Add a lattice instance for Map
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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Lattice.agda
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Lattice.agda
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@ -42,7 +42,7 @@ record IsLattice {a} (A : Set a)
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absorb-⊓-⊔ : (x y : A) → (x ⊓ (x ⊔ y)) ≈ x
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open IsSemilattice joinSemilattice public
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open IsSemilattice meetSemilattice public renaming
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open IsSemilattice meetSemilattice public hiding (≈-equiv; ≈-refl; ≈-sym; ≈-trans) renaming
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( ⊔-assoc to ⊓-assoc
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; ⊔-comm to ⊓-comm
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; ⊔-idemp to ⊓-idemp
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@ -350,3 +350,41 @@ module IsLatticeInstances where
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; absorb-⊔-⊓ = absorb-⊔-⊓
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; absorb-⊓-⊔ = absorb-⊓-⊔
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}
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module ForMap {a} {A B : Set a}
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(≡-dec-A : Decidable (_≡_ {a} {A}))
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(_≈₂_ : B → B → Set a)
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(_⊔₂_ : B → B → B)
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(_⊓₂_ : B → B → B)
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(lB : IsLattice B _≈₂_ _⊔₂_ _⊓₂_) where
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open import Map A B ≡-dec-A
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open IsLattice lB renaming
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( ≈-refl to ≈₂-refl; ≈-sym to ≈₂-sym
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; ⊔-idemp to ⊔₂-idemp; ⊓-idemp to ⊓₂-idemp
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; absorb-⊔-⊓ to absorb-⊔₂-⊓₂; absorb-⊓-⊔ to absorb-⊓₂-⊔₂
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)
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private
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module MapJoin = IsSemilatticeInstances.ForMap ≡-dec-A _≈₂_ _⊔₂_ (IsLattice.joinSemilattice lB)
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module MapMeet = IsSemilatticeInstances.ForMap ≡-dec-A _≈₂_ _⊓₂_ (IsLattice.meetSemilattice lB)
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infix 4 _≈_
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infixl 20 _⊔_
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_≈_ : Map → Map → Set a
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_≈_ = lift (_≈₂_)
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_⊔_ : Map → Map → Map
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m₁ ⊔ m₂ = union _⊔₂_ m₁ m₂
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_⊓_ : Map → Map → Map
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m₁ ⊓ m₂ = intersect _⊓₂_ m₁ m₂
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MapIsLattice : IsLattice Map _≈_ _⊔_ _⊓_
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MapIsLattice = record
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{ joinSemilattice = MapJoin.MapIsUnionSemilattice
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; meetSemilattice = MapMeet.MapIsIntersectSemilattice
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; absorb-⊔-⊓ = union-intersect-absorb _≈₂_ ≈₂-refl ≈₂-sym _⊔₂_ _⊓₂_ ⊔₂-idemp ⊓₂-idemp absorb-⊔₂-⊓₂ absorb-⊓₂-⊔₂
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; absorb-⊓-⊔ = intersect-union-absorb _≈₂_ ≈₂-refl ≈₂-sym _⊔₂_ _⊓₂_ ⊔₂-idemp ⊓₂-idemp absorb-⊔₂-⊓₂ absorb-⊓₂-⊔₂
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}
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12
Map.agda
12
Map.agda
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@ -682,8 +682,8 @@ module _ (_≈_ : B → B → Set b) where
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(_⊔₂_ : B → B → B) (_⊓₂_ : B → B → B)
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(⊔₂-idemp : ∀ (b : B) → (b ⊔₂ b) ≈ b)
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(⊓₂-idemp : ∀ (b : B) → (b ⊓₂ b) ≈ b)
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(⊔₂-⊓₂-absorb : ∀ {b₁ b₂ : B} → (b₁ ⊔₂ (b₁ ⊓₂ b₂)) ≈ b₁)
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(⊓₂-⊔₂-absorb : ∀ {b₁ b₂ : B} → (b₁ ⊓₂ (b₁ ⊔₂ b₂)) ≈ b₁)
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(⊔₂-⊓₂-absorb : ∀ (b₁ b₂ : B) → (b₁ ⊔₂ (b₁ ⊓₂ b₂)) ≈ b₁)
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(⊓₂-⊔₂-absorb : ∀ (b₁ b₂ : B) → (b₁ ⊓₂ (b₁ ⊔₂ b₂)) ≈ b₁)
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where
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private module I⊔ = ImplInsert _⊔₂_
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private module I⊓ = ImplInsert _⊓₂_
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@ -703,7 +703,7 @@ module _ (_≈_ : B → B → Set b) where
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(single {v₂} v₂∈m₂)) , v₁v₁'v₂∈m₁m₁₂))
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rewrite Map-functional {m = m₁} k,v₁∈m₁ k,v₁'∈m₁
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rewrite Map-functional {m = m₁ ⊓ (m₁ ⊔ m₂)} k,v∈m₁m₁₂ v₁v₁'v₂∈m₁m₁₂ =
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(v₁' , (⊓₂-⊔₂-absorb , k,v₁'∈m₁))
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(v₁' , (⊓₂-⊔₂-absorb v₁' v₂ , k,v₁'∈m₁))
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... | (_ , (bothⁱ (single {v₁} k,v₁∈m₁)
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(in₁ (single {v₁'} k,v₁'∈m₁) _) , v₁v₁'∈m₁m₁₂))
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rewrite Map-functional {m = m₁} k,v₁∈m₁ k,v₁'∈m₁
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@ -717,7 +717,7 @@ module _ (_≈_ : B → B → Set b) where
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with ∈k-dec k l₂
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... | yes k∈km₂ =
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let (v₂ , k,v₂∈m₂) = locate k∈km₂
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in (v ⊓₂ (v ⊔₂ v₂) , (≈-sym ⊓₂-⊔₂-absorb , 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)))
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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)))
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... | 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)))
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union-intersect-absorb : ∀ (m₁ m₂ : Map) → lift (m₁ ⊔ (m₁ ⊓ m₂)) m₁
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@ -731,7 +731,7 @@ module _ (_≈_ : B → B → Set b) where
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(single {v₂} k,v₂∈m₂)) , v₁v₁'v₂∈m₁m₁₂))
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rewrite Map-functional {m = m₁} k,v₁∈m₁ k,v₁'∈m₁
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rewrite Map-functional {m = m₁ ⊔ (m₁ ⊓ m₂)} k,v∈m₁m₁₂ v₁v₁'v₂∈m₁m₁₂ =
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(v₁' , (⊔₂-⊓₂-absorb , k,v₁'∈m₁))
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(v₁' , (⊔₂-⊓₂-absorb v₁' v₂ , k,v₁'∈m₁))
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... | (_ , (in₁ (single {v₁} k,v₁∈m₁) k∉km₁₂ , k,v₁∈m₁m₁₂))
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rewrite Map-functional {m = m₁ ⊔ (m₁ ⊓ m₂)} k,v∈m₁m₁₂ k,v₁∈m₁m₁₂ =
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(v₁ , (≈-refl , k,v₁∈m₁))
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@ -744,5 +744,5 @@ module _ (_≈_ : B → B → Set b) where
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with ∈k-dec k l₂
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... | yes k∈km₂ =
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let (v₂ , k,v₂∈m₂) = locate k∈km₂
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in (v ⊔₂ (v ⊓₂ v₂) , (≈-sym ⊔₂-⊓₂-absorb , 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)))
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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)))
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... | no k∉km₂ = (v , (≈-refl , I⊔.union-preserves-∈₁ u₁ k,v∈m₁ (I⊓.intersect-preserves-∉₂ {k} {l₁} {l₂} k∉km₂)))
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