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Add definitions for function-like relations #3001

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Add definitions for function-like relations #3001

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20 changes: 20 additions & 0 deletions src/Relation/Binary/Definitions.agda
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Expand Up @@ -262,3 +262,23 @@ record NonEmpty {A : Set a} {B : Set b}
{x} : A
{y} : B
proof : T x y

-- LeftUnique - the relation is injective,
-- i.e. every element of the codomain has at most one preimage.
LeftUnique : Rel A ℓ₁ → Rel B ℓ₂ → REL A B ℓ → Set _
LeftUnique _≈₁_ _≈₂_ _↦_ = ∀ {x y u v} → x ↦ u → y ↦ v → u ≈₂ v → x ≈₁ y

-- RightUnique - the relation is a (partial) function,
-- i.e. every element of the domain has at most one image.
RightUnique : Rel A ℓ₁ → Rel B ℓ₂ → REL A B ℓ → Set _
RightUnique _≈₁_ _≈₂_ _↦_ = ∀ {x y u v} → x ↦ u → y ↦ v → x ≈₁ y → u ≈₂ v

-- LeftTotal - the relation is total (as in a total function),
-- i.e. every element of the domain has at least one image.
LeftTotal : Rel A ℓ₁ → REL A B ℓ → Set _
LeftTotal _≈_ _↦_ = ∀ x → ∃[ u ] x ↦ u

-- RightTotal - the relation is surjective,
-- i.e. every element of the codomain has at least one preimage.
RightTotal : Rel B ℓ₂ → REL A B ℓ → Set _
RightTotal _≈_ _↦_ = ∀ u → ∃[ x ] x ↦ u
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