{-# OPTIONS --safe #-}
module Cubical.Algebra.DistLattice.Basis where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Function
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.Equiv.HalfAdjoint
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Univalence
open import Cubical.Foundations.Transport
open import Cubical.Foundations.SIP
open import Cubical.Foundations.Powerset
open import Cubical.Functions.Embedding
open import Cubical.Data.Sigma
open import Cubical.Data.Nat using (ℕ ; zero ; suc)
open import Cubical.Data.FinData
open import Cubical.Data.Bool
open import Cubical.Structures.Axioms
open import Cubical.Structures.Auto
open import Cubical.Structures.Macro
open import Cubical.Algebra.Semigroup
open import Cubical.Algebra.Monoid
open import Cubical.Algebra.Monoid.BigOp
open import Cubical.Algebra.CommMonoid
open import Cubical.Algebra.Semilattice
open import Cubical.Algebra.Lattice
open import Cubical.Algebra.DistLattice
open import Cubical.Algebra.DistLattice.BigOps
private
variable
ℓ : Level
module _ (L' : DistLattice ℓ) where
private L = fst L'
open DistLatticeStr (snd L')
open Join L'
record IsGenSublattice (M : Semilattice ℓ) (e : fst M → L) : Type ℓ where
constructor
isgensublattice
open SemilatticeStr (snd M) renaming (ε to 0s ; _·_ to _∧s_)
field
isInj : ∀ x y → e x ≡ e y → x ≡ y
pres0 : e 0s ≡ 0l
resp∧ : ∀ x y → e (x ∧s y) ≡ e x ∧l e y
⋁Gen : ∀ (x : L) → ∃[ n ∈ ℕ ] Σ[ α ∈ FinVec (fst M) n ] (⋁ (e ∘ α) ≡ x)
record IsBasis (S : ℙ L) : Type ℓ where
constructor
isbasis
field
contains1 : 1l ∈ S
∧lClosed : ∀ (x y : L) → x ∈ S → y ∈ S → x ∧l y ∈ S
⋁Basis : ∀ (x : L) → ∃[ n ∈ ℕ ] Σ[ α ∈ FinVec L n ] (∀ i → α i ∈ S) × (⋁ α ≡ x)
open IsBasis
open SemilatticeStr hiding (is-set)
Basis→MeetSemilattice : (S : ℙ L) → IsBasis S → Semilattice ℓ
fst (Basis→MeetSemilattice S isBasisS) = Σ[ l ∈ L ] (l ∈ S)
ε (snd (Basis→MeetSemilattice S isBasisS)) = 1l , isBasisS .contains1
_·_ (snd (Basis→MeetSemilattice S isBasisS)) x y = fst x ∧l fst y
, isBasisS .∧lClosed _ _ (snd x) (snd y)
isSemilattice (snd (Basis→MeetSemilattice S isBasisS)) = makeIsSemilattice
(isSetΣ is-set λ _ → isProp→isSet (S _ .snd))
(λ _ _ _ → Σ≡Prop (λ _ → S _ .snd) (∧lAssoc _ _ _))
(λ _ → Σ≡Prop (λ _ → S _ .snd) (∧lRid _))
(λ _ _ → Σ≡Prop (λ _ → S _ .snd) (∧lComm _ _))
λ _ → Σ≡Prop (λ _ → S _ .snd) (∧lIdem _)