Dependent products large preorders

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

Created on 2023-05-09.
Last modified on 2024-04-11.

module order-theory.dependent-products-large-preorders where
open import foundation.large-binary-relations
open import foundation.propositions
open import foundation.universe-levels

open import order-theory.large-preorders


Given a family P : I → Large-Preorder α β of large preorders indexed by a type I : UU l, the dependent prodcut of the large preorders P i is again a large preorder.


module _
  {α : Level  Level} {β : Level  Level  Level}
  {l : Level} {I : UU l} (P : I  Large-Preorder α β)

  type-Π-Large-Preorder : (l1 : Level)  UU (α l1  l)
  type-Π-Large-Preorder l1 = (i : I)  type-Large-Preorder (P i) l1

  leq-prop-Π-Large-Preorder :
      ( λ l1 l2  β l1 l2  l)
      ( type-Π-Large-Preorder)
  leq-prop-Π-Large-Preorder x y =
    Π-Prop I  i  leq-prop-Large-Preorder (P i) (x i) (y i))

  leq-Π-Large-Preorder :
      ( λ l1 l2  β l1 l2  l)
      ( type-Π-Large-Preorder)
  leq-Π-Large-Preorder x y =
    type-Prop (leq-prop-Π-Large-Preorder x y)

  is-prop-leq-Π-Large-Preorder :
    is-prop-Large-Relation type-Π-Large-Preorder leq-Π-Large-Preorder
  is-prop-leq-Π-Large-Preorder x y =
    is-prop-type-Prop (leq-prop-Π-Large-Preorder x y)

  refl-leq-Π-Large-Preorder :
    is-reflexive-Large-Relation type-Π-Large-Preorder leq-Π-Large-Preorder
  refl-leq-Π-Large-Preorder x i = refl-leq-Large-Preorder (P i) (x i)

  transitive-leq-Π-Large-Preorder :
    is-transitive-Large-Relation type-Π-Large-Preorder leq-Π-Large-Preorder
  transitive-leq-Π-Large-Preorder x y z H K i =
    transitive-leq-Large-Preorder (P i) (x i) (y i) (z i) (H i) (K i)

  Π-Large-Preorder : Large-Preorder  l1  α l1  l)  l1 l2  β l1 l2  l)
  type-Large-Preorder Π-Large-Preorder =
  leq-prop-Large-Preorder Π-Large-Preorder =
  refl-leq-Large-Preorder Π-Large-Preorder =
  transitive-leq-Large-Preorder Π-Large-Preorder =

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