteMap
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lattice = record
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{ _≈_ = _≈_
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@ -104,3 +131,46 @@ module _ (ks : List A) where
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; _⊓_ = _⊓_
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; isLattice = isLattice
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}
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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 lL using () renaming (_≼_ to _≼ˡ_)
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module _ (f : L → FiniteMap) (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 → FiniteMap
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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'-Monotonicᵐ lL (proj₁ ∘ f) f-Monotonic g g-Monotonicʳ ks l₁≼l₂
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all-equal-keys : ∀ (fm₁ fm₂ : FiniteMap) → (Map.keys (proj₁ fm₁) ≡ Map.keys (proj₁ fm₂))
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all-equal-keys (fm₁ , km₁≡ks) (fm₂ , km₂≡ks) = trans km₁≡ks (sym km₂≡ks)
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∈k-exclusive : ∀ (fm₁ fm₂ : FiniteMap) {k : A} → ¬ ((k ∈k fm₁) × (¬ k ∈k fm₂))
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∈k-exclusive fm₁ fm₂ {k} (k∈kfm₁ , k∉kfm₂) =
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let
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k∈kfm₂ = subst (λ l → k ∈ˡ l) (all-equal-keys fm₁ fm₂) k∈kfm₁
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in
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k∉kfm₂ k∈kfm₂
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m₁≼m₂⇒m₁[ks]≼m₂[ks] : ∀ (fm₁ fm₂ : FiniteMap) (ks' : List A) →
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fm₁ ≼ fm₂ → Pairwise _≼₂_ (fm₁ [ ks' ]) (fm₂ [ ks' ])
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m₁≼m₂⇒m₁[ks]≼m₂[ks] _ _ [] _ = []
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m₁≼m₂⇒m₁[ks]≼m₂[ks] fm₁@(m₁ , km₁≡ks) fm₂@(m₂ , km₂≡ks) (k ∷ ks'') m₁≼m₂
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with ∈k-dec k (proj₁ m₁) | ∈k-dec k (proj₁ m₂)
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... | yes k∈km₁ | yes k∈km₂ =
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let
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(v₁ , k,v₁∈m₁) = locateᵐ {m = m₁} k∈km₁
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(v₂ , k,v₂∈m₂) = locateᵐ {m = m₂} k∈km₂
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in
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(m₁≼m₂⇒m₁[k]≼m₂[k] m₁ m₂ m₁≼m₂ k,v₁∈m₁ k,v₂∈m₂) ∷ m₁≼m₂⇒m₁[ks]≼m₂[ks] fm₁ fm₂ ks'' m₁≼m₂
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... | no k∉km₁ | no k∉km₂ = m₁≼m₂⇒m₁[ks]≼m₂[ks] fm₁ fm₂ ks'' m₁≼m₂
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... | yes k∈km₁ | no k∉km₂ = ⊥-elim (∈k-exclusive fm₁ fm₂ (k∈km₁ , k∉km₂))
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... | no k∉km₁ | yes k∈km₂ = ⊥-elim (∈k-exclusive fm₂ fm₁ (k∈km₂ , k∉km₁))
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