```------------------------------------------------------------------------
-- The Agda standard library
--
-- Properties satisfied by posets
------------------------------------------------------------------------

{-# OPTIONS --without-K --safe #-}

open import Function.Base using (flip; _∘_)
open import Relation.Binary
import Relation.Binary.Consequences as Consequences
open import Relation.Nullary using (¬_)

module Relation.Binary.Properties.Poset
{p₁ p₂ p₃} (P : Poset p₁ p₂ p₃) where

open Poset P renaming (Carrier to A)

import Relation.Binary.Construct.NonStrictToStrict _≈_ _≤_ as ToStrict
import Relation.Binary.Properties.Preorder preorder as PreorderProperties
open Eq using (_≉_)

------------------------------------------------------------------------
-- The _≥_ relation is also a poset.

infix 4 _≥_

_≥_ : Rel A p₃
x ≥ y = y ≤ x

open PreorderProperties public
using ()
renaming
( invIsPreorder to ≥-isPreorder
; invPreorder   to ≥-preorder
)

≥-isPartialOrder : IsPartialOrder _≈_ _≥_
≥-isPartialOrder = record
{ isPreorder   = PreorderProperties.invIsPreorder
; antisym      = flip antisym
}

≥-poset : Poset p₁ p₂ p₃
≥-poset = record
{ isPartialOrder = ≥-isPartialOrder
}

open Poset ≥-poset public
using ()
renaming
( refl      to ≥-refl
; reflexive to ≥-reflexive
; trans     to ≥-trans
; antisym   to ≥-antisym
)

------------------------------------------------------------------------
-- Negated order

infix 4 _≰_

_≰_ : Rel A p₃
x ≰ y = ¬ (x ≤ y)

≰-respˡ-≈ : _≰_ Respectsˡ _≈_
≰-respˡ-≈ x≈y = _∘ ≤-respˡ-≈ (Eq.sym x≈y)

≰-respʳ-≈ : _≰_ Respectsʳ _≈_
≰-respʳ-≈ x≈y = _∘ ≤-respʳ-≈ (Eq.sym x≈y)

------------------------------------------------------------------------
-- Partial orders can be turned into strict partial orders

infix 4 _<_

_<_ : Rel A _
_<_ = ToStrict._<_

<-isStrictPartialOrder : IsStrictPartialOrder _≈_ _<_
<-isStrictPartialOrder = ToStrict.<-isStrictPartialOrder isPartialOrder

<-strictPartialOrder : StrictPartialOrder _ _ _
<-strictPartialOrder = record
{ isStrictPartialOrder = <-isStrictPartialOrder
}

open StrictPartialOrder <-strictPartialOrder public
using ( <-resp-≈; <-respʳ-≈; <-respˡ-≈)
renaming
( irrefl to <-irrefl
; asym   to <-asym
; trans  to <-trans
)

<⇒≉ : ∀ {x y} → x < y → x ≉ y
<⇒≉ = ToStrict.<⇒≉

≤∧≉⇒< : ∀ {x y} → x ≤ y → x ≉ y → x < y
≤∧≉⇒< = ToStrict.≤∧≉⇒<

<⇒≱ : ∀ {x y} → x < y → ¬ (y ≤ x)
<⇒≱ = ToStrict.<⇒≱ antisym

≤⇒≯ : ∀ {x y} → x ≤ y → ¬ (y < x)
≤⇒≯ = ToStrict.≤⇒≯ antisym

------------------------------------------------------------------------
-- Other properties

mono⇒cong : ∀ {f} → f Preserves _≤_ ⟶ _≤_ → f Preserves _≈_ ⟶ _≈_
mono⇒cong = Consequences.mono⇒cong _≈_ _≈_ Eq.sym reflexive antisym

antimono⇒cong : ∀ {f} → f Preserves _≤_ ⟶ _≥_ → f Preserves _≈_ ⟶ _≈_
antimono⇒cong = Consequences.antimono⇒cong _≈_ _≈_ Eq.sym reflexive antisym

------------------------------------------------------------------------
-- DEPRECATED NAMES
------------------------------------------------------------------------
-- Please use the new names as continuing support for the old names is
-- not guaranteed.

-- Version 1.2

invIsPartialOrder = ≥-isPartialOrder
{-# WARNING_ON_USAGE invIsPartialOrder
"Warning: invIsPartialOrder was deprecated in v1.2.
#-}

invPoset = ≥-poset
{-# WARNING_ON_USAGE invPoset
"Warning: invPoset was deprecated in v1.2.
#-}

strictPartialOrder = <-strictPartialOrder
{-# WARNING_ON_USAGE strictPartialOrder
"Warning: strictPartialOrder was deprecated in v1.2.