Mere logical equivalences

Content created by Egbert Rijke and Fredrik Bakke.

Created on 2024-04-11.
Last modified on 2024-04-11.

module foundation.mere-logical-equivalences where
Imports 
open import foundation.conjunction
open import foundation.dependent-pair-types
open import foundation.inhabited-types
open import foundation.logical-equivalences
open import foundation.mere-functions
open import foundation.propositional-truncations
open import foundation.universe-levels

open import foundation-core.cartesian-product-types
open import foundation-core.equivalences
open import foundation-core.function-types
open import foundation-core.propositions

Idea

Two types A and B are merely logically equivalent if the type of logical equivalences between A and B is inhabited.

  mere-iff A B := ║(A ↔ B)║₋₁

Definitions

Mere logical equivalence of types

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

  prop-mere-iff : Prop (l1 ⊔ l2)
  prop-mere-iff = trunc-Prop (A ↔ B)

  mere-iff : UU (l1 ⊔ l2)
  mere-iff = type-Prop prop-mere-iff

  is-prop-mere-iff : is-prop mere-iff
  is-prop-mere-iff = is-prop-type-Prop prop-mere-iff

Properties

Mere logical equivalence is a reflexive relation

module _
  {l : Level} (A : UU l)
  where

  refl-mere-iff : mere-iff A A
  refl-mere-iff = unit-trunc-Prop id-iff

Mere logical equivalence is a transitive relation

module _
  {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3}
  where

  transitive-mere-iff : mere-iff B C → mere-iff A B → mere-iff A C
  transitive-mere-iff |g| =
    rec-trunc-Prop
      ( prop-mere-iff A C)
      ( λ f →
        rec-trunc-Prop
          ( prop-mere-iff A C)
          ( λ g → unit-trunc-Prop (g ∘iff f))
          ( |g|))

Mere logical equivalence is a symmetric relation

module _
  {l1 l2 : Level} {A : UU l1} {B : UU l2}
  where

  symmetric-mere-iff : mere-iff A B → mere-iff B A
  symmetric-mere-iff =
    rec-trunc-Prop (prop-mere-iff B A) (unit-trunc-Prop ∘ inv-iff)

Merely logically equivalent types are coinhabited

If A and B are merely logically equivalent then they are coinhabited.

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

  ev-forward-mere-iff' : mere-iff A B → A → is-inhabited B
  ev-forward-mere-iff' |f| a =
    rec-trunc-Prop
      ( trunc-Prop B)
      ( λ f → unit-trunc-Prop (forward-implication f a))
      ( |f|)

  ev-forward-mere-iff : mere-iff A B → is-inhabited A → is-inhabited B
  ev-forward-mere-iff |f| =
    rec-trunc-Prop (trunc-Prop B) (ev-forward-mere-iff' |f|)

  ev-backward-mere-iff' : mere-iff A B → B → is-inhabited A
  ev-backward-mere-iff' |f| b =
    rec-trunc-Prop
      ( trunc-Prop A)
      ( λ f → unit-trunc-Prop (backward-implication f b))
      ( |f|)

  ev-backward-mere-iff : mere-iff A B → is-inhabited B → is-inhabited A
  ev-backward-mere-iff |f| =
    rec-trunc-Prop (trunc-Prop A) (ev-backward-mere-iff' |f|)

  is-coinhabited-mere-iff : mere-iff A B → is-inhabited A ↔ is-inhabited B
  is-coinhabited-mere-iff |f| =
    ( ev-forward-mere-iff |f| , ev-backward-mere-iff |f|)

Merely logically equivalent types have bidirectional mere functions

If A and B are merely logically equivalent then we have a mere function from A to B and a mere function from B to A.

module _
  {l1 l2 : Level} {A : UU l1} {B : UU l2}
  where

  forward-mere-function-mere-iff : mere-iff A B → mere-function A B
  forward-mere-function-mere-iff =
    rec-trunc-Prop
      ( prop-mere-function A B)
      ( unit-trunc-Prop ∘ forward-implication)

  backward-mere-function-mere-iff : mere-iff A B → mere-function B A
  backward-mere-function-mere-iff =
    rec-trunc-Prop
      ( prop-mere-function B A)
      ( unit-trunc-Prop ∘ backward-implication)

Mere logical equivalence is equivalent to having bidirectional mere functions

For all types we have the equivalence

  (mere-iff A B) ≃ (mere-function A B × mere-function B A).

module _
  {l1 l2 : Level} {A : UU l1} {B : UU l2}
  where

  compute-mere-iff :
    mere-iff A B ≃ mere-function A B × mere-function B A
  compute-mere-iff = equiv-product-trunc-Prop (A → B) (B → A)

See also

Recent changes