Lower types of elements in W-types

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

Created on 2023-01-26.
Last modified on 2023-03-14.

module trees.lower-types-w-types where
open import foundation.existential-quantification
open import foundation.universe-levels

open import trees.ranks-of-elements-w-types
open import trees.w-types


We define by induction a type family over W A B in a way that generalizes the construction of the standard finite types over ℕ to arbitrary W-types.


module _
  {l1 l2 : Level} {A : UU l1} {B : A  UU l2}

    lower-𝕎 : 𝕎 A B  UU (l1  l2)
    lower-tree-𝕎 :
      {x : A} {f : B x  𝕎 A B} 
      ((y : B x)  lower-𝕎 (f y))  lower-𝕎 (tree-𝕎 x f)

  inclusion-lower-𝕎 : {x : 𝕎 A B}  lower-𝕎 x  𝕎 A B
  inclusion-lower-𝕎 (lower-tree-𝕎 {a} {f} g) =
    tree-𝕎 a  y  inclusion-lower-𝕎 (g y))

  upper-bound-rank-inclusion-lower-𝕎 :
    {x : 𝕎 A B} (y : lower-𝕎 x)  inclusion-lower-𝕎 y ≼-𝕎 x
  upper-bound-rank-inclusion-lower-𝕎 (lower-tree-𝕎 g) y =
    intro-∃ y (upper-bound-rank-inclusion-lower-𝕎 (g y))

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