------------------------------------------------------------------------
-- The Agda standard library
--
-- Definitions of properties over distance functions
------------------------------------------------------------------------

-- The contents of this module should be accessed via `Function.Metric`.

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

module Function.Metric.Definitions where

open import Algebra.Core using (Op₂)
open import Data.Product using ()
open import Function.Metric.Core using (DistanceFunction)
open import Level using (Level)
open import Relation.Binary.Core using (Rel; _Preserves₂_⟶_⟶_)
open import Relation.Nullary using (¬_)

private
  variable
    a i  ℓ₁ ℓ₂ : Level
    A : Set a
    I : Set i

-----------------------------------------------------------------------
-- Properties

Congruent : Rel A ℓ₁  Rel I ℓ₂  DistanceFunction A I  Set _
Congruent _≈ₐ_ _≈ᵢ_ d = d Preserves₂ _≈ₐ_  _≈ₐ_  _≈ᵢ_

Indiscernable : Rel A ℓ₁  Rel I ℓ₂  DistanceFunction A I  I  Set _
Indiscernable _≈ₐ_ _≈ᵢ_ d 0# =  {x y}  d x y ≈ᵢ 0#  x ≈ₐ y

Definite : Rel A ℓ₁  Rel I ℓ₂  DistanceFunction A I  I  Set _
Definite _≈ₐ_ _≈ᵢ_ d 0# =  {x y}  x ≈ₐ y  d x y ≈ᵢ 0#

NonNegative : Rel I ℓ₂  DistanceFunction A I  I  Set _
NonNegative _≤_ d 0# =  {x y}  0#  d x y

Symmetric : Rel I   DistanceFunction A I  Set _
Symmetric _≈_ d =  x y  d x y  d y x

TriangleInequality : Rel I   Op₂ I  DistanceFunction A I  _
TriangleInequality _≤_ _∙_ d =  x y z  d x z  (d x y  d y z)

Bounded : Rel I   DistanceFunction A I  Set _
Bounded _≤_ d =  λ n   x y  d x y  n

TranslationInvariant : Rel I ℓ₂  Op₂ A  DistanceFunction A I  Set _
TranslationInvariant _≈_ _∙_ d =  {x y a}  d (x  a) (y  a)  d x y

Contracting : Rel I   (A  A)  DistanceFunction A I  Set _
Contracting _≤_ f d =  x y  d (f x) (f y)  d x y

ContractingOnOrbits : Rel I   (A  A)  DistanceFunction A I  Set _
ContractingOnOrbits _≤_ f d =  x  d (f x) (f (f x))  d x (f x)

StrictlyContracting : Rel A ℓ₁  Rel I ℓ₂  (A  A)  DistanceFunction A I  Set _
StrictlyContracting _≈_ _<_ f d =  {x y}  ¬ (y  x)  d (f x) (f y) < d x y

StrictlyContractingOnOrbits : Rel A ℓ₁  Rel I ℓ₂  (A  A)  DistanceFunction A I  Set _
StrictlyContractingOnOrbits _≈_ _<_ f d =  {x}  ¬ (f x  x)  d (f x) (f (f x)) < d x (f x)