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.large-frames where
open import foundation.dependent-pair-types
open import foundation.identity-types
open import foundation.large-binary-relations
open import foundation.sets
open import foundation.universe-levels

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


A large frame is a large meet-suplattice satisfying the distributive law for meets over suprema.


Large frames

  Large-Frame (α : Level  Level) (β : Level  Level  Level) (γ : Level) : UUω
    large-poset-Large-Frame :
      Large-Poset α β
    is-large-meet-semilattice-Large-Frame :
      is-large-meet-semilattice-Large-Poset large-poset-Large-Frame
    is-large-suplattice-Large-Frame :
      is-large-suplattice-Large-Poset γ large-poset-Large-Frame
    distributive-meet-sup-Large-Frame :
      {l1 l2 l3 : Level}
      (x : type-Large-Poset large-poset-Large-Frame l1)
      {I : UU l2} (y : I  type-Large-Poset large-poset-Large-Frame l3) 
        ( large-poset-Large-Frame)
        ( is-large-meet-semilattice-Large-Frame)
        ( x)
        ( sup-has-least-upper-bound-family-of-elements-Large-Poset
          ( is-large-suplattice-Large-Frame y)) 
        ( is-large-suplattice-Large-Frame
          ( λ i 
              ( large-poset-Large-Frame)
              ( is-large-meet-semilattice-Large-Frame)
              ( x)
              ( y i)))

open Large-Frame public

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

  large-preorder-Large-Frame : Large-Preorder α β
  large-preorder-Large-Frame =
    large-preorder-Large-Poset (large-poset-Large-Frame L)

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

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

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

  leq-prop-Large-Frame : Large-Relation-Prop β type-Large-Frame
  leq-prop-Large-Frame = leq-prop-Large-Poset (large-poset-Large-Frame L)

  leq-Large-Frame : Large-Relation β type-Large-Frame
  leq-Large-Frame = leq-Large-Poset (large-poset-Large-Frame L)

  is-prop-leq-Large-Frame :
    is-prop-Large-Relation type-Large-Frame leq-Large-Frame
  is-prop-leq-Large-Frame =
    is-prop-leq-Large-Poset (large-poset-Large-Frame L)

  leq-eq-Large-Frame :
    {l1 : Level} {x y : type-Large-Frame l1} 
    (x  y)  leq-Large-Frame x y
  leq-eq-Large-Frame =
    leq-eq-Large-Poset (large-poset-Large-Frame L)

  refl-leq-Large-Frame :
    is-reflexive-Large-Relation type-Large-Frame leq-Large-Frame
  refl-leq-Large-Frame = refl-leq-Large-Poset (large-poset-Large-Frame L)

  antisymmetric-leq-Large-Frame :
    is-antisymmetric-Large-Relation type-Large-Frame leq-Large-Frame
  antisymmetric-leq-Large-Frame =
    antisymmetric-leq-Large-Poset (large-poset-Large-Frame L)

  transitive-leq-Large-Frame :
    is-transitive-Large-Relation type-Large-Frame leq-Large-Frame
  transitive-leq-Large-Frame =
    transitive-leq-Large-Poset (large-poset-Large-Frame L)

  large-meet-semilattice-Large-Frame :
    Large-Meet-Semilattice α β
  large-poset-Large-Meet-Semilattice large-meet-semilattice-Large-Frame =
    large-poset-Large-Frame L
    large-meet-semilattice-Large-Frame =
    is-large-meet-semilattice-Large-Frame L

  has-meets-Large-Frame :
    has-meets-Large-Poset (large-poset-Large-Frame L)
  has-meets-Large-Frame =
    has-meets-Large-Meet-Semilattice large-meet-semilattice-Large-Frame

  meet-Large-Frame :
    {l1 l2 : Level} 
    type-Large-Frame l1  type-Large-Frame l2  type-Large-Frame (l1  l2)
  meet-Large-Frame =
      ( large-poset-Large-Frame L)
      ( is-large-meet-semilattice-Large-Frame L)

  is-greatest-binary-lower-bound-meet-Large-Frame :
    {l1 l2 : Level} 
    (x : type-Large-Frame l1) (y : type-Large-Frame l2) 
      ( large-poset-Large-Frame L)
      ( x)
      ( y)
      ( meet-Large-Frame x y)
  is-greatest-binary-lower-bound-meet-Large-Frame =
      ( large-poset-Large-Frame L)
      ( is-large-meet-semilattice-Large-Frame L)

  ap-meet-Large-Frame :
    {l1 l2 : Level}
    {x x' : type-Large-Frame l1} {y y' : type-Large-Frame l2} 
    (x  x')  (y  y')  (meet-Large-Frame x y  meet-Large-Frame x' y')
  ap-meet-Large-Frame =
    ap-meet-Large-Meet-Semilattice large-meet-semilattice-Large-Frame

  has-top-element-Large-Frame :
    has-top-element-Large-Poset (large-poset-Large-Frame L)
  has-top-element-Large-Frame =

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

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

  sup-Large-Frame :
    {l1 l2 : Level} {I : UU l1} 
    (I  type-Large-Frame l2)  type-Large-Frame (γ  l1  l2)
  sup-Large-Frame =
    sup-is-large-suplattice-Large-Poset γ
      ( large-poset-Large-Frame L)
      ( is-large-suplattice-Large-Frame L)

  is-least-upper-bound-sup-Large-Frame :
    {l1 l2 : Level} {I : UU l1} (x : I  type-Large-Frame l2) 
      ( large-poset-Large-Frame L)
      ( x)
      ( sup-Large-Frame x)
  is-least-upper-bound-sup-Large-Frame =
    is-least-upper-bound-sup-is-large-suplattice-Large-Poset γ
      ( large-poset-Large-Frame L)
      ( is-large-suplattice-Large-Frame L)

  large-suplattice-Large-Frame : Large-Suplattice α β γ
  large-poset-Large-Suplattice large-suplattice-Large-Frame =
    large-poset-Large-Frame L
  is-large-suplattice-Large-Suplattice large-suplattice-Large-Frame =
    is-large-suplattice-Large-Frame L

  is-upper-bound-sup-Large-Frame :
    {l1 l2 : Level} {I : UU l1} (x : I  type-Large-Frame l2) 
      ( large-poset-Large-Frame L)
      ( x)
      ( sup-Large-Frame x)
  is-upper-bound-sup-Large-Frame =
    is-upper-bound-sup-Large-Suplattice large-suplattice-Large-Frame


Small constructions from large frames

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

  preorder-Large-Frame : (l : Level)  Preorder (α l) (β l l)
  preorder-Large-Frame = preorder-Large-Preorder (large-preorder-Large-Frame L)

  poset-Large-Frame : (l : Level)  Poset (α l) (β l l)
  poset-Large-Frame = poset-Large-Poset (large-poset-Large-Frame L)

  is-suplattice-poset-Large-Frame :
    (l1 l2 : Level)  is-suplattice-Poset l1 (poset-Large-Frame (γ  l1  l2))
  pr1 (is-suplattice-poset-Large-Frame l1 l2 I y) =
    sup-Large-Frame L y
  pr2 (is-suplattice-poset-Large-Frame l1 l2 I y) =
    is-least-upper-bound-sup-Large-Frame L y

  suplattice-Large-Frame :
    (l1 l2 : Level) 
    Suplattice (α (γ  l1  l2)) (β (γ  l1  l2) (γ  l1  l2)) l1
  pr1 (suplattice-Large-Frame l1 l2) = poset-Large-Frame (γ  l1  l2)
  pr2 (suplattice-Large-Frame l1 l2) = is-suplattice-poset-Large-Frame l1 l2

  is-meet-semilattice-poset-Large-Frame :
    (l : Level)  is-meet-semilattice-Poset (poset-Large-Frame l)
  pr1 (is-meet-semilattice-poset-Large-Frame l x y) =
    meet-Large-Frame L x y
  pr2 (is-meet-semilattice-poset-Large-Frame l x y) =
    is-greatest-binary-lower-bound-meet-Large-Frame L x y

  order-theoretic-meet-semilattice-Large-Frame :
    (l : Level)  Order-Theoretic-Meet-Semilattice (α l) (β l l)
  pr1 (order-theoretic-meet-semilattice-Large-Frame l) =
    poset-Large-Frame l
  pr2 (order-theoretic-meet-semilattice-Large-Frame l) =
    is-meet-semilattice-poset-Large-Frame l

  meet-semilattice-Large-Frame : (l : Level)  Meet-Semilattice (α l)
  meet-semilattice-Large-Frame =
      ( large-meet-semilattice-Large-Frame L)

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