Unordered tuples of types

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

Created on 2022-05-01.
Last modified on 2024-01-11.

module foundation.unordered-tuples-of-types where
open import elementary-number-theory.natural-numbers

open import foundation.dependent-pair-types
open import foundation.fundamental-theorem-of-identity-types
open import foundation.structure-identity-principle
open import foundation.univalence
open import foundation.universe-levels
open import foundation.unordered-tuples

open import foundation-core.equivalences
open import foundation-core.identity-types
open import foundation-core.torsorial-type-families

open import univalent-combinatorics.finite-types


An unordered tuple of types is an unordered tuple of elements in a universe


Unordered tuple of types

unordered-tuple-types : (l : Level)    UU (lsuc l)
unordered-tuple-types l n = unordered-tuple n (UU l)

Equivalences of unordered pairs of types

equiv-unordered-tuple-types :
  {l1 l2 : Level} (n : ) 
  unordered-tuple-types l1 n  unordered-tuple-types l2 n  UU (l1  l2)
equiv-unordered-tuple-types n A B =
  Σ ( type-unordered-tuple n A  type-unordered-tuple n B)
    ( λ e 
      (i : type-unordered-tuple n A) 
      element-unordered-tuple n A i 
      element-unordered-tuple n B (map-equiv e i))

module _
  {l1 l2 : Level} (n : )
  (A : unordered-tuple-types l1 n) (B : unordered-tuple-types l2 n)
  (e : equiv-unordered-tuple-types n A B)

  equiv-type-equiv-unordered-tuple-types :
    type-unordered-tuple n A  type-unordered-tuple n B
  equiv-type-equiv-unordered-tuple-types = pr1 e

  map-equiv-type-equiv-unordered-tuple-types :
    type-unordered-tuple n A  type-unordered-tuple n B
  map-equiv-type-equiv-unordered-tuple-types =
    map-equiv equiv-type-equiv-unordered-tuple-types

  equiv-element-equiv-unordered-tuple-types :
    (i : type-unordered-tuple n A) 
    ( element-unordered-tuple n A i) 
    ( element-unordered-tuple n B
      ( map-equiv-type-equiv-unordered-tuple-types i))
  equiv-element-equiv-unordered-tuple-types = pr2 e


Univalence for unordered pairs of types

module _
  {l : Level} {n : } (A : unordered-tuple-types l n)

  id-equiv-unordered-tuple-types : equiv-unordered-tuple-types n A A
  pr1 id-equiv-unordered-tuple-types = id-equiv
  pr2 id-equiv-unordered-tuple-types i = id-equiv

  equiv-eq-unordered-tuple-types :
    (B : unordered-tuple-types l n)  A  B  equiv-unordered-tuple-types n A B
  equiv-eq-unordered-tuple-types .A refl = id-equiv-unordered-tuple-types

  is-torsorial-equiv-unordered-tuple-types :
    is-torsorial (equiv-unordered-tuple-types n A)
  is-torsorial-equiv-unordered-tuple-types =
      ( is-torsorial-equiv-UU-Fin {k = n} (type-unordered-tuple-UU-Fin n A))
      ( pair (type-unordered-tuple-UU-Fin n A) id-equiv)
      ( is-torsorial-equiv-fam (element-unordered-tuple n A))

  is-equiv-equiv-eq-unordered-tuple-types :
    (B : unordered-tuple-types l n) 
    is-equiv (equiv-eq-unordered-tuple-types B)
  is-equiv-equiv-eq-unordered-tuple-types =

  extensionality-unordered-tuple-types :
    (B : unordered-tuple-types l n) 
    (A  B)  equiv-unordered-tuple-types n A B
  pr1 (extensionality-unordered-tuple-types B) =
    equiv-eq-unordered-tuple-types B
  pr2 (extensionality-unordered-tuple-types B) =
    is-equiv-equiv-eq-unordered-tuple-types B

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