Products of equivalence relataions

Content created by Fredrik Bakke, Egbert Rijke, Fernando Chu, Julian KG, fernabnor and louismntnu.

Created on 2023-03-26.
Last modified on 2024-02-06.

module foundation.products-equivalence-relations where
Imports 
open import foundation.binary-relations
open import foundation.dependent-pair-types
open import foundation.products-binary-relations
open import foundation.universe-levels

open import foundation-core.cartesian-product-types
open import foundation-core.equivalence-relations

Idea

Given two equivalence relations R and S, their product is an equivalence relation.

Definition

The product of two equivalence relations

module _
  {l1 l2 l3 l4 : Level}
  {A : UU l1} (R : equivalence-relation l2 A)
  {B : UU l3} (S : equivalence-relation l4 B)
  where

  reflexive-product-Relation-Prop :
    is-reflexive-Relation-Prop
      ( product-Relation-Prop
        ( prop-equivalence-relation R)
        ( prop-equivalence-relation S))
  pr1 (reflexive-product-Relation-Prop x) = refl-equivalence-relation R (pr1 x)
  pr2 (reflexive-product-Relation-Prop x) = refl-equivalence-relation S (pr2 x)

  symmetric-product-Relation-Prop :
    is-symmetric-Relation-Prop
      ( product-Relation-Prop
        ( prop-equivalence-relation R)
        ( prop-equivalence-relation S))
  pr1 (symmetric-product-Relation-Prop x y (p , q)) =
    symmetric-equivalence-relation R (pr1 x) (pr1 y) p
  pr2 (symmetric-product-Relation-Prop x y (p , q)) =
    symmetric-equivalence-relation S (pr2 x) (pr2 y) q

  transitive-product-Relation-Prop :
    is-transitive-Relation-Prop
      ( product-Relation-Prop
        ( prop-equivalence-relation R)
        ( prop-equivalence-relation S))
  pr1 (transitive-product-Relation-Prop x y z (p , q) (p' , q')) =
    transitive-equivalence-relation R (pr1 x) (pr1 y) (pr1 z) p p'
  pr2 (transitive-product-Relation-Prop x y z (p , q) (p' , q')) =
    transitive-equivalence-relation S (pr2 x) (pr2 y) (pr2 z) q q'

  product-equivalence-relation : equivalence-relation (l2 ⊔ l4) (A × B)
  pr1 product-equivalence-relation =
    product-Relation-Prop
      ( prop-equivalence-relation R)
      ( prop-equivalence-relation S)
  pr1 (pr2 product-equivalence-relation) = reflexive-product-Relation-Prop
  pr1 (pr2 (pr2 product-equivalence-relation)) = symmetric-product-Relation-Prop
  pr2 (pr2 (pr2 product-equivalence-relation)) =
    transitive-product-Relation-Prop

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