From 5d54e62c3ac5599402348daacaded0a9936a2943 Mon Sep 17 00:00:00 2001 From: Danila Fedorin Date: Sat, 23 Sep 2023 15:08:04 -0700 Subject: [PATCH] Move the lattice etc. instances into Lattice.Map Signed-off-by: Danila Fedorin --- Equivalence.agda | 41 --- Lattice.agda | 69 ----- Lattice/Map.agda | 783 ++++++++++++++++++++++++----------------------- 3 files changed, 398 insertions(+), 495 deletions(-) diff --git a/Equivalence.agda b/Equivalence.agda index edac885..82abed5 100644 --- a/Equivalence.agda +++ b/Equivalence.agda @@ -35,44 +35,3 @@ module IsEquivalenceInstances where , IsEquivalence.≈-trans eB b₁≈b₂ b₂≈b₃ ) } - - module ForMap {a b} (A : Set a) (B : Set b) - (≡-dec-A : Decidable (_≡_ {a} {A})) - (_≈₂_ : B → B → Set b) - (eB : IsEquivalence B _≈₂_) where - - open import Lattice.Map A B ≡-dec-A using (Map; lift; subset) - open import Data.List using (_∷_; []) -- TODO: re-export these with nicer names from map - - open IsEquivalence eB renaming - ( ≈-refl to ≈₂-refl - ; ≈-sym to ≈₂-sym - ; ≈-trans to ≈₂-trans - ) - - _≈_ : Map → Map → Set (Agda.Primitive._⊔_ a b) - _≈_ = lift _≈₂_ - - _⊆_ : Map → Map → Set (Agda.Primitive._⊔_ a b) - _⊆_ = subset _≈₂_ - - private - ⊆-refl : (m : Map) → m ⊆ m - ⊆-refl _ k v k,v∈m = (v , (≈₂-refl , k,v∈m)) - - ⊆-trans : (m₁ m₂ m₃ : Map) → m₁ ⊆ m₂ → m₂ ⊆ m₃ → m₁ ⊆ m₃ - ⊆-trans _ _ _ m₁⊆m₂ m₂⊆m₃ k v k,v∈m₁ = - let - (v' , (v≈v' , k,v'∈m₂)) = m₁⊆m₂ k v k,v∈m₁ - (v'' , (v'≈v'' , k,v''∈m₃)) = m₂⊆m₃ k v' k,v'∈m₂ - in (v'' , (≈₂-trans v≈v' v'≈v'' , k,v''∈m₃)) - - LiftEquivalence : IsEquivalence Map _≈_ - LiftEquivalence = record - { ≈-refl = λ {m} → (⊆-refl m , ⊆-refl m) - ; ≈-sym = λ {m₁} {m₂} (m₁⊆m₂ , m₂⊆m₁) → (m₂⊆m₁ , m₁⊆m₂) - ; ≈-trans = λ {m₁} {m₂} {m₃} (m₁⊆m₂ , m₂⊆m₁) (m₂⊆m₃ , m₃⊆m₂) → - ( ⊆-trans m₁ m₂ m₃ m₁⊆m₂ m₂⊆m₃ - , ⊆-trans m₃ m₂ m₁ m₃⊆m₂ m₂⊆m₁ - ) - } diff --git a/Lattice.agda b/Lattice.agda index 03b61b6..731e80c 100644 --- a/Lattice.agda +++ b/Lattice.agda @@ -215,48 +215,6 @@ module IsSemilatticeInstances where ) } - module ForMap {a} {A B : Set a} - (≡-dec-A : Decidable (_≡_ {a} {A})) - (_≈₂_ : B → B → Set a) - (_⊔₂_ : B → B → B) - (sB : IsSemilattice B _≈₂_ _⊔₂_) where - - open import Lattice.Map A B ≡-dec-A - open IsSemilattice sB renaming - ( ≈-refl to ≈₂-refl; ≈-sym to ≈₂-sym; ≈-⊔-cong to ≈₂-⊔₂-cong - ; ⊔-assoc to ⊔₂-assoc; ⊔-comm to ⊔₂-comm; ⊔-idemp to ⊔₂-idemp - ) - - module MapEquiv = IsEquivalenceInstances.ForMap A B ≡-dec-A _≈₂_ (IsSemilattice.≈-equiv sB) - open MapEquiv using (_≈_) public - - infixl 20 _⊔_ - infixl 20 _⊓_ - - _⊔_ : Map → Map → Map - m₁ ⊔ m₂ = union _⊔₂_ m₁ m₂ - - _⊓_ : Map → Map → Map - m₁ ⊓ m₂ = intersect _⊔₂_ m₁ m₂ - - MapIsUnionSemilattice : IsSemilattice Map _≈_ _⊔_ - MapIsUnionSemilattice = record - { ≈-equiv = MapEquiv.LiftEquivalence - ; ≈-⊔-cong = λ {m₁} {m₂} {m₃} {m₄} → union-cong _≈₂_ _⊔₂_ ≈₂-⊔₂-cong {m₁} {m₂} {m₃} {m₄} - ; ⊔-assoc = union-assoc _≈₂_ ≈₂-refl ≈₂-sym _⊔₂_ ⊔₂-assoc - ; ⊔-comm = union-comm _≈₂_ ≈₂-refl ≈₂-sym _⊔₂_ ⊔₂-comm - ; ⊔-idemp = union-idemp _≈₂_ ≈₂-refl ≈₂-sym _⊔₂_ ⊔₂-idemp - } - - MapIsIntersectSemilattice : IsSemilattice Map _≈_ _⊓_ - MapIsIntersectSemilattice = record - { ≈-equiv = MapEquiv.LiftEquivalence - ; ≈-⊔-cong = λ {m₁} {m₂} {m₃} {m₄} → intersect-cong _≈₂_ _⊔₂_ ≈₂-⊔₂-cong {m₁} {m₂} {m₃} {m₄} - ; ⊔-assoc = intersect-assoc _≈₂_ ≈₂-refl ≈₂-sym _⊔₂_ ⊔₂-assoc - ; ⊔-comm = intersect-comm _≈₂_ ≈₂-refl ≈₂-sym _⊔₂_ ⊔₂-comm - ; ⊔-idemp = intersect-idemp _≈₂_ ≈₂-refl ≈₂-sym _⊔₂_ ⊔₂-idemp - } - module IsLatticeInstances where module ForNat where open Nat @@ -329,33 +287,6 @@ module IsLatticeInstances where ) } - module ForMap {a} {A B : Set a} - (≡-dec-A : Decidable (_≡_ {a} {A})) - (_≈₂_ : B → B → Set a) - (_⊔₂_ : B → B → B) - (_⊓₂_ : B → B → B) - (lB : IsLattice B _≈₂_ _⊔₂_ _⊓₂_) where - - open import Lattice.Map A B ≡-dec-A - open IsLattice lB renaming - ( ≈-refl to ≈₂-refl; ≈-sym to ≈₂-sym - ; ⊔-idemp to ⊔₂-idemp; ⊓-idemp to ⊓₂-idemp - ; absorb-⊔-⊓ to absorb-⊔₂-⊓₂; absorb-⊓-⊔ to absorb-⊓₂-⊔₂ - ) - - module MapJoin = IsSemilatticeInstances.ForMap ≡-dec-A _≈₂_ _⊔₂_ (IsLattice.joinSemilattice lB) - open MapJoin using (_⊔_; _≈_) public - - module MapMeet = IsSemilatticeInstances.ForMap ≡-dec-A _≈₂_ _⊓₂_ (IsLattice.meetSemilattice lB) - open MapMeet using (_⊓_) public - - MapIsLattice : IsLattice Map _≈_ _⊔_ _⊓_ - MapIsLattice = record - { joinSemilattice = MapJoin.MapIsUnionSemilattice - ; meetSemilattice = MapMeet.MapIsIntersectSemilattice - ; absorb-⊔-⊓ = union-intersect-absorb _≈₂_ ≈₂-refl ≈₂-sym _⊔₂_ _⊓₂_ ⊔₂-idemp ⊓₂-idemp absorb-⊔₂-⊓₂ absorb-⊓₂-⊔₂ - ; absorb-⊓-⊔ = intersect-union-absorb _≈₂_ ≈₂-refl ≈₂-sym _⊔₂_ _⊓₂_ ⊔₂-idemp ⊓₂-idemp absorb-⊔₂-⊓₂ absorb-⊓₂-⊔₂ - } module IsFiniteHeightLatticeInstances where module ForProd {a} {A B : Set a} diff --git a/Lattice/Map.agda b/Lattice/Map.agda index d7b79ee..d89dd97 100644 --- a/Lattice/Map.agda +++ b/Lattice/Map.agda @@ -1,3 +1,4 @@ +open import Lattice 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) @@ -5,8 +6,10 @@ open import Relation.Nullary using (Dec; yes; no; Reflects; ofʸ; ofⁿ) open import Agda.Primitive using (Level) renaming (_⊔_ to _⊔ℓ_) module Lattice.Map {a b : Level} (A : Set a) (B : Set b) + (_≈₂_ : B → B → Set b) + (_⊔₂_ : B → B → B) (_⊓₂_ : B → B → B) (≡-dec-A : Decidable (_≡_ {a} {A})) - where + (lB : IsLattice B _≈₂_ _⊔₂_ _⊓₂_) where import Data.List.Membership.Propositional as MemProp @@ -16,7 +19,16 @@ 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 (⊥) +open import Data.Empty using (⊥; ⊥-elim) +open import Equivalence + +open IsLattice lB using () renaming + ( ≈-refl to ≈₂-refl; ≈-sym to ≈₂-sym; ≈-trans to ≈₂-trans + ; ≈-⊔-cong to ≈₂-⊔₂-cong; ≈-⊓-cong to ≈₂-⊓₂-cong + ; ⊔-idemp to ⊔₂-idemp; ⊔-comm to ⊔₂-comm; ⊔-assoc to ⊔₂-assoc + ; ⊓-idemp to ⊓₂-idemp; ⊓-comm to ⊓₂-comm; ⊓-assoc to ⊓₂-assoc + ; absorb-⊔-⊓ to absorb-⊔₂-⊓₂; absorb-⊓-⊔ to absorb-⊓₂-⊔₂ + ) keys : List (A × B) → List A keys = map proj₁ @@ -45,9 +57,6 @@ All¬-¬Any : ∀ {p c} {C : Set c} {P : C → Set p} {l : List C} → All (λ x 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 (_∈_) @@ -63,9 +72,9 @@ private module _ where 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)) + rewrite sym k,v≡x = ⊥-elim (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)) + rewrite sym k,v'≡x = ⊥-elim (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 @@ -91,12 +100,12 @@ private module _ where 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 +private module ImplRelation 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)) + Σ B (λ v' → v ≈₂ v' × ((k , v') ∈ m₂)) private module ImplInsert (f : B → B → B) where open import Data.List using (map) @@ -124,18 +133,18 @@ private module ImplInsert (f : B → B → B) where 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') + ... | no k≢k' = ⊥-elim (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)} → + 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')) + ... | yes k≡k' = ⊥-elim (k∉kl (here k≡k')) ... | no _ = cong (λ xs' → k' ∷ xs') (insert-keys-∉ (λ k∈kxs → k∉kl (there k∈kxs))) @@ -157,7 +166,7 @@ private module ImplInsert (f : B → B → B) where 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')) + ... | yes k≡k' = ⊥-elim (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)} → @@ -179,7 +188,7 @@ private module ImplInsert (f : B → B → B) where 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')) + ... | yes k≡k' = ⊥-elim (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)} → @@ -212,9 +221,9 @@ private module ImplInsert (f : B → B → B) where ¬ 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)) + ... | yes k∈kl₁ = ⊥-elim (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)) + ... | yes k∈kl₂ = ⊥-elim (k∉l₁l₂ (union-preserves-∈k₂ {l₁ = l₁} k∈kl₂)) ... | no k∉kl₂ = (k∉kl₁ , k∉kl₂) @@ -222,7 +231,7 @@ private module ImplInsert (f : B → B → B) where ¬ 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)) + ... | yes k'≡k = ⊥-elim (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'' @@ -245,7 +254,7 @@ private module ImplInsert (f : B → B → B) 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,v∈xs₁)) + ... | yes k≡k' rewrite k≡k' = ⊥-elim (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' = @@ -257,10 +266,10 @@ private module ImplInsert (f : B → B → B) where 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) + ... | no k≢k' = ⊥-elim (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)) + ... | yes k≡k' rewrite k≡k' = ⊥-elim (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)} → @@ -274,7 +283,7 @@ private module ImplInsert (f : B → B → B) where 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)) + ... | yes k≡k' rewrite k≡k' = ⊥-elim (All¬-¬Any k'≢xs₁ (∈-cong proj₁ k,v₁∈xs₁)) ... | no k≢k' = k≢k' update : A → B → List (A × B) → List (A × B) @@ -383,7 +392,7 @@ private module ImplInsert (f : B → B → B) where 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) + ... | no k'∉kl₁ = ⊥-elim (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) @@ -394,7 +403,7 @@ private module ImplInsert (f : B → B → B) where 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'')) + ... | yes k'≡k'' = ⊥-elim (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'' @@ -413,10 +422,10 @@ private module ImplInsert (f : B → B → B) where 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) + ... | no k'≢k' = ⊥-elim (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)) + ... | yes k≡k' rewrite k≡k' = ⊥-elim (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)} → @@ -430,7 +439,7 @@ private module ImplInsert (f : B → B → B) where 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)) + ... | yes k≡k' rewrite k≡k' = ⊥-elim (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)} → @@ -452,400 +461,404 @@ _∈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 +open ImplRelation renaming (subset to subset-impl) + +_⊆_ : Map → Map → Set (a ⊔ℓ b) +_⊆_ (kvs₁ , _) (kvs₂ , _) = subset-impl kvs₁ kvs₂ + +⊆-refl : (m : Map) → m ⊆ m +⊆-refl _ k v k,v∈m = (v , (≈₂-refl , k,v∈m)) + +⊆-trans : (m₁ m₂ m₃ : Map) → m₁ ⊆ m₂ → m₂ ⊆ m₃ → m₁ ⊆ m₃ +⊆-trans _ _ _ m₁⊆m₂ m₂⊆m₃ k v k,v∈m₁ = + let + (v' , (v≈v' , k,v'∈m₂)) = m₁⊆m₂ k v k,v∈m₁ + (v'' , (v'≈v'' , k,v''∈m₃)) = m₂⊆m₃ k v' k,v'∈m₂ + in (v'' , (≈₂-trans v≈v' v'≈v'' , k,v''∈m₃)) + +_≈_ : Map → Map → Set (a ⊔ℓ b) +_≈_ m₁ m₂ = m₁ ⊆ m₂ × m₂ ⊆ m₁ + +≈-equiv : IsEquivalence Map _≈_ +≈-equiv = record + { ≈-refl = λ {m} → (⊆-refl m , ⊆-refl m) + ; ≈-sym = λ {m₁} {m₂} (m₁⊆m₂ , m₂⊆m₁) → (m₂⊆m₁ , m₁⊆m₂) + ; ≈-trans = λ {m₁} {m₂} {m₃} (m₁⊆m₂ , m₂⊆m₁) (m₂⊆m₃ , m₃⊆m₂) → + ( ⊆-trans m₁ m₂ m₃ m₁⊆m₂ m₂⊆m₃ + , ⊆-trans m₃ m₂ m₁ m₃⊆m₂ m₂⊆m₁ + ) + } + 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 - ) +open ImplInsert _⊔₂_ using (union-preserves-Unique) renaming (insert to insert-impl; union to union-impl) +open ImplInsert _⊓₂_ using (intersect-preserves-Unique) renaming (intersect to intersect-impl) - union : Map → Map → Map - union (kvs₁ , _) (kvs₂ , uks₂) = (union-impl kvs₁ kvs₂ , union-preserves-Unique kvs₁ kvs₂ uks₂) +_⊔_ : Map → Map → Map +_⊔_ (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) +_⊓_ : Map → Map → Map +_⊓_ (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 _⊔₂_ 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 - ) +open ImplInsert _⊓₂_ 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 ⟧ +⟦_⟧ : Expr -> Map +⟦ ` m ⟧ = m +⟦ e₁ ∪ e₂ ⟧ = ⟦ e₁ ⟧ ⊔ ⟦ e₂ ⟧ +⟦ e₁ ∩ e₂ ⟧ = ⟦ 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) +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 (v₁ ⊔₂ v₂) (e₁ ∪ e₂) + bothⁱ : ∀ {v₁ v₂ : B} {e₁ e₂ : Expr} → Provenance k v₁ e₁ → Provenance k v₂ e₂ → Provenance k (v₁ ⊓₂ 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 , (in₁ 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 , (in₂ 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) +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 (v₁ ⊔₂ 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 , (in₁ 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 , (in₂ k∉ke₁ g₂ , union-preserves-∈₂ k∉ke₁ k,v₂∈e₂)) +... | no k∉ke₁ | no k∉ke₂ = ⊥-elim (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 (v₁ ⊓₂ v₂ , (bothⁱ g₁ g₂ , intersect-combines (proj₂ ⟦ e₁ ⟧) (proj₂ ⟦ e₂ ⟧) k,v₁∈e₁ k,v₂∈e₂)) +... | yes k∈ke₁ | no k∉ke₂ = ⊥-elim (intersect-preserves-∉₂ {l₁ = proj₁ ⟦ e₁ ⟧} k∉ke₂ k∈ke₁e₂) +... | no k∉ke₁ | yes k∈ke₂ = ⊥-elim (intersect-preserves-∉₁ {l₂ = proj₁ ⟦ e₂ ⟧} k∉ke₁ k∈ke₁e₂) +... | no k∉ke₁ | no k∉ke₂ = ⊥-elim (intersect-preserves-∉₂ {l₁ = proj₁ ⟦ e₁ ⟧} k∉ke₂ k∈ke₁e₂) +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 ≈₂ v₂ → SubsetInfo m₁ m₂ + fine : m₁ ⊆ m₂ → SubsetInfo m₁ m₂ -module _ (_≈_ : B → B → Set b) where - open ImplRelation _≈_ renaming (subset to subset-impl) +SubsetInfo-to-dec : ∀ {m₁ m₂ : Map} → SubsetInfo m₁ m₂ → Dec (m₁ ⊆ m₂) +SubsetInfo-to-dec (extra k k∈km₁ k∉km₂) = + let (v , k,v∈m₁) = locate k∈km₁ + in no (λ m₁⊆m₂ → + let (v' , (_ , k,v'∈m₂)) = m₁⊆m₂ k v k,v∈m₁ + in k∉km₂ (∈-cong proj₁ k,v'∈m₂)) +SubsetInfo-to-dec {m₁} {m₂} (mismatch k v₁ v₂ k,v₁∈m₁ k,v₂∈m₂ v₁̷≈v₂) = + no (λ m₁⊆m₂ → + let (v' , (v₁≈v' , k,v'∈m₂)) = m₁⊆m₂ k v₁ k,v₁∈m₁ + in v₁̷≈v₂ (subst (λ v'' → v₁ ≈₂ v'') (Map-functional {k} {v'} {v₂} {m₂} k,v'∈m₂ k,v₂∈m₂) v₁≈v')) -- for some reason, can't just use subst... +SubsetInfo-to-dec (fine m₁⊆m₂) = yes m₁⊆m₂ - subset : Map → Map → Set (a ⊔ℓ b) - subset (kvs₁ , _) (kvs₂ , _) = subset-impl kvs₁ kvs₂ +module _ (≈₂-dec : ∀ (b₁ b₂ : B) → Dec (b₁ ≈₂ b₂)) where + compute-SubsetInfo : ∀ m₁ m₂ → SubsetInfo m₁ m₂ + compute-SubsetInfo ([] , _) m₂ = fine (λ k v ()) + compute-SubsetInfo m₁@((k , v) ∷ xs₁ , push k≢xs₁ uxs₁) m₂@(l₂ , u₂) + with compute-SubsetInfo (xs₁ , uxs₁) m₂ + ... | extra k' k'∈kxs₁ k'∉km₂ = extra k' (there k'∈kxs₁) k'∉km₂ + ... | mismatch k' v₁ v₂ k',v₁∈xs₁ k',v₂∈m₂ v₁̷≈v₂ = + mismatch k' v₁ v₂ (there k',v₁∈xs₁) k',v₂∈m₂ v₁̷≈v₂ + ... | fine xs₁⊆m₂ with ∈k-dec k l₂ + ... | no k∉km₂ = extra k (here refl) k∉km₂ + ... | yes k∈km₂ with locate k∈km₂ + ... | (v' , k,v'∈m₂) with ≈₂-dec v v' + ... | no v̷≈v' = mismatch k v v' (here refl) (k,v'∈m₂) v̷≈v' + ... | yes v≈v' = fine m₁⊆m₂ + where + m₁⊆m₂ : m₁ ⊆ m₂ + m₁⊆m₂ k' v'' (here k,v≡k',v'') + rewrite cong proj₁ k,v≡k',v'' + rewrite cong proj₂ k,v≡k',v'' = + (v' , (v≈v' , k,v'∈m₂)) + m₁⊆m₂ k' v'' (there k,v≡k',v'') = + xs₁⊆m₂ k' v'' k,v≡k',v'' - lift : Map → Map → Set (a ⊔ℓ b) - lift m₁ m₂ = subset m₁ m₂ × subset m₂ m₁ + ⊆-dec : ∀ m₁ m₂ → Dec (m₁ ⊆ m₂) + ⊆-dec m₁ m₂ = SubsetInfo-to-dec (compute-SubsetInfo m₁ m₂) - 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 + ≈-dec : ∀ m₁ m₂ → Dec (m₁ ≈ m₂) + ≈-dec m₁ m₂ + with ⊆-dec m₁ m₂ | ⊆-dec m₂ m₁ + ... | yes m₁⊆m₂ | yes m₂⊆m₁ = yes (m₁⊆m₂ , m₂⊆m₁) + ... | _ | no m₂̷⊆m₁ = no (λ (_ , m₂⊆m₁) → m₂̷⊆m₁ m₂⊆m₁) + ... | no m₁̷⊆m₂ | _ = no (λ (m₁⊆m₂ , _) → m₁̷⊆m₂ m₁⊆m₂) - SubsetInfo-to-dec : ∀ {m₁ m₂ : Map} → SubsetInfo m₁ m₂ → Dec (subset m₁ m₂) - SubsetInfo-to-dec (extra k k∈km₁ k∉km₂) = - let (v , k,v∈m₁) = locate k∈km₁ - in no (λ m₁⊆m₂ → - let (v' , (_ , k,v'∈m₂)) = m₁⊆m₂ k v k,v∈m₁ - in k∉km₂ (∈-cong proj₁ k,v'∈m₂)) - SubsetInfo-to-dec {m₁} {m₂} (mismatch k v₁ v₂ k,v₁∈m₁ k,v₂∈m₂ v₁̷≈v₂) = - no (λ m₁⊆m₂ → - let (v' , (v₁≈v' , k,v'∈m₂)) = m₁⊆m₂ k v₁ k,v₁∈m₁ - in v₁̷≈v₂ (subst (λ v'' → v₁ ≈ v'') (Map-functional {k} {v'} {v₂} {m₂} k,v'∈m₂ k,v₂∈m₂) v₁≈v')) -- for some reason, can't just use subst... - SubsetInfo-to-dec (fine m₁⊆m₂) = yes m₁⊆m₂ +private module I⊔ = ImplInsert _⊔₂_ +private module I⊓ = ImplInsert _⊓₂_ +≈-⊔-cong : ∀ {m₁ m₂ m₃ m₄ : Map} → m₁ ≈ m₂ → m₃ ≈ m₄ → (m₁ ⊔ m₃) ≈ (m₂ ⊔ m₄) +≈-⊔-cong {m₁} {m₂} {m₃} {m₄} (m₁⊆m₂ , m₂⊆m₁) (m₃⊆m₄ , m₄⊆m₃) = + ( ⊔-⊆ m₁ m₂ m₃ m₄ (m₁⊆m₂ , m₂⊆m₁) (m₃⊆m₄ , m₄⊆m₃) + , ⊔-⊆ m₂ m₁ m₄ m₃ (m₂⊆m₁ , m₁⊆m₂) (m₄⊆m₃ , m₃⊆m₄) + ) + where + ≈-∉-cong : ∀ {m₁ m₂ : Map} {k : A} → m₁ ≈ m₂ → ¬ k ∈k m₁ → ¬ k ∈k m₂ + ≈-∉-cong (m₁⊆m₂ , m₂⊆m₁) k∉km₁ k∈km₂ = + let (v₂ , k,v₂∈m₂) = locate k∈km₂ + (_ , (_ , k,v₁∈m₁)) = m₂⊆m₁ _ v₂ k,v₂∈m₂ + in k∉km₁ (∈-cong proj₁ k,v₁∈m₁) - module _ (≈-dec : ∀ (b₁ b₂ : B) → Dec (b₁ ≈ b₂)) where - private - compute-SubsetInfo : ∀ m₁ m₂ → SubsetInfo m₁ m₂ - compute-SubsetInfo ([] , _) m₂ = fine (λ k v ()) - compute-SubsetInfo m₁@((k , v) ∷ xs₁ , push k≢xs₁ uxs₁) m₂@(l₂ , u₂) - with compute-SubsetInfo (xs₁ , uxs₁) m₂ - ... | extra k' k'∈kxs₁ k'∉km₂ = extra k' (there k'∈kxs₁) k'∉km₂ - ... | mismatch k' v₁ v₂ k',v₁∈xs₁ k',v₂∈m₂ v₁̷≈v₂ = - mismatch k' v₁ v₂ (there k',v₁∈xs₁) k',v₂∈m₂ v₁̷≈v₂ - ... | fine xs₁⊆m₂ with ∈k-dec k l₂ - ... | no k∉km₂ = extra k (here refl) k∉km₂ - ... | yes k∈km₂ with locate k∈km₂ - ... | (v' , k,v'∈m₂) with ≈-dec v v' - ... | no v̷≈v' = mismatch k v v' (here refl) (k,v'∈m₂) v̷≈v' - ... | yes v≈v' = fine m₁⊆m₂ - where - m₁⊆m₂ : subset m₁ m₂ - m₁⊆m₂ k' v'' (here k,v≡k',v'') - rewrite cong proj₁ k,v≡k',v'' - rewrite cong proj₂ k,v≡k',v'' = - (v' , (v≈v' , k,v'∈m₂)) - m₁⊆m₂ k' v'' (there k,v≡k',v'') = - xs₁⊆m₂ k' v'' k,v≡k',v'' + ⊔-⊆ : ∀ (m₁ m₂ m₃ m₄ : Map) → m₁ ≈ m₂ → m₃ ≈ m₄ → (m₁ ⊔ m₃) ⊆ (m₂ ⊔ m₄) + ⊔-⊆ m₁ m₂ m₃ m₄ m₁≈m₂ m₃≈m₄ k v k,v∈m₁m₃ + with Expr-Provenance k ((` m₁) ∪ (` m₃)) (∈-cong proj₁ k,v∈m₁m₃) + ... | (_ , (bothᵘ (single {v₁} v₁∈m₁) (single {v₃} v₃∈m₃) , v₁v₃∈m₁m₃)) + rewrite Map-functional {m = m₁ ⊔ m₃} k,v∈m₁m₃ v₁v₃∈m₁m₃ = + let (v₂ , (v₁≈v₂ , k,v₂∈m₂)) = proj₁ m₁≈m₂ k v₁ v₁∈m₁ + (v₄ , (v₃≈v₄ , k,v₄∈m₄)) = proj₁ m₃≈m₄ k v₃ v₃∈m₃ + in (v₂ ⊔₂ v₄ , (≈₂-⊔₂-cong v₁≈v₂ v₃≈v₄ , I⊔.union-combines (proj₂ m₂) (proj₂ m₄) k,v₂∈m₂ k,v₄∈m₄)) + ... | (_ , (in₁ (single {v₁} v₁∈m₁) k∉km₃ , v₁∈m₁m₃)) + rewrite Map-functional {m = m₁ ⊔ m₃} k,v∈m₁m₃ v₁∈m₁m₃ = + let (v₂ , (v₁≈v₂ , k,v₂∈m₂)) = proj₁ m₁≈m₂ k v₁ v₁∈m₁ + k∉km₄ = ≈-∉-cong {m₃} {m₄} m₃≈m₄ k∉km₃ + in (v₂ , (v₁≈v₂ , I⊔.union-preserves-∈₁ (proj₂ m₂) k,v₂∈m₂ k∉km₄)) + ... | (_ , (in₂ k∉km₁ (single {v₃} v₃∈m₃) , v₃∈m₁m₃)) + rewrite Map-functional {m = m₁ ⊔ m₃} k,v∈m₁m₃ v₃∈m₁m₃ = + let (v₄ , (v₃≈v₄ , k,v₄∈m₄)) = proj₁ m₃≈m₄ k v₃ v₃∈m₃ + k∉km₂ = ≈-∉-cong {m₁} {m₂} m₁≈m₂ k∉km₁ + in (v₄ , (v₃≈v₄ , I⊔.union-preserves-∈₂ k∉km₂ k,v₄∈m₄)) - subset-dec : ∀ m₁ m₂ → Dec (subset m₁ m₂) - subset-dec m₁ m₂ = SubsetInfo-to-dec (compute-SubsetInfo m₁ m₂) +≈-⊓-cong : ∀ {m₁ m₂ m₃ m₄ : Map} → m₁ ≈ m₂ → m₃ ≈ m₄ → (m₁ ⊓ m₃) ≈ (m₂ ⊓ m₄) +≈-⊓-cong {m₁} {m₂} {m₃} {m₄} (m₁⊆m₂ , m₂⊆m₁) (m₃⊆m₄ , m₄⊆m₃) = + ( ⊓-⊆ m₁ m₂ m₃ m₄ (m₁⊆m₂ , m₂⊆m₁) (m₃⊆m₄ , m₄⊆m₃) + , ⊓-⊆ m₂ m₁ m₄ m₃ (m₂⊆m₁ , m₁⊆m₂) (m₄⊆m₃ , m₃⊆m₄) + ) + where + ⊓-⊆ : ∀ (m₁ m₂ m₃ m₄ : Map) → m₁ ≈ m₂ → m₃ ≈ m₄ → (m₁ ⊓ m₃) ⊆ (m₂ ⊓ m₄) + ⊓-⊆ m₁ m₂ m₃ m₄ m₁≈m₂ m₃≈m₄ k v k,v∈m₁m₃ + with Expr-Provenance k ((` m₁) ∩ (` m₃)) (∈-cong proj₁ k,v∈m₁m₃) + ... | (_ , (bothⁱ (single {v₁} v₁∈m₁) (single {v₃} v₃∈m₃) , v₁v₃∈m₁m₃)) + rewrite Map-functional {m = m₁ ⊓ m₃} k,v∈m₁m₃ v₁v₃∈m₁m₃ = + let (v₂ , (v₁≈v₂ , k,v₂∈m₂)) = proj₁ m₁≈m₂ k v₁ v₁∈m₁ + (v₄ , (v₃≈v₄ , k,v₄∈m₄)) = proj₁ m₃≈m₄ k v₃ v₃∈m₃ + in (v₂ ⊓₂ v₄ , (≈₂-⊓₂-cong v₁≈v₂ v₃≈v₄ , I⊓.intersect-combines (proj₂ m₂) (proj₂ m₄) k,v₂∈m₂ k,v₄∈m₄)) - lift-dec : ∀ m₁ m₂ → Dec (lift m₁ m₂) - lift-dec m₁ m₂ - with subset-dec m₁ m₂ | subset-dec m₂ m₁ - ... | yes m₁⊆m₂ | yes m₂⊆m₁ = yes (m₁⊆m₂ , m₂⊆m₁) - ... | _ | no m₂̷⊆m₁ = no (λ (_ , m₂⊆m₁) → m₂̷⊆m₁ m₂⊆m₁) - ... | no m₁̷⊆m₂ | _ = no (λ (m₁⊆m₂ , _) → m₁̷⊆m₂ m₁⊆m₂) +⊔-idemp : ∀ (m : Map) → (m ⊔ m) ≈ m +⊔-idemp m@(l , u) = (mm-m-⊆ , m-mm-⊆) + where + mm-m-⊆ : (m ⊔ m) ⊆ m + mm-m-⊆ k v k,v∈mm + with Expr-Provenance k ((` m) ∪ (` m)) (∈-cong proj₁ k,v∈mm) + ... | (_ , (bothᵘ (single {v'} v'∈m) (single {v''} v''∈m) , v'v''∈mm)) + rewrite Map-functional {m = m} v'∈m v''∈m + rewrite Map-functional {m = m ⊔ m} k,v∈mm v'v''∈mm = + (v'' , (⊔₂-idemp v'' , v''∈m)) + ... | (_ , (in₁ (single {v'} v'∈m) k∉km , _)) = ⊥-elim (k∉km (∈-cong proj₁ v'∈m)) + ... | (_ , (in₂ k∉km (single {v''} v''∈m) , _)) = ⊥-elim (k∉km (∈-cong proj₁ v''∈m)) - -- The Provenance type requires both union and intersection functions, - -- but sometimes here we're working with one operation only. Just use the - -- union/intersection function for both -- it doesn't matter, since we don't - -- use the dual operations in these proofs. + m-mm-⊆ : m ⊆ (m ⊔ m) + m-mm-⊆ k v k,v∈m = (v ⊔₂ v , (≈₂-sym (⊔₂-idemp v) , I⊔.union-combines u u k,v∈m k,v∈m)) - module _ (f : B → B → B) - (≈-f-cong : ∀ {b₁ b₂ b₃ b₄} → b₁ ≈ b₂ → b₃ ≈ b₄ → f b₁ b₃ ≈ f b₂ b₄) where - private module I = ImplInsert f +⊔-comm : ∀ (m₁ m₂ : Map) → (m₁ ⊔ m₂) ≈ (m₂ ⊔ m₁) +⊔-comm m₁ m₂ = (⊔-comm-⊆ m₁ m₂ , ⊔-comm-⊆ m₂ m₁) + where + ⊔-comm-⊆ : ∀ (m₁ m₂ : Map) → (m₁ ⊔ m₂) ⊆ (m₂ ⊔ m₁) + ⊔-comm-⊆ m₁@(l₁ , u₁) m₂@(l₂ , u₂) k v k,v∈m₁m₂ + with Expr-Provenance k ((` m₁) ∪ (` m₂)) (∈-cong proj₁ k,v∈m₁m₂) + ... | (_ , (bothᵘ {v₁} {v₂} (single v₁∈m₁) (single v₂∈m₂) , v₁v₂∈m₁m₂)) + rewrite Map-functional {m = m₁ ⊔ m₂} k,v∈m₁m₂ v₁v₂∈m₁m₂ = + (v₂ ⊔₂ v₁ , (⊔₂-comm v₁ v₂ , I⊔.union-combines u₂ u₁ v₂∈m₂ v₁∈m₁)) + ... | (_ , (in₁ {v₁} (single v₁∈m₁) k∉km₂ , v₁∈m₁m₂)) + rewrite Map-functional {m = m₁ ⊔ m₂} k,v∈m₁m₂ v₁∈m₁m₂ = + (v₁ , (≈₂-refl , I⊔.union-preserves-∈₂ k∉km₂ v₁∈m₁)) + ... | (_ , (in₂ {v₂} k∉km₁ (single v₂∈m₂) , v₂∈m₁m₂)) + rewrite Map-functional {m = m₁ ⊔ m₂} k,v∈m₁m₂ v₂∈m₁m₂ = + (v₂ , (≈₂-refl , I⊔.union-preserves-∈₁ u₂ v₂∈m₂ k∉km₁)) - union-cong : ∀ {m₁ m₂ m₃ m₄ : Map} → lift m₁ m₂ → lift m₃ m₄ → lift (union f m₁ m₃) (union f m₂ m₄) - union-cong {m₁} {m₂} {m₃} {m₄} (m₁⊆m₂ , m₂⊆m₁) (m₃⊆m₄ , m₄⊆m₃) = - ( union-subset m₁ m₂ m₃ m₄ (m₁⊆m₂ , m₂⊆m₁) (m₃⊆m₄ , m₄⊆m₃) - , union-subset m₂ m₁ m₄ m₃ (m₂⊆m₁ , m₁⊆m₂) (m₄⊆m₃ , m₃⊆m₄) - ) - where - ≈-∉-cong : ∀ {m₁ m₂ : Map} {k : A} → lift m₁ m₂ → ¬ k ∈k m₁ → ¬ k ∈k m₂ - ≈-∉-cong (m₁⊆m₂ , m₂⊆m₁) k∉km₁ k∈km₂ = - let (v₂ , k,v₂∈m₂) = locate k∈km₂ - (_ , (_ , k,v₁∈m₁)) = m₂⊆m₁ _ v₂ k,v₂∈m₂ - in k∉km₁ (∈-cong proj₁ k,v₁∈m₁) +⊔-assoc : ∀ (m₁ m₂ m₃ : Map) → ((m₁ ⊔ m₂) ⊔ m₃) ≈ (m₁ ⊔ (m₂ ⊔ m₃)) +⊔-assoc m₁@(l₁ , u₁) m₂@(l₂ , u₂) m₃@(l₃ , u₃) = (⊔-assoc₁ , ⊔-assoc₂) + where + ⊔-assoc₁ : ((m₁ ⊔ m₂) ⊔ m₃) ⊆ (m₁ ⊔ (m₂ ⊔ m₃)) + ⊔-assoc₁ k v k,v∈m₁₂m₃ + with Expr-Provenance k (((` m₁) ∪ (` m₂)) ∪ (` m₃)) (∈-cong proj₁ k,v∈m₁₂m₃) + ... | (_ , (in₂ k∉ke₁₂ (single {v₃} v₃∈e₃) , v₃∈m₁₂m₃)) + rewrite Map-functional {m = (m₁ ⊔ m₂) ⊔ m₃} k,v∈m₁₂m₃ v₃∈m₁₂m₃ = + let (k∉ke₁ , k∉ke₂) = I⊔.∉-union-∉-either {l₁ = l₁} {l₂ = l₂} k∉ke₁₂ + in (v₃ , (≈₂-refl , I⊔.union-preserves-∈₂ k∉ke₁ (I⊔.union-preserves-∈₂ k∉ke₂ v₃∈e₃))) + ... | (_ , (in₁ (in₂ k∉ke₁ (single {v₂} v₂∈e₂)) k∉ke₃ , v₂∈m₁₂m₃)) + rewrite Map-functional {m = (m₁ ⊔ m₂) ⊔ m₃} k,v∈m₁₂m₃ v₂∈m₁₂m₃ = + (v₂ , (≈₂-refl , I⊔.union-preserves-∈₂ k∉ke₁ (I⊔.union-preserves-∈₁ u₂ v₂∈e₂ k∉ke₃))) + ... | (_ , (bothᵘ (in₂ k∉ke₁ (single {v₂} v₂∈e₂)) (single {v₃} v₃∈e₃) , v₂v₃∈m₁₂m₃)) + rewrite Map-functional {m = (m₁ ⊔ m₂) ⊔ m₃} k,v∈m₁₂m₃ v₂v₃∈m₁₂m₃ = + (v₂ ⊔₂ v₃ , (≈₂-refl , I⊔.union-preserves-∈₂ k∉ke₁ (I⊔.union-combines u₂ u₃ v₂∈e₂ v₃∈e₃))) + ... | (_ , (in₁ (in₁ (single {v₁} v₁∈e₁) k∉ke₂) k∉ke₃ , v₁∈m₁₂m₃)) + rewrite Map-functional {m = (m₁ ⊔ m₂) ⊔ m₃} k,v∈m₁₂m₃ v₁∈m₁₂m₃ = + (v₁ , (≈₂-refl , I⊔.union-preserves-∈₁ u₁ v₁∈e₁ (I⊔.union-preserves-∉ k∉ke₂ k∉ke₃))) + ... | (_ , (bothᵘ (in₁ (single {v₁} v₁∈e₁) k∉ke₂) (single {v₃} v₃∈e₃) , v₁v₃∈m₁₂m₃)) + rewrite Map-functional {m = (m₁ ⊔ m₂) ⊔ m₃} k,v∈m₁₂m₃ v₁v₃∈m₁₂m₃ = + (v₁ ⊔₂ v₃ , (≈₂-refl , I⊔.union-combines u₁ (I⊔.union-preserves-Unique l₂ l₃ u₃) v₁∈e₁ (I⊔.union-preserves-∈₂ k∉ke₂ v₃∈e₃))) + ... | (_ , (in₁ (bothᵘ (single {v₁} v₁∈e₁) (single {v₂} v₂∈e₂)) k∉ke₃), v₁v₂∈m₁₂m₃) + rewrite Map-functional {m = (m₁ ⊔ m₂) ⊔ m₃} k,v∈m₁₂m₃ v₁v₂∈m₁₂m₃ = + (v₁ ⊔₂ v₂ , (≈₂-refl , I⊔.union-combines u₁ (I⊔.union-preserves-Unique l₂ l₃ u₃) v₁∈e₁ (I⊔.union-preserves-∈₁ u₂ v₂∈e₂ k∉ke₃))) + ... | (_ , (bothᵘ (bothᵘ (single {v₁} v₁∈e₁) (single {v₂} v₂∈e₂)) (single {v₃} v₃∈e₃) , v₁v₂v₃∈m₁₂m₃)) + rewrite Map-functional {m = (m₁ ⊔ m₂) ⊔ m₃} k,v∈m₁₂m₃ v₁v₂v₃∈m₁₂m₃ = + (v₁ ⊔₂ (v₂ ⊔₂ v₃) , (⊔₂-assoc v₁ v₂ v₃ , I⊔.union-combines u₁ (I⊔.union-preserves-Unique l₂ l₃ u₃) v₁∈e₁ (I⊔.union-combines u₂ u₃ v₂∈e₂ v₃∈e₃))) - union-subset : ∀ (m₁ m₂ m₃ m₄ : Map) → lift m₁ m₂ → lift m₃ m₄ → subset (union f m₁ m₃) (union f m₂ m₄) - union-subset m₁ m₂ m₃ m₄ m₁≈m₂ m₃≈m₄ k v k,v∈m₁m₃ - with Expr-Provenance f f k ((` m₁) ∪ (` m₃)) (∈-cong proj₁ k,v∈m₁m₃) - ... | (_ , (bothᵘ (single {v₁} v₁∈m₁) (single {v₃} v₃∈m₃) , v₁v₃∈m₁m₃)) - rewrite Map-functional {m = union f m₁ m₃} k,v∈m₁m₃ v₁v₃∈m₁m₃ = - let (v₂ , (v₁≈v₂ , k,v₂∈m₂)) = proj₁ m₁≈m₂ k v₁ v₁∈m₁ - (v₄ , (v₃≈v₄ , k,v₄∈m₄)) = proj₁ m₃≈m₄ k v₃ v₃∈m₃ - in (f v₂ v₄ , (≈-f-cong v₁≈v₂ v₃≈v₄ , I.union-combines (proj₂ m₂) (proj₂ m₄) k,v₂∈m₂ k,v₄∈m₄)) - ... | (_ , (in₁ (single {v₁} v₁∈m₁) k∉km₃ , v₁∈m₁m₃)) - rewrite Map-functional {m = union f m₁ m₃} k,v∈m₁m₃ v₁∈m₁m₃ = - let (v₂ , (v₁≈v₂ , k,v₂∈m₂)) = proj₁ m₁≈m₂ k v₁ v₁∈m₁ - k∉km₄ = ≈-∉-cong {m₃} {m₄} m₃≈m₄ k∉km₃ - in (v₂ , (v₁≈v₂ , I.union-preserves-∈₁ (proj₂ m₂) k,v₂∈m₂ k∉km₄)) - ... | (_ , (in₂ k∉km₁ (single {v₃} v₃∈m₃) , v₃∈m₁m₃)) - rewrite Map-functional {m = union f m₁ m₃} k,v∈m₁m₃ v₃∈m₁m₃ = - let (v₄ , (v₃≈v₄ , k,v₄∈m₄)) = proj₁ m₃≈m₄ k v₃ v₃∈m₃ - k∉km₂ = ≈-∉-cong {m₁} {m₂} m₁≈m₂ k∉km₁ - in (v₄ , (v₃≈v₄ , I.union-preserves-∈₂ k∉km₂ k,v₄∈m₄)) + ⊔-assoc₂ : (m₁ ⊔ (m₂ ⊔ m₃)) ⊆ ((m₁ ⊔ m₂) ⊔ m₃) + ⊔-assoc₂ k v k,v∈m₁m₂₃ + with Expr-Provenance k ((` m₁) ∪ ((` m₂) ∪ (` m₃))) (∈-cong proj₁ k,v∈m₁m₂₃) + ... | (_ , (in₂ k∉ke₁ (in₂ k∉ke₂ (single {v₃} v₃∈e₃)) , v₃∈m₁m₂₃)) + rewrite Map-functional {m = m₁ ⊔ (m₂ ⊔ m₃)} k,v∈m₁m₂₃ v₃∈m₁m₂₃ = + (v₃ , (≈₂-refl , I⊔.union-preserves-∈₂ (I⊔.union-preserves-∉ k∉ke₁ k∉ke₂) v₃∈e₃)) + ... | (_ , (in₂ k∉ke₁ (in₁ (single {v₂} v₂∈e₂) k∉ke₃) , v₂∈m₁m₂₃)) + rewrite Map-functional {m = m₁ ⊔ (m₂ ⊔ m₃)} k,v∈m₁m₂₃ v₂∈m₁m₂₃ = + (v₂ , (≈₂-refl , I⊔.union-preserves-∈₁ (I⊔.union-preserves-Unique l₁ l₂ u₂) (I⊔.union-preserves-∈₂ k∉ke₁ v₂∈e₂) k∉ke₃)) + ... | (_ , (in₂ k∉ke₁ (bothᵘ (single {v₂} v₂∈e₂) (single {v₃} v₃∈e₃)) , v₂v₃∈m₁m₂₃)) + rewrite Map-functional {m = m₁ ⊔ (m₂ ⊔ m₃)} k,v∈m₁m₂₃ v₂v₃∈m₁m₂₃ = + (v₂ ⊔₂ v₃ , (≈₂-refl , I⊔.union-combines (I⊔.union-preserves-Unique l₁ l₂ u₂) u₃ (I⊔.union-preserves-∈₂ k∉ke₁ v₂∈e₂) v₃∈e₃)) + ... | (_ , (in₁ (single {v₁} v₁∈e₁) k∉ke₂₃ , v₁∈m₁m₂₃)) + rewrite Map-functional {m = m₁ ⊔ (m₂ ⊔ m₃)} k,v∈m₁m₂₃ v₁∈m₁m₂₃ = + let (k∉ke₂ , k∉ke₃) = I⊔.∉-union-∉-either {l₁ = l₂} {l₂ = l₃} k∉ke₂₃ + in (v₁ , (≈₂-refl , I⊔.union-preserves-∈₁ (I⊔.union-preserves-Unique l₁ l₂ u₂) (I⊔.union-preserves-∈₁ u₁ v₁∈e₁ k∉ke₂) k∉ke₃)) + ... | (_ , (bothᵘ (single {v₁} v₁∈e₁) (in₂ k∉ke₂ (single {v₃} v₃∈e₃)) , v₁v₃∈m₁m₂₃)) + rewrite Map-functional {m = m₁ ⊔ (m₂ ⊔ m₃)} k,v∈m₁m₂₃ v₁v₃∈m₁m₂₃ = + (v₁ ⊔₂ v₃ , (≈₂-refl , I⊔.union-combines (I⊔.union-preserves-Unique l₁ l₂ u₂) u₃ (I⊔.union-preserves-∈₁ u₁ v₁∈e₁ k∉ke₂) v₃∈e₃)) + ... | (_ , (bothᵘ (single {v₁} v₁∈e₁) (in₁ (single {v₂} v₂∈e₂) k∉ke₃) , v₁v₂∈m₁m₂₃)) + rewrite Map-functional {m = m₁ ⊔ (m₂ ⊔ m₃)} k,v∈m₁m₂₃ v₁v₂∈m₁m₂₃ = + (v₁ ⊔₂ v₂ , (≈₂-refl , I⊔.union-preserves-∈₁ (I⊔.union-preserves-Unique l₁ l₂ u₂) (I⊔.union-combines u₁ u₂ v₁∈e₁ v₂∈e₂) k∉ke₃)) + ... | (_ , (bothᵘ (single {v₁} v₁∈e₁) (bothᵘ (single {v₂} v₂∈e₂) (single {v₃} v₃∈e₃)) , v₁v₂v₃∈m₁m₂₃)) + rewrite Map-functional {m = m₁ ⊔ (m₂ ⊔ m₃)} k,v∈m₁m₂₃ v₁v₂v₃∈m₁m₂₃ = + ((v₁ ⊔₂ v₂) ⊔₂ v₃ , (≈₂-sym (⊔₂-assoc v₁ v₂ v₃) , I⊔.union-combines (I⊔.union-preserves-Unique l₁ l₂ u₂) u₃ (I⊔.union-combines u₁ u₂ v₁∈e₁ v₂∈e₂) v₃∈e₃)) - intersect-cong : ∀ {m₁ m₂ m₃ m₄ : Map} → lift m₁ m₂ → lift m₃ m₄ → lift (intersect f m₁ m₃) (intersect f m₂ m₄) - intersect-cong {m₁} {m₂} {m₃} {m₄} (m₁⊆m₂ , m₂⊆m₁) (m₃⊆m₄ , m₄⊆m₃) = - ( intersect-subset m₁ m₂ m₃ m₄ (m₁⊆m₂ , m₂⊆m₁) (m₃⊆m₄ , m₄⊆m₃) - , intersect-subset m₂ m₁ m₄ m₃ (m₂⊆m₁ , m₁⊆m₂) (m₄⊆m₃ , m₃⊆m₄) - ) - where - intersect-subset : ∀ (m₁ m₂ m₃ m₄ : Map) → lift m₁ m₂ → lift m₃ m₄ → subset (intersect f m₁ m₃) (intersect f m₂ m₄) - intersect-subset m₁ m₂ m₃ m₄ m₁≈m₂ m₃≈m₄ k v k,v∈m₁m₃ - with Expr-Provenance f f k ((` m₁) ∩ (` m₃)) (∈-cong proj₁ k,v∈m₁m₃) - ... | (_ , (bothⁱ (single {v₁} v₁∈m₁) (single {v₃} v₃∈m₃) , v₁v₃∈m₁m₃)) - rewrite Map-functional {m = intersect f m₁ m₃} k,v∈m₁m₃ v₁v₃∈m₁m₃ = - let (v₂ , (v₁≈v₂ , k,v₂∈m₂)) = proj₁ m₁≈m₂ k v₁ v₁∈m₁ - (v₄ , (v₃≈v₄ , k,v₄∈m₄)) = proj₁ m₃≈m₄ k v₃ v₃∈m₃ - in (f v₂ v₄ , (≈-f-cong v₁≈v₂ v₃≈v₄ , I.intersect-combines (proj₂ m₂) (proj₂ m₄) k,v₂∈m₂ k,v₄∈m₄)) +⊓-idemp : ∀ (m : Map) → (m ⊓ m) ≈ m +⊓-idemp m@(l , u) = (mm-m-⊆ , m-mm-⊆) + where + mm-m-⊆ : (m ⊓ m) ⊆ m + mm-m-⊆ k v k,v∈mm + with Expr-Provenance k ((` m) ∩ (` m)) (∈-cong proj₁ k,v∈mm) + ... | (_ , (bothⁱ (single {v'} v'∈m) (single {v''} v''∈m) , v'v''∈mm)) + rewrite Map-functional {m = m} v'∈m v''∈m + rewrite Map-functional {m = m ⊓ m} k,v∈mm v'v''∈mm = + (v'' , (⊓₂-idemp v'' , v''∈m)) - module _ (≈-refl : ∀ {b : B} → b ≈ b) - (≈-sym : ∀ {b₁ b₂ : B} → b₁ ≈ b₂ → b₂ ≈ b₁) - (f : B → B → B) where - private module I = ImplInsert f + m-mm-⊆ : m ⊆ (m ⊓ m) + m-mm-⊆ k v k,v∈m = (v ⊓₂ v , (≈₂-sym (⊓₂-idemp v) , I⊓.intersect-combines u u k,v∈m k,v∈m)) - module _ (f-idemp : ∀ (b : B) → f b b ≈ b) where - union-idemp : ∀ (m : Map) → lift (union f m m) m - union-idemp m@(l , u) = (mm-m-subset , m-mm-subset) - where - mm-m-subset : subset (union f m m) m - mm-m-subset k v k,v∈mm - with Expr-Provenance f f k ((` m) ∪ (` m)) (∈-cong proj₁ k,v∈mm) - ... | (_ , (bothᵘ (single {v'} v'∈m) (single {v''} v''∈m) , v'v''∈mm)) - rewrite Map-functional {m = m} v'∈m v''∈m - rewrite Map-functional {m = union f m m} k,v∈mm v'v''∈mm = - (v'' , (f-idemp v'' , v''∈m)) - ... | (_ , (in₁ (single {v'} v'∈m) k∉km , _)) = absurd (k∉km (∈-cong proj₁ v'∈m)) - ... | (_ , (in₂ k∉km (single {v''} v''∈m) , _)) = absurd (k∉km (∈-cong proj₁ v''∈m)) +⊓-comm : ∀ (m₁ m₂ : Map) → (m₁ ⊓ m₂) ≈ (m₂ ⊓ m₁) +⊓-comm m₁ m₂ = (⊓-comm-⊆ m₁ m₂ , ⊓-comm-⊆ m₂ m₁) + where + ⊓-comm-⊆ : ∀ (m₁ m₂ : Map) → (m₁ ⊓ m₂) ⊆ (m₂ ⊓ m₁) + ⊓-comm-⊆ m₁@(l₁ , u₁) m₂@(l₂ , u₂) k v k,v∈m₁m₂ + with Expr-Provenance k ((` m₁) ∩ (` m₂)) (∈-cong proj₁ k,v∈m₁m₂) + ... | (_ , (bothⁱ {v₁} {v₂} (single v₁∈m₁) (single v₂∈m₂) , v₁v₂∈m₁m₂)) + rewrite Map-functional {m = m₁ ⊓ m₂} k,v∈m₁m₂ v₁v₂∈m₁m₂ = + (v₂ ⊓₂ v₁ , (⊓₂-comm v₁ v₂ , I⊓.intersect-combines u₂ u₁ v₂∈m₂ v₁∈m₁)) - m-mm-subset : subset m (union f m m) - m-mm-subset k v k,v∈m = (f v v , (≈-sym (f-idemp v) , I.union-combines u u k,v∈m k,v∈m)) +⊓-assoc : ∀ (m₁ m₂ m₃ : Map) → ((m₁ ⊓ m₂) ⊓ m₃) ≈ (m₁ ⊓ (m₂ ⊓ m₃)) +⊓-assoc m₁@(l₁ , u₁) m₂@(l₂ , u₂) m₃@(l₃ , u₃) = (⊓-assoc₁ , ⊓-assoc₂) + where + ⊓-assoc₁ : ((m₁ ⊓ m₂) ⊓ m₃) ⊆ (m₁ ⊓ (m₂ ⊓ m₃)) + ⊓-assoc₁ k v k,v∈m₁₂m₃ + with Expr-Provenance k (((` m₁) ∩ (` m₂)) ∩ (` m₃)) (∈-cong proj₁ k,v∈m₁₂m₃) + ... | (_ , (bothⁱ (bothⁱ (single {v₁} v₁∈e₁) (single {v₂} v₂∈e₂)) (single {v₃} v₃∈e₃) , v₁v₂v₃∈m₁₂m₃)) + rewrite Map-functional {m = (m₁ ⊓ m₂) ⊓ m₃} k,v∈m₁₂m₃ v₁v₂v₃∈m₁₂m₃ = + (v₁ ⊓₂ (v₂ ⊓₂ v₃) , (⊓₂-assoc v₁ v₂ v₃ , I⊓.intersect-combines u₁ (I⊓.intersect-preserves-Unique {l₂} {l₃} u₃) v₁∈e₁ (I⊓.intersect-combines u₂ u₃ v₂∈e₂ v₃∈e₃))) - module _ (f-comm : ∀ (b₁ b₂ : B) → f b₁ b₂ ≈ f b₂ b₁) where - union-comm : ∀ (m₁ m₂ : Map) → lift (union f m₁ m₂) (union f m₂ m₁) - union-comm m₁ m₂ = (union-comm-subset m₁ m₂ , union-comm-subset m₂ m₁) - where - union-comm-subset : ∀ (m₁ m₂ : Map) → subset (union f m₁ m₂) (union f m₂ m₁) - union-comm-subset m₁@(l₁ , u₁) m₂@(l₂ , u₂) k v k,v∈m₁m₂ - with Expr-Provenance f f k ((` m₁) ∪ (` m₂)) (∈-cong proj₁ k,v∈m₁m₂) - ... | (_ , (bothᵘ {v₁} {v₂} (single v₁∈m₁) (single v₂∈m₂) , v₁v₂∈m₁m₂)) - rewrite Map-functional {m = union f m₁ m₂} k,v∈m₁m₂ v₁v₂∈m₁m₂ = - (f v₂ v₁ , (f-comm v₁ v₂ , I.union-combines u₂ u₁ v₂∈m₂ v₁∈m₁)) - ... | (_ , (in₁ {v₁} (single v₁∈m₁) k∉km₂ , v₁∈m₁m₂)) - rewrite Map-functional {m = union f m₁ m₂} k,v∈m₁m₂ v₁∈m₁m₂ = - (v₁ , (≈-refl , I.union-preserves-∈₂ k∉km₂ v₁∈m₁)) - ... | (_ , (in₂ {v₂} k∉km₁ (single v₂∈m₂) , v₂∈m₁m₂)) - rewrite Map-functional {m = union f m₁ m₂} k,v∈m₁m₂ v₂∈m₁m₂ = - (v₂ , (≈-refl , I.union-preserves-∈₁ u₂ v₂∈m₂ k∉km₁)) + ⊓-assoc₂ : (m₁ ⊓ (m₂ ⊓ m₃)) ⊆ ((m₁ ⊓ m₂) ⊓ m₃) + ⊓-assoc₂ k v k,v∈m₁m₂₃ + with Expr-Provenance k ((` m₁) ∩ ((` m₂) ∩ (` m₃))) (∈-cong proj₁ k,v∈m₁m₂₃) + ... | (_ , (bothⁱ (single {v₁} v₁∈e₁) (bothⁱ (single {v₂} v₂∈e₂) (single {v₃} v₃∈e₃)) , v₁v₂v₃∈m₁m₂₃)) + rewrite Map-functional {m = m₁ ⊓ (m₂ ⊓ m₃)} k,v∈m₁m₂₃ v₁v₂v₃∈m₁m₂₃ = + ((v₁ ⊓₂ v₂) ⊓₂ v₃ , (≈₂-sym (⊓₂-assoc v₁ v₂ v₃) , I⊓.intersect-combines (I⊓.intersect-preserves-Unique {l₁} {l₂} u₂) u₃ (I⊓.intersect-combines u₁ u₂ v₁∈e₁ v₂∈e₂) v₃∈e₃)) - module _ (f-assoc : ∀ (b₁ b₂ b₃ : B) → f (f b₁ b₂) b₃ ≈ f b₁ (f b₂ b₃)) where - union-assoc : ∀ (m₁ m₂ m₃ : Map) → lift (union f (union f m₁ m₂) m₃) (union f m₁ (union f m₂ m₃)) - union-assoc m₁@(l₁ , u₁) m₂@(l₂ , u₂) m₃@(l₃ , u₃) = (union-assoc₁ , union-assoc₂) - where - union-assoc₁ : subset (union f (union f m₁ m₂) m₃) (union f m₁ (union f m₂ m₃)) - union-assoc₁ k v k,v∈m₁₂m₃ - with Expr-Provenance f f k (((` m₁) ∪ (` m₂)) ∪ (` m₃)) (∈-cong proj₁ k,v∈m₁₂m₃) - ... | (_ , (in₂ k∉ke₁₂ (single {v₃} v₃∈e₃) , v₃∈m₁₂m₃)) - rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₃∈m₁₂m₃ = - let (k∉ke₁ , k∉ke₂) = I.∉-union-∉-either {l₁ = l₁} {l₂ = l₂} k∉ke₁₂ - in (v₃ , (≈-refl , I.union-preserves-∈₂ k∉ke₁ (I.union-preserves-∈₂ k∉ke₂ v₃∈e₃))) - ... | (_ , (in₁ (in₂ k∉ke₁ (single {v₂} v₂∈e₂)) k∉ke₃ , v₂∈m₁₂m₃)) - rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₂∈m₁₂m₃ = - (v₂ , (≈-refl , I.union-preserves-∈₂ k∉ke₁ (I.union-preserves-∈₁ u₂ v₂∈e₂ k∉ke₃))) - ... | (_ , (bothᵘ (in₂ k∉ke₁ (single {v₂} v₂∈e₂)) (single {v₃} v₃∈e₃) , v₂v₃∈m₁₂m₃)) - rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₂v₃∈m₁₂m₃ = - (f v₂ v₃ , (≈-refl , I.union-preserves-∈₂ k∉ke₁ (I.union-combines u₂ u₃ v₂∈e₂ v₃∈e₃))) - ... | (_ , (in₁ (in₁ (single {v₁} v₁∈e₁) k∉ke₂) k∉ke₃ , v₁∈m₁₂m₃)) - rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁∈m₁₂m₃ = - (v₁ , (≈-refl , I.union-preserves-∈₁ u₁ v₁∈e₁ (I.union-preserves-∉ k∉ke₂ k∉ke₃))) - ... | (_ , (bothᵘ (in₁ (single {v₁} v₁∈e₁) k∉ke₂) (single {v₃} v₃∈e₃) , v₁v₃∈m₁₂m₃)) - rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁v₃∈m₁₂m₃ = - (f v₁ v₃ , (≈-refl , I.union-combines u₁ (I.union-preserves-Unique l₂ l₃ u₃) v₁∈e₁ (I.union-preserves-∈₂ k∉ke₂ v₃∈e₃))) - ... | (_ , (in₁ (bothᵘ (single {v₁} v₁∈e₁) (single {v₂} v₂∈e₂)) k∉ke₃), v₁v₂∈m₁₂m₃) - rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁v₂∈m₁₂m₃ = - (f v₁ v₂ , (≈-refl , I.union-combines u₁ (I.union-preserves-Unique l₂ l₃ u₃) v₁∈e₁ (I.union-preserves-∈₁ u₂ v₂∈e₂ k∉ke₃))) - ... | (_ , (bothᵘ (bothᵘ (single {v₁} v₁∈e₁) (single {v₂} v₂∈e₂)) (single {v₃} v₃∈e₃) , v₁v₂v₃∈m₁₂m₃)) - rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁v₂v₃∈m₁₂m₃ = - (f v₁ (f v₂ v₃) , (f-assoc v₁ v₂ v₃ , I.union-combines u₁ (I.union-preserves-Unique l₂ l₃ u₃) v₁∈e₁ (I.union-combines u₂ u₃ v₂∈e₂ v₃∈e₃))) +absorb-⊓-⊔ : ∀ (m₁ m₂ : Map) → (m₁ ⊓ (m₁ ⊔ m₂)) ≈ m₁ +absorb-⊓-⊔ m₁@(l₁ , u₁) m₂@(l₂ , u₂) = (absorb-⊓-⊔¹ , absorb-⊓-⊔²) + where + absorb-⊓-⊔¹ : (m₁ ⊓ (m₁ ⊔ m₂)) ⊆ m₁ + absorb-⊓-⊔¹ k v k,v∈m₁m₁₂ + with Expr-Provenance k ((` m₁) ∩ ((` m₁) ∪ (` m₂))) (∈-cong proj₁ k,v∈m₁m₁₂) + ... | (_ , (bothⁱ (single {v₁} k,v₁∈m₁) + (bothᵘ (single {v₁'} k,v₁'∈m₁) + (single {v₂} v₂∈m₂)) , v₁v₁'v₂∈m₁m₁₂)) + rewrite Map-functional {m = m₁} k,v₁∈m₁ k,v₁'∈m₁ + rewrite Map-functional {m = m₁ ⊓ (m₁ ⊔ m₂)} k,v∈m₁m₁₂ v₁v₁'v₂∈m₁m₁₂ = + (v₁' , (absorb-⊓₂-⊔₂ v₁' v₂ , k,v₁'∈m₁)) + ... | (_ , (bothⁱ (single {v₁} k,v₁∈m₁) + (in₁ (single {v₁'} k,v₁'∈m₁) _) , v₁v₁'∈m₁m₁₂)) + rewrite Map-functional {m = m₁} k,v₁∈m₁ k,v₁'∈m₁ + rewrite Map-functional {m = m₁ ⊓ (m₁ ⊔ m₂)} k,v∈m₁m₁₂ v₁v₁'∈m₁m₁₂ = + (v₁' , (⊓₂-idemp v₁' , k,v₁'∈m₁)) + ... | (_ , (bothⁱ (single {v₁} k,v₁∈m₁) + (in₂ k∉m₁ _ ) , _)) = ⊥-elim (k∉m₁ (∈-cong proj₁ k,v₁∈m₁)) - union-assoc₂ : subset (union f m₁ (union f m₂ m₃)) (union f (union f m₁ m₂) m₃) - union-assoc₂ k v k,v∈m₁m₂₃ - with Expr-Provenance f f k ((` m₁) ∪ ((` m₂) ∪ (` m₃))) (∈-cong proj₁ k,v∈m₁m₂₃) - ... | (_ , (in₂ k∉ke₁ (in₂ k∉ke₂ (single {v₃} v₃∈e₃)) , v₃∈m₁m₂₃)) - rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₃∈m₁m₂₃ = - (v₃ , (≈-refl , I.union-preserves-∈₂ (I.union-preserves-∉ k∉ke₁ k∉ke₂) v₃∈e₃)) - ... | (_ , (in₂ k∉ke₁ (in₁ (single {v₂} v₂∈e₂) k∉ke₃) , v₂∈m₁m₂₃)) - rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₂∈m₁m₂₃ = - (v₂ , (≈-refl , I.union-preserves-∈₁ (I.union-preserves-Unique l₁ l₂ u₂) (I.union-preserves-∈₂ k∉ke₁ v₂∈e₂) k∉ke₃)) - ... | (_ , (in₂ k∉ke₁ (bothᵘ (single {v₂} v₂∈e₂) (single {v₃} v₃∈e₃)) , v₂v₃∈m₁m₂₃)) - rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₂v₃∈m₁m₂₃ = - (f v₂ v₃ , (≈-refl , I.union-combines (I.union-preserves-Unique l₁ l₂ u₂) u₃ (I.union-preserves-∈₂ k∉ke₁ v₂∈e₂) v₃∈e₃)) - ... | (_ , (in₁ (single {v₁} v₁∈e₁) k∉ke₂₃ , v₁∈m₁m₂₃)) - rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁∈m₁m₂₃ = - let (k∉ke₂ , k∉ke₃) = I.∉-union-∉-either {l₁ = l₂} {l₂ = l₃} k∉ke₂₃ - in (v₁ , (≈-refl , I.union-preserves-∈₁ (I.union-preserves-Unique l₁ l₂ u₂) (I.union-preserves-∈₁ u₁ v₁∈e₁ k∉ke₂) k∉ke₃)) - ... | (_ , (bothᵘ (single {v₁} v₁∈e₁) (in₂ k∉ke₂ (single {v₃} v₃∈e₃)) , v₁v₃∈m₁m₂₃)) - rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁v₃∈m₁m₂₃ = - (f v₁ v₃ , (≈-refl , I.union-combines (I.union-preserves-Unique l₁ l₂ u₂) u₃ (I.union-preserves-∈₁ u₁ v₁∈e₁ k∉ke₂) v₃∈e₃)) - ... | (_ , (bothᵘ (single {v₁} v₁∈e₁) (in₁ (single {v₂} v₂∈e₂) k∉ke₃) , v₁v₂∈m₁m₂₃)) - rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁v₂∈m₁m₂₃ = - (f v₁ v₂ , (≈-refl , I.union-preserves-∈₁ (I.union-preserves-Unique l₁ l₂ u₂) (I.union-combines u₁ u₂ v₁∈e₁ v₂∈e₂) k∉ke₃)) - ... | (_ , (bothᵘ (single {v₁} v₁∈e₁) (bothᵘ (single {v₂} v₂∈e₂) (single {v₃} v₃∈e₃)) , v₁v₂v₃∈m₁m₂₃)) - rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁v₂v₃∈m₁m₂₃ = - (f (f v₁ v₂) v₃ , (≈-sym (f-assoc v₁ v₂ v₃) , I.union-combines (I.union-preserves-Unique l₁ l₂ u₂) u₃ (I.union-combines u₁ u₂ v₁∈e₁ v₂∈e₂) v₃∈e₃)) + absorb-⊓-⊔² : m₁ ⊆ (m₁ ⊓ (m₁ ⊔ m₂)) + absorb-⊓-⊔² k v k,v∈m₁ + with ∈k-dec k l₂ + ... | yes k∈km₂ = + let (v₂ , k,v₂∈m₂) = locate k∈km₂ + in (v ⊓₂ (v ⊔₂ v₂) , (≈₂-sym (absorb-⊓₂-⊔₂ v v₂) , I⊓.intersect-combines u₁ (I⊔.union-preserves-Unique l₁ l₂ u₂) k,v∈m₁ (I⊔.union-combines u₁ u₂ k,v∈m₁ k,v₂∈m₂))) + ... | no k∉km₂ = (v ⊓₂ v , (≈₂-sym (⊓₂-idemp v) , I⊓.intersect-combines u₁ (I⊔.union-preserves-Unique l₁ l₂ u₂) k,v∈m₁ (I⊔.union-preserves-∈₁ u₁ k,v∈m₁ k∉km₂))) - module _ (f-idemp : ∀ (b : B) → f b b ≈ b) where - intersect-idemp : ∀ (m : Map) → lift (intersect f m m) m - intersect-idemp m@(l , u) = (mm-m-subset , m-mm-subset) - where - mm-m-subset : subset (intersect f m m) m - mm-m-subset k v k,v∈mm - with Expr-Provenance f f k ((` m) ∩ (` m)) (∈-cong proj₁ k,v∈mm) - ... | (_ , (bothⁱ (single {v'} v'∈m) (single {v''} v''∈m) , v'v''∈mm)) - rewrite Map-functional {m = m} v'∈m v''∈m - rewrite Map-functional {m = intersect f m m} k,v∈mm v'v''∈mm = - (v'' , (f-idemp v'' , v''∈m)) +absorb-⊔-⊓ : ∀ (m₁ m₂ : Map) → (m₁ ⊔ (m₁ ⊓ m₂)) ≈ m₁ +absorb-⊔-⊓ m₁@(l₁ , u₁) m₂@(l₂ , u₂) = (absorb-⊔-⊓¹ , absorb-⊔-⊓²) + where + absorb-⊔-⊓¹ : (m₁ ⊔ (m₁ ⊓ m₂)) ⊆ m₁ + absorb-⊔-⊓¹ k v k,v∈m₁m₁₂ + with Expr-Provenance k ((` m₁) ∪ ((` m₁) ∩ (` m₂))) (∈-cong proj₁ k,v∈m₁m₁₂) + ... | (_ , (bothᵘ (single {v₁} k,v₁∈m₁) + (bothⁱ (single {v₁'} k,v₁'∈m₁) + (single {v₂} k,v₂∈m₂)) , v₁v₁'v₂∈m₁m₁₂)) + rewrite Map-functional {m = m₁} k,v₁∈m₁ k,v₁'∈m₁ + rewrite Map-functional {m = m₁ ⊔ (m₁ ⊓ m₂)} k,v∈m₁m₁₂ v₁v₁'v₂∈m₁m₁₂ = + (v₁' , (absorb-⊔₂-⊓₂ v₁' v₂ , k,v₁'∈m₁)) + ... | (_ , (in₁ (single {v₁} k,v₁∈m₁) k∉km₁₂ , k,v₁∈m₁m₁₂)) + rewrite Map-functional {m = m₁ ⊔ (m₁ ⊓ m₂)} k,v∈m₁m₁₂ k,v₁∈m₁m₁₂ = + (v₁ , (≈₂-refl , k,v₁∈m₁)) + ... | (_ , (in₂ k∉km₁ (bothⁱ (single {v₁'} k,v₁'∈m₁) + (single {v₂} k,v₂∈m₂)) , _)) = + ⊥-elim (k∉km₁ (∈-cong proj₁ k,v₁'∈m₁)) - m-mm-subset : subset m (intersect f m m) - m-mm-subset k v k,v∈m = (f v v , (≈-sym (f-idemp v) , I.intersect-combines u u k,v∈m k,v∈m)) + absorb-⊔-⊓² : m₁ ⊆ (m₁ ⊔ (m₁ ⊓ m₂)) + absorb-⊔-⊓² k v k,v∈m₁ + with ∈k-dec k l₂ + ... | yes k∈km₂ = + let (v₂ , k,v₂∈m₂) = locate k∈km₂ + in (v ⊔₂ (v ⊓₂ v₂) , (≈₂-sym (absorb-⊔₂-⊓₂ v v₂) , I⊔.union-combines u₁ (I⊓.intersect-preserves-Unique {l₁} {l₂} u₂) k,v∈m₁ (I⊓.intersect-combines u₁ u₂ k,v∈m₁ k,v₂∈m₂))) + ... | no k∉km₂ = (v , (≈₂-refl , I⊔.union-preserves-∈₁ u₁ k,v∈m₁ (I⊓.intersect-preserves-∉₂ {k} {l₁} {l₂} k∉km₂))) - module _ (f-comm : ∀ (b₁ b₂ : B) → f b₁ b₂ ≈ f b₂ b₁) where - intersect-comm : ∀ (m₁ m₂ : Map) → lift (intersect f m₁ m₂) (intersect f m₂ m₁) - intersect-comm m₁ m₂ = (intersect-comm-subset m₁ m₂ , intersect-comm-subset m₂ m₁) - where - intersect-comm-subset : ∀ (m₁ m₂ : Map) → subset (intersect f m₁ m₂) (intersect f m₂ m₁) - intersect-comm-subset m₁@(l₁ , u₁) m₂@(l₂ , u₂) k v k,v∈m₁m₂ - with Expr-Provenance f f k ((` m₁) ∩ (` m₂)) (∈-cong proj₁ k,v∈m₁m₂) - ... | (_ , (bothⁱ {v₁} {v₂} (single v₁∈m₁) (single v₂∈m₂) , v₁v₂∈m₁m₂)) - rewrite Map-functional {m = intersect f m₁ m₂} k,v∈m₁m₂ v₁v₂∈m₁m₂ = - (f v₂ v₁ , (f-comm v₁ v₂ , I.intersect-combines u₂ u₁ v₂∈m₂ v₁∈m₁)) +isUnionSemilattice : IsSemilattice Map _≈_ _⊔_ +isUnionSemilattice = record + { ≈-equiv = ≈-equiv + ; ≈-⊔-cong = λ {m₁} {m₂} {m₃} {m₄} m₁≈m₂ m₃≈m₄ → ≈-⊔-cong {m₁} {m₂} {m₃} {m₄} m₁≈m₂ m₃≈m₄ + ; ⊔-assoc = ⊔-assoc + ; ⊔-comm = ⊔-comm + ; ⊔-idemp = ⊔-idemp + } - module _ (f-assoc : ∀ (b₁ b₂ b₃ : B) → f (f b₁ b₂) b₃ ≈ f b₁ (f b₂ b₃)) where - intersect-assoc : ∀ (m₁ m₂ m₃ : Map) → lift (intersect f (intersect f m₁ m₂) m₃) (intersect f m₁ (intersect f m₂ m₃)) - intersect-assoc m₁@(l₁ , u₁) m₂@(l₂ , u₂) m₃@(l₃ , u₃) = (intersect-assoc₁ , intersect-assoc₂) - where - intersect-assoc₁ : subset (intersect f (intersect f m₁ m₂) m₃) (intersect f m₁ (intersect f m₂ m₃)) - intersect-assoc₁ k v k,v∈m₁₂m₃ - with Expr-Provenance f f k (((` m₁) ∩ (` m₂)) ∩ (` m₃)) (∈-cong proj₁ k,v∈m₁₂m₃) - ... | (_ , (bothⁱ (bothⁱ (single {v₁} v₁∈e₁) (single {v₂} v₂∈e₂)) (single {v₃} v₃∈e₃) , v₁v₂v₃∈m₁₂m₃)) - rewrite Map-functional {m = intersect f (intersect f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁v₂v₃∈m₁₂m₃ = - (f v₁ (f v₂ v₃) , (f-assoc v₁ v₂ v₃ , I.intersect-combines u₁ (I.intersect-preserves-Unique {l₂} {l₃} u₃) v₁∈e₁ (I.intersect-combines u₂ u₃ v₂∈e₂ v₃∈e₃))) +isIntersectSemilattice : IsSemilattice Map _≈_ _⊓_ +isIntersectSemilattice = record + { ≈-equiv = ≈-equiv + ; ≈-⊔-cong = λ {m₁} {m₂} {m₃} {m₄} m₁≈m₂ m₃≈m₄ → ≈-⊓-cong {m₁} {m₂} {m₃} {m₄} m₁≈m₂ m₃≈m₄ + ; ⊔-assoc = ⊓-assoc + ; ⊔-comm = ⊓-comm + ; ⊔-idemp = ⊓-idemp + } - intersect-assoc₂ : subset (intersect f m₁ (intersect f m₂ m₃)) (intersect f (intersect f m₁ m₂) m₃) - intersect-assoc₂ k v k,v∈m₁m₂₃ - with Expr-Provenance f f k ((` m₁) ∩ ((` m₂) ∩ (` m₃))) (∈-cong proj₁ k,v∈m₁m₂₃) - ... | (_ , (bothⁱ (single {v₁} v₁∈e₁) (bothⁱ (single {v₂} v₂∈e₂) (single {v₃} v₃∈e₃)) , v₁v₂v₃∈m₁m₂₃)) - rewrite Map-functional {m = intersect f m₁ (intersect f m₂ m₃)} k,v∈m₁m₂₃ v₁v₂v₃∈m₁m₂₃ = - (f (f v₁ v₂) v₃ , (≈-sym (f-assoc v₁ v₂ v₃) , I.intersect-combines (I.intersect-preserves-Unique {l₁} {l₂} u₂) u₃ (I.intersect-combines u₁ u₂ v₁∈e₁ v₂∈e₂) v₃∈e₃)) - - module _ (≈-refl : ∀ {b : B} → b ≈ b) - (≈-sym : ∀ {b₁ b₂ : B} → b₁ ≈ b₂ → b₂ ≈ b₁) - (_⊔₂_ : B → B → B) (_⊓₂_ : B → B → B) - (⊔₂-idemp : ∀ (b : B) → (b ⊔₂ b) ≈ b) - (⊓₂-idemp : ∀ (b : B) → (b ⊓₂ b) ≈ b) - (⊔₂-⊓₂-absorb : ∀ (b₁ b₂ : B) → (b₁ ⊔₂ (b₁ ⊓₂ b₂)) ≈ b₁) - (⊓₂-⊔₂-absorb : ∀ (b₁ b₂ : B) → (b₁ ⊓₂ (b₁ ⊔₂ b₂)) ≈ b₁) - where - private module I⊔ = ImplInsert _⊔₂_ - private module I⊓ = ImplInsert _⊓₂_ - - private - _⊔_ = union _⊔₂_ - _⊓_ = intersect _⊓₂_ - - intersect-union-absorb : ∀ (m₁ m₂ : Map) → lift (m₁ ⊓ (m₁ ⊔ m₂)) m₁ - intersect-union-absorb m₁@(l₁ , u₁) m₂@(l₂ , u₂) = (intersect-union-absorb₁ , intersect-union-absorb₂) - where - intersect-union-absorb₁ : subset (m₁ ⊓ (m₁ ⊔ m₂)) m₁ - intersect-union-absorb₁ k v k,v∈m₁m₁₂ - with Expr-Provenance _ _ k ((` m₁) ∩ ((` m₁) ∪ (` m₂))) (∈-cong proj₁ k,v∈m₁m₁₂) - ... | (_ , (bothⁱ (single {v₁} k,v₁∈m₁) - (bothᵘ (single {v₁'} k,v₁'∈m₁) - (single {v₂} v₂∈m₂)) , v₁v₁'v₂∈m₁m₁₂)) - rewrite Map-functional {m = m₁} k,v₁∈m₁ k,v₁'∈m₁ - rewrite Map-functional {m = m₁ ⊓ (m₁ ⊔ m₂)} k,v∈m₁m₁₂ v₁v₁'v₂∈m₁m₁₂ = - (v₁' , (⊓₂-⊔₂-absorb v₁' v₂ , k,v₁'∈m₁)) - ... | (_ , (bothⁱ (single {v₁} k,v₁∈m₁) - (in₁ (single {v₁'} k,v₁'∈m₁) _) , v₁v₁'∈m₁m₁₂)) - rewrite Map-functional {m = m₁} k,v₁∈m₁ k,v₁'∈m₁ - rewrite Map-functional {m = m₁ ⊓ (m₁ ⊔ m₂)} k,v∈m₁m₁₂ v₁v₁'∈m₁m₁₂ = - (v₁' , (⊓₂-idemp v₁' , k,v₁'∈m₁)) - ... | (_ , (bothⁱ (single {v₁} k,v₁∈m₁) - (in₂ k∉m₁ _ ) , _)) = absurd (k∉m₁ (∈-cong proj₁ k,v₁∈m₁)) - - intersect-union-absorb₂ : subset m₁ (m₁ ⊓ (m₁ ⊔ m₂)) - intersect-union-absorb₂ k v k,v∈m₁ - with ∈k-dec k l₂ - ... | yes k∈km₂ = - let (v₂ , k,v₂∈m₂) = locate k∈km₂ - in (v ⊓₂ (v ⊔₂ v₂) , (≈-sym (⊓₂-⊔₂-absorb v v₂) , I⊓.intersect-combines u₁ (I⊔.union-preserves-Unique l₁ l₂ u₂) k,v∈m₁ (I⊔.union-combines u₁ u₂ k,v∈m₁ k,v₂∈m₂))) - ... | no k∉km₂ = (v ⊓₂ v , (≈-sym (⊓₂-idemp v) , I⊓.intersect-combines u₁ (I⊔.union-preserves-Unique l₁ l₂ u₂) k,v∈m₁ (I⊔.union-preserves-∈₁ u₁ k,v∈m₁ k∉km₂))) - - union-intersect-absorb : ∀ (m₁ m₂ : Map) → lift (m₁ ⊔ (m₁ ⊓ m₂)) m₁ - union-intersect-absorb m₁@(l₁ , u₁) m₂@(l₂ , u₂) = (union-intersect-absorb₁ , union-intersect-absorb₂) - where - union-intersect-absorb₁ : subset (m₁ ⊔ (m₁ ⊓ m₂)) m₁ - union-intersect-absorb₁ k v k,v∈m₁m₁₂ - with Expr-Provenance _ _ k ((` m₁) ∪ ((` m₁) ∩ (` m₂))) (∈-cong proj₁ k,v∈m₁m₁₂) - ... | (_ , (bothᵘ (single {v₁} k,v₁∈m₁) - (bothⁱ (single {v₁'} k,v₁'∈m₁) - (single {v₂} k,v₂∈m₂)) , v₁v₁'v₂∈m₁m₁₂)) - rewrite Map-functional {m = m₁} k,v₁∈m₁ k,v₁'∈m₁ - rewrite Map-functional {m = m₁ ⊔ (m₁ ⊓ m₂)} k,v∈m₁m₁₂ v₁v₁'v₂∈m₁m₁₂ = - (v₁' , (⊔₂-⊓₂-absorb v₁' v₂ , k,v₁'∈m₁)) - ... | (_ , (in₁ (single {v₁} k,v₁∈m₁) k∉km₁₂ , k,v₁∈m₁m₁₂)) - rewrite Map-functional {m = m₁ ⊔ (m₁ ⊓ m₂)} k,v∈m₁m₁₂ k,v₁∈m₁m₁₂ = - (v₁ , (≈-refl , k,v₁∈m₁)) - ... | (_ , (in₂ k∉km₁ (bothⁱ (single {v₁'} k,v₁'∈m₁) - (single {v₂} k,v₂∈m₂)) , _)) = - absurd (k∉km₁ (∈-cong proj₁ k,v₁'∈m₁)) - - union-intersect-absorb₂ : subset m₁ (m₁ ⊔ (m₁ ⊓ m₂)) - union-intersect-absorb₂ k v k,v∈m₁ - with ∈k-dec k l₂ - ... | yes k∈km₂ = - let (v₂ , k,v₂∈m₂) = locate k∈km₂ - in (v ⊔₂ (v ⊓₂ v₂) , (≈-sym (⊔₂-⊓₂-absorb v v₂) , I⊔.union-combines u₁ (I⊓.intersect-preserves-Unique {l₁} {l₂} u₂) k,v∈m₁ (I⊓.intersect-combines u₁ u₂ k,v∈m₁ k,v₂∈m₂))) - ... | no k∉km₂ = (v , (≈-refl , I⊔.union-preserves-∈₁ u₁ k,v∈m₁ (I⊓.intersect-preserves-∉₂ {k} {l₁} {l₂} k∉km₂))) +isLattice : IsLattice Map _≈_ _⊔_ _⊓_ +isLattice = record + { joinSemilattice = isUnionSemilattice + ; meetSemilattice = isIntersectSemilattice + ; absorb-⊔-⊓ = absorb-⊔-⊓ + ; absorb-⊓-⊔ = absorb-⊓-⊔ + }