module Relation.Binary.List.NonStrictLex where
open import Data.Empty
open import Function
open import Data.Unit using (⊤; tt)
open import Data.Product
open import Data.List
open import Level
open import Relation.Nullary
open import Relation.Binary
import Relation.Binary.NonStrictToStrict as Conv
import Relation.Binary.List.StrictLex as Strict
open import Relation.Binary.List.Pointwise as Pointwise using ([])
module _ {A : Set} where
Lex : (P : Set) → (_≈_ _≤_ : Rel A zero) → Rel (List A) zero
Lex P _≈_ _≤_ = Strict.Lex P _≈_ (Conv._<_ _≈_ _≤_)
Lex-< : (_≈_ _≤_ : Rel A zero) → Rel (List A) zero
Lex-< = Lex ⊥
Lex-≤ : (_≈_ _≤_ : Rel A zero) → Rel (List A) zero
Lex-≤ = Lex ⊤
≤-reflexive : ∀ _≈_ _≤_ → Pointwise.Rel _≈_ ⇒ Lex-≤ _≈_ _≤_
≤-reflexive _≈_ _≤_ = Strict.≤-reflexive _≈_ (Conv._<_ _≈_ _≤_)
<-irreflexive : ∀ {_≈_ _≤_} →
Irreflexive (Pointwise.Rel _≈_) (Lex-< _≈_ _≤_)
<-irreflexive {_≈_} {_≤_} =
Strict.<-irreflexive {_<_ = Conv._<_ _≈_ _≤_} (Conv.irrefl _≈_ _≤_)
transitive : ∀ {P _≈_ _≤_} →
IsPartialOrder _≈_ _≤_ → Transitive (Lex P _≈_ _≤_)
transitive po =
Strict.transitive
isEquivalence
(Conv.<-resp-≈ _ _ isEquivalence ≤-resp-≈)
(Conv.trans _ _ po)
where open IsPartialOrder po
antisymmetric : ∀ {P _≈_ _≤_} →
Symmetric _≈_ → Antisymmetric _≈_ _≤_ →
Antisymmetric (Pointwise.Rel _≈_) (Lex P _≈_ _≤_)
antisymmetric {_≈_ = _≈_} {_≤_} sym antisym =
Strict.antisymmetric sym (Conv.irrefl _≈_ _≤_)
(Conv.antisym⟶asym _≈_ _ antisym)
<-asymmetric : ∀ {_≈_ _≤_} →
IsEquivalence _≈_ → _≤_ Respects₂ _≈_ →
Antisymmetric _≈_ _≤_ → Asymmetric (Lex-< _≈_ _≤_)
<-asymmetric {_≈_} eq resp antisym =
Strict.<-asymmetric sym (Conv.<-resp-≈ _ _ eq resp)
(Conv.antisym⟶asym _≈_ _ antisym)
where open IsEquivalence eq
respects₂ : ∀ {P _≈_ _≤_} →
IsEquivalence _≈_ → _≤_ Respects₂ _≈_ →
Lex P _≈_ _≤_ Respects₂ Pointwise.Rel _≈_
respects₂ eq resp = Strict.respects₂ eq (Conv.<-resp-≈ _ _ eq resp)
decidable : ∀ {P _≈_ _≤_} →
Dec P → Decidable _≈_ → Decidable _≤_ →
Decidable (Lex P _≈_ _≤_)
decidable dec-P dec-≈ dec-≤ =
Strict.decidable dec-P dec-≈ (Conv.decidable _ _ dec-≈ dec-≤)
<-decidable :
∀ {_≈_ _≤_} →
Decidable _≈_ → Decidable _≤_ → Decidable (Lex-< _≈_ _≤_)
<-decidable = decidable (no id)
≤-decidable :
∀ {_≈_ _≤_} →
Decidable _≈_ → Decidable _≤_ → Decidable (Lex-≤ _≈_ _≤_)
≤-decidable = decidable (yes tt)
≤-total : ∀ {_≈_ _≤_} → Symmetric _≈_ → Decidable _≈_ →
Antisymmetric _≈_ _≤_ → Total _≤_ →
Total (Lex-≤ _≈_ _≤_)
≤-total sym dec-≈ antisym tot =
Strict.≤-total sym (Conv.trichotomous _ _ sym dec-≈ antisym tot)
<-compare : ∀ {_≈_ _≤_} → Symmetric _≈_ → Decidable _≈_ →
Antisymmetric _≈_ _≤_ → Total _≤_ →
Trichotomous (Pointwise.Rel _≈_) (Lex-< _≈_ _≤_)
<-compare sym dec-≈ antisym tot =
Strict.<-compare sym (Conv.trichotomous _ _ sym dec-≈ antisym tot)
≤-isPreorder : ∀ {_≈_ _≤_} →
IsPartialOrder _≈_ _≤_ →
IsPreorder (Pointwise.Rel _≈_) (Lex-≤ _≈_ _≤_)
≤-isPreorder po =
Strict.≤-isPreorder
isEquivalence (Conv.trans _ _ po)
(Conv.<-resp-≈ _ _ isEquivalence ≤-resp-≈)
where open IsPartialOrder po
≤-isPartialOrder : ∀ {_≈_ _≤_} →
IsPartialOrder _≈_ _≤_ →
IsPartialOrder (Pointwise.Rel _≈_) (Lex-≤ _≈_ _≤_)
≤-isPartialOrder po =
Strict.≤-isPartialOrder
(Conv.isPartialOrder⟶isStrictPartialOrder _ _ po)
≤-isTotalOrder : ∀ {_≈_ _≤_} →
Decidable _≈_ → IsTotalOrder _≈_ _≤_ →
IsTotalOrder (Pointwise.Rel _≈_) (Lex-≤ _≈_ _≤_)
≤-isTotalOrder dec tot =
Strict.≤-isTotalOrder
(Conv.isTotalOrder⟶isStrictTotalOrder _ _ dec tot)
≤-isDecTotalOrder :
∀ {_≈_ _≤_} →
IsDecTotalOrder _≈_ _≤_ →
IsDecTotalOrder (Pointwise.Rel _≈_) (Lex-≤ _≈_ _≤_)
≤-isDecTotalOrder dtot =
Strict.≤-isDecTotalOrder
(Conv.isDecTotalOrder⟶isStrictTotalOrder _ _ dtot)
<-isStrictPartialOrder : ∀ {_≈_ _≤_} →
IsPartialOrder _≈_ _≤_ →
IsStrictPartialOrder (Pointwise.Rel _≈_) (Lex-< _≈_ _≤_)
<-isStrictPartialOrder po =
Strict.<-isStrictPartialOrder
(Conv.isPartialOrder⟶isStrictPartialOrder _ _ po)
<-isStrictTotalOrder : ∀ {_≈_ _≤_} →
Decidable _≈_ → IsTotalOrder _≈_ _≤_ →
IsStrictTotalOrder (Pointwise.Rel _≈_) (Lex-< _≈_ _≤_)
<-isStrictTotalOrder dec tot =
Strict.<-isStrictTotalOrder
(Conv.isTotalOrder⟶isStrictTotalOrder _ _ dec tot)
≤-preorder : Poset _ _ _ → Preorder _ _ _
≤-preorder po = record
{ isPreorder = ≤-isPreorder isPartialOrder
} where open Poset po
≤-partialOrder : Poset _ _ _ → Poset _ _ _
≤-partialOrder po = record
{ isPartialOrder = ≤-isPartialOrder isPartialOrder
} where open Poset po
≤-decTotalOrder : DecTotalOrder _ _ _ → DecTotalOrder _ _ _
≤-decTotalOrder dtot = record
{ isDecTotalOrder = ≤-isDecTotalOrder isDecTotalOrder
} where open DecTotalOrder dtot
<-strictPartialOrder : Poset _ _ _ → StrictPartialOrder _ _ _
<-strictPartialOrder po = record
{ isStrictPartialOrder = <-isStrictPartialOrder isPartialOrder
} where open Poset po
<-strictTotalOrder : DecTotalOrder _ _ _ → StrictTotalOrder _ _ _
<-strictTotalOrder dtot = record
{ isStrictTotalOrder = <-isStrictTotalOrder _≟_ isTotalOrder
} where open IsDecTotalOrder (DecTotalOrder.isDecTotalOrder dtot)