```------------------------------------------------------------------------
-- The Agda standard library
--
-- Properties of binary relations
------------------------------------------------------------------------

-- The contents of this module should be accessed via `Relation.Binary`.

{-# OPTIONS --cubical-compatible --safe #-}

module Relation.Binary.Definitions where

open import Agda.Builtin.Equality using (_≡_)

open import Data.Product.Base using (_×_; ∃-syntax)
open import Data.Sum.Base using (_⊎_)
open import Function.Base using (_on_; flip)
open import Level
open import Relation.Binary.Core
open import Relation.Nullary as Nullary using (¬_; Dec)

private
variable
a b c ℓ ℓ₁ ℓ₂ ℓ₃ : Level
A : Set a
B : Set b
C : Set c

------------------------------------------------------------------------
-- Definitions
------------------------------------------------------------------------

-- Reflexivity - defined without an underlying equality. It could
-- alternatively be defined as `_≈_ ⇒ _∼_` for some equality `_≈_`.

-- Confusingly the convention in the library is to use the name "refl"
-- for proofs of Reflexive and `reflexive` for proofs of type `_≈_ ⇒ _∼_`,
-- e.g. in the definition of `IsEquivalence` later in this file. This
-- convention is a legacy from the early days of the library.

Reflexive : Rel A ℓ → Set _
Reflexive _∼_ = ∀ {x} → x ∼ x

-- Generalised symmetry.

Sym : REL A B ℓ₁ → REL B A ℓ₂ → Set _
Sym P Q = P ⇒ flip Q

-- Symmetry.

Symmetric : Rel A ℓ → Set _
Symmetric _∼_ = Sym _∼_ _∼_

-- Generalised transitivity.

Trans : REL A B ℓ₁ → REL B C ℓ₂ → REL A C ℓ₃ → Set _
Trans P Q R = ∀ {i j k} → P i j → Q j k → R i k

RightTrans : REL A B ℓ₁ → REL B B ℓ₂ → Set _
RightTrans R S = Trans R S R

LeftTrans : REL A A ℓ₁ → REL A B ℓ₂ → Set _
LeftTrans S R = Trans S R R

-- A flipped variant of generalised transitivity.

TransFlip : REL A B ℓ₁ → REL B C ℓ₂ → REL A C ℓ₃ → Set _
TransFlip P Q R = ∀ {i j k} → Q j k → P i j → R i k

-- Transitivity.

Transitive : Rel A ℓ → Set _
Transitive _∼_ = Trans _∼_ _∼_ _∼_

-- Generalised antisymmetry

Antisym : REL A B ℓ₁ → REL B A ℓ₂ → REL A B ℓ₃ → Set _
Antisym R S E = ∀ {i j} → R i j → S j i → E i j

-- Antisymmetry.

Antisymmetric : Rel A ℓ₁ → Rel A ℓ₂ → Set _
Antisymmetric _≈_ _≤_ = Antisym _≤_ _≤_ _≈_

-- Irreflexivity - this is defined terms of the underlying equality.

Irreflexive : REL A B ℓ₁ → REL A B ℓ₂ → Set _
Irreflexive _≈_ _<_ = ∀ {x y} → x ≈ y → ¬ (x < y)

-- Asymmetry.

Asymmetric : Rel A ℓ → Set _
Asymmetric _<_ = ∀ {x y} → x < y → ¬ (y < x)

-- Density

Dense : Rel A ℓ → Set _
Dense _<_ = ∀ {x y} → x < y → ∃[ z ] x < z × z < y

-- Generalised connex - at least one of the two relations holds.

Connex : REL A B ℓ₁ → REL B A ℓ₂ → Set _
Connex P Q = ∀ x y → P x y ⊎ Q y x

-- Totality.

Total : Rel A ℓ → Set _
Total _∼_ = Connex _∼_ _∼_

-- Generalised trichotomy - exactly one of three types has a witness.

data Tri (A : Set a) (B : Set b) (C : Set c) : Set (a ⊔ b ⊔ c) where
tri< : ( a :   A) (¬b : ¬ B) (¬c : ¬ C) → Tri A B C
tri≈ : (¬a : ¬ A) ( b :   B) (¬c : ¬ C) → Tri A B C
tri> : (¬a : ¬ A) (¬b : ¬ B) ( c :   C) → Tri A B C

-- Trichotomy.

Trichotomous : Rel A ℓ₁ → Rel A ℓ₂ → Set _
Trichotomous _≈_ _<_ = ∀ x y → Tri (x < y) (x ≈ y) (x > y)
where _>_ = flip _<_

-- Generalised maximum element.

Max : REL A B ℓ → B → Set _
Max _≤_ T = ∀ x → x ≤ T

-- Maximum element.

Maximum : Rel A ℓ → A → Set _
Maximum = Max

-- Generalised minimum element.

Min : REL A B ℓ → A → Set _
Min R = Max (flip R)

-- Minimum element.

Minimum : Rel A ℓ → A → Set _
Minimum = Min

-- Definitions for apartness relations

-- Note that Cotransitive's arguments are permuted with respect to Transitive's.
Cotransitive : Rel A ℓ → Set _
Cotransitive _#_ = ∀ {x y} → x # y → ∀ z → (x # z) ⊎ (z # y)

Tight : Rel A ℓ₁ → Rel A ℓ₂ → Set _
Tight _≈_ _#_ = ∀ x y → (¬ x # y → x ≈ y) × (x ≈ y → ¬ x # y)

-- Properties of order morphisms, aka order-preserving maps

Monotonic₁ : Rel A ℓ₁ → Rel B ℓ₂ → (A → B) → Set _
Monotonic₁ _≤_ _⊑_ f = f Preserves _≤_ ⟶ _⊑_

Antitonic₁ : Rel A ℓ₁ → Rel B ℓ₂ → (A → B) → Set _
Antitonic₁ _≤_ _⊑_ f = f Preserves (flip _≤_) ⟶ _⊑_

Monotonic₂ : Rel A ℓ₁ → Rel B ℓ₂ → Rel C ℓ₃ → (A → B → C) → Set _
Monotonic₂ _≤_ _⊑_ _≼_ ∙ = ∙ Preserves₂ _≤_ ⟶ _⊑_ ⟶ _≼_

MonotonicAntitonic : Rel A ℓ₁ → Rel B ℓ₂ → Rel C ℓ₃ → (A → B → C) → Set _
MonotonicAntitonic _≤_ _⊑_ _≼_ ∙ = ∙ Preserves₂ _≤_ ⟶ (flip _⊑_) ⟶ _≼_

AntitonicMonotonic : Rel A ℓ₁ → Rel B ℓ₂ → Rel C ℓ₃ → (A → B → C) → Set _
AntitonicMonotonic _≤_ _⊑_ _≼_ ∙ = ∙ Preserves₂ (flip _≤_) ⟶ _⊑_ ⟶ _≼_

Antitonic₂ : Rel A ℓ₁ → Rel B ℓ₂ → Rel C ℓ₃ → (A → B → C) → Set _
Antitonic₂ _≤_ _⊑_ _≼_ ∙ = ∙ Preserves₂ (flip _≤_) ⟶ (flip _⊑_) ⟶ _≼_

Adjoint : Rel A ℓ₁ → Rel B ℓ₂ → (A → B) → (B → A) → Set _
Adjoint _≤_ _⊑_ f g = ∀ {x y} → (f x ⊑ y → x ≤ g y) × (x ≤ g y → f x ⊑ y)

-- Unary relations respecting a binary relation.

_⟶_Respects_ : (A → Set ℓ₁) → (B → Set ℓ₂) → REL A B ℓ₃ → Set _
P ⟶ Q Respects _∼_ = ∀ {x y} → x ∼ y → P x → Q y

-- Unary relation respects a binary relation.

_Respects_ : (A → Set ℓ₁) → Rel A ℓ₂ → Set _
P Respects _∼_ = P ⟶ P Respects _∼_

-- Right respecting - relatedness is preserved on the right by equality.

_Respectsʳ_ : REL A B ℓ₁ → Rel B ℓ₂ → Set _
_∼_ Respectsʳ _≈_ = ∀ {x} → (x ∼_) Respects _≈_

-- Left respecting - relatedness is preserved on the left by equality.

_Respectsˡ_ : REL A B ℓ₁ → Rel A ℓ₂ → Set _
P Respectsˡ _∼_ = ∀ {y} → (flip P y) Respects _∼_

-- Respecting - relatedness is preserved on both sides by equality

_Respects₂_ : Rel A ℓ₁ → Rel A ℓ₂ → Set _
P Respects₂ _∼_ = (P Respectsʳ _∼_) × (P Respectsˡ _∼_)

-- Substitutivity - any two related elements satisfy exactly the same
-- set of unary relations. Note that only the various derivatives
-- of propositional equality can satisfy this property.

Substitutive : Rel A ℓ₁ → (ℓ₂ : Level) → Set _
Substitutive {A = A} _∼_ p = (P : A → Set p) → P Respects _∼_

-- Irrelevancy - all proofs that a given pair of elements are related
-- are indistinguishable.

Irrelevant : REL A B ℓ → Set _
Irrelevant _∼_ = ∀ {x y} → Nullary.Irrelevant (x ∼ y)

-- Recomputability - we can rebuild a relevant proof given an
-- irrelevant one.

Recomputable : REL A B ℓ → Set _
Recomputable _∼_ = ∀ {x y} → Nullary.Recomputable (x ∼ y)

-- Stability

Stable : REL A B ℓ → Set _
Stable _∼_ = ∀ x y → Nullary.Stable (x ∼ y)

-- Weak decidability - it is sometimes possible to determine if a given
-- pair of elements are related.

WeaklyDecidable : REL A B ℓ → Set _
WeaklyDecidable _∼_ = ∀ x y → Nullary.WeaklyDecidable (x ∼ y)

-- Decidability - it is possible to determine whether a given pair of
-- elements are related.

Decidable : REL A B ℓ → Set _
Decidable _∼_ = ∀ x y → Dec (x ∼ y)

-- Propositional equality is decidable for the type.

DecidableEquality : (A : Set a) → Set _
DecidableEquality A = Decidable {A = A} _≡_

-- Universal - all pairs of elements are related

Universal : REL A B ℓ → Set _
Universal _∼_ = ∀ x y → x ∼ y

-- Non-emptiness - at least one pair of elements are related.

record NonEmpty {A : Set a} {B : Set b}
(T : REL A B ℓ) : Set (a ⊔ b ⊔ ℓ) where
constructor nonEmpty
field
{x}   : A
{y}   : B
proof : T x y
```