Multivariable decidable relations

Content created by Jonathan Prieto-Cubides, Fredrik Bakke and Egbert Rijke.

Created on 2022-09-09.
Last modified on 2023-06-08.

module foundation.multivariable-decidable-relations where
Imports 
open import elementary-number-theory.natural-numbers

open import foundation.decidable-subtypes
open import foundation.multivariable-correspondences
open import foundation.multivariable-relations
open import foundation.universe-levels

open import univalent-combinatorics.standard-finite-types

Idea

Consider a family of types A i indexed by i : Fin n. An n-ary decidable relation on the tuples of elements of the A i is a decidable subtype of the product of the A i.

Definition

multivariable-decidable-relation :
  {l1 : Level} (l2 : Level) (n : ℕ) (A : Fin n → UU l1) → UU (l1 ⊔ lsuc l2)
multivariable-decidable-relation l2 n A =
  decidable-subtype l2 ((i : Fin n) → A i)

module _
  {l1 l2 : Level} {n : ℕ} {A : Fin n → UU l1}
  (R : multivariable-decidable-relation l2 n A)
  where

  multivariable-relation-multivariable-decidable-relation :
    multivariable-relation l2 n A
  multivariable-relation-multivariable-decidable-relation =
    subtype-decidable-subtype R

  multivariable-correspondence-multivariable-decidable-relation :
    multivariable-correspondence l2 n A
  multivariable-correspondence-multivariable-decidable-relation =
    is-in-decidable-subtype R

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