Upper bounds in large posets

module order-theory.upper-bounds-large-posets where

open import foundation.dependent-pair-types
open import foundation.logical-equivalences
open import foundation.propositions
open import foundation.universe-levels

open import order-theory.dependent-products-large-posets
open import order-theory.large-posets

Idea

A binary upper bound of two elements a and b of a large poset P is an element x of P such that a ≤ x and b ≤ x both hold. Similarly, an upper bound of a family a : I → P of elements of P is an element x of P such that the inequality (a i) ≤ x holds for every i : I.

Definitions

The predicate of being an upper bound of a family of elements

module _
  {α : Level → Level} {β : Level → Level → Level}
  (P : Large-Poset α β)
  {l1 l2 : Level} {I : UU l1} (x : I → type-Large-Poset P l2)
  where

  is-upper-bound-family-of-elements-Large-Poset-Prop :
    {l3 : Level} (y : type-Large-Poset P l3) → Prop (β l2 l3 ⊔ l1)
  is-upper-bound-family-of-elements-Large-Poset-Prop y =
    Π-Prop I (λ i → leq-Large-Poset-Prop P (x i) y)

  is-upper-bound-family-of-elements-Large-Poset :
    {l3 : Level} (y : type-Large-Poset P l3) → UU (β l2 l3 ⊔ l1)
  is-upper-bound-family-of-elements-Large-Poset y =
    type-Prop (is-upper-bound-family-of-elements-Large-Poset-Prop y)

  is-prop-is-upper-bound-family-of-elements-Large-Poset :
    {l3 : Level} (y : type-Large-Poset P l3) →
    is-prop (is-upper-bound-family-of-elements-Large-Poset y)
  is-prop-is-upper-bound-family-of-elements-Large-Poset y =
    is-prop-type-Prop (is-upper-bound-family-of-elements-Large-Poset-Prop y)

Properties

An element x : Π-Large-Poset P of a dependent product of large posets P i indexed by i : I is an upper bound of a family a : J → Π-Large-Poset P if and only if x i is an upper bound of the family (j ↦ a j i) : J → P i of elements of P i

module _
  {α : Level → Level} {β : Level → Level → Level}
  {l1 : Level} {I : UU l1} (P : I → Large-Poset α β)
  {l2 l3 : Level} {J : UU l2} (a : J → type-Π-Large-Poset P l3)
  {l4 : Level} (x : type-Π-Large-Poset P l4)
  where

  is-upper-bound-family-of-elements-Π-Large-Poset :
    ( (i : I) →
      is-upper-bound-family-of-elements-Large-Poset (P i) (λ j → a j i) (x i)) →
    is-upper-bound-family-of-elements-Large-Poset (Π-Large-Poset P) a x
  is-upper-bound-family-of-elements-Π-Large-Poset H j i = H i j

  map-inv-is-upper-bound-family-of-elements-Π-Large-Poset :
    is-upper-bound-family-of-elements-Large-Poset (Π-Large-Poset P) a x →
    (i : I) →
    is-upper-bound-family-of-elements-Large-Poset (P i) (λ j → a j i) (x i)
  map-inv-is-upper-bound-family-of-elements-Π-Large-Poset H i j = H j i

  logical-equivalence-is-upper-bound-family-of-elements-Π-Large-Poset :
    ( (i : I) →
      is-upper-bound-family-of-elements-Large-Poset (P i) (λ j → a j i) (x i)) ↔
    is-upper-bound-family-of-elements-Large-Poset (Π-Large-Poset P) a x
  pr1 logical-equivalence-is-upper-bound-family-of-elements-Π-Large-Poset =
    is-upper-bound-family-of-elements-Π-Large-Poset
  pr2 logical-equivalence-is-upper-bound-family-of-elements-Π-Large-Poset =
    map-inv-is-upper-bound-family-of-elements-Π-Large-Poset