Add a lattice instance for products
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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Lattice.agda
129
Lattice.agda
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@ -256,53 +256,98 @@ module SemilatticeInstances where
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}
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}
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private module NatInstances where
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open Nat
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open NatProps
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open Eq
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open SemilatticeInstances.ForNat
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open Data.Product
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module LatticeInstances where
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module ForNat where
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open Nat
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open NatProps
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open Eq
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open SemilatticeInstances.ForNat
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open Data.Product
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private
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minmax-absorb : {x y : ℕ} → x ⊓ (x ⊔ y) ≡ x
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minmax-absorb {x} {y} = ≤-antisym x⊓x⊔y≤x (helper x⊓x≤x⊓x⊔y (⊓-idem x))
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where
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x⊓x⊔y≤x = proj₁ (Semilattice.⊔-bound NatMinSemilattice x (x ⊔ y) (x ⊓ (x ⊔ y)) refl)
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x⊓x≤x⊓x⊔y = ⊓-mono-≤ {x} {x} ≤-refl (proj₁ (Semilattice.⊔-bound NatMaxSemilattice x y (x ⊔ y) refl))
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private
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minmax-absorb : {x y : ℕ} → x ⊓ (x ⊔ y) ≡ x
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minmax-absorb {x} {y} = ≤-antisym x⊓x⊔y≤x (helper x⊓x≤x⊓x⊔y (⊓-idem x))
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where
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x⊓x⊔y≤x = proj₁ (Semilattice.⊔-bound NatMinSemilattice x (x ⊔ y) (x ⊓ (x ⊔ y)) refl)
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x⊓x≤x⊓x⊔y = ⊓-mono-≤ {x} {x} ≤-refl (proj₁ (Semilattice.⊔-bound NatMaxSemilattice x y (x ⊔ y) refl))
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-- >:(
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helper : x ⊓ x ≤ x ⊓ (x ⊔ y) → x ⊓ x ≡ x → x ≤ x ⊓ (x ⊔ y)
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helper x⊓x≤x⊓x⊔y x⊓x≡x rewrite x⊓x≡x = x⊓x≤x⊓x⊔y
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-- >:(
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helper : x ⊓ x ≤ x ⊓ (x ⊔ y) → x ⊓ x ≡ x → x ≤ x ⊓ (x ⊔ y)
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helper x⊓x≤x⊓x⊔y x⊓x≡x rewrite x⊓x≡x = x⊓x≤x⊓x⊔y
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maxmin-absorb : {x y : ℕ} → x ⊔ (x ⊓ y) ≡ x
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maxmin-absorb {x} {y} = ≤-antisym (helper x⊔x⊓y≤x⊔x (⊔-idem x)) x≤x⊔x⊓y
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where
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x≤x⊔x⊓y = proj₁ (Semilattice.⊔-bound NatMaxSemilattice x (x ⊓ y) (x ⊔ (x ⊓ y)) refl)
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x⊔x⊓y≤x⊔x = ⊔-mono-≤ {x} {x} ≤-refl (proj₁ (Semilattice.⊔-bound NatMinSemilattice x y (x ⊓ y) refl))
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maxmin-absorb : {x y : ℕ} → x ⊔ (x ⊓ y) ≡ x
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maxmin-absorb {x} {y} = ≤-antisym (helper x⊔x⊓y≤x⊔x (⊔-idem x)) x≤x⊔x⊓y
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where
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x≤x⊔x⊓y = proj₁ (Semilattice.⊔-bound NatMaxSemilattice x (x ⊓ y) (x ⊔ (x ⊓ y)) refl)
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x⊔x⊓y≤x⊔x = ⊔-mono-≤ {x} {x} ≤-refl (proj₁ (Semilattice.⊔-bound NatMinSemilattice x y (x ⊓ y) refl))
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-- >:(
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helper : x ⊔ (x ⊓ y) ≤ x ⊔ x → x ⊔ x ≡ x → x ⊔ (x ⊓ y) ≤ x
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helper x⊔x⊓y≤x⊔x x⊔x≡x rewrite x⊔x≡x = x⊔x⊓y≤x⊔x
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-- >:(
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helper : x ⊔ (x ⊓ y) ≤ x ⊔ x → x ⊔ x ≡ x → x ⊔ (x ⊓ y) ≤ x
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helper x⊔x⊓y≤x⊔x x⊔x≡x rewrite x⊔x≡x = x⊔x⊓y≤x⊔x
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NatLattice : Lattice ℕ
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NatLattice = record
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{ _≼_ = _≤_
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; _⊔_ = _⊔_
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; _⊓_ = _⊓_
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; isLattice = record
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{ joinSemilattice = Semilattice.isSemilattice NatMaxSemilattice
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; meetSemilattice = Semilattice.isSemilattice NatMinSemilattice
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; absorb-⊔-⊓ = λ x y → maxmin-absorb {x} {y}
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; absorb-⊓-⊔ = λ x y → minmax-absorb {x} {y}
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NatLattice : Lattice ℕ
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NatLattice = record
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{ _≼_ = _≤_
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; _⊔_ = _⊔_
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; _⊓_ = _⊓_
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; isLattice = record
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{ joinSemilattice = Semilattice.isSemilattice NatMaxSemilattice
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; meetSemilattice = Semilattice.isSemilattice NatMinSemilattice
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; absorb-⊔-⊓ = λ x y → maxmin-absorb {x} {y}
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; absorb-⊓-⊔ = λ x y → minmax-absorb {x} {y}
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}
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}
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}
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-- ProdSemilattice : {a : Level} → {A B : Set a} → {{ Semilattice A }} → {{ Semilattice B }} → Semilattice (A × B)
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-- ProdSemilattice {a} {A} {B} {{slA}} {{slB}} = record
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-- { _≼_ = λ (a₁ , b₁) (a₂ , b₂) → Semilattice._≼_ slA a₁ a₂ × Semilattice._≼_ slB b₁ b₂
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-- ; _⊔_ = λ (a₁ , b₁) (a₂ , b₂) → (Semilattice._⊔_ slA a₁ a₂ , Semilattice._⊔_ slB b₁ b₂)
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-- ; isSemilattice = record
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-- {
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-- }
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-- }
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module ForProd {a} {A B : Set a} (lA : Lattice A) (lB : Lattice B) where
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private
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_≼₁_ = Lattice._≼_ lA
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_≼₂_ = Lattice._≼_ lB
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_⊔₁_ = Lattice._⊔_ lA
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_⊔₂_ = Lattice._⊔_ lB
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_⊓₁_ = Lattice._⊓_ lA
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_⊓₂_ = Lattice._⊓_ lB
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joinA = record { _≼_ = _≼₁_; _⊔_ = _⊔₁_; isSemilattice = Lattice.joinSemilattice lA }
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joinB = record { _≼_ = _≼₂_; _⊔_ = _⊔₂_; isSemilattice = Lattice.joinSemilattice lB }
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meetA = record { _≼_ = λ a b → b ≼₁ a; _⊔_ = _⊓₁_; isSemilattice = Lattice.meetSemilattice lA }
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meetB = record { _≼_ = λ a b → b ≼₂ a; _⊔_ = _⊓₂_; isSemilattice = Lattice.meetSemilattice lB }
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module ProdJoin = SemilatticeInstances.ForProd joinA joinB
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module ProdMeet = SemilatticeInstances.ForProd meetA meetB
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_≼_ = Semilattice._≼_ ProdJoin.ProdSemilattice
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_⊔_ = Semilattice._⊔_ ProdJoin.ProdSemilattice
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_⊓_ = Semilattice._⊔_ ProdMeet.ProdSemilattice
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open Eq
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open Data.Product
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private
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absorb-⊔-⊓ : (p₁ p₂ : A × B) → p₁ ⊔ (p₁ ⊓ p₂) ≡ p₁
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absorb-⊔-⊓ (a₁ , b₁) (a₂ , b₂)
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rewrite Lattice.absorb-⊔-⊓ lA a₁ a₂
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rewrite Lattice.absorb-⊔-⊓ lB b₁ b₂ = refl
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absorb-⊓-⊔ : (p₁ p₂ : A × B) → p₁ ⊓ (p₁ ⊔ p₂) ≡ p₁
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absorb-⊓-⊔ (a₁ , b₁) (a₂ , b₂)
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rewrite Lattice.absorb-⊓-⊔ lA a₁ a₂
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rewrite Lattice.absorb-⊓-⊔ lB b₁ b₂ = refl
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ProdLattice : Lattice (A × B)
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ProdLattice = record
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{ _≼_ = _≼_
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; _⊔_ = _⊔_
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; _⊓_ = _⊓_
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; isLattice = record
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{ joinSemilattice = Semilattice.isSemilattice ProdJoin.ProdSemilattice
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; meetSemilattice = Semilattice.isSemilattice ProdMeet.ProdSemilattice
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; absorb-⊔-⊓ = absorb-⊔-⊓
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; absorb-⊓-⊔ = absorb-⊓-⊔
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}
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}
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