# Propositions

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

Created on 2022-02-07.

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 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.

## Definitions

### The predicate of being a proposition

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


### The type of propositions

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

## Properties

### To show that a type is a proposition we may assume it has an element

abstract
is-prop-has-element :
{l1 : Level} {X : UU l1} → (X → is-prop X) → is-prop X
is-prop-has-element 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


### 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) (λ 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 (λ x → K)

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

type-product-Prop : UU (l1 ⊔ l2)
type-product-Prop = type-Prop P × type-Prop Q

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

product-Prop : Prop (l1 ⊔ l2)
product-Prop = (type-product-Prop , is-prop-product-Prop)


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

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


We now 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 (λ 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-implicit-Π-Prop : is-prop type-implicit-Π-Prop
is-prop-implicit-Π-Prop =
is-prop-implicit-Π (λ 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-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 (λ _ → 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-function-Prop :
{l1 l2 : Level} {A : UU l1} (P : Prop l2) →
is-prop (type-function-Prop A P)
is-prop-function-Prop 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-function-Prop 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-hom-Prop :
{l1 l2 : Level} (P : Prop l1) (Q : Prop l2) →
is-prop (type-hom-Prop P Q)
is-prop-hom-Prop P = is-prop-function-Prop

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-hom-Prop P Q

infixr 5 _⇒_
_⇒_ = hom-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 (λ y → K (f (g y)) y))))
( is-prop-Σ
( is-prop-function-type H)
( λ h → is-prop-is-contr (is-contr-Π (λ 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 (λ 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 (λ x → is-prop-Π (λ 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


### Operations on propositions

There is a wide range of operations on propositions due to the rich structure of intuitionistic logic. Below we give a structured overview of a notable selection of such operations and their notation in the library.

The list is split into two sections, the first consists of operations that generalize to arbitrary types and even sufficiently nice subuniverses, such as -types.

NameOperator on typesOperator on propositions/subtypes
Dependent sumΣΣ-Prop
Dependent productΠΠ-Prop
Functions→⇒
Logical equivalence↔⇔
Product×product-Prop
Join*join-Prop
Exclusive sumexclusive-sumexclusive-sum-Prop
Coproduct+N/A

Note that for many operations in the second section, there is an equivalent operation on propositions in the first.

NameOperator on typesOperator on propositions/subtypes
Initial objectemptyempty-Prop
Terminal objectunitunit-Prop
Existential quantificationexists-structure∃
Unique existential quantificationuniquely-exists-structure∃!
Universal quantification∀' (equivalent to Π-Prop)
Conjunction∧ (equivalent to product-Prop)
Disjunctiondisjunction-type∨ (equivalent to join-Prop)
Exclusive disjunctionxor-type⊻ (equivalent to exclusive-sum-Prop)
Negation¬¬'
Double negation¬¬¬¬'

We can also organize these operations by indexed and binary variants, giving us the following table:

NameBinaryIndexed
Product×Π
Conjunction∧∀'
Constructive existence+Σ
Existence∨∃
Unique existence⊻∃!

### Table of files about propositional logic

The following table gives an overview of basic constructions in propositional logic and related considerations.

ConceptFile
Propositions (foundation-core)foundation-core.propositions
Propositions (foundation)foundation.propositions
Subterminal typesfoundation.subterminal-types
Subsingleton inductionfoundation.subsingleton-induction
Empty types (foundation-core)foundation-core.empty-types
Empty types (foundation)foundation.empty-types
Unit typefoundation.unit-type
Logical equivalencesfoundation.logical-equivalences
Propositional extensionalityfoundation.propositional-extensionality
Mere logical equivalencesfoundation.mere-logical-equivalences
Conjunctionfoundation.conjunction
Disjunctionfoundation.disjunction
Exclusive disjunctionfoundation.exclusive-disjunction
Existential quantificationfoundation.existential-quantification
Uniqueness quantificationfoundation.uniqueness-quantification
Universal quantificationfoundation.universal-quantification
Negationfoundation.negation
Double negationfoundation.double-negation
Propositional truncationsfoundation.propositional-truncations
Universal property of propositional truncationsfoundation.universal-property-propositional-truncation
The induction principle of propositional truncationsfoundation.induction-principle-propositional-truncation
Functoriality of propositional truncationsfoundation.functoriality-propositional-truncations
Propositional resizingfoundation.propositional-resizing
Impredicative encodings of the logical operationsfoundation.impredicative-encodings