Dependent products of large frames

Content created by Egbert Rijke, Fredrik Bakke, Julian KG, Maša Žaucer, fernabnor, Gregor Perčič and louismntnu.

Created on 2023-05-12.
Last modified on 2024-04-11.

module order-theory.dependent-products-large-frames where
open import foundation.function-extensionality
open import foundation.identity-types
open import foundation.large-binary-relations
open import foundation.sets
open import foundation.universe-levels

open import order-theory.dependent-products-large-meet-semilattices
open import order-theory.dependent-products-large-posets
open import order-theory.dependent-products-large-suplattices
open import order-theory.greatest-lower-bounds-large-posets
open import order-theory.large-frames
open import order-theory.large-meet-semilattices
open import order-theory.large-posets
open import order-theory.large-suplattices
open import order-theory.least-upper-bounds-large-posets
open import

Given a family L : I → Large-Frame α β of large frames indexed by a type I : UU l, the product of the large frame L i is again a large frame.

module _
  {α : Level  Level} {β : Level  Level  Level} {γ : Level}
  {l1 : Level} {I : UU l1} (L : I  Large-Frame α β γ)

  large-poset-Π-Large-Frame :
    Large-Poset  l2  α l2  l1)  l2 l3  β l2 l3  l1)
  large-poset-Π-Large-Frame =
    Π-Large-Poset  i  large-poset-Large-Frame (L i))

  large-meet-semilattice-Π-Large-Frame :
    Large-Meet-Semilattice  l2  α l2  l1)  l2 l3  β l2 l3  l1)
  large-meet-semilattice-Π-Large-Frame =
    Π-Large-Meet-Semilattice  i  large-meet-semilattice-Large-Frame (L i))

  has-meets-Π-Large-Frame :
    has-meets-Large-Poset large-poset-Π-Large-Frame
  has-meets-Π-Large-Frame =
      ( λ i  large-poset-Large-Frame (L i))
      ( λ i  has-meets-Large-Frame (L i))

  large-suplattice-Π-Large-Frame :
    Large-Suplattice  l2  α l2  l1)  l2 l3  β l2 l3  l1) γ
  large-suplattice-Π-Large-Frame =
    Π-Large-Suplattice  i  large-suplattice-Large-Frame (L i))

  is-large-suplattice-Π-Large-Frame :
    is-large-suplattice-Large-Poset γ large-poset-Π-Large-Frame
  is-large-suplattice-Π-Large-Frame =
      ( λ i  large-suplattice-Large-Frame (L i))

  set-Π-Large-Frame : (l : Level)  Set (α l  l1)
  set-Π-Large-Frame = set-Large-Poset large-poset-Π-Large-Frame

  type-Π-Large-Frame : (l : Level)  UU (α l  l1)
  type-Π-Large-Frame = type-Large-Poset large-poset-Π-Large-Frame

  is-set-type-Π-Large-Frame : {l : Level}  is-set (type-Π-Large-Frame l)
  is-set-type-Π-Large-Frame =
    is-set-type-Large-Poset large-poset-Π-Large-Frame

  leq-prop-Π-Large-Frame :
      ( λ l2 l3  β l2 l3  l1)
      ( type-Π-Large-Frame)
  leq-prop-Π-Large-Frame =
    leq-prop-Large-Poset large-poset-Π-Large-Frame

  leq-Π-Large-Frame :
      ( λ l2 l3  β l2 l3  l1)
      ( type-Π-Large-Frame)
  leq-Π-Large-Frame = leq-Large-Poset large-poset-Π-Large-Frame

  is-prop-leq-Π-Large-Frame :
    is-prop-Large-Relation type-Π-Large-Frame leq-Π-Large-Frame
  is-prop-leq-Π-Large-Frame =
    is-prop-leq-Large-Poset large-poset-Π-Large-Frame

  refl-leq-Π-Large-Frame :
    is-reflexive-Large-Relation type-Π-Large-Frame leq-Π-Large-Frame
  refl-leq-Π-Large-Frame = refl-leq-Large-Poset large-poset-Π-Large-Frame

  antisymmetric-leq-Π-Large-Frame :
    is-antisymmetric-Large-Relation type-Π-Large-Frame leq-Π-Large-Frame
  antisymmetric-leq-Π-Large-Frame =
    antisymmetric-leq-Large-Poset large-poset-Π-Large-Frame

  transitive-leq-Π-Large-Frame :
    is-transitive-Large-Relation type-Π-Large-Frame leq-Π-Large-Frame
  transitive-leq-Π-Large-Frame =
    transitive-leq-Large-Poset large-poset-Π-Large-Frame

  meet-Π-Large-Frame :
    {l2 l3 : Level} 
    type-Π-Large-Frame l2 
    type-Π-Large-Frame l3 
    type-Π-Large-Frame (l2  l3)
  meet-Π-Large-Frame =
    meet-has-meets-Large-Poset has-meets-Π-Large-Frame

  is-greatest-binary-lower-bound-meet-Π-Large-Frame :
    {l2 l3 : Level}
    (x : type-Π-Large-Frame l2)
    (y : type-Π-Large-Frame l3) 
      ( large-poset-Π-Large-Frame)
      ( x)
      ( y)
      ( meet-Π-Large-Frame x y)
  is-greatest-binary-lower-bound-meet-Π-Large-Frame =

  top-Π-Large-Frame : type-Π-Large-Frame lzero
  top-Π-Large-Frame =
    top-Large-Meet-Semilattice large-meet-semilattice-Π-Large-Frame

  is-top-element-top-Π-Large-Frame :
    {l1 : Level} (x : type-Π-Large-Frame l1) 
    leq-Π-Large-Frame x top-Π-Large-Frame
  is-top-element-top-Π-Large-Frame =

  has-top-element-Π-Large-Frame :
    has-top-element-Large-Poset large-poset-Π-Large-Frame
  has-top-element-Π-Large-Frame =

  is-large-meet-semilattice-Π-Large-Frame :
    is-large-meet-semilattice-Large-Poset large-poset-Π-Large-Frame
  is-large-meet-semilattice-Π-Large-Frame =

  sup-Π-Large-Frame :
    {l2 l3 : Level} {J : UU l2} (x : J  type-Π-Large-Frame l3) 
    type-Π-Large-Frame (γ  l2  l3)
  sup-Π-Large-Frame =
    sup-is-large-suplattice-Large-Poset γ
      ( large-poset-Π-Large-Frame)
      ( is-large-suplattice-Π-Large-Frame)

  is-least-upper-bound-sup-Π-Large-Frame :
    {l2 l3 : Level} {J : UU l2} (x : J  type-Π-Large-Frame l3) 
      ( large-poset-Π-Large-Frame)
      ( x)
      ( sup-Π-Large-Frame x)
  is-least-upper-bound-sup-Π-Large-Frame =
    is-least-upper-bound-sup-is-large-suplattice-Large-Poset γ
      ( large-poset-Π-Large-Frame)
      ( is-large-suplattice-Π-Large-Frame)

  distributive-meet-sup-Π-Large-Frame :
    {l2 l3 l4 : Level}
    (x : type-Π-Large-Frame l2)
    {J : UU l3} (y : J  type-Π-Large-Frame l4) 
    meet-Π-Large-Frame x (sup-Π-Large-Frame y) 
    sup-Π-Large-Frame  j  meet-Π-Large-Frame x (y j))
  distributive-meet-sup-Π-Large-Frame x y =
      ( λ i  distributive-meet-sup-Large-Frame (L i) (x i)  j  y j i))

  Π-Large-Frame : Large-Frame  l2  α l2  l1)  l2 l3  β l2 l3  l1) γ
  large-poset-Large-Frame Π-Large-Frame =
  is-large-meet-semilattice-Large-Frame Π-Large-Frame =
  is-large-suplattice-Large-Frame Π-Large-Frame =
  distributive-meet-sup-Large-Frame Π-Large-Frame =

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