Conjunction of propositions

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

Created on 2022-02-08.
Last modified on 2024-02-06.

module foundation.conjunction where
open import foundation.decidable-types
open import foundation.dependent-pair-types
open import foundation.logical-equivalences
open import foundation.universe-levels

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


The conjunction of two propositions P and Q is the proposition that both P and Q hold.


conjunction-Prop = product-Prop

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

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

infixr 15 _∧_
_∧_ = type-conjunction-Prop

Note: The symbol used for the conjunction _∧_ is the logical and (agda-input: \wedge \and).

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


Introduction rule for conjunction

intro-conjunction-Prop :
  {l1 l2 : Level} (P : Prop l1) (Q : Prop l2) 
  type-Prop P  type-Prop Q  type-conjunction-Prop P Q
pr1 (intro-conjunction-Prop P Q p q) = p
pr2 (intro-conjunction-Prop P Q p q) = q

The universal property of conjunction

iff-universal-property-conjunction-Prop :
  {l1 l2 : Level} (P : Prop l1) (Q : Prop l2)
  {l3 : Level} (R : Prop l3) 
  ( type-hom-Prop R P × type-hom-Prop R Q) 
  ( type-hom-Prop R (conjunction-Prop P Q))
pr1 (pr1 (iff-universal-property-conjunction-Prop P Q R) (f , g) r) = f r
pr2 (pr1 (iff-universal-property-conjunction-Prop P Q R) (f , g) r) = g r
pr1 (pr2 (iff-universal-property-conjunction-Prop P Q R) h) r = pr1 (h r)
pr2 (pr2 (iff-universal-property-conjunction-Prop P Q R) h) r = pr2 (h r)

equiv-universal-property-conjunction-Prop :
  {l1 l2 : Level} (P : Prop l1) (Q : Prop l2)
  {l3 : Level} (R : Prop l3) 
  ( type-hom-Prop R P × type-hom-Prop R Q) 
  ( type-hom-Prop R (conjunction-Prop P Q))
equiv-universal-property-conjunction-Prop P Q R =
    ( conjunction-Prop (hom-Prop R P) (hom-Prop R Q))
    ( hom-Prop R (conjunction-Prop P Q))
    ( iff-universal-property-conjunction-Prop P Q R)

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