Pairs of distinct elements

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

Created on 2022-04-11.
Last modified on 2024-01-31.

module foundation.pairs-of-distinct-elements where
open import foundation.dependent-pair-types
open import foundation.embeddings
open import foundation.equivalences
open import foundation.fundamental-theorem-of-identity-types
open import foundation.negated-equality
open import foundation.negation
open import foundation.structure-identity-principle
open import foundation.subtype-identity-principle
open import foundation.universe-levels

open import foundation-core.cartesian-product-types
open import foundation-core.contractible-types
open import foundation-core.identity-types
open import foundation-core.injective-maps
open import foundation-core.torsorial-type-families


Pairs of distinct elements in a type A consist of two elements x and y of A equipped with an element of type x ≠ y.


pair-of-distinct-elements : {l : Level}  UU l  UU l
pair-of-distinct-elements A =
  Σ A  x  Σ A  y  x  y))

module _
  {l : Level} {A : UU l} (p : pair-of-distinct-elements A)

  first-pair-of-distinct-elements : A
  first-pair-of-distinct-elements = pr1 p

  second-pair-of-distinct-elements : A
  second-pair-of-distinct-elements = pr1 (pr2 p)

  distinction-pair-of-distinct-elements :
    first-pair-of-distinct-elements  second-pair-of-distinct-elements
  distinction-pair-of-distinct-elements = pr2 (pr2 p)


Characterization of the identity type of the type of pairs of distinct elements

module _
  {l : Level} {A : UU l}

  Eq-pair-of-distinct-elements :
    (p q : pair-of-distinct-elements A)  UU l
  Eq-pair-of-distinct-elements p q =
    (first-pair-of-distinct-elements p  first-pair-of-distinct-elements q) ×
    (second-pair-of-distinct-elements p  second-pair-of-distinct-elements q)

  refl-Eq-pair-of-distinct-elements :
    (p : pair-of-distinct-elements A)  Eq-pair-of-distinct-elements p p
  pr1 (refl-Eq-pair-of-distinct-elements p) = refl
  pr2 (refl-Eq-pair-of-distinct-elements p) = refl

  Eq-eq-pair-of-distinct-elements :
    (p q : pair-of-distinct-elements A) 
    p  q  Eq-pair-of-distinct-elements p q
  Eq-eq-pair-of-distinct-elements p .p refl =
    refl-Eq-pair-of-distinct-elements p

  is-torsorial-Eq-pair-of-distinct-elements :
    (p : pair-of-distinct-elements A) 
    is-torsorial (Eq-pair-of-distinct-elements p)
  is-torsorial-Eq-pair-of-distinct-elements p =
      ( is-torsorial-Id (first-pair-of-distinct-elements p))
      ( pair (first-pair-of-distinct-elements p) refl)
      ( is-torsorial-Eq-subtype
        ( is-torsorial-Id (second-pair-of-distinct-elements p))
        ( λ x  is-prop-neg)
        ( second-pair-of-distinct-elements p)
        ( refl)
        ( distinction-pair-of-distinct-elements p))

  is-equiv-Eq-eq-pair-of-distinct-elements :
    (p q : pair-of-distinct-elements A) 
    is-equiv (Eq-eq-pair-of-distinct-elements p q)
  is-equiv-Eq-eq-pair-of-distinct-elements p =
      ( is-torsorial-Eq-pair-of-distinct-elements p)
      ( Eq-eq-pair-of-distinct-elements p)

  eq-Eq-pair-of-distinct-elements :
    {p q : pair-of-distinct-elements A} 
    first-pair-of-distinct-elements p  first-pair-of-distinct-elements q 
    second-pair-of-distinct-elements p  second-pair-of-distinct-elements q 
    p  q
  eq-Eq-pair-of-distinct-elements {p} {q} α β =
    map-inv-is-equiv (is-equiv-Eq-eq-pair-of-distinct-elements p q) (pair α β)

Equivalences map pairs of distinct elements to pairs of distinct elements

module _
  {l1 l2 : Level} {A : UU l1} {B : UU l2} (e : A  B)

  map-equiv-pair-of-distinct-elements :
    pair-of-distinct-elements A  pair-of-distinct-elements B
  pr1 (map-equiv-pair-of-distinct-elements p) =
    map-equiv e (first-pair-of-distinct-elements p)
  pr1 (pr2 (map-equiv-pair-of-distinct-elements p)) =
    map-equiv e (second-pair-of-distinct-elements p)
  pr2 (pr2 (map-equiv-pair-of-distinct-elements p)) q =
    distinction-pair-of-distinct-elements p
      ( is-injective-is-equiv (is-equiv-map-equiv e) q)

  map-inv-equiv-pair-of-distinct-elements :
    pair-of-distinct-elements B  pair-of-distinct-elements A
  pr1 (map-inv-equiv-pair-of-distinct-elements q) =
    map-inv-equiv e (first-pair-of-distinct-elements q)
  pr1 (pr2 (map-inv-equiv-pair-of-distinct-elements q)) =
    map-inv-equiv e (second-pair-of-distinct-elements q)
  pr2 (pr2 (map-inv-equiv-pair-of-distinct-elements q)) p =
    distinction-pair-of-distinct-elements q
      ( is-injective-is-equiv (is-equiv-map-inv-equiv e) p)

  is-section-map-inv-equiv-pair-of-distinct-elements :
    (q : pair-of-distinct-elements B) 
    ( map-equiv-pair-of-distinct-elements
      ( map-inv-equiv-pair-of-distinct-elements q)) 
    ( q)
  is-section-map-inv-equiv-pair-of-distinct-elements q =
      ( is-section-map-inv-equiv e (first-pair-of-distinct-elements q))
      ( is-section-map-inv-equiv e (second-pair-of-distinct-elements q))

  is-retraction-map-inv-equiv-pair-of-distinct-elements :
    (p : pair-of-distinct-elements A) 
    ( map-inv-equiv-pair-of-distinct-elements
      ( map-equiv-pair-of-distinct-elements p)) 
    ( p)
  is-retraction-map-inv-equiv-pair-of-distinct-elements p =
      ( is-retraction-map-inv-equiv e (first-pair-of-distinct-elements p))
      ( is-retraction-map-inv-equiv e (second-pair-of-distinct-elements p))

  is-equiv-map-equiv-pair-of-distinct-elements :
    is-equiv map-equiv-pair-of-distinct-elements
  is-equiv-map-equiv-pair-of-distinct-elements =

  equiv-pair-of-distinct-elements :
    pair-of-distinct-elements A  pair-of-distinct-elements B
  pr1 equiv-pair-of-distinct-elements = map-equiv-pair-of-distinct-elements
  pr2 equiv-pair-of-distinct-elements =

Embeddings map pairs of distinct elements to pairs of distinct elements

module _
  {l1 l2 : Level} {A : UU l1} {B : UU l2} (e : A  B)

  emb-pair-of-distinct-elements :
    pair-of-distinct-elements A  pair-of-distinct-elements B
  emb-pair-of-distinct-elements =
      ( λ x  Σ B  y  x  y))
      ( e)
      ( λ x 
          ( λ y  map-emb e x  y)
          ( e)
          ( λ _  emb-equiv (equiv-neg (equiv-ap-emb e))))

  map-emb-pair-of-distinct-elements :
    pair-of-distinct-elements A  pair-of-distinct-elements B
  map-emb-pair-of-distinct-elements =
    map-emb emb-pair-of-distinct-elements

  is-emb-map-emb-pair-of-distinct-elements :
    is-emb map-emb-pair-of-distinct-elements
  is-emb-map-emb-pair-of-distinct-elements =
    is-emb-map-emb emb-pair-of-distinct-elements

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