{-# OPTIONS --cubical-compatible --safe #-}
module Relation.Binary.Construct.Flip.EqAndOrd where
open import Data.Product.Base using (_,_)
open import Function.Base using (flip; _∘_)
open import Level using (Level)
open import Relation.Binary.Core using (Rel; REL; _⇒_)
open import Relation.Binary.Bundles
using (Setoid; DecSetoid; Preorder; Poset; TotalOrder; DecTotalOrder
; StrictPartialOrder; StrictTotalOrder; TotalPreorder)
open import Relation.Binary.Structures
using (IsEquivalence; IsDecEquivalence; IsPreorder; IsPartialOrder
; IsTotalOrder; IsDecTotalOrder; IsStrictPartialOrder
; IsStrictTotalOrder; IsTotalPreorder)
open import Relation.Binary.Definitions
using (Reflexive; Symmetric; Transitive; Asymmetric; Total; _Respects_
; _Respects₂_; Minimum; Maximum; Irreflexive; Antisymmetric
; Trichotomous; Decidable; tri<; tri>; tri≈)
private
variable
a b p ℓ ℓ₁ ℓ₂ : Level
A B : Set a
≈ ∼ ≤ < : Rel A ℓ
module _ (∼ : Rel A ℓ) where
refl : Reflexive ∼ → Reflexive (flip ∼)
refl refl = refl
sym : Symmetric ∼ → Symmetric (flip ∼)
sym sym = sym
trans : Transitive ∼ → Transitive (flip ∼)
trans trans = flip trans
asym : Asymmetric ∼ → Asymmetric (flip ∼)
asym asym = asym
total : Total ∼ → Total (flip ∼)
total total x y = total y x
resp : ∀ {p} (P : A → Set p) → Symmetric ∼ →
P Respects ∼ → P Respects (flip ∼)
resp _ sym resp ∼ = resp (sym ∼)
max : ∀ {⊥} → Minimum ∼ ⊥ → Maximum (flip ∼) ⊥
max min = min
min : ∀ {⊤} → Maximum ∼ ⊤ → Minimum (flip ∼) ⊤
min max = max
module _ {≈ : Rel A ℓ₁} (∼ : Rel A ℓ₂) where
reflexive : Symmetric ≈ → (≈ ⇒ ∼) → (≈ ⇒ flip ∼)
reflexive sym impl = impl ∘ sym
irrefl : Symmetric ≈ → Irreflexive ≈ ∼ → Irreflexive ≈ (flip ∼)
irrefl sym irrefl x≈y y∼x = irrefl (sym x≈y) y∼x
antisym : Antisymmetric ≈ ∼ → Antisymmetric ≈ (flip ∼)
antisym antisym = flip antisym
compare : Trichotomous ≈ ∼ → Trichotomous ≈ (flip ∼)
compare cmp x y with cmp x y
... | tri< x<y x≉y y≮x = tri> y≮x x≉y x<y
... | tri≈ x≮y x≈y y≮x = tri≈ y≮x x≈y x≮y
... | tri> x≮y x≉y y<x = tri< y<x x≉y x≮y
module _ (∼₁ : Rel A ℓ₁) (∼₂ : Rel A ℓ₂) where
resp₂ : ∼₁ Respects₂ ∼₂ → (flip ∼₁) Respects₂ ∼₂
resp₂ (resp₁ , resp₂) = resp₂ , resp₁
module _ (∼ : REL A B ℓ) where
dec : Decidable ∼ → Decidable (flip ∼)
dec dec = flip dec
isEquivalence : IsEquivalence ≈ → IsEquivalence (flip ≈)
isEquivalence {≈ = ≈} eq = record
{ refl = refl ≈ Eq.refl
; sym = sym ≈ Eq.sym
; trans = trans ≈ Eq.trans
} where module Eq = IsEquivalence eq
isDecEquivalence : IsDecEquivalence ≈ → IsDecEquivalence (flip ≈)
isDecEquivalence {≈ = ≈} eq = record
{ isEquivalence = isEquivalence Dec.isEquivalence
; _≟_ = dec ≈ Dec._≟_
} where module Dec = IsDecEquivalence eq
isPreorder : IsPreorder ≈ ∼ → IsPreorder ≈ (flip ∼)
isPreorder {≈ = ≈} {∼ = ∼} O = record
{ isEquivalence = O.isEquivalence
; reflexive = reflexive ∼ O.Eq.sym O.reflexive
; trans = trans ∼ O.trans
} where module O = IsPreorder O
isTotalPreorder : IsTotalPreorder ≈ ∼ → IsTotalPreorder ≈ (flip ∼)
isTotalPreorder O = record
{ isPreorder = isPreorder O.isPreorder
; total = total _ O.total
} where module O = IsTotalPreorder O
isPartialOrder : IsPartialOrder ≈ ≤ → IsPartialOrder ≈ (flip ≤)
isPartialOrder {≤ = ≤} O = record
{ isPreorder = isPreorder O.isPreorder
; antisym = antisym ≤ O.antisym
} where module O = IsPartialOrder O
isTotalOrder : IsTotalOrder ≈ ≤ → IsTotalOrder ≈ (flip ≤)
isTotalOrder O = record
{ isPartialOrder = isPartialOrder O.isPartialOrder
; total = total _ O.total
} where module O = IsTotalOrder O
isDecTotalOrder : IsDecTotalOrder ≈ ≤ → IsDecTotalOrder ≈ (flip ≤)
isDecTotalOrder O = record
{ isTotalOrder = isTotalOrder O.isTotalOrder
; _≟_ = O._≟_
; _≤?_ = dec _ O._≤?_
} where module O = IsDecTotalOrder O
isStrictPartialOrder : IsStrictPartialOrder ≈ < →
IsStrictPartialOrder ≈ (flip <)
isStrictPartialOrder {< = <} O = record
{ isEquivalence = O.isEquivalence
; irrefl = irrefl < O.Eq.sym O.irrefl
; trans = trans < O.trans
; <-resp-≈ = resp₂ _ _ O.<-resp-≈
} where module O = IsStrictPartialOrder O
isStrictTotalOrder : IsStrictTotalOrder ≈ < →
IsStrictTotalOrder ≈ (flip <)
isStrictTotalOrder {< = <} O = record
{ isStrictPartialOrder = isStrictPartialOrder O.isStrictPartialOrder
; compare = compare _ O.compare
} where module O = IsStrictTotalOrder O
setoid : Setoid a ℓ → Setoid a ℓ
setoid S = record
{ isEquivalence = isEquivalence S.isEquivalence
} where module S = Setoid S
decSetoid : DecSetoid a ℓ → DecSetoid a ℓ
decSetoid S = record
{ isDecEquivalence = isDecEquivalence S.isDecEquivalence
} where module S = DecSetoid S
preorder : Preorder a ℓ₁ ℓ₂ → Preorder a ℓ₁ ℓ₂
preorder O = record
{ isPreorder = isPreorder O.isPreorder
} where module O = Preorder O
totalPreorder : TotalPreorder a ℓ₁ ℓ₂ → TotalPreorder a ℓ₁ ℓ₂
totalPreorder O = record
{ isTotalPreorder = isTotalPreorder O.isTotalPreorder
} where module O = TotalPreorder O
poset : Poset a ℓ₁ ℓ₂ → Poset a ℓ₁ ℓ₂
poset O = record
{ isPartialOrder = isPartialOrder O.isPartialOrder
} where module O = Poset O
totalOrder : TotalOrder a ℓ₁ ℓ₂ → TotalOrder a ℓ₁ ℓ₂
totalOrder O = record
{ isTotalOrder = isTotalOrder O.isTotalOrder
} where module O = TotalOrder O
decTotalOrder : DecTotalOrder a ℓ₁ ℓ₂ → DecTotalOrder a ℓ₁ ℓ₂
decTotalOrder O = record
{ isDecTotalOrder = isDecTotalOrder O.isDecTotalOrder
} where module O = DecTotalOrder O
strictPartialOrder : StrictPartialOrder a ℓ₁ ℓ₂ →
StrictPartialOrder a ℓ₁ ℓ₂
strictPartialOrder O = record
{ isStrictPartialOrder = isStrictPartialOrder O.isStrictPartialOrder
} where module O = StrictPartialOrder O
strictTotalOrder : StrictTotalOrder a ℓ₁ ℓ₂ →
StrictTotalOrder a ℓ₁ ℓ₂
strictTotalOrder O = record
{ isStrictTotalOrder = isStrictTotalOrder O.isStrictTotalOrder
} where module O = StrictTotalOrder O