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------------------------------------------------------------------------
-- Agda Helium
--
-- Construct algebras of vectors in a pointwise manner
------------------------------------------------------------------------
{-# OPTIONS --safe --without-K #-}
module Helium.Algebra.Decidable.Construct.Pointwise where
open import Algebra.Lattice.Bundles using (RawLattice)
open import Algebra.Core
open import Data.Nat using (ℕ)
open import Data.Product using (_,_)
open import Data.Vec.Functional using (replicate; map; zipWith)
open import Data.Vec.Functional.Relation.Binary.Pointwise using (Pointwise)
import Data.Vec.Functional.Relation.Binary.Pointwise.Properties as Pwₚ
open import Function using (_$_)
open import Helium.Algebra.Decidable.Bundles
open import Helium.Algebra.Decidable.Structures
open import Relation.Binary.Bundles using (Setoid)
open import Relation.Binary.Core using (Rel)
module _ {a ℓ} {A : Set a} {_≈_ : Rel A ℓ} where
private
_≋_ = Pointwise _≈_
module _ {∨ ∧ : Op₂ A} where
isLattice : IsLattice _≈_ ∨ ∧ → ∀ n → IsLattice (_≋_ {n}) (zipWith ∨) (zipWith ∧)
isLattice L n = record
{ isDecEquivalence = Pwₚ.isDecEquivalence isDecEquivalence n
; ∨-comm = λ x y i → ∨-comm (x i) (y i)
; ∨-assoc = λ x y z i → ∨-assoc (x i) (y i) (z i)
; ∨-cong = λ x≈y u≈v i → ∨-cong (x≈y i) (u≈v i)
; ∧-comm = λ x y i → ∧-comm (x i) (y i)
; ∧-assoc = λ x y z i → ∧-assoc (x i) (y i) (z i)
; ∧-cong = λ x≈y u≈v i → ∧-cong (x≈y i) (u≈v i)
; absorptive = (λ x y i → ∨-absorbs-∧ (x i) (y i))
, (λ x y i → ∧-absorbs-∨ (x i) (y i))
}
where open IsLattice L
isDistributiveLattice : IsDistributiveLattice _≈_ ∨ ∧ → ∀ n → IsDistributiveLattice (_≋_ {n}) (zipWith ∨) (zipWith ∧)
isDistributiveLattice L n = record
{ isLattice = isLattice L.isLattice n
; ∨-distrib-∧ = (λ x y z i → L.∨-distribˡ-∧ (x i) (y i) (z i))
, (λ x y z i → L.∨-distribʳ-∧ (x i) (y i) (z i))
}
where module L = IsDistributiveLattice L
module _ {∨ ∧ : Op₂ A} {¬ : Op₁ A} {⊤ ⊥ : A} where
isBooleanAlgebra : IsBooleanAlgebra _≈_ ∨ ∧ ¬ ⊤ ⊥ → ∀ n → IsBooleanAlgebra (_≋_ {n}) (zipWith ∨) (zipWith ∧) (map ¬) (replicate ⊤) (replicate ⊥)
isBooleanAlgebra B n = record
{ isDistributiveLattice = isDistributiveLattice B.isDistributiveLattice n
; ∨-complement = (λ x i → B.∨-complementˡ (x i))
, (λ x i → B.∨-complementʳ (x i))
; ∧-complement = (λ x i → B.∧-complementˡ (x i))
, (λ x i → B.∧-complementʳ (x i))
; ¬-cong = λ x≈y i → B.¬-cong (x≈y i)
}
where module B = IsBooleanAlgebra B
rawLattice : ∀ {a ℓ} → RawLattice a ℓ → ℕ → RawLattice a ℓ
rawLattice L n = record
{ _≈_ = Pointwise _≈_ {n}
; _∧_ = zipWith _∧_
; _∨_ = zipWith _∨_
}
where open RawLattice L
lattice : ∀ {a ℓ} → Lattice a ℓ → ℕ → Lattice a ℓ
lattice L n = record { isLattice = isLattice (Lattice.isLattice L) n }
distributiveLattice : ∀ {a ℓ} → DistributiveLattice a ℓ → ℕ → DistributiveLattice a ℓ
distributiveLattice L n = record
{ isDistributiveLattice =
isDistributiveLattice (DistributiveLattice.isDistributiveLattice L) n
}
rawBooleanAlgebra : ∀ {a ℓ} → RawBooleanAlgebra a ℓ → ℕ → RawBooleanAlgebra a ℓ
rawBooleanAlgebra B n = record
{ _≈_ = Pointwise _≈_ {n}
; _∧_ = zipWith _∧_
; _∨_ = zipWith _∨_
; ¬_ = map ¬_
; ⊤ = replicate ⊤
; ⊥ = replicate ⊥
}
where open RawBooleanAlgebra B
booleanAlgebra : ∀ {a ℓ} → BooleanAlgebra a ℓ → ℕ → BooleanAlgebra a ℓ
booleanAlgebra B n = record
{ isBooleanAlgebra = isBooleanAlgebra (BooleanAlgebra.isBooleanAlgebra B) n }
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