Monotonic rational neighborhood relations

Content created by Louis Wasserman and malarbol.

Created on 2025-08-18.
Last modified on 2025-08-18.

module metric-spaces.monotonic-rational-neighborhood-relations where
Imports 
open import elementary-number-theory.positive-rational-numbers

open import foundation.propositions
open import foundation.universe-levels

open import metric-spaces.rational-neighborhood-relations

Idea

A rational neighborhood relation is monotonic if, for all d₁ < d₂, all d₁-neighborhoods are d₂-neighborhoods.

Definitions

The property of being a monotonic rational neighborhood relation

module _
  {l1 l2 : Level} {A : UU l1} (B : Rational-Neighborhood-Relation l2 A)
  where

  is-monotonic-prop-Rational-Neighborhood-Relation : Prop (l1 ⊔ l2)
  is-monotonic-prop-Rational-Neighborhood-Relation =
    Π-Prop
      ( A)
      ( λ x →
        ( Π-Prop
          ( A)
          ( λ y →
            ( Π-Prop
              ( ℚ⁺)
              ( λ d₁ →
                ( Π-Prop
                  ( ℚ⁺)
                  ( λ d₂ →
                    ( Π-Prop
                      ( le-ℚ⁺ d₁ d₂)
                      ( λ H →
                        hom-Prop (B d₁ x y) (B d₂ x y))))))))))

  is-monotonic-Rational-Neighborhood-Relation : UU (l1 ⊔ l2)
  is-monotonic-Rational-Neighborhood-Relation =
    type-Prop is-monotonic-prop-Rational-Neighborhood-Relation

  is-prop-is-monotonic-Rational-Neighborhood-Relation :
    is-prop is-monotonic-Rational-Neighborhood-Relation
  is-prop-is-monotonic-Rational-Neighborhood-Relation =
    is-prop-type-Prop is-monotonic-prop-Rational-Neighborhood-Relation

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