636 lines
33 KiB
Agda
636 lines
33 KiB
Agda
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.Unit using (⊤)
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module Lattice.FiniteMap (A : Set) (B : Set)
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{_≈₂_ : B → B → Set}
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{_⊔₂_ : B → B → B} {_⊓₂_ : B → B → B}
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{{≡-Decidable-A : IsDecidable {_} {A} _≡_}}
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{{lB : IsLattice B _≈₂_ _⊔₂_ _⊓₂_}} (ks : List A) where
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open IsLattice lB using () renaming (_≼_ to _≼₂_)
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open import Lattice.Map A B _ as Map
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using
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( Map
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; ⊔-equal-keys
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; ⊓-equal-keys
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; subset-impl
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; Map-functional
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; Expr-Provenance
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; Expr-Provenance-≡
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; `_; _∪_; _∩_
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; in₁; in₂; bothᵘ; single
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; ⊔-combines
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)
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renaming
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( _≈_ to _≈ᵐ_
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; _⊔_ to _⊔ᵐ_
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; _⊓_ to _⊓ᵐ_
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; ≈-equiv to ≈ᵐ-equiv
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; ≈-⊔-cong to ≈ᵐ-⊔ᵐ-cong
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; ⊔-assoc to ⊔ᵐ-assoc
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; ⊔-comm to ⊔ᵐ-comm
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; ⊔-idemp to ⊔ᵐ-idemp
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; ≈-⊓-cong to ≈ᵐ-⊓ᵐ-cong
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; ⊓-assoc to ⊓ᵐ-assoc
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; ⊓-comm to ⊓ᵐ-comm
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; ⊓-idemp to ⊓ᵐ-idemp
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; absorb-⊔-⊓ to absorb-⊔ᵐ-⊓ᵐ
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; absorb-⊓-⊔ to absorb-⊓ᵐ-⊔ᵐ
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; ≈-Decidable to ≈ᵐ-Decidable
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; _[_] to _[_]ᵐ
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; []-∈ to []ᵐ-∈
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; m₁≼m₂⇒m₁[k]≼m₂[k] to m₁≼m₂⇒m₁[k]ᵐ≼m₂[k]ᵐ
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; m₁≈m₂⇒k∈m₁⇒k∈km₂⇒v₁≈v₂ to m₁≈m₂⇒k∈m₁⇒k∈km₂⇒v₁≈v₂ᵐ
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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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; updating-via-k∈ks to updatingᵐ-via-k∈ks
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; updating-via-k∈ks-≡ to updatingᵐ-via-k∈ks-≡
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; updating-via-∈k-forward to updatingᵐ-via-∈k-forward
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; updating-via-k∉ks-forward to updatingᵐ-via-k∉ks-forward
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; updating-via-k∉ks-backward to updatingᵐ-via-k∉ks-backward
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; f'-Monotonic to f'-Monotonicᵐ
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; _≼_ to _≼ᵐ_
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; ∈k-dec to ∈k-decᵐ
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)
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open import Data.Empty using (⊥-elim)
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open import Data.List using (List; length; []; _∷_; map)
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open import Data.List.Membership.Propositional using () renaming (_∈_ to _∈ˡ_)
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open import Data.List.Properties using (∷-injectiveʳ)
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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)
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open import Data.Nat using (ℕ)
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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; _∷_; []; Unique; push; empty; All¬-¬Any)
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open import Showable using (Showable; show)
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open import Isomorphism using (IsInverseˡ; IsInverseʳ)
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open import Chain using (Height)
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private module WithKeys (ks : List A) where
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FiniteMap : Set
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FiniteMap = Σ Map (λ m → Map.keys m ≡ ks)
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instance
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showable : {{ showableA : Showable A }} {{ showableB : Showable B }} →
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Showable FiniteMap
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showable = record { show = λ (m₁ , _) → show m₁ }
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_≈_ : FiniteMap → FiniteMap → Set
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_≈_ (m₁ , _) (m₂ , _) = m₁ ≈ᵐ m₂
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instance
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≈-Decidable : {{ IsDecidable _≈₂_ }} → IsDecidable _≈_
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≈-Decidable {{≈₂-Decidable}} = record
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{ R-dec = λ fm₁ fm₂ → IsDecidable.R-dec (≈ᵐ-Decidable {{≈₂-Decidable}})
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(proj₁ fm₁) (proj₁ fm₂)
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}
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_⊔_ : FiniteMap → FiniteMap → FiniteMap
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_⊔_ (m₁ , km₁≡ks) (m₂ , km₂≡ks) =
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( m₁ ⊔ᵐ m₂
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, trans (sym (⊔-equal-keys {m₁} {m₂} (trans (km₁≡ks) (sym km₂≡ks))))
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km₁≡ks
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)
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_⊓_ : FiniteMap → FiniteMap → FiniteMap
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_⊓_ (m₁ , km₁≡ks) (m₂ , km₂≡ks) =
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( m₁ ⊓ᵐ m₂
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, trans (sym (⊓-equal-keys {m₁} {m₂} (trans (km₁≡ks) (sym km₂≡ks))))
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km₁≡ks
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)
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_∈_ : A × B → FiniteMap → Set
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_∈_ k,v (m₁ , _) = k,v ∈ˡ (proj₁ m₁)
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_∈k_ : A → FiniteMap → Set
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_∈k_ k (m₁ , _) = k ∈ˡ (keysᵐ m₁)
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open Map using (forget) public
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∈k-dec = ∈k-decᵐ
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locate : ∀ {k : A} {fm : FiniteMap} → k ∈k fm → Σ B (λ v → (k , v) ∈ fm)
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locate {k} {fm = (m , _)} k∈kfm = locateᵐ {k} {m} k∈kfm
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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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[]-∈ : ∀ {k : A} {v : B} {ks' : List A} (fm : FiniteMap) →
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k ∈ˡ ks' → (k , v) ∈ fm → v ∈ˡ (fm [ ks' ])
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[]-∈ {k} {v} {ks'} (m , _) k∈ks' k,v∈fm = []ᵐ-∈ m k,v∈fm k∈ks'
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≈-equiv : IsEquivalence FiniteMap _≈_
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≈-equiv = record
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{ ≈-refl =
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λ {(m , _)} → IsEquivalence.≈-refl ≈ᵐ-equiv {m}
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; ≈-sym =
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λ {(m₁ , _)} {(m₂ , _)} → IsEquivalence.≈-sym ≈ᵐ-equiv {m₁} {m₂}
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; ≈-trans =
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λ {(m₁ , _)} {(m₂ , _)} {(m₃ , _)} →
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IsEquivalence.≈-trans ≈ᵐ-equiv {m₁} {m₂} {m₃}
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}
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open IsEquivalence ≈-equiv public
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instance
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isUnionSemilattice : IsSemilattice FiniteMap _≈_ _⊔_
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isUnionSemilattice = record
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{ ≈-equiv = ≈-equiv
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; ≈-⊔-cong =
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λ {(m₁ , _)} {(m₂ , _)} {(m₃ , _)} {(m₄ , _)} m₁≈m₂ m₃≈m₄ →
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≈ᵐ-⊔ᵐ-cong {m₁} {m₂} {m₃} {m₄} m₁≈m₂ m₃≈m₄
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; ⊔-assoc = λ (m₁ , _) (m₂ , _) (m₃ , _) → ⊔ᵐ-assoc m₁ m₂ m₃
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; ⊔-comm = λ (m₁ , _) (m₂ , _) → ⊔ᵐ-comm m₁ m₂
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; ⊔-idemp = λ (m , _) → ⊔ᵐ-idemp m
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}
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isIntersectSemilattice : IsSemilattice FiniteMap _≈_ _⊓_
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isIntersectSemilattice = record
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{ ≈-equiv = ≈-equiv
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; ≈-⊔-cong =
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λ {(m₁ , _)} {(m₂ , _)} {(m₃ , _)} {(m₄ , _)} m₁≈m₂ m₃≈m₄ →
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≈ᵐ-⊓ᵐ-cong {m₁} {m₂} {m₃} {m₄} m₁≈m₂ m₃≈m₄
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; ⊔-assoc = λ (m₁ , _) (m₂ , _) (m₃ , _) → ⊓ᵐ-assoc m₁ m₂ m₃
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; ⊔-comm = λ (m₁ , _) (m₂ , _) → ⊓ᵐ-comm m₁ m₂
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; ⊔-idemp = λ (m , _) → ⊓ᵐ-idemp m
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}
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isLattice : IsLattice FiniteMap _≈_ _⊔_ _⊓_
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isLattice = record
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{ joinSemilattice = isUnionSemilattice
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; meetSemilattice = isIntersectSemilattice
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; absorb-⊔-⊓ = λ (m₁ , _) (m₂ , _) → absorb-⊔ᵐ-⊓ᵐ m₁ m₂
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; absorb-⊓-⊔ = λ (m₁ , _) (m₂ , _) → absorb-⊓ᵐ-⊔ᵐ m₁ m₂
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}
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lattice : Lattice FiniteMap
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lattice = record
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{ _≈_ = _≈_
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; _⊔_ = _⊔_
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; _⊓_ = _⊓_
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; isLattice = isLattice
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}
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open IsLattice isLattice using (_≼_; ⊔-Monotonicˡ; ⊔-Monotonicʳ) public
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m₁≼m₂⇒m₁[k]≼m₂[k] : ∀ (fm₁ fm₂ : FiniteMap) {k : A} {v₁ v₂ : B} →
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fm₁ ≼ fm₂ → (k , v₁) ∈ fm₁ → (k , v₂) ∈ fm₂ → v₁ ≼₂ v₂
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m₁≼m₂⇒m₁[k]≼m₂[k] (m₁ , _) (m₂ , _) m₁≼m₂ k,v₁∈m₁ k,v₂∈m₂ = m₁≼m₂⇒m₁[k]ᵐ≼m₂[k]ᵐ m₁ m₂ m₁≼m₂ k,v₁∈m₁ k,v₂∈m₂
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m₁≈m₂⇒k∈m₁⇒k∈km₂⇒v₁≈v₂ : ∀ (fm₁ fm₂ : FiniteMap) {k : A} →
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fm₁ ≈ fm₂ → ∀ (k∈kfm₁ : k ∈k fm₁) (k∈kfm₂ : k ∈k fm₂) →
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proj₁ (locate {fm = fm₁} k∈kfm₁) ≈₂ proj₁ (locate {fm = fm₂} k∈kfm₂)
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m₁≈m₂⇒k∈m₁⇒k∈km₂⇒v₁≈v₂ (m₁ , _) (m₂ , _) = m₁≈m₂⇒k∈m₁⇒k∈km₂⇒v₁≈v₂ᵐ m₁ m₂
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module GeneralizedUpdate
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{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_}}
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(f : L → FiniteMap) (f-Monotonic : Monotonic (IsLattice._≼_ lL) _≼_ f)
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(g : A → L → B) (g-Monotonicʳ : ∀ k → Monotonic (IsLattice._≼_ lL) _≼₂_ (g k))
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(ks : List A) where
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open IsLattice lL using () renaming (_≼_ to _≼ˡ_)
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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 (proj₁ ∘ f) f-Monotonic g g-Monotonicʳ ks l₁≼l₂
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f'-∈k-forward : ∀ {k l} → k ∈k (f l) → k ∈k (f' l)
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f'-∈k-forward {k} {l} = updatingᵐ-via-∈k-forward (proj₁ (f l)) ks (updater l)
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f'-k∈ks : ∀ {k l} → k ∈ˡ ks → k ∈k (f' l) → (k , updater l k) ∈ (f' l)
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f'-k∈ks {k} {l} = updatingᵐ-via-k∈ks (proj₁ (f l)) (updater l)
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f'-k∈ks-≡ : ∀ {k v l} → k ∈ˡ ks → (k , v) ∈ (f' l) → v ≡ updater l k
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f'-k∈ks-≡ {k} {v} {l} = updatingᵐ-via-k∈ks-≡ (proj₁ (f l)) (updater l)
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f'-k∉ks-forward : ∀ {k v l} → ¬ k ∈ˡ ks → (k , v) ∈ (f l) → (k , v) ∈ (f' l)
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f'-k∉ks-forward {k} {v} {l} = updatingᵐ-via-k∉ks-forward (proj₁ (f l)) (updater l)
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f'-k∉ks-backward : ∀ {k v l} → ¬ k ∈ˡ ks → (k , v) ∈ (f' l) → (k , v) ∈ (f l)
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f'-k∉ks-backward {k} {v} {l} = updatingᵐ-via-k∉ks-backward (proj₁ (f l)) (updater l)
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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-decᵐ k (proj₁ m₁) | ∈k-decᵐ 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]ᵐ≼m₂[k]ᵐ 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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private
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_⊆ᵐ_ : ∀ {ks₁ ks₂ : List A} → WithKeys.FiniteMap ks₁ → WithKeys.FiniteMap ks₂ → Set
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_⊆ᵐ_ fm₁ fm₂ = subset-impl (proj₁ (proj₁ fm₁)) (proj₁ (proj₁ fm₂))
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_∈ᵐ_ : ∀ {ks : List A} → A × B → WithKeys.FiniteMap ks → Set
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_∈ᵐ_ {ks} = WithKeys._∈_ ks
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FromBothMaps : ∀ (k : A) (v : B) {ks : List A} (fm₁ fm₂ : WithKeys.FiniteMap ks) → Set
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FromBothMaps k v fm₁ fm₂ =
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Σ (B × B)
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(λ (v₁ , v₂) → ( (v ≡ v₁ ⊔₂ v₂) × ((k , v₁) ∈ᵐ fm₁ × (k , v₂) ∈ᵐ fm₂)))
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Provenance-union : ∀ {ks : List A} (fm₁ fm₂ : WithKeys.FiniteMap ks) {k : A} {v : B} →
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(k , v) ∈ᵐ (WithKeys._⊔_ ks fm₁ fm₂) → FromBothMaps k v fm₁ fm₂
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Provenance-union fm₁@(m₁ , ks₁≡ks) fm₂@(m₂ , ks₂≡ks) {k} {v} k,v∈fm₁fm₂
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with Expr-Provenance-≡ ((` m₁) ∪ (` m₂)) k,v∈fm₁fm₂
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... | in₁ (single k,v∈m₁) k∉km₂
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with k∈km₁ ← (WithKeys.forget k,v∈m₁)
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rewrite trans ks₁≡ks (sym ks₂≡ks) =
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⊥-elim (k∉km₂ k∈km₁)
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... | in₂ k∉km₁ (single k,v∈m₂)
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with k∈km₂ ← (WithKeys.forget k,v∈m₂)
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rewrite trans ks₁≡ks (sym ks₂≡ks) =
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⊥-elim (k∉km₁ k∈km₂)
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... | bothᵘ {v₁} {v₂} (single k,v₁∈m₁) (single k,v₂∈m₂) =
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((v₁ , v₂) , (refl , (k,v₁∈m₁ , k,v₂∈m₂)))
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private module IterProdIsomorphism where
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open WithKeys
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open import Data.Unit using (tt)
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open import Lattice.Unit using ()
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renaming
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( _≈_ to _≈ᵘ_
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; _⊔_ to _⊔ᵘ_
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; _⊓_ to _⊓ᵘ_
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; ≈-Decidable to ≈ᵘ-Decidable
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; isLattice to isLatticeᵘ
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; ≈-equiv to ≈ᵘ-equiv
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; fixedHeight to fixedHeightᵘ
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)
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open import Lattice.IterProd B ⊤ _
|
||
as IP
|
||
using (IterProd)
|
||
open IsLattice lB using ()
|
||
renaming
|
||
( ≈-trans to ≈₂-trans
|
||
; ≈-sym to ≈₂-sym
|
||
; FixedHeight to FixedHeight₂
|
||
)
|
||
|
||
from : ∀ {ks : List A} → FiniteMap ks → IterProd (length ks)
|
||
from {[]} (([] , _) , _) = tt
|
||
from {k ∷ ks'} (((k' , v) ∷ fm' , push _ uks') , refl) =
|
||
(v , from ((fm' , uks'), refl))
|
||
|
||
to : ∀ {ks : List A} → Unique ks → IterProd (length ks) → FiniteMap ks
|
||
to {[]} _ ⊤ = (([] , empty) , refl)
|
||
to {k ∷ ks'} (push k≢ks' uks') (v , rest) =
|
||
let
|
||
((fm' , ufm') , fm'≡ks') = to uks' rest
|
||
|
||
-- This would be easier if we pattern matched on the equiality proof
|
||
-- to get refl, but that makes it harder to reason about 'to' when
|
||
-- the arguments are not known to be refl.
|
||
k≢fm' = subst (λ ks → All (λ k' → ¬ k ≡ k') ks) (sym fm'≡ks') k≢ks'
|
||
kvs≡ks = cong (k ∷_) fm'≡ks'
|
||
in
|
||
(((k , v) ∷ fm' , push k≢fm' ufm') , kvs≡ks)
|
||
|
||
_≈ⁱᵖ_ : ∀ {n : ℕ} → IterProd n → IterProd n → Set
|
||
_≈ⁱᵖ_ {n} = IP._≈_ {n}
|
||
|
||
_⊔ⁱᵖ_ : ∀ {ks : List A} →
|
||
IterProd (length ks) → IterProd (length ks) → IterProd (length ks)
|
||
_⊔ⁱᵖ_ {ks} = IP._⊔_ {length ks}
|
||
|
||
to-build : ∀ {b : B} {ks : List A} (uks : Unique ks) →
|
||
let fm = to uks (IP.build b tt (length ks))
|
||
in ∀ (k : A) (v : B) → (k , v) ∈ᵐ fm → v ≡ b
|
||
to-build {b} {k ∷ ks'} (push _ uks') k v (here refl) = refl
|
||
to-build {b} {k ∷ ks'} (push _ uks') k' v (there k',v∈m') =
|
||
to-build {ks = ks'} uks' k' v k',v∈m'
|
||
|
||
|
||
-- The left inverse is: from (to x) = x
|
||
from-to-inverseˡ : ∀ {ks : List A} (uks : Unique ks) →
|
||
IsInverseˡ (_≈_ ks) (_≈ⁱᵖ_ {length ks})
|
||
(from {ks}) (to {ks} uks)
|
||
from-to-inverseˡ {[]} _ _ = IsEquivalence.≈-refl (IP.≈-equiv {0})
|
||
from-to-inverseˡ {k ∷ ks'} (push k≢ks' uks') (v , rest)
|
||
with ((fm' , ufm') , refl) ← to uks' rest in p rewrite sym p =
|
||
(IsLattice.≈-refl lB , from-to-inverseˡ {ks'} uks' rest)
|
||
-- the rewrite here is needed because the IH is in terms of `to uks' rest`,
|
||
-- but we end up with the 'unpacked' form (fm', ...). So, put it back
|
||
-- in the 'packed' form after we've performed enough inspection
|
||
-- to know we take the cons branch of `to`.
|
||
|
||
-- The map has its own uniqueness proof, but the call to 'to' needs a standalone
|
||
-- uniqueness proof too. Work with both proofs as needed to thread things through.
|
||
--
|
||
-- The right inverse is: to (from x) = x
|
||
from-to-inverseʳ : ∀ {ks : List A} (uks : Unique ks) →
|
||
IsInverseʳ (_≈_ ks) (_≈ⁱᵖ_ {length ks})
|
||
(from {ks}) (to {ks} uks)
|
||
from-to-inverseʳ {[]} _ (([] , empty) , kvs≡ks) rewrite kvs≡ks =
|
||
( (λ k v ())
|
||
, (λ k v ())
|
||
)
|
||
from-to-inverseʳ {k ∷ ks'} uks@(push _ uks'₁) fm₁@(((k , v) ∷ fm'₁ , push _ uks'₂) , refl)
|
||
with to uks'₁ (from ((fm'₁ , uks'₂) , refl))
|
||
| from-to-inverseʳ {ks'} uks'₁ ((fm'₁ , uks'₂) , refl)
|
||
... | ((fm'₂ , ufm'₂) , _)
|
||
| (fm'₂⊆fm'₁ , fm'₁⊆fm'₂) = (m₂⊆m₁ , m₁⊆m₂)
|
||
where
|
||
kvs₁ = (k , v) ∷ fm'₁
|
||
kvs₂ = (k , v) ∷ fm'₂
|
||
|
||
m₁⊆m₂ : subset-impl kvs₁ kvs₂
|
||
m₁⊆m₂ k' v' (here refl) =
|
||
(v' , (IsLattice.≈-refl lB , here refl))
|
||
m₁⊆m₂ k' v' (there k',v'∈fm'₁) =
|
||
let (v'' , (v'≈v'' , k',v''∈fm'₂)) =
|
||
fm'₁⊆fm'₂ k' v' k',v'∈fm'₁
|
||
in (v'' , (v'≈v'' , there k',v''∈fm'₂))
|
||
|
||
m₂⊆m₁ : subset-impl kvs₂ kvs₁
|
||
m₂⊆m₁ k' v' (here refl) =
|
||
(v' , (IsLattice.≈-refl lB , here refl))
|
||
m₂⊆m₁ k' v' (there k',v'∈fm'₂) =
|
||
let (v'' , (v'≈v'' , k',v''∈fm'₁)) =
|
||
fm'₂⊆fm'₁ k' v' k',v'∈fm'₂
|
||
in (v'' , (v'≈v'' , there k',v''∈fm'₁))
|
||
|
||
private
|
||
first-key-in-map : ∀ {k : A} {ks : List A} (fm : FiniteMap (k ∷ ks)) →
|
||
Σ B (λ v → (k , v) ∈ᵐ fm)
|
||
first-key-in-map (((k , v) ∷ _ , _) , refl) = (v , here refl)
|
||
|
||
from-first-value : ∀ {k : A} {ks : List A} (fm : FiniteMap (k ∷ ks)) →
|
||
proj₁ (from fm) ≡ proj₁ (first-key-in-map fm)
|
||
from-first-value {k} {ks} (((k , v) ∷ _ , push _ _) , refl) = refl
|
||
|
||
-- We need pop because reasoning about two distinct 'refl' pattern
|
||
-- matches is giving us unification errors. So, stash the 'refl' pattern
|
||
-- matching into a helper functions, and write solutions in terms
|
||
-- of that.
|
||
pop : ∀ {k : A} {ks : List A} → FiniteMap (k ∷ ks) → FiniteMap ks
|
||
pop (((_ ∷ fm') , push _ ufm') , refl) = ((fm' , ufm') , refl)
|
||
|
||
pop-≈ : ∀ {k : A} {ks : List A} (fm₁ fm₂ : FiniteMap (k ∷ ks)) →
|
||
_≈_ _ fm₁ fm₂ → _≈_ _ (pop fm₁) (pop fm₂)
|
||
pop-≈ {k} {ks} fm₁ fm₂ (fm₁⊆fm₂ , fm₂⊆fm₁) =
|
||
(narrow fm₁⊆fm₂ , narrow fm₂⊆fm₁)
|
||
where
|
||
narrow₁ : ∀ {fm₁ fm₂ : FiniteMap (k ∷ ks)} →
|
||
fm₁ ⊆ᵐ fm₂ → pop fm₁ ⊆ᵐ fm₂
|
||
narrow₁ {(_ ∷ _ , push _ _) , refl} kvs₁⊆kvs₂ k' v' k',v'∈fm'₁ =
|
||
kvs₁⊆kvs₂ k' v' (there k',v'∈fm'₁)
|
||
|
||
narrow₂ : ∀ {fm₁ : FiniteMap ks} {fm₂ : FiniteMap (k ∷ ks)} →
|
||
fm₁ ⊆ᵐ fm₂ → fm₁ ⊆ᵐ pop fm₂
|
||
narrow₂ {fm₁} {fm₂ = (_ ∷ fm'₂ , push k≢ks _) , kvs≡ks@refl} kvs₁⊆kvs₂ k' v' k',v'∈fm'₁
|
||
with kvs₁⊆kvs₂ k' v' k',v'∈fm'₁
|
||
... | (v'' , (v'≈v'' , here refl)) rewrite sym (proj₂ fm₁) =
|
||
⊥-elim (All¬-¬Any k≢ks (forget k',v'∈fm'₁))
|
||
... | (v'' , (v'≈v'' , there k',v'∈fm'₂)) =
|
||
(v'' , (v'≈v'' , k',v'∈fm'₂))
|
||
|
||
narrow : ∀ {fm₁ fm₂ : FiniteMap (k ∷ ks)} →
|
||
fm₁ ⊆ᵐ fm₂ → pop fm₁ ⊆ᵐ pop fm₂
|
||
narrow {fm₁} {fm₂} x = narrow₂ {pop fm₁} (narrow₁ {fm₂ = fm₂} x)
|
||
|
||
k,v∈pop⇒k,v∈ : ∀ {k : A} {ks : List A} {k' : A} {v : B} (fm : FiniteMap (k ∷ ks)) →
|
||
(k' , v) ∈ᵐ pop fm → (¬ k ≡ k' × ((k' , v) ∈ᵐ fm))
|
||
k,v∈pop⇒k,v∈ {k} {ks} {k'} {v} (m@((k , _) ∷ fm' , push k≢ks uks') , refl) k',v∈fm =
|
||
( (λ { refl → All¬-¬Any k≢ks (forget k',v∈fm) })
|
||
, there k',v∈fm
|
||
)
|
||
|
||
k,v∈⇒k,v∈pop : ∀ {k : A} {ks : List A} {k' : A} {v : B} (fm : FiniteMap (k ∷ ks)) →
|
||
¬ k ≡ k' → (k' , v) ∈ᵐ fm → (k' , v) ∈ᵐ pop fm
|
||
k,v∈⇒k,v∈pop (m@(_ ∷ _ , push k≢ks _) , refl) k≢k' (here refl) = ⊥-elim (k≢k' refl)
|
||
k,v∈⇒k,v∈pop (m@(_ ∷ _ , push k≢ks _) , refl) k≢k' (there k,v'∈fm') = k,v'∈fm'
|
||
|
||
pop-⊔-distr : ∀ {k : A} {ks : List A} (fm₁ fm₂ : FiniteMap (k ∷ ks)) →
|
||
_≈_ _ (pop (_⊔_ _ fm₁ fm₂)) ((_⊔_ _ (pop fm₁) (pop fm₂)))
|
||
pop-⊔-distr {k} {ks} fm₁@(m₁ , _) fm₂@(m₂ , _) =
|
||
(pfm₁fm₂⊆pfm₁pfm₂ , pfm₁pfm₂⊆pfm₁fm₂)
|
||
where
|
||
-- pfm₁fm₂⊆pfm₁pfm₂ = {!!}
|
||
pfm₁fm₂⊆pfm₁pfm₂ : pop (_⊔_ _ fm₁ fm₂) ⊆ᵐ (_⊔_ _ (pop fm₁) (pop fm₂))
|
||
pfm₁fm₂⊆pfm₁pfm₂ k' v' k',v'∈pfm₁fm₂
|
||
with (k≢k' , k',v'∈fm₁fm₂) ← k,v∈pop⇒k,v∈ (_⊔_ _ fm₁ fm₂) k',v'∈pfm₁fm₂
|
||
with ((v₁ , v₂) , (refl , (k,v₁∈fm₁ , k,v₂∈fm₂)))
|
||
← Provenance-union fm₁ fm₂ k',v'∈fm₁fm₂
|
||
with k',v₁∈pfm₁ ← k,v∈⇒k,v∈pop fm₁ k≢k' k,v₁∈fm₁
|
||
with k',v₂∈pfm₂ ← k,v∈⇒k,v∈pop fm₂ k≢k' k,v₂∈fm₂
|
||
=
|
||
( v₁ ⊔₂ v₂
|
||
, (IsLattice.≈-refl lB
|
||
, ⊔-combines {m₁ = proj₁ (pop fm₁)}
|
||
{m₂ = proj₁ (pop fm₂)}
|
||
k',v₁∈pfm₁ k',v₂∈pfm₂
|
||
)
|
||
)
|
||
|
||
pfm₁pfm₂⊆pfm₁fm₂ : (_⊔_ _ (pop fm₁) (pop fm₂)) ⊆ᵐ pop (_⊔_ _ fm₁ fm₂)
|
||
pfm₁pfm₂⊆pfm₁fm₂ k' v' k',v'∈pfm₁pfm₂
|
||
with ((v₁ , v₂) , (refl , (k,v₁∈pfm₁ , k,v₂∈pfm₂)))
|
||
← Provenance-union (pop fm₁) (pop fm₂) k',v'∈pfm₁pfm₂
|
||
with (k≢k' , k',v₁∈fm₁) ← k,v∈pop⇒k,v∈ fm₁ k,v₁∈pfm₁
|
||
with (_ , k',v₂∈fm₂) ← k,v∈pop⇒k,v∈ fm₂ k,v₂∈pfm₂
|
||
=
|
||
( v₁ ⊔₂ v₂
|
||
, ( IsLattice.≈-refl lB
|
||
, k,v∈⇒k,v∈pop (_⊔_ _ fm₁ fm₂) k≢k'
|
||
(⊔-combines {m₁ = m₁} {m₂ = m₂}
|
||
k',v₁∈fm₁ k',v₂∈fm₂)
|
||
)
|
||
)
|
||
|
||
from-rest : ∀ {k : A} {ks : List A} (fm : FiniteMap (k ∷ ks)) →
|
||
proj₂ (from fm) ≡ from (pop fm)
|
||
from-rest (((_ ∷ fm') , push _ ufm') , refl) = refl
|
||
|
||
from-preserves-≈ : ∀ {ks : List A} → {fm₁ fm₂ : FiniteMap ks} →
|
||
_≈_ _ fm₁ fm₂ → (_≈ⁱᵖ_ {length ks}) (from fm₁) (from fm₂)
|
||
from-preserves-≈ {[]} {_} {_} _ = IsEquivalence.≈-refl ≈ᵘ-equiv
|
||
from-preserves-≈ {k ∷ ks'} {fm₁@(m₁ , _)} {fm₂@(m₂ , _)} fm₁≈fm₂@(kvs₁⊆kvs₂ , kvs₂⊆kvs₁)
|
||
with first-key-in-map fm₁
|
||
| first-key-in-map fm₂
|
||
| from-first-value fm₁
|
||
| from-first-value fm₂
|
||
... | (v₁ , k,v₁∈fm₁) | (v₂ , k,v₂∈fm₂) | refl | refl
|
||
with kvs₁⊆kvs₂ _ _ k,v₁∈fm₁
|
||
... | (v₁' , (v₁≈v₁' , k,v₁'∈fm₂))
|
||
rewrite Map-functional {m = m₂} k,v₂∈fm₂ k,v₁'∈fm₂
|
||
rewrite from-rest fm₁ rewrite from-rest fm₂
|
||
=
|
||
( v₁≈v₁'
|
||
, from-preserves-≈ {ks'} {pop fm₁} {pop fm₂}
|
||
(pop-≈ fm₁ fm₂ fm₁≈fm₂)
|
||
)
|
||
|
||
to-preserves-≈ : ∀ {ks : List A} (uks : Unique ks) {ip₁ ip₂ : IterProd (length ks)} →
|
||
_≈ⁱᵖ_ {length ks} ip₁ ip₂ → _≈_ _ (to uks ip₁) (to uks ip₂)
|
||
to-preserves-≈ {[]} empty {tt} {tt} _ = ((λ k v ()), (λ k v ()))
|
||
to-preserves-≈ {k ∷ ks'} uks@(push k≢ks' uks') {ip₁@(v₁ , rest₁)} {ip₂@(v₂ , rest₂)} (v₁≈v₂ , rest₁≈rest₂) = (fm₁⊆fm₂ , fm₂⊆fm₁)
|
||
where
|
||
inductive-step : ∀ {v₁ v₂ : B} {rest₁ rest₂ : IterProd (length ks')} →
|
||
v₁ ≈₂ v₂ → _≈ⁱᵖ_ {length ks'} rest₁ rest₂ →
|
||
to uks (v₁ , rest₁) ⊆ᵐ to uks (v₂ , rest₂)
|
||
inductive-step {v₁} {v₂} {rest₁} {rest₂} v₁≈v₂ rest₁≈rest₂ k v k,v∈kvs₁
|
||
with ((fm'₁ , ufm'₁) , fm'₁≡ks') ← to uks' rest₁ in p₁
|
||
with ((fm'₂ , ufm'₂) , fm'₂≡ks') ← to uks' rest₂ in p₂
|
||
with k,v∈kvs₁
|
||
... | here refl = (v₂ , (v₁≈v₂ , here refl))
|
||
... | there k,v∈fm'₁ with refl ← p₁ with refl ← p₂ =
|
||
let
|
||
(fm'₁⊆fm'₂ , _) = to-preserves-≈ uks' {rest₁} {rest₂}
|
||
rest₁≈rest₂
|
||
(v' , (v≈v' , k,v'∈kvs₁)) = fm'₁⊆fm'₂ k v k,v∈fm'₁
|
||
in
|
||
(v' , (v≈v' , there k,v'∈kvs₁))
|
||
|
||
fm₁⊆fm₂ : to uks ip₁ ⊆ᵐ to uks ip₂
|
||
fm₁⊆fm₂ = inductive-step v₁≈v₂ rest₁≈rest₂
|
||
|
||
fm₂⊆fm₁ : to uks ip₂ ⊆ᵐ to uks ip₁
|
||
fm₂⊆fm₁ = inductive-step (≈₂-sym v₁≈v₂)
|
||
(IP.≈-sym {length ks'} rest₁≈rest₂)
|
||
|
||
from-⊔-distr : ∀ {ks : List A} → (fm₁ fm₂ : FiniteMap ks) →
|
||
_≈ⁱᵖ_ {length ks} (from (_⊔_ _ fm₁ fm₂))
|
||
(_⊔ⁱᵖ_ {ks} (from fm₁) (from fm₂))
|
||
from-⊔-distr {[]} fm₁ fm₂ = IsEquivalence.≈-refl ≈ᵘ-equiv
|
||
from-⊔-distr {k ∷ ks} fm₁@(m₁ , _) fm₂@(m₂ , _)
|
||
with first-key-in-map (_⊔_ _ fm₁ fm₂)
|
||
| first-key-in-map fm₁
|
||
| first-key-in-map fm₂
|
||
| from-first-value (_⊔_ _ fm₁ fm₂)
|
||
| from-first-value fm₁ | from-first-value fm₂
|
||
... | (v , k,v∈fm₁fm₂) | (v₁ , k,v₁∈fm₁) | (v₂ , k,v₂∈fm₂) | refl | refl | refl
|
||
with Expr-Provenance k ((` m₁) ∪ (` m₂)) (forget k,v∈fm₁fm₂)
|
||
... | (_ , (in₁ _ k∉km₂ , _)) = ⊥-elim (k∉km₂ (forget k,v₂∈fm₂))
|
||
... | (_ , (in₂ k∉km₁ _ , _)) = ⊥-elim (k∉km₁ (forget k,v₁∈fm₁))
|
||
... | (v₁⊔v₂ , (bothᵘ {v₁'} {v₂'} (single k,v₁'∈m₁) (single k,v₂'∈m₂) , k,v₁⊔v₂∈m₁m₂))
|
||
rewrite Map-functional {m = m₁} k,v₁∈fm₁ k,v₁'∈m₁
|
||
rewrite Map-functional {m = m₂} k,v₂∈fm₂ k,v₂'∈m₂
|
||
rewrite Map-functional {m = proj₁ (_⊔_ _ fm₁ fm₂)} k,v∈fm₁fm₂ k,v₁⊔v₂∈m₁m₂
|
||
rewrite from-rest (_⊔_ _ fm₁ fm₂) rewrite from-rest fm₁ rewrite from-rest fm₂
|
||
= ( IsLattice.≈-refl lB
|
||
, IsEquivalence.≈-trans
|
||
(IP.≈-equiv {length ks})
|
||
(from-preserves-≈ {_} {pop (_⊔_ _ fm₁ fm₂)}
|
||
{_⊔_ _ (pop fm₁) (pop fm₂)}
|
||
(pop-⊔-distr fm₁ fm₂))
|
||
((from-⊔-distr (pop fm₁) (pop fm₂)))
|
||
)
|
||
|
||
|
||
to-⊔-distr : ∀ {ks : List A} (uks : Unique ks) → (ip₁ ip₂ : IterProd (length ks)) →
|
||
_≈_ _ (to uks (_⊔ⁱᵖ_ {ks} ip₁ ip₂)) ((_⊔_ _ (to uks ip₁) (to uks ip₂)))
|
||
to-⊔-distr {[]} empty tt tt = ((λ k v ()), (λ k v ()))
|
||
to-⊔-distr {ks@(k ∷ ks')} uks@(push k≢ks' uks') ip₁@(v₁ , rest₁) ip₂@(v₂ , rest₂) = (fm⊆fm₁fm₂ , fm₁fm₂⊆fm)
|
||
where
|
||
fm₁ = to uks ip₁
|
||
fm₁' = to uks' rest₁
|
||
fm₂ = to uks ip₂
|
||
fm₂' = to uks' rest₂
|
||
fm = to uks (_⊔ⁱᵖ_ {k ∷ ks'} ip₁ ip₂)
|
||
|
||
fm⊆fm₁fm₂ : fm ⊆ᵐ (_⊔_ _ fm₁ fm₂)
|
||
fm⊆fm₁fm₂ k v (here refl) =
|
||
(v₁ ⊔₂ v₂
|
||
, (IsLattice.≈-refl lB
|
||
, ⊔-combines {k} {v₁} {v₂} {proj₁ fm₁} {proj₁ fm₂}
|
||
(here refl) (here refl)
|
||
)
|
||
)
|
||
fm⊆fm₁fm₂ k' v (there k',v∈fm')
|
||
with (fm'⊆fm'₁fm'₂ , _) ← to-⊔-distr uks' rest₁ rest₂
|
||
with (v' , (v₁⊔v₂≈v' , k',v'∈fm'₁fm'₂))
|
||
← fm'⊆fm'₁fm'₂ k' v k',v∈fm'
|
||
with (_ , (refl , (v₁∈fm'₁ , v₂∈fm'₂)))
|
||
← Provenance-union fm₁' fm₂' k',v'∈fm'₁fm'₂ =
|
||
( v'
|
||
, ( v₁⊔v₂≈v'
|
||
, ⊔-combines {m₁ = proj₁ fm₁} {m₂ = proj₁ fm₂}
|
||
(there v₁∈fm'₁) (there v₂∈fm'₂)
|
||
)
|
||
)
|
||
|
||
fm₁fm₂⊆fm : (_⊔_ _ fm₁ fm₂) ⊆ᵐ fm
|
||
fm₁fm₂⊆fm k' v k',v∈fm₁fm₂
|
||
with (_ , fm'₁fm'₂⊆fm')
|
||
← to-⊔-distr uks' rest₁ rest₂
|
||
with (_ , (refl , (v₁∈fm₁ , v₂∈fm₂)))
|
||
← Provenance-union fm₁ fm₂ k',v∈fm₁fm₂
|
||
with v₁∈fm₁ | v₂∈fm₂
|
||
... | here refl | here refl =
|
||
(v , (IsLattice.≈-refl lB , here refl))
|
||
... | here refl | there k',v₂∈fm₂' =
|
||
⊥-elim (All¬-¬Any k≢ks' (subst (k' ∈ˡ_) (proj₂ fm₂')
|
||
(forget k',v₂∈fm₂')))
|
||
... | there k',v₁∈fm₁' | here refl =
|
||
⊥-elim (All¬-¬Any k≢ks' (subst (k' ∈ˡ_) (proj₂ fm₁')
|
||
(forget k',v₁∈fm₁')))
|
||
... | there k',v₁∈fm₁' | there k',v₂∈fm₂' =
|
||
let
|
||
k',v₁v₂∈fm₁'fm₂' =
|
||
⊔-combines {m₁ = proj₁ fm₁'} {m₂ = proj₁ fm₂'}
|
||
k',v₁∈fm₁' k',v₂∈fm₂'
|
||
(v' , (v₁⊔v₂≈v' , v'∈fm')) =
|
||
fm'₁fm'₂⊆fm' _ _ k',v₁v₂∈fm₁'fm₂'
|
||
in
|
||
(v' , (v₁⊔v₂≈v' , there v'∈fm'))
|
||
|
||
module FixedHeight {ks : List A} {{≈₂-Decidable : IsDecidable _≈₂_}} {h₂ : ℕ} {{fhB : FixedHeight₂ h₂}} (uks : Unique ks) where
|
||
import Isomorphism
|
||
open Isomorphism.TransportFiniteHeight
|
||
(IP.isFiniteHeightLattice {k = length ks} {{fhB = fixedHeightᵘ}}) (isLattice ks)
|
||
{f = to uks} {g = from {ks}}
|
||
(to-preserves-≈ uks) (from-preserves-≈ {ks})
|
||
(to-⊔-distr uks) (from-⊔-distr {ks})
|
||
(from-to-inverseʳ uks) (from-to-inverseˡ uks)
|
||
using (isFiniteHeightLattice; finiteHeightLattice; fixedHeight) public
|
||
|
||
-- Helpful lemma: all entries of the 'bottom' map are assigned to bottom.
|
||
|
||
open Height (IsFiniteHeightLattice.fixedHeight isFiniteHeightLattice) using (⊥)
|
||
|
||
⊥-contains-bottoms : ∀ {k : A} {v : B} → (k , v) ∈ᵐ ⊥ → v ≡ (Height.⊥ fhB)
|
||
⊥-contains-bottoms {k} {v} k,v∈⊥
|
||
rewrite IP.⊥-built {length ks} {{fhB = fixedHeightᵘ}} =
|
||
to-build uks k v k,v∈⊥
|
||
|
||
open WithKeys ks public
|
||
module FixedHeight = IterProdIsomorphism.FixedHeight
|