Logical equivalences

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

Created on 2022-01-29.
Last modified on 2024-02-06.

module foundation.logical-equivalences where
Imports
open import foundation.dependent-pair-types
open import foundation.equality-cartesian-product-types
open import foundation.equivalence-extensionality
open import foundation.function-extensionality
open import foundation.functoriality-cartesian-product-types
open import foundation.type-arithmetic-dependent-pair-types
open import foundation.universe-levels

open import foundation-core.cartesian-product-types
open import foundation-core.contractible-types
open import foundation-core.equivalences
open import foundation-core.fibers-of-maps
open import foundation-core.function-types
open import foundation-core.functoriality-dependent-pair-types
open import foundation-core.homotopies
open import foundation-core.identity-types
open import foundation-core.injective-maps
open import foundation-core.propositions

Idea

Logical equivalences between two types A and B consist of a map A → B and a map B → A. The type of logical equivalences between types is the Curry-Howard interpretation of logical equivalences between propositions.

Definition

Logical equivalences between types

infix 6 _↔_

_↔_ : {l1 l2 : Level}  UU l1  UU l2  UU (l1  l2)
A  B = (A  B) × (B  A)

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

  forward-implication : A  B
  forward-implication = pr1 H

  backward-implication : B  A
  backward-implication = pr2 H

Logical equivalences between propositions

module _
  {l1 l2 : Level} (P : Prop l1) (Q : Prop l2)
  where

  type-iff-Prop : UU (l1  l2)
  type-iff-Prop = type-Prop P  type-Prop Q

  is-prop-iff-Prop : is-prop type-iff-Prop
  is-prop-iff-Prop =
    is-prop-product
      ( is-prop-function-type (is-prop-type-Prop Q))
      ( is-prop-function-type (is-prop-type-Prop P))

  iff-Prop : Prop (l1  l2)
  pr1 iff-Prop = type-iff-Prop
  pr2 iff-Prop = is-prop-iff-Prop

  infix 6 _⇔_

  _⇔_ : UU (l1  l2)
  _⇔_ = type-iff-Prop

Composition of logical equivalences

infixr 15 _∘iff_

_∘iff_ :
  {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} 
  (B  C)  (A  B)  (A  C)
pr1 ((g1 , g2) ∘iff (f1 , f2)) = g1  f1
pr2 ((g1 , g2) ∘iff (f1 , f2)) = f2  g2

Inverting a logical equivalence

inv-iff :
  {l1 l2 : Level} {A : UU l1} {B : UU l2}  (A  B)  (B  A)
pr1 (inv-iff (f , g)) = g
pr2 (inv-iff (f , g)) = f

Properties

Characterizing equality of logical equivalences

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

  htpy-iff : (f g : A  B)  UU (l1  l2)
  htpy-iff f g =
    ( forward-implication f ~ forward-implication g) ×
    ( backward-implication f ~ backward-implication g)

  ext-iff : (f g : A  B)  (f  g)  htpy-iff f g
  ext-iff f g = equiv-product equiv-funext equiv-funext ∘e equiv-pair-eq f g

  refl-htpy-iff : (f : A  B)  htpy-iff f f
  pr1 (refl-htpy-iff f) = refl-htpy
  pr2 (refl-htpy-iff f) = refl-htpy

  htpy-eq-iff : {f g : A  B}  f  g  htpy-iff f g
  htpy-eq-iff {f} {g} = map-equiv (ext-iff f g)

  eq-htpy-iff : (f g : A  B)  htpy-iff f g  (f  g)
  eq-htpy-iff f g = map-inv-equiv (ext-iff f g)

Logical equivalences between propositions induce equivalences

module _
  {l1 l2 : Level} (P : Prop l1) (Q : Prop l2)
  where

  equiv-iff' : (P  Q)  (type-Prop P  type-Prop Q)
  pr1 (equiv-iff' t) = pr1 t
  pr2 (equiv-iff' t) = is-equiv-is-prop (pr2 P) (pr2 Q) (pr2 t)

  equiv-iff :
    (type-Prop P  type-Prop Q)  (type-Prop Q  type-Prop P) 
    type-Prop P  type-Prop Q
  equiv-iff f g = equiv-iff' (f , g)

Equivalences are logical equivalences

iff-equiv : {l1 l2 : Level} {A : UU l1} {B : UU l2}  (A  B)  (A  B)
pr1 (iff-equiv e) = map-equiv e
pr2 (iff-equiv e) = map-inv-equiv e

is-injective-iff-equiv :
  {l1 l2 : Level} {A : UU l1} {B : UU l2}  is-injective (iff-equiv {A = A} {B})
is-injective-iff-equiv p = eq-htpy-equiv (pr1 (htpy-eq-iff p))

compute-fiber-iff-equiv :
  {l1 l2 : Level} {A : UU l1} {B : UU l2} ((f , g) : A  B) 
  fiber (iff-equiv) (f , g)  Σ (is-equiv f)  f'  map-inv-is-equiv f' ~ g)
compute-fiber-iff-equiv {A = A} {B} (f , g) =
  ( equiv-tot  _  equiv-funext)) ∘e
  ( left-unit-law-Σ-is-contr (is-torsorial-Id' f) (f , refl)) ∘e
  ( inv-associative-Σ (A  B) (_= f) _) ∘e
  ( equiv-tot  _  equiv-left-swap-Σ)) ∘e
  ( associative-Σ (A  B) _ _) ∘e
  ( equiv-tot  e  equiv-pair-eq (iff-equiv e) (f , g)))

Two equal propositions are logically equivalent

iff-eq : {l1 : Level} {P Q : Prop l1}  P  Q  P  Q
pr1 (iff-eq refl) = id
pr2 (iff-eq refl) = id

Logical equivalence of propositions is equivalent to equivalence of propositions

abstract
  is-equiv-equiv-iff :
    {l1 l2 : Level} (P : Prop l1) (Q : Prop l2) 
    is-equiv (equiv-iff' P Q)
  is-equiv-equiv-iff P Q =
    is-equiv-is-prop
      ( is-prop-iff-Prop P Q)
      ( is-prop-type-equiv-Prop P Q)
      ( iff-equiv)

equiv-equiv-iff :
  {l1 l2 : Level} (P : Prop l1) (Q : Prop l2) 
  (P  Q)  (type-Prop P  type-Prop Q)
pr1 (equiv-equiv-iff P Q) = equiv-iff' P Q
pr2 (equiv-equiv-iff P Q) = is-equiv-equiv-iff P Q

Logical equivalences between dependent function types

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

  iff-Π-iff-family : ((i : I)  A i  B i)  ((i : I)  A i)  ((i : I)  B i)
  pr1 (iff-Π-iff-family e) a i = forward-implication (e i) (a i)
  pr2 (iff-Π-iff-family e) b i = backward-implication (e i) (b i)

Reasoning with logical equivalences

Logical equivalences can be constructed by equational reasoning in the following way:

logical-equivalence-reasoning
  X ↔ Y by equiv-1
    ↔ Z by equiv-2
    ↔ V by equiv-3
infixl 1 logical-equivalence-reasoning_
infixl 0 step-logical-equivalence-reasoning

logical-equivalence-reasoning_ :
  {l1 : Level} (X : UU l1)  X  X
pr1 (logical-equivalence-reasoning X) = id
pr2 (logical-equivalence-reasoning X) = id

step-logical-equivalence-reasoning :
  {l1 l2 l3 : Level} {X : UU l1} {Y : UU l2} 
  (X  Y)  (Z : UU l3)  (Y  Z)  (X  Z)
step-logical-equivalence-reasoning e Z f = f ∘iff e

syntax step-logical-equivalence-reasoning e Z f = e  Z by f

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