------------------------------------------------------------------------
-- The Agda standard library
--
-- Core definitions for metrics over ℕ
------------------------------------------------------------------------
 
{-# OPTIONS --cubical-compatible --safe #-}
 
module Function.Metric.Nat.Structures where
 
open import Data.Nat.Base hiding (suc)
open import Function.Base using (const)
open import Level using (Level; suc)
open import Relation.Binary.Core using (Rel)
open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 
open import Function.Metric.Nat.Core
open import Function.Metric.Nat.Definitions
import Function.Metric.Structures as Base
 
private
  variable
    a ℓ : Level
    A   : Set a
 
------------------------------------------------------------------------
-- Proto-metrics
 
IsProtoMetric : Rel A ℓ → DistanceFunction A → Set _
IsProtoMetric _≈_ = Base.IsProtoMetric _≈_ _≡_ _≤_ 
 
open Base using (module IsProtoMetric) public
 
------------------------------------------------------------------------
-- Pre-metrics
 
IsPreMetric : Rel A ℓ → DistanceFunction A → Set _
IsPreMetric _≈_ = Base.IsPreMetric _≈_ _≡_ _≤_ 
 
open Base using (module IsPreMetric) public
 
------------------------------------------------------------------------
-- Quasi-semi-metrics
 
IsQuasiSemiMetric : Rel A ℓ → DistanceFunction A → Set _
IsQuasiSemiMetric _≈_ = Base.IsQuasiSemiMetric _≈_ _≡_ _≤_ 
 
open Base using (module IsQuasiSemiMetric) public
 
------------------------------------------------------------------------
-- Semi-metrics
 
IsSemiMetric : Rel A ℓ → DistanceFunction A → Set _
IsSemiMetric _≈_ = Base.IsSemiMetric _≈_ _≡_ _≤_ 
 
open Base using (module IsSemiMetric) public
 
------------------------------------------------------------------------
-- Metrics
 
IsMetric : Rel A ℓ → DistanceFunction A → Set _
IsMetric _≈_ = Base.IsGeneralMetric _≈_ _≡_ _≤_  _+_
 
module IsMetric {_≈_ : Rel A ℓ} {d : DistanceFunction A}
                (M : IsMetric _≈_ d) where
  open Base.IsGeneralMetric M public
 
------------------------------------------------------------------------
-- Ultra-metrics
 
IsUltraMetric : Rel A ℓ → DistanceFunction A → Set _
IsUltraMetric _≈_ = Base.IsGeneralMetric _≈_ _≡_ _≤_  _⊔_
 
module IsUltraMetric {_≈_ : Rel A ℓ} {d : DistanceFunction A}
                     (UM : IsUltraMetric _≈_ d) where
  open Base.IsGeneralMetric UM public