Propositions

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

Created on 2022-02-07.
Last modified on 2024-02-06.

module foundation-core.propositions where
Imports
open import foundation.dependent-pair-types
open import foundation.function-extensionality
open import foundation.universe-levels

open import foundation-core.cartesian-product-types
open import foundation-core.contractible-types
open import foundation-core.equality-dependent-pair-types
open import foundation-core.equivalences
open import foundation-core.function-types
open import foundation-core.homotopies
open import foundation-core.identity-types
open import foundation-core.transport-along-identifications

Idea

A type is considered to be a proposition if its identity types are contractible. This condition is equivalent to the condition that it has up to identification at most one element.

Definition

is-prop : {l : Level} (A : UU l)  UU l
is-prop A = (x y : A)  is-contr (x  y)

Prop :
  (l : Level)  UU (lsuc l)
Prop l = Σ (UU l) is-prop

module _
  {l : Level}
  where

  type-Prop : Prop l  UU l
  type-Prop P = pr1 P

  abstract
    is-prop-type-Prop : (P : Prop l)  is-prop (type-Prop P)
    is-prop-type-Prop P = pr2 P

Examples

We prove here only that any contractible type is a proposition. The fact that the empty type and the unit type are propositions can be found in

foundation.empty-types
foundation.unit-type

Properties

To show that a type is a proposition, we may assume it is inhabited

abstract
  is-prop-is-inhabited :
    {l1 : Level} {X : UU l1}  (X  is-prop X)  is-prop X
  is-prop-is-inhabited f x y = f x x y

Equivalent characterizations of propositions

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

  all-elements-equal : UU l
  all-elements-equal = (x y : A)  x  y

  is-proof-irrelevant : UU l
  is-proof-irrelevant = A  is-contr A

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

  abstract
    is-prop-all-elements-equal : all-elements-equal A  is-prop A
    pr1 (is-prop-all-elements-equal H x y) = (inv (H x x))  (H x y)
    pr2 (is-prop-all-elements-equal H x .x) refl = left-inv (H x x)

  abstract
    eq-is-prop' : is-prop A  all-elements-equal A
    eq-is-prop' H x y = pr1 (H x y)

  abstract
    eq-is-prop : is-prop A  {x y : A}  x  y
    eq-is-prop H {x} {y} = eq-is-prop' H x y

  abstract
    is-proof-irrelevant-all-elements-equal :
      all-elements-equal A  is-proof-irrelevant A
    pr1 (is-proof-irrelevant-all-elements-equal H a) = a
    pr2 (is-proof-irrelevant-all-elements-equal H a) = H a

  abstract
    is-proof-irrelevant-is-prop : is-prop A  is-proof-irrelevant A
    is-proof-irrelevant-is-prop =
      is-proof-irrelevant-all-elements-equal  eq-is-prop'

  abstract
    is-prop-is-proof-irrelevant : is-proof-irrelevant A  is-prop A
    is-prop-is-proof-irrelevant H x y = is-prop-is-contr (H x) x y

  abstract
    eq-is-proof-irrelevant : is-proof-irrelevant A  all-elements-equal A
    eq-is-proof-irrelevant = eq-is-prop'  is-prop-is-proof-irrelevant

A map between propositions is an equivalence if there is a map in the reverse direction

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

  abstract
    is-equiv-is-prop :
      is-prop A  is-prop B  {f : A  B}  (B  A)  is-equiv f
    is-equiv-is-prop is-prop-A is-prop-B {f} g =
      is-equiv-is-invertible
        ( g)
        ( λ y  eq-is-prop is-prop-B)
        ( λ x  eq-is-prop is-prop-A)

  abstract
    equiv-prop : is-prop A  is-prop B  (A  B)  (B  A)  A  B
    pr1 (equiv-prop is-prop-A is-prop-B f g) = f
    pr2 (equiv-prop is-prop-A is-prop-B f g) =
      is-equiv-is-prop is-prop-A is-prop-B g

Propositions are closed under equivalences

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

  abstract
    is-prop-is-equiv : {f : A  B}  is-equiv f  is-prop B  is-prop A
    is-prop-is-equiv {f} E H =
      is-prop-is-proof-irrelevant
        ( λ a  is-contr-is-equiv B f E (is-proof-irrelevant-is-prop H (f a)))

  abstract
    is-prop-equiv : A  B  is-prop B  is-prop A
    is-prop-equiv (f , is-equiv-f) = is-prop-is-equiv is-equiv-f

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

  abstract
    is-prop-is-equiv' : {f : A  B}  is-equiv f  is-prop A  is-prop B
    is-prop-is-equiv' E H =
      is-prop-is-equiv (is-equiv-map-inv-is-equiv E) H

  abstract
    is-prop-equiv' : A  B  is-prop A  is-prop B
    is-prop-equiv' (f , is-equiv-f) = is-prop-is-equiv' is-equiv-f

Propositions are closed under dependent pair types

abstract
  is-prop-Σ :
    {l1 l2 : Level} {A : UU l1} {B : A  UU l2} 
    is-prop A  ((x : A)  is-prop (B x))  is-prop (Σ A B)
  is-prop-Σ H K x y =
    is-contr-equiv'
      ( Eq-Σ x y)
      ( equiv-eq-pair-Σ x y)
      ( is-contr-Σ'
        ( H (pr1 x) (pr1 y))
        ( λ p  K (pr1 y) (tr _ p (pr2 x)) (pr2 y)))

Σ-Prop :
  {l1 l2 : Level} (P : Prop l1) (Q : type-Prop P  Prop l2) 
  Prop (l1  l2)
pr1 (Σ-Prop P Q) = Σ (type-Prop P)  p  type-Prop (Q p))
pr2 (Σ-Prop P Q) =
  is-prop-Σ
    ( is-prop-type-Prop P)
    ( λ p  is-prop-type-Prop (Q p))

Propositions are closed under cartesian product types

abstract
  is-prop-product :
    {l1 l2 : Level} {A : UU l1} {B : UU l2} 
    is-prop A  is-prop B  is-prop (A × B)
  is-prop-product H K = is-prop-Σ H  x  K)

product-Prop : {l1 l2 : Level}  Prop l1  Prop l2  Prop (l1  l2)
pr1 (product-Prop P Q) = type-Prop P × type-Prop Q
pr2 (product-Prop P Q) =
  is-prop-product (is-prop-type-Prop P) (is-prop-type-Prop Q)

Products of families of propositions are propositions

abstract
  is-prop-Π :
    {l1 l2 : Level} {A : UU l1} {B : A  UU l2} 
    ((x : A)  is-prop (B x))  is-prop ((x : A)  B x)
  is-prop-Π H =
    is-prop-is-proof-irrelevant
      ( λ f  is-contr-Π  x  is-proof-irrelevant-is-prop (H x) (f x)))

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

  type-Π-Prop : UU (l1  l2)
  type-Π-Prop = (x : A)  type-Prop (P x)

  is-prop-type-Π-Prop : is-prop type-Π-Prop
  is-prop-type-Π-Prop = is-prop-Π  x  is-prop-type-Prop (P x))

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

We repeat the above for implicit Π-types.

abstract
  is-prop-implicit-Π :
    {l1 l2 : Level} {A : UU l1} {B : A  UU l2} 
    ((x : A)  is-prop (B x))  is-prop ({x : A}  B x)
  is-prop-implicit-Π H =
    is-prop-equiv
      ( ( λ f x  f {x}) ,
        ( is-equiv-is-invertible  g {x}  g x) (refl-htpy) (refl-htpy)))
      ( is-prop-Π H)

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

  type-implicit-Π-Prop : UU (l1  l2)
  type-implicit-Π-Prop = {x : A}  type-Prop (P x)

  is-prop-type-implicit-Π-Prop : is-prop type-implicit-Π-Prop
  is-prop-type-implicit-Π-Prop =
    is-prop-implicit-Π  x  is-prop-type-Prop (P x))

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

The type of functions into a proposition is a proposition

abstract
  is-prop-function-type :
    {l1 l2 : Level} {A : UU l1} {B : UU l2} 
    is-prop B  is-prop (A  B)
  is-prop-function-type H = is-prop-Π  _  H)

type-function-Prop :
  {l1 l2 : Level}  UU l1  Prop l2  UU (l1  l2)
type-function-Prop A P = A  type-Prop P

is-prop-type-function-Prop :
  {l1 l2 : Level} (A : UU l1) (P : Prop l2) 
  is-prop (type-function-Prop A P)
is-prop-type-function-Prop A P =
  is-prop-function-type (is-prop-type-Prop P)

function-Prop :
  {l1 l2 : Level}  UU l1  Prop l2  Prop (l1  l2)
pr1 (function-Prop A P) = type-function-Prop A P
pr2 (function-Prop A P) = is-prop-type-function-Prop A P

type-hom-Prop :
  {l1 l2 : Level} (P : Prop l1) (Q : Prop l2)  UU (l1  l2)
type-hom-Prop P = type-function-Prop (type-Prop P)

is-prop-type-hom-Prop :
  {l1 l2 : Level} (P : Prop l1) (Q : Prop l2) 
  is-prop (type-hom-Prop P Q)
is-prop-type-hom-Prop P = is-prop-type-function-Prop (type-Prop P)

hom-Prop :
  {l1 l2 : Level}  Prop l1  Prop l2  Prop (l1  l2)
pr1 (hom-Prop P Q) = type-hom-Prop P Q
pr2 (hom-Prop P Q) = is-prop-type-hom-Prop P Q

implication-Prop :
  {l1 l2 : Level}  Prop l1  Prop l2  Prop (l1  l2)
implication-Prop = hom-Prop

type-implication-Prop :
  {l1 l2 : Level}  Prop l1  Prop l2  UU (l1  l2)
type-implication-Prop = type-hom-Prop

infixr 5 _⇒_
_⇒_ = type-implication-Prop

The type of equivalences between two propositions is a proposition

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

  is-prop-equiv-is-prop : is-prop A  is-prop B  is-prop (A  B)
  is-prop-equiv-is-prop H K =
    is-prop-Σ
      ( is-prop-function-type K)
      ( λ f 
        is-prop-product
          ( is-prop-Σ
            ( is-prop-function-type H)
            ( λ g  is-prop-is-contr (is-contr-Π  y  K (f (g y)) y))))
          ( is-prop-Σ
            ( is-prop-function-type H)
            ( λ h  is-prop-is-contr (is-contr-Π  x  H (h (f x)) x)))))

type-equiv-Prop :
  { l1 l2 : Level} (P : Prop l1) (Q : Prop l2)  UU (l1  l2)
type-equiv-Prop P Q = (type-Prop P)  (type-Prop Q)

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

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

A type is a proposition if and only if the type of its endomaps is contractible

abstract
  is-prop-is-contr-endomaps :
    {l : Level} (P : UU l)  is-contr (P  P)  is-prop P
  is-prop-is-contr-endomaps P H =
    is-prop-all-elements-equal  x  htpy-eq (eq-is-contr H))

abstract
  is-contr-endomaps-is-prop :
    {l : Level} (P : UU l)  is-prop P  is-contr (P  P)
  is-contr-endomaps-is-prop P is-prop-P =
    is-proof-irrelevant-is-prop (is-prop-function-type is-prop-P) id

Being a proposition is a proposition

abstract
  is-prop-is-prop :
    {l : Level} (A : UU l)  is-prop (is-prop A)
  is-prop-is-prop A = is-prop-Π  x  is-prop-Π  y  is-property-is-contr))

is-prop-Prop : {l : Level} (A : UU l)  Prop l
pr1 (is-prop-Prop A) = is-prop A
pr2 (is-prop-Prop A) = is-prop-is-prop A

Recent changes